The Hybrid Grand Unified Theory

ATLANTIS S TUDIES IN M ATHEMATICS FOR E NGINEERING AND S CIENCE VOLUME 3 S ERIES E DITOR : C.K. C HUI

Atlantis Studies in Mathematics for Engineering and Science Series Editor: C. K. Chui, Stanford University, USA (ISSN: 1875-7642)

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c ATLANTIS PRESS / WORLD SCIENTIFIC 

The Hybrid Grand Unified Theory

V. LAKSHMIKANTHAM Florida Institute of Technology Department of Mathematical Sciences Melbourne, USA

E. ESCULTURA Lakshmikantham Institute for Advanced Studies G.V.P. College of Engineering Madhuravada Visakhapatnam, (A.P.) India

S. LEELA Florida Institute of Technology Department of Mathematical Sciences Melbourne, USA

A MSTERDAM – PARIS

Atlantis Press 29, avenue Laumi`ere 75019 Paris, France For information on all Atlantis Press publications, visit our website at: www.atlantis-press.com Copyright This book, or any parts thereof, may not be reproduced for commercial purposes in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system known or to be invented, without prior permission from the Publisher.

ISBN: 978-90-78677-21-5 ISSN: 1875-7642

e-ISBN: 978-94-91216-23-7

c 2009 ATLANTIS PRESS / WORLD SCIENTIFIC 

Preface

The physicist is interested in discovering the laws of inanimate nature and the mathematician uses the depth of his thought into exploring the mathematical concepts. But, the symbiotic connection between physics and mathematics and the enormous usefulness of mathematics in the natural sciences is something quite mysterious. No rational justification appears to be satisfactory to understand the uncanny success of mathematics and its role in physical theories. A possible explanation is that the laws of nature are written in the language of mathematics. A complete mechanism, natural or man-made, is understood fully by taking it apart, studying its parts or components and their properties and learn how these parts fit to make the whole. It is really a natural way to understand any object or concept. The path of progress in knowledge is similarly a combination of both exposition of parts and their synthesis. Natural sciences and other sciences such as biology, psychology, etc. are reminders of these strides, taken in order to comprehend the world around us and there have been attempts to catch a glimpse of the higher reality and understand it. The human mind is limited and cannot go beyond certain limitations of time, space and causation. Still, the universe is experienced as a dynamic inseparable whole, including the observer in an essential way. In this experience, the traditional concepts of time and space of isolated objects and of cause and effect lose their meaning. Such an experience is very similar to that of the ancient scientists (Rishis). However, they repeatedly insisted that the ultimate reality can never be an object of reasoning, deduction or demonstrable knowledge. It can never be adequately described by words and Max Planck of the modern scientific era observed that science cannot solve the ultimate mystery of nature and that is because in the final analysis, we ourselves are part of nature and therefore part of the mystery we are trying to solve. The scientists in physics are happy with physical concepts that can be visualized or observed directly. But most of the time they are forced to utilize mathematical concepts to describe the physical concepts. Take for example, the case of the black hole. It is not entirely observable but it is a physical concept because there is an overwhelming evidence of its existence. Although they have verified its existence in the core of galaxies including our milky way, it is not known exactly what it is and therefore, they use the mathematical concept of a singularity to describe it. However that is not amenable to computation or studying its structure. Another such important physical concept is that of the basic conv

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stituent of matter. It is very important to know what constitutes matter and it cannot be ignored because of our deep conviction in the order of the universe. This is our attempt to provide a hybrid grand unified theory to understand the universe, both in its micro/quantum aspects as well as macro/galactic aspects. It is truly a hybrid theory as it tries to encompass both the modern and ancient theories of the universe, together with its functioning at all levels of human comprehension. During this attempt it becomes necessary to acknowledge the ambiguity and limitations of mathematics concerning the fundamental concepts of very large and very small numbers, infinity and the limiting process in general. Although the mathematical modeling is the most advanced methodology of physics and it owes all its tremendous achievements to the former, the existence of unsolved problems and unanswered questions suggest the need for improvement. It is a fact that the origin of the concepts such as natural numbers, rational and irrational numbers, zero, infinity and the place value system, among others, originated in Veda Samhitas, ancient scientific texts of India. However, because of the quirks of our narrated human history, our second hand reception of these concepts from the Arabs and the belief that everything originated in Greece, the true original source appears to have been obscured and lost. It is therefore necessary to return to the source to clarify and understand the basics to shed some light on the existing real number system. In Chapter 1, we revisit the fundamentals of mathematics to bring out sources of ambiguity in certain basic mathematical concepts. We also raise many questions that are challenging the physicists and discuss the relevance of mathematical methodology in physics. Certain related concepts such as cosmic waves are also considered. Chapter 2 deals with the mathematics that is essential in aiding the description of physical concepts that are discussed in Chapter 3. In particular, the existing real number system is revised with the purpose of bringing out the useful nature of decimal numeration system in the current digital era of high accuracy computation with the aid of technology. The ambiguity involved in dealing with infinity, limiting process and the computation of very large and very small numbers is minimized. The mathematics of generalized curves, generalized fractals and chaos provides the mathematical modeling of physical concepts that arise in this grand unified theory. The integrated Pontrjagin’s maximum principle is briefly discussed as it has been instrumental in solving the famous n-body problem. The introduction of dark numbers as part of the refined real number system paves the way to quantize the basic constituent of matter and in general, dark matter. In chapter 3, the qualitative modeling is employed to study the search for basic constituent of matter and its ramifications in explaining quantum gravity and macro gravity. The ten natural laws of nature are enumerated and the mathematical model of a superstring is introduced to represent the structure of the basic constituent of matter. Verification of this grand unified theory (GUT) is carried out by explaining various natural phenomena such as ultra-energetic indexcosmic waves, supernova and the rare hit on earth by asteroids inspite of being so close to the astreoid belt, etc. In Chapter 4, the theoretical consideration of n-body problem and turbulence are described. Also, physics of the mind is investigated since in this grand unified theory, mind

Preface

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is also an important part in the study of the universe around us. Finally, in Chapter 5, the hybrid nature of this book is really brought to the fore as it deals with matters of mind and consciousness, evolution and involution, creation and dissolution, etc. according to the view of ancient scientists (Rishis). It is really like coming full circle to consider how much alike are the theories of ancient and modern scientists, relative to the universe around us. Some of the important features of the book are as follows: (i) It puts the real number system on solid foundations without inconsistency and emphasizes the appropriateness of the decimal system as a computation tool; (ii) It improves the existing real number system by incorporating the notion of dark numbers and their duals, personal and impersonal infinity; (iii) It introduces the superstring as the basic constituent of matter and the fractal nature of the superstring is modeled by the Cauchy representation of dark number; (iv) It relates the dark matter of physics and dark numbers of mathematics; (v) It employs the truly hybrid approach of combining qualitative mathematics and computation to discover the natural laws in order to explain in a unified way several natural phenomena, at micro/quantum and macro/galactic levels; (vi) It attempts to show the common hybrid interaction between modern and ancient scientific theories of nature and search for higher reality. We wish to express our deepest appreciation to Professors G.S. Osipenko, V.V. Gudkov, C.G. Jesdudason and T. Gnana Bhaskar for the various scientific discussions. We are immensely thankful to Ms. Sally Ellingson for the excellent typing of the manuscript in all its stages.

Contents

Preface

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Basic Problems of Mathematics and Physics 1.1 1.2 1.3 1.4

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Introduction . . . . . . . . . . . . . . Retrospection of Fundamentals . . . . The Problems of Physics . . . . . . . Related Concepts . . . . . . . . . . . 1.4.1 Waves and Vibration . . . . . 1.4.2 Cosmic Waves . . . . . . . . 1.4.3 The Sine and Related Curves 1.4.4 Generalized Fractal . . . . .

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Mathematics of Grand Unified Theory 2.1 2.2

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1 1 5 8 8 9 10 10 13

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . Hybrid Real Number System . . . . . . . . . . . . . . . . 2.2.1 The Construction of Hybrid Real Number System 2.2.2 The Nonterminating Decimals . . . . . . . . . . 2.2.3 The Structure of R∗ . . . . . . . . . . . . . . . . 2.2.4 More Properties of Nonstandard Numbers . . . . 2.2.5 g-Limit and the Limit . . . . . . . . . . . . . . . Special Functions and the Generalized Curves . . . . . . . 2.3.1 Some Mischievous Functions . . . . . . . . . . . 2.3.2 The Infinitesimal Zigzag . . . . . . . . . . . . . . 2.3.3 Significance of the Infinitesimal Zigzag . . . . . . 2.3.4 The Wild Oscillation sin 1x . . . . . . . . . . . . . 2.3.5 Rapid Spiral and Oscillation . . . . . . . . . . . . 2.3.6 Rectification . . . . . . . . . . . . . . . . . . . . 2.3.7 Application of the Infinitesimal Zigzag . . . . . . 2.3.8 The Generalized Curves . . . . . . . . . . . . . . Generalized Fractal and Chaos . . . . . . . . . . . . . . . 2.4.1 Definitions and Examples . . . . . . . . . . . . . 2.4.2 The Peano Space-Filling Curve . . . . . . . . . . 2.4.3 Filling up the Unit Cube . . . . . . . . . . . . . . 2.4.4 The Infinitesimal Zigzag as Limit Set of Fractal . 2.4.5 A More General Fractal . . . . . . . . . . . . . . The Integrated Pontrjagin Maximum Principle . . . . . . . ix

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2.5.1 2.5.2 2.5.3 2.5.4 2.5.5 3.

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Grand Unified Theory (GUT)

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3.1 3.2

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3.3

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The Integrated Version . . . . . . . . . . . . . . . . . . . . Formulation of the Problem . . . . . . . . . . . . . . . . . Existence Theorems . . . . . . . . . . . . . . . . . . . . . The Pontrjagin Maximum Principle . . . . . . . . . . . . . The Integrated Form of the Pontrjagin Maximum Principle

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Search for the Basic Constituent of Matter . . . . . . . . . . . . . . . . . . . 3.2.1 The Primum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Primal Polarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Primum, Photon and Wave-Particle Duality . . . . . . . . . . . . . . . . . 3.2.4 Matter- Anti-Matter Interaction . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 Qualitative and Computational Models of the Primum and Photon . . . . . Quantum Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Primal Coupling and Interaction . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Do Coupled Prima have Anti-Matter? . . . . . . . . . . . . . . . . . . . . 3.3.3 Genesis of the Atom and Formation of Heavy Isotope . . . . . . . . . . . 3.3.4 Superconductivity and the Bose-Einstein Condensate . . . . . . . . . . . 3.3.5 Electric Current, Generation and Conduction . . . . . . . . . . . . . . . . 3.3.6 Thermonuclear Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.7 Further Verification of Quantum Gravity . . . . . . . . . . . . . . . . . . 3.3.8 Updates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Macro Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 The Cosmology of our Universe . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Early Cosmological History of the Earth . . . . . . . . . . . . . . . . . . 3.4.3 Cosmological Vortex Interaction and the Trek Back Home to Dark Matter 3.4.4 Cosmic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . More Verification of GUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Issues, Facts and Internal Dynamics of our Universe . . . . . . . . . . . . 3.5.2 The Milky Way and Andromeda and their “Cannibalistic” Activity . . . . 3.5.3 Point of No Return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.4 Stable Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.5 A Paradox No More . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.6 The Transitory Natural Laws . . . . . . . . . . . . . . . . . . . . . . . . 3.5.7 Ultra-Energetic Cosmic Waves . . . . . . . . . . . . . . . . . . . . . . . 3.5.8 Celestial Spectacle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.9 More Verification in the Cosmos . . . . . . . . . . . . . . . . . . . . . . 3.5.10 Verification in the Solar System . . . . . . . . . . . . . . . . . . . . . . . An Overview of GUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 The Mathematics of GUT . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 The Laws of Nature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3 Formation of a Universe . . . . . . . . . . . . . . . . . . . . . . . . . . .

Theoretical Applications 4.1 4.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . The Solution of the Gravitational n-Body Problem . . . . . 4.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . 4.2.2 The Solution of the Gravitational n-Body Problem

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Contents

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4.2.3 General Case of the Problem . . . . . . . . . . . . . . . . 4.2.4 Solution for the Other Categories of Bodies in the Cosmos 4.2.5 Extension . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.6 Concluding Remark . . . . . . . . . . . . . . . . . . . . . Theory of Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Qualitative Modeling . . . . . . . . . . . . . . . . . . . . 4.3.2 Geological Turbulence . . . . . . . . . . . . . . . . . . . . 4.3.3 Earthquake . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Volcanic Activity . . . . . . . . . . . . . . . . . . . . . . 4.3.5 Atmospheric Turbulence . . . . . . . . . . . . . . . . . . 4.3.6 Tornado . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.7 Lightning . . . . . . . . . . . . . . . . . . . . . . . . . . The Physics of the Mind . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 The Mind . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 The Human Brain . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Brain Waves . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.4 Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.5 Sensation and Concepts . . . . . . . . . . . . . . . . . . . 4.4.6 Value, Perception and Cognition . . . . . . . . . . . . . .

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Hybrid Unified Theory 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9

Introduction . . . . . . . . . . . . . . . . Mind and Consciousness . . . . . . . . . Inanimate and Animate Matter . . . . . . Evolution and Involution . . . . . . . . . Big Bang and Big Crunch . . . . . . . . . Sristi and Laya (Creation and Dissolution) Infinite Void and Infinite Light . . . . . . Phenomenon and Noumenon . . . . . . . Conclusion . . . . . . . . . . . . . . . .

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Bibliography

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Index

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Chapter 1

Basic Problems of Mathematics and Physics

1.1 Introduction We know that an attempt to explain all concepts can hardly be called scientific. Some concepts require acceptance without explanation. Whatever, we the scientists say, is expressed in terms of limited and approximate descriptions, which are improved in successive steps. We progress from truth to truth and from lesser truth to higher truth. We view the truth, get and absorb as much of it as the circumstances permit, color it with our feelings, understand it with our intellect and grasp with our minds. This makes the difference among human beings and sometimes generates contradictory ideas. Nonetheless, we all belong to the same universal truth. In this preliminary chapter, we shall attempt to assemble many of the questions and paradoxes that have been raised and observed in the present real number system, set theory and theoretical physics. These are mainly due to not having a clear understanding of the concepts like infinity, large and small numbers and the set axioms. In physics, the raised questions essentially deal with non-observable dark matter, the Cosmos, the basic constituent of matter, Big Bang, and many other related notions. For example, Section 1.1 deals with the retrospection of fundamentals in analysis where a clear exposition of some important concepts is presented. Section 1.2 is concerned with the problems in physics and several basic unanswered questions are incorporated. The final section 1.3 discusses some related notions such as indexcosmic waves, generalized fractals and vibration that are useful later.

1.2 Retrospection of Fundamentals As we are aware, much of mathematical work in the 20th century has been devoted to examining the logical foundations and structure of the subject. One of the major influences on 20th century mathematics is Cantor’s introduction of infinite sets into the vocabulary of mathematics. The interest in the set theory developed rapidly until virtually every field of mathematics has felt its impact. Under its influence, a considerable unification of traditional mathematics did occur and new mathematics has been created at an explosive rate. Another interesting development is Alan M. Turing’s change of definition of numbers which concentrated on what a machine could produce using programs. Thus a computable number is a number for which there is a program to compute it in some Turing machine, to as many digits as we may specify. Thus Turing changed the concept of a number. For example, the number π is now a program which generates some six billion digits and is no longer the original infinite (non-ending) representation. This takes away our attention from actual infinite back to a finite, though large but V. Lakshmikantham et al., The Hybrid Grand Unified Theory, Atlantis Studies in Mathematics for Engineering and Science 3, DOI 10.1007/978-94-91216-23-7_1, © 2009 Atlantis Press/World Scientific

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unbounded number which is nearer to potential infinite (personal infinity, a concept we introduce in Chapter 2) of ancient times. Most mathematicians are aware of this change in the definition of a number but those involved with computation always make use of it. The crisis in the foundation of mathematics, that is brought about by the discovery of paradoxes in the wake of Cantor’s general theory of sets, has resulted in numerous attempts at the resolution. These have given rise to the three main schools of thought or philosophy. They are logistic, intuitionist and formalist schools, each of which dealt with paradoxes of general set theory in its own way. Let us list below a few examples to illustrate the dificulty involved in defining certain sets. (i) (Bertrand Russell) Let M be the set of all sets where each element does not belong to itself, i.e. M = {m : m ∈ m}. Then it must be either M ∈ M or M ∈ M. If M ∈ M, then the defining conditions for M holds and M ∈ M. On the other hand, if M ∈ M, then M satisfies the defining condition and therefore M ∈ M. A way out of this dilemma is to agree that M is not really a set and this kind of self reference is a source of ambiguity and should not be used in defining a set. (ii) The famous Russell antonym: A Cretan (native to Crete) saying “All Cretan’s are liars.” Is he telling the truth? (iii) The barber paradox: The barber of Seville shaves those and only those who do not shave themselves. Who shaves the barber? Another source of ambiguity in mathematics are statements involving the universal or existential quantifier on an infinite set. Such statements are unverifiable. For example, to verify that every element of an infinite set has property A, we check an element to see if it has the property and keep doing the checking process for each element. Obviously, we cannot exhaust all of the elements of the infinite set and thus we cannot verify if the property A holds for all elements of the set. In mathematics, particularly theory of numbers, there are many statements involving the infinite set of natural numbers which raise questions that remain unanswered i.e. the statements are neither proved nor disproved. Let us list the following: (i) A perfect number has the sum of its proper factors equalling the number itself. The first few known perfect numbers are 6, 28, 496, 8128, and 33, 550, 336. The question is, are all perfect numbers even? (ii) Twin primes are prime numbers that differ by 2, like 3 and 5 or 11 and 13. The questions is, are there twin primes that are arbitrarily large? Does there exist an infinite number of twin primes? (iii) Goldbach’s conjecture which says that every even number except 2, is the sum of two primes. For example 4 = 2 + 2, 6 = 3 + 3, 8 = 5 + 3 etc. Question is, is the conjecture true? Consider the statement with the existential quantifier: The decimal expansion of π has no row of one hundred threes. True or not, is not known although extensive calculation on the decimal expansion of π has not yielded such row of threes. The probability that this statement is true is  9 100 1 − ( 10 ) which is almost 1. This shows that even if the probability that a statement is true is near 1, it is not a total certainty. Among the field axioms which deal with the properties of real numbers R, the following two are important for us: (i) trichotomy axiom which says that for any a, b ∈ R, exactly one of the following is true, a = b, a < b or a > b; (ii) completeness axiom that says every non empty subset S of R that is bounded above (has an upper bound) has the least upper bound.

Basic Problems of Mathematics and Physics

3

We also require the axiom of choice related to sets: Given any nonempty set A, whose members are pairwise disjoint nonempty sets, there exists a set B consisting of exactly one element taken from each set belonging to A. B is called the choice set. Though many useful results in mathematics depend on the axiom of choice, this axiom has been seriously challenged by some mathematicians since they feel it is meaningless though not false and many others accept it since it seems reasonable. However, Banach and Tarski proved a most disconcerting result using the axiom of choice, which goes against our intuition and so is considered a paradox (the Banach-Tarski Paradox). It essentially says that if a soft ball is suitably sliced into infinitely small little pieces, then the pieces can be suitably rearranged, without distortion, and reconstructed into a ball, the size of the earth. This is a contradiction in R3 inherited from the reals and attributed to axiom of choice. The specific source of the problem is the Archimedian property of the reals which says given any real ε > 0 no matter how small and any number M, no matter how large, there exists some number N such that N ε > M. This allows us to form a arbitrarily large object from arbitrarily large number of arbitrarily small pieces [59]. A common feature of elements of any well defined set is to have the property of potentially exhibiting certain qualitative or quantitative activities which result in the production of certain static or dynamic structures that are essentially for the continued existence or the survival of the elements and therefore for the set itself. This implies certain order relations between elements. The order need not always be a fixed one. The temporal sequence of activities induces temporal partial ordering. For example, biochemical reactions in a cell occur in a certain order. Remember every part of a cell, every structure in it, is the result of the primary activity of the genes, which control the production and functioning of every structure in the cell. Sometimes only a subset of the set exhibits potentially possible activities and may wait for the need to arise. Thus, we can always find at least one monotone sequence in any given set and so there exists always an algorithm how to choose an element. Consequently, we replace the axiom of choice by the strong axiom of choice and successfully construct the choice set. If we find a set whose elements are neither known to us nor have any property we can find in order to induce certain ordering, then the weak axiom of choice is employed by some inherit property of the set to find a choice set. The concept of an irrational number was introduced into the set of fractions (quotients of the form y = 0, x, y are integers). It is known that fractions (rationals) have terminating decimal expansions or a pattern of repetition of digits. While defining irrational number as one that does not have a fractional representation, we encounter decimal expansions which are nonterminating and have no periodic repetition of digits. This fact that such numbers have an infinite number of digits in their decimal expansion make it an ambiguous, hence non-verifiable, concept. There is no way to verify if the decimal expansion of π has indeed no periodic pattern of digits, since we cannot compute all if its digits. x y,

Another example of encountering difficulty when dealing with ambiguous or vaguely defined concepts is the following traditional construction of an irrational number as the limit of a sequence of rationals. This also provides a modification of Felix Brouwer’s counter example to the trichotomy axiom [5]. Let C be a given irrational number (whose decimal representation is known up to only first n digits). We want to isolate C in an interval such that all the decimals to the left of C are less than C and all decimals to the right of C are greater than C. We do this by constructing a sequence of smaller and smaller rational intervals such that each interval in the sequence is contained in the preceding one and such a sequence is called a nested sequence. An interval is rational if their left and right end points are rationals. In the construction, we skip those rationals that do not satisfy the condition of providing the end points of intervals of this nested sequence.

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The Hybrid Grand Unified Theory

Given two rationals x, y we can tell if x < y or x > y. Even then, we cannot line up all the rationals on the real line under the ordering <, since there is an infinite number of rationals between any two given rationals and this is due to the undefinable nature of the concept infinity. However, we can proceed with the following scenario: Start with a certain rational interval [A, B] with A < C < B, and find a nested sequence of rational intervals [An , Bn ], with A < An < C < Bn < B, for each n = 1, 2, 3, . . . , n. At each stage, we want to make sure that An  C − 10−n , Bn  C + 10−n , and [An , Bn ] ⊂ [An−1 , Bn−1 ] ⊂ [A, B]. Since the irrational number C is defined to be the limit of sequences of rationals, we can choose the end points An , Bn , of intervals [An , Bn ] as members of two sequences {An }, {Bn } where {An } is a monotonic increasing sequence and {Bn } is a monotonic decreasing sequence of rationals satisfying A  A1  A2  · · ·  An < C < Bn  · · ·  B2  B1  B, and for each n,

C − An  10−n and Bn −C  10−n

This process can be continued as long as we are able to identify An , Bn to be such that An < C < Bn i.e. as far as we know the representation of C with its n decimal digits. It cannot be taken further since we are unable to find An+1 , Bn+1 with error of 10−(n+1) and establish An+1 < C < Bn+1 , with C being known only to n places. No matter how large the number n is, we still have the disadvantage of not getting the next interval [An+1 , Bn+1 ]. Consequently we have to acknowledge the inherent trouble involved with understanding and dealing with irrational numbers and with the concept of infinity. This example shows that the real number system has no ordering under the relation < and the trichotomy axiom which says, given two real numbers x, y, only one of the following holds: x < y, x = y, x > y, is unverifiable. Next we shall give a proof to show that rational and irrational numbers in the reals are not dense. Let p ∈ R be any irrational number and {qn } be a sequence of rationals converging to p from the left in the natural ordering of reals. Let dn be the distance from qn to p and take an open ball of radius dn /10n , with center at qn . Note that qn tends to p but distinct from it for any n. Take an open ball of the same radius dn /10n , centered at p and take the union of open balls, centered at qn , as n → ∞ and call it U. If r is any real, rational or irrational, to the left of p, then r is separated from p by two disjoint open balls, one in U and the other in its complement, center at p. If p is rational, then we take {qn } as a sequence of irrationals that tend to p, which is allowed by the Axiom of Choice. The same result would hold for any r distinct from and to the right of p. Mathematics is a universal language form that is well suited to talk about concepts that are abstractions of the human mind as well as concepts that relate to and attempt to describe the physical universe around us, in terms of several laws of nature. For purposes of logical rigor and consistency, we have to start with certain set of symbols, concepts and premises (these may change as available knowledge advances) and proceed to develop mathematics as a deductive system. Its obvious tools are measurement and computation which are limited. In order to represent the external world through mathematics, just measurement and computation alone would not be adequate. It requires abstraction, intuition, imagination, visualization, trial and error in order to sift out what is more appropriate, thought experimentation, creativity, thinking backwards and the art of making inferences

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and drawing conclusions. These activities are collectively present in qualitative mathematics, which is complementary to measurement and computation. The skill of thinking backwards involves figuring out what we need as premises or boundary conditions to obtain a desired result. As example, consider the famous inverse (ill-posed) problem known as gravitational n-body problem: Given n bodies in cosmos at time T , of given masses, positions and velocities and subject to their mutual gravitational attraction, find their positions, velocities and paths at later time t > T . This problem is ill-posed because the bodies have cosmological history and initial, boundary conditions belong to the past and we do not know what they were. In spite of the fact that mathematics is known for its universality, the power of its abstractions and growing usefulness in all fields of sciences and arts, we have to admit that there are certain sources of uncertainty or ambiguity in mathematics. These essentially deal with the ideas of infinity and infinitesimal and consequently, with very large and very small numbers. Scientists and practitioners of computation have tried to cope with this problem by approximation and use of scientific notation to represent any order of magnitude, large or small, by using powers of ten. For example, the radius of our visible universe is of order of magnitude 1010 light years while the order of magnitude of the basic constituent of matter (a non-agitated superstring) is less than 10−14 . The ambiguity involved in approximation can be minimized by emphasizing the order of magnitude of the error, at every level of approximation. Even then certain concepts like infinity and infinitesimal remain mainly intuitive and ideal. We need to employ some new concepts to denote the pragmatic level at which these can be handled, in computations and deliberations. Also, we have to make sure to avoid vacuous concepts and statements because these invariably lead to contradictions. For example, consider the statement “The largest integer is 1” and its proof: Let N be the largest integer and by ordering axiom, N < 1, N = 1 or N > 1. First option is ruled out because of the definition of “largest”: Take N > 1 which gives N 2 > N, contradicting the assumption that N is largest. Therefore N can only be equal to 1. In this formulation of Perron paradox, the culprit is the vacuous concept “largest integer”. Having pointed out certain challenges and ambiguities that are present in mathematics, we shall revisit the real number systems in Chapter 2, with a purpose to deal with some of the questions raised here and discuss the development of the decimal system of numeration which is more suitable for the present era of high accuracy computation with the aid of technology.

1.3 The Problems of Physics Report from the Hubble says: matter forms in the supposedly empty space between cosmological bodies at the staggering rate of one star per minute capped by the recent discovery of two baby galaxies and indication of more in the last four years [4], [108], [105]. First cosmic dust forms then it gets entangled into cosmological vortices and collects at their cores (core: collected mass around the eye) and become cosmological bodies like galaxy, planet and moon. There are places in the cosmos called star nurseries that produce stars at quite a rapid rate [105]. While these findings resolve the puzzle of the missing 95% of matter in the Cosmos, that, after all, it is there but for cannot be detected, physics is faced with the unprecedented challenge of how to study matter whose existence has no direct evidence whatsoever. And yet it exists in view of the first law of thermodynamics that says energy, therefore, also matter, cannot be created or destroyed. Since then that missing matter has been called dark matter because light, our medium for observation, cannot detect it and the question is how to deal with the nonobservable like dark or invisible matter. At the same time, there are long-standing unsolved problems of physics such as the gravita-

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The Hybrid Grand Unified Theory

tional n-body and the turbulence problems as well as some old fundamental questions that remain unanswered to this day. Physics has abandoned some of them, for example, the gravitational n-body problem and the structure of the electron, but is in hot pursuit of others such as the 5,000-year-old quest for the basic constituent of matter and the turbulence problem. The pursuit of the former has absorbed staggering amount of resources during the atom-smashing frenzy of the last half century and for good reason. Unless we know that basic constituent we really do not know what matter is and physics has correctly assessed that this question is the key to the resolution of its fundamental questions and longstanding unsolved problems. Let us list down the others. (1) What is gravity? (2) What is a black hole? (3) What is the so-called elementary particle and what is its structure? (4) What is superconductivity? (5) What are cosmic waves and what are their source and medium? (6) What is charge? (7) Explain magnetic levitation (the basis of the development of the magnetic train). (8) What was the Big Bang? (9) How do galaxies and other cosmological bodies form? (10) Why do they spin? (11) What is the destiny of our universe? The questions spill over to the applications of physics. In biology we have these questions: (1) What distinguishes living from non-living organism? (2) How does the brain work? (3) What is cosmic energy and what is the nature of its interaction with the brain? (4) How does a mutant spread in the body? In psychology, there is a need for physics-based theory of intelligence to explain: (1) Intelligence, (2) Learning and creativity, and (3) Cognition, i.e., the human ability to know the real world and express that knowledge as physical theory. Then there are astonishing and paranormal phenomena some of which now have physical explanation. Paranormal refers to natural phenomena having no physical explanation yet. They include: (1) Kerlian photography, (2) Human aura, (3) Human levitation, and (4) Telekinesis.

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The present methodology of physics has been to study the modeling of physical concepts computationally and mathematically, but measurement and computation cannot answer these questions. At best they can only describe them. Nor can they solve long-standing problems of physics such as the gravitational n-body problem for, as we noted earlier, some basic concepts like mass and gravity are not known. Newton’s law of gravitation only describes the behavior of two bodies subject to their mutual gravitational attraction. Quantum physics has a number of differential equations. Their solutions are supposed to reveal the behavior and properties of physical systems. However, they are external to them and do not reveal their internal structure or dynamics and, therefore, the solutions lack the capability to predict the course of further development. Moreover, a lot of differential equations of physics are unsolvable. One of them is the Navier-Stokes equation of fluid mechanics which is supposed to describe the behavior of turbulence in fluids and solve the turbulence problem. At any rate, we summarize the inadequacy of computation and measurement as a tool to solve problems in physics: (1) It has no predictive capability because it only describes behavior and properties mathematically and statistically. (2) The mathematical model of a physical system belongs to some mathematical system. The physical system belongs to the physical world but the model is man-made and represents thought. Therefore, they are independent, each one determined by its own premises. Therefore, the mathematical model cannot provide the solution of problem belonging to another system independent of it. (3) A mathematical model is static and unaffected by and not sensitive to emerging physical conditions, e.g., interaction with other physical systems, which are dynamic. (4) Computation on the mathematical model reveals the behavior of the mathematical system the model belongs to and is independent of the physical system it models; in general computation provides information about some mathematical system; any inference about the physical system derived from it amounts to reasoning by analogy. (5) Global behavior of a physical system and its model is generally independent of local behavior; e.g., a differential equation and its solution have only local validity; global phenomenon, like gas turbulence cannot be derived from local behavior such as properties of individual molecules. (6) Most differential equations are unsolvable. (7) Complicated configurations of matter are difficult if not impossible to model mathematically, e.g., the nucleus of an atom; even a model that takes its constituent parts into account yields only limited information; no amount of computation, for example, can explain why protons coexist in the nucleus despite their tremendous repulsion. (8) Computation is inadequate for the study of chaos, e.g., limit set of fractal [17], [53] and dark matter as fundamental chaos. To be able to know the nature of any physical system, i.e., its internal structure and dynamics, it is necessary to know how nature works and that requires knowing its laws of motion and behavior called natural laws. They are revealed by the behavior of physical systems or natural phenomena. But they have to be articulated to be able to explain them. The articulation is the task of the scientist. Such knowledge of nature, articulation of its laws and explanation of natural phenomena cannot be accomplished by computation and measurement. We need qualitative mathematics, their complement, to do the quintuple task of knowing nature, articulating its laws, using them to explain natural phenomena, predicting their future development and solving scientific problems including technological problems. This is what we call qualitative modeling, the complement of mathematical modeling;

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The Hybrid Grand Unified Theory

therefore, it is non-computational. Since we are introducing qualitative mathematics and qualitative modeling, we shall refer to them from now on as computational mathematics and computational modeling. Qualitative analysis goes with computational modeling in solving scientific problems. Recall that the earliest known law of nature is the first law of thermodynamics that says energy cannot be created or destroyed (note that neither computation nor measurement is involved in its statement) [74]. Since energy is motion of matter the first law can be stated equivalently as: matter cannot be created or destroyed. However, as we noted earlier, the first law does not take latent energy into account. Thus, it has inadequacy. We give one more example of its inadequacy with this thought experiment: shoot a beam of light into a vacuum and turn it off. What happens to the energy of light? It vanishes without trace because no atoms or molecules absorb and convert it to kinetic energy. Otherwise, it would have meant rise in temperature along the beam. At any rate, we shall upgrade the first law, call it by another name and consider it the first milestone in the search for the basic constituent of matter, in Chapter 3.

1.4 Related Concepts We shall begin to identify patterns of behavior and regularity in the motion of matter from which we shall draw out the laws of nature, which help to maintain stability and order in our universe.

1.4.1 Waves and Vibration Wave is synchronized motion of opposite forces in the medium. When one drops a piece of rock into a pool of water it pushes its molecules down. At the same time, water pressure pushes them up beyond the surface due to momentum which gravity pulls down and gravity pulls them down below the surface, again, by momentum, etc., and the cycle continues. This synchronized cyclic motion of opposite forces induces molecular vibration in and around this spot and, by resonance, imparts it on the molecules around this point causing the wave to propagate outward along concentric circles. Note that the vibrating molecules of water remain in their positions. By energy conservation the radial profile of the wave is sinusoidal, being the optimal balance between symmetry and motion (perfect symmetry precludes motion, motion breaks symmetry). The sine function is its mathematical model and our analysis is the qualitative model. Resonance between waves or vibrations occurs when they have the same order of magnitude of arc length or frequency. We can actually experiment with it. We put two tuning forks with the same pitch (same frequency) nearby. If we strike one it will produce a sound of that pitch; in a few seconds the other tuning fork will also produce the same sound and pitch. That is resonance. Consider a guitar string made of strong elastic material like steel. It is pulled tight by two pegs at its ends. We pull and stretch at the middle away from normal position and release it. The pegs pull it towards normal position but its momentum pushes itself beyond and towards the other side of the normal position. Again, the pegs pull it back toward normal position but goes beyond because of momentum, etc. The constant pull by the pegs has dampening effect and the energy imparted by the stretching eventually dissipates and soon the string is in normal position at rest. If we take the configuration of the string at any time away from the normal it will be sinusoidal, one of the regular patterns in nature that we shall synthesize and integrate into and state as a law of nature later. Moreover, if we plot the path of the midpoint of the string against time it will be sinusoidal that flattens along the normal position as its energy dissipates. The swinging of the clock’s pendulum if plotted against time is also sinusoidal and the striving for

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perfection in clock making opened up a field of mathematics about sinusoidal motion called harmonic analysis. Electromagnetic waves, guitar vibration and the swinging of the pendulum are harmonic motion. We can just imagine how much mathematics went into the development of stereo music and radio and television broadcasting as well as clock making. The characteristics of the vibration of the string revealed by the sound it makes in terms of pitch and tone depends on its internal structure such as kind of material, diameter, length and tightness, etc. The law of nature it reveals follows: Internal-External Dichotomy. The interactions and dynamics of a physical system are shaped by the internal and external factors; in general the internal is principal over the external and the latter works through the former. There are many examples of how this law is realized in nature. The chicken egg has internal structure and properties but the principal internal factor is the gene and the external factor is the right temperature around the egg. The same is true of the germination of a seed. Even in human society this law applies. The internal factor is principally the totality of its socio-political-economic forces and the external factor its external relations with the rest of the world. That is why it is very difficult for a country to impose its rule on another.

1.4.2 Cosmic Waves Everything vibrates due to the impact of cosmic waves from all directions in the Cosmos. The vibration of the atom vibrates its nucleus that generates Type I or basic cosmic or electromagnetic waves. Basic cosmic wave serves as carrier of photon, radio or television signals and thought or signal from the sense organ or the gene, i.e., information encoded by neural vibration, or converted signals from a sense organ or vibration characteristics of the gene. In the last case, the encoded basic cosmic waves are called brain waves. In the latter neural vibration is superposed or encoded on basic cosmic wave. Waves as we know them ride on some medium. For now we do not know what that medium is for basic cosmic waves but we shall find out later. Suffice it to say that it is the same stuff as the flux around the nucleus of the atom on which the orbital electrons ride as well as the flux on which the electrons ride through the conductor in electrical transmission. The nucleus is very small, its vibration vigorous. The finer the wave it generates the more energetic it is because its kinetic energy is given by E = h f , where f is frequency and h is Planck’s constant, one of the constants of nature; its value is h = 6.6 × 10−34 joules [135]. That is only a minor part of its energy because as we shall see later the nucleus of an atom is fractal consisting of sequences of finer and finer vibrating parts so that when we add up all the products h f , the sum is huge amount of kinetic energy. Radio wave is basic cosmic wave that carries radio signals. It is so energetic it penetrates solid obstacles that we can turn on our radio in our room. But it is distorted by metal and concrete metal reinforcement. Thus, an outdoor antenna is needed for fine music. When two huge bodies press against each other (compression) with tremendous force, e.g., at interfacing tectonic plates (hard rocks that cover the Earths interior) some 40 km below sea level the atoms at their interface vibrate vigorously. It is similar to the vigorous vibration of the guitar string except that tension is involved instead of compression. When the tension is raised by tightening the string its vibration rises and produces higher pitch (the same effect when the finger presses the guitar string). At any rate, the vibrating atomic nuclei at the interface generate two-layered nested fractal sequences of basic cosmic waves (i.e., having finer and finer wave structure, as we have seen earlier, that stores huge amounts of energy) [25], [51], [21]. They are called seismic waves. Seismic wave

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The Hybrid Grand Unified Theory

has two components each coming from the interfacing bodies. They are extremely energetic due their fractal astructures and are known to soften or melt metal and pulverize concrete that cause so much destruction during an earthquake. They irritate sea creatures like whales and dolphins before an impending earthquake when their intensity rises. This is the explanation of the so-called mass suicide of these mammals when they jump off the seacoast into shallow water or dry land. Tectonic earthquake is the result of the steady dis-alignment of the interfacing tectonic plates due to the tidal cycles, where the oceanic plate tends to move down because of the Sun’s and Moon’s push on the ocean during low tide. When the plates break due to uneven compression of dis-alignment the ocean plate goes under, causing devastating tremor (earthquake). Jelly fish also rise from the depths when there is raised intensity of seismic waves. In 1999 the water filter of a power plant’s cooling system in the Philippines was clogged by jellyfish causing blackout over the Northern Island of Luzon on the eve of a devastating earthquake. During intense seismic wave propagation of impending earthquake, roaches fly off the floor and wall to escape irritation, dogs howl in distress and horses jump and run erratically. The Chinese are quite adept at interpreting the behavior of animals for predicting earthquake. Every time there is strong compression, e.g., nuclear energy released in nuclear explosion that compresses the nuclear fuel against the bomb casing, seismic waves are generated. Lightning also generates seismic waves due to compression of ions against the surrounding atmosphere.

1.4.3 The Sine and Related Curves The sine curve is a very important curve in both mathematics and physics because of its central role in harmonic analysis and the universality of oscillation in nature. It has optimal properties such as optimal balance between symmetry and motion, since motion breaks symmetry and perfect symmetry precludes motion. In a physical system symmetry expresses energy conservation; so does smoothness that the sine curve possesses. The sine curve is everywhere in nature: profile of water wave, electromagnetic wave, photon and vibrating guitar string since its graph against time is the sine curve. The configuration of the electron and all other simple elementary particles is sinusoidal helix and the helix and circle are related to the sine curve, all of them universal configuration of nature. The projection of a helix on the plane through its axis is the sine curve. The helix can be drawn quickly by tracing an imaginary horizontal circle and lifting it.

1.4.4 Generalized Fractal Geometrical fractal is a sequence of affine transformations (combination of contraction and translation) of a geometrical figure at the decreasing scale. Since the images of both contraction and translation are similar to their respective pre-images the image of affine transformation is also similar to its pre-image. Therefore, affine transformation is called self-similar transformation and its essential feature is self-similarity. We generalize fractal to include, in addition to contraction and translation, sliding along a curve, rotation and taking mirror image. Moreover, each term in the sequence is similar to the other terms. When each term in the sequence except the first belongs to the preceding term it is called nested fractal. We generalize this concept to include general properties not just geometrical properties. For example, consider the root of a tree. The trunk extends to several primary roots, each primary root splits to several secondary roots, each secondary root splits to tertiary roots, etc. Here the branching property of the primary roots is replicated in each term in the sequence and every term is a branch of the preceding term except the first term attached to the trunk.

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Moreover, the trunk splits into branches, each branch splits into smaller branches, each small branch splits into twigs, each twig grows leaves, each stem of the leaf splits into primary vein, each primary vein splits into secondary veins, then tertiary veins and each tertiary vein ends up into the stomata. Both examples are nested fractal. We call them generalized fractal. Fractal is nature’s way of packing huge energy in a physical system or allowing a physical process to proceed at optimal efficiency. In the above example the fractal roots collects nutrients from the soil most efficiently and pass them on to the fractal branches, secondary branches, twigs, and the fractal veins of the leaf through the stomata manufacture food. Then food is distributed through the same fractal to where it is needed, e.g., to the flowers for fruit bearing. Thus, fractal like oscillation is universal. Other examples of generalized fractal some of which we shall take up later are: (1) Cosmic waves. The three types are: (a) type I, i.e., basic cosmic or electromagnetic waves, (b) type II seismic waves and type III, i.e., seismic waves generated by such explosions as lightning, supernova and nuclear explosion [25], [51], [17], [47]. Basic cosmic waves encoded with information from the brain, i.e., superposed with neural vibration, are called brain waves. (2) Our universe as a super...super galaxy. It is a generalized fractal with our universe as first term, then galaxies, stars, planets and planetoids, moons, and cosmic dust. They form nested fractal sequences of macro vortex fluxes of superstrings. (3) River. It splits into tributaries, springs and rapids in the mountain. Water from the mountain flows into the rapids and on to the springs and tributaries that join a river; then the latter empties into the sea.

Chapter 2

Mathematics of Grand Unified Theory

2.1 Introduction An improved real number system, devoid of the paradoxes of the present system, that was effective in resolving certain fundamental problems in Physics, leading to a grand unified theory, was discussed by Escultura in a number of papers [32]-[49]. For example, the notion of dark numbers was introduced to explain dark matter. On the other hand, the fact that the well known notions of natural numbers, rational and irrational numbers , zero, infinity and the place value system of decimal numeration among others originated in the Vedas, the ancient scientific texts of India, seems to have been lost and later ignored totally. Utilizing the ideas originating from the Vedas, [83, 60] a hybrid real number system is proposed to strengthen the existing real number system. In Section 2.2, we make an effort to clarify the construction of the hybrid real number system and the evolution of ideas concerning the nonterminating decimals and dark numbers. Also, the notion of personal and impersonal infinities are incorporated to bring out the duality between“ideal” and “pragmatic” and some applications of the new number system are given. In Section 2.3, we describe some needed special functions and the generalized curves. We introduce in section 2.4, the generalized fractals and chaos. Finally in Section 2.5, we deal with the integrated Pontriagin’s maximum principle which applies to generalized curves.

2.2 Hybrid Real Number System Pursuit of truth is the goal of all types of scientific and intellectual endeavors and is carried out in many different disciplines. However, a common tool for many of these pursuits is Mathematics. It is a matter of common acceptance that mathematics is the creative genius of the human mind and intellect and the concept of numbers is at the foundation of mathematics. It is interesting to dwell on the origin of numbers which takes us to the ancient scientific texts of India, called Vedas, [7, 122, 123]. The notion of “oneness” is often alluded to in the Vedasamhitas, in reference to the one entity which forms the substratum of everything that exists. “Ekam sat viprah bahudha vadanti” is a famous vedic declaration that says “That which exists is one; seers call it by various names”. Some of these names include That One (Ekam), Absolute Reality, Impersonal Being, etc. This philosophical perception of unity leads to the notion of the number one that modern mathematics represents by the numeral 1. We accept the above vedic declaration as a fundamental postulate. i.e. there exists the natural number one. Isopanishad proclaims the following: “The invisible (un-manifest Impersonal Being) is infinite, the visible (manifest) too is infinite. From that infinite Impersonal being, the visible universe of infinite expanse has emerged. The infinite (un-manifest) remains the same, even though V. Lakshmikantham et al., The Hybrid Grand Unified Theory, Atlantis Studies in Mathematics for Engineering and Science 3, DOI 10.1007/978-94-91216-23-7_2, © 2009 Atlantis Press/World Scientific

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an infinite (visible) universe has come out of it”. This does not imply that the visible universe is disconnected with the Impersonal being. It is still connected so that it can be absorbed back into itself. In other words, we can say that this ONE underlying unmanifest reality has manifested as the visible universe without any distortions or aberrations to itself. In this visible universe, all human cognitive experiences involve the perceiver and the perceived. This corresponds to the notion of duality (Dvaita) , the number two, represented by the numeral 2 and the algebraic process of addition (+) yielding 1 + 1 = 2. Proceeding in this manner, the set of basic numerals 1 through 9, are generated. On the other hand, the nonperceivability of the Unmanifest and the negative way of realizing the Impersonal being is known as “Sunya” and this leads to the concept of zero, represented by the numeral 0. With this symbol, the genius of the place-value system of ancient India, gives 1 + 9 = 10 which is utilized further to create the numbers 11, 12, . . . , 19, 20, . . . , 99, 100, . . . , 999, 1000 and so on. The number 1031 (1 followed by thirty one zeros) is known as Parardha and 1062 (1 followed by sixty two zeros) is known as Mohogha. We denote by the symbol , N, the set of all these natural numbers. That is, N = {0, 1, 2, . . . } [60, 82, 83]. This raises the question of how far we can go in keeping count of such numbers and the concept of infinity to describe that which is not finite. Drawing a philosophical parallel, we see that the visible universe as an infinite extension of that one Impersonal Being, which is a mixture of existence and nonexistence relative to our perception since it depends on our senses. As our senses improve, i.e. as our true knowledge of the universe increases, our perception changes. Though our visible universe is complete in the sense that it has everything that is needed, we can grasp or observe only a portion of it and we have been constantly improving in that effort. Nonetheless, there is a large percentage of dark matter as well as antimatter, regarding which we have no clue yet. The concepts relating to the system of numbers are highly dependent on our perspective. Thus, the concept of infinity and of the infinitesimal are entirely based on our perception (of largeness and smallness) and changes as our knowledge changes. It is up to the individual or group of observers and their computational context to find personal infinities and personal infinitesimals [121, 123]. For example, to a physicist, infinity may mean letting an experiment run for a very long duration of time while an astronomer may see infinity as the distance to the end of the universe. Ideas such as infinity and unboundedness are always relative to one’s personal visible framework of time and space, culture and society. Thus, we distinguish between personal infinities (comprehensibly large numbers) and impersonal infinity (ideal infinity) which goes beyond the idea of all personal infinities. We shall denote them by ∞ p and ∞I respectively. Note that ∞ p may not be necessarily infinite and it is a context dependent notion of arbitrary largeness.

2.2.1 The Construction of Hybrid Real Number System The computational requirements of the modern scientific world demand that we deal with numbers of all sizes, from extremely large numbers to very small numbers. In order to proceed with the construction of the hybrid real number system, we accept the following [38, 44, 60]: (1) The use of primary natural numbers 1, 2, . . . , 9 together with 0 and the use of 10 as the base for representing all decimal numerals. The decimal representation of numbers is highly significant and beneficial.

Mathematics of Grand Unified Theory

15

(2) The addition operation (+) and its natural extension multiplication (·); the dual operation, subtraction (−). The usual rules of sign are valid in all these arithmetic operations. (3) Subtraction leads to the notion of additive inverse. Each real number has a unique additive inverse, n + (−n) = 0. Equipped with these essentials, a number an an−1 · · · a1 a0 can be expressed in the form, an an−1 · · · a1 a0 = an (10)n + an−1 (10)n−1 + · · · + a0 (10)0 , where ai ’s , i = 0, 1, . . . , n are basic numerals of the set {0, 1, . . . , 9} and n ∈ N. Division by a nonzero number leads to the notion of multiplicative inverse. Division of an integer x by a nonzero integer y, or quotient, denoted by xy and defined by: xy = x 1y . Even though a rational number can be defined as the quotient xy , y = 0 of two integers it is not satisfactory since the meaning or representation of a number that is not rational is vague. Instead, we define, a terminating decimal by, an an−1 · · · a1 a0 .b1 b2 · · · bk =

n

k

i=0

i=1

∑ ai (10)i + ∑ bi (10)−i

= an (10)n + an−1 (10)n−1 + · · · + a0 +b1 (0.1) + b2 (0.1)2 + · · · + bk (0.1)k , where n, k ∈ N. Here, an an−1 · · · a0 is the integral part, and b1 b2 · · ·bk is the decimal part. Note that each terminating decimal is clearly defined. We formally define an integer as the integral part of a terminating decimal.The use of base 10 and place value system defines such an integer clearly. It is well known that the quotient xy results in a terminating decimal only if y has no prime factor other than 2 or 5. Keeping the right summation index k fixed and finite, letting n tend to a personal infinity ∞ p , which here is a large finite number, we get ∞p

k

i=0

i=1

∑ ai (10)i + ∑ bi (10)−i

which is a terminating decimal with a large integral part. Further, based on the property of the existence of the additive inverse, we also have negative terminating decimals. We denote by R the set of all terminating decimals. The usual algebraic operations of addition, subtraction and multiplication are well defined in this system. However, some nonzero integers do not have multiplicative inverses. A terminating decimal with an integral part that has a prime factor other than 2 or 5, does not have an inverse in R. Thus, R is not a ring nor is a field. The mapping, N → N.000 · · · which is an isomorphism that embeds the natural numbers in the system of terminating decimals. In xy , if y has any factor other than 2 and 5, it does not result in a terminating decimal. For example, 7 = 2.3333, 3

1 = 0.052631578947368421 19

16

The Hybrid Grand Unified Theory

Formally, one encounters nonterminating decimals, when considering an expression of the form n

∞p

i=0

i=1

∑ ai (10)i + ∑ bi (10)−i ,

where ∞ p is a personal infinity that is equinumerous to a infinite countable set. We may also consider expressions of the form ∞p

∞p

i=0

i=1

∑ ai (10)i + ∑ bi (10)−i .

However, the term nonterminating decimal is vague and it needs to be clarified further. In the next section, we define non-terminating decimals, decimals that are not terminating, in terms of elements of R and hence propose a formal extension of R.

2.2.2 The Nonterminating Decimals Usually, a nonterminating decimal refers to an infinite array of digits in the decimal part of the number. Most often, all the digits that occur are not known. For example, in the decimal representations of the quotients xy , y = 0 if y has prime factors other than 2 or 5, then we have a nonterminating decimal with features of repetition of a single or group of digits. Also, the numbers known as irrational numbers in traditional mathematics are known to have nonterminating decimal representations with no particular pattern in the occurrence of digits. In fact, we only know the digits to the extent that we compute them. This description, however, does not constitute a formal definition and we wish to present a precise notion of a nonterminating decimal keeping the well defined terminating decimals as a central point of reference. With the notion of nonterminating decimals in place, we propose a hybrid real number system R∗ based on R, the set of all terminating decimals. Consider a sequence of terminating decimals of the form, N.a1 , N.a1 a2 , . . . , N.a1 a2 · · · an , . . .

(2.1)

where N is an integer and the a j ’s are basic numerals. We call the sequence (2.1) a standard generating sequence or g-sequence. The non-terminating decimal, N.a1 a2 · · · an · · · is defined and approximated by its

nth

(2.2)

g-term, N.a1 a2 · · · an ,

(2.3)

which is a terminating decimal. The non-terminating decimal (2.2) is referred to as the g-limit of the g-sequence (2.1). The nth g-term (2.3) approximates the g-limit at margin of error (maximum error) 10−n . Note that the n th g-term repeats all the previous digits of the decimal in the same order so that even if a finite number of terms of the g-sequence (2.1) are deleted, the nonterminating decimal it defines, i.e., its g-limit remains unchanged. We introduce a norm, called the g-norm and define that the g-norm of a decimal is itself. The g-limit is defined in terms of the g-norm. Define the nth distance dn between two non-terminating decimals a, b as the numerical value of the difference between their nth g-terms, an , bn , i.e., dn = |an −bn | and the distance between a and b is the g-limit of the g-sequence {dn }; Note that, at each step

Mathematics of Grand Unified Theory

17

only terminating decimals are involved and only the g-limit will involve a non-terminating decimal. We denote by R∗ the closure of R in the g-norm, (R∗ is the g-closure of R) and it includes the non-terminating decimals. The algebraic operations like addition and multiplication for non-terminating decimals are defined via their nth g-terms, which are always terminating decimals. If a and b are two nonterminating decimals (i) a + b is defined to be the g-limit of the g-sequence {an + bn }, (ii) the product ab is defined to be the g-limit of the sequence {an bn }, where an , bn are nth g-terms of a, b respectively. This is exactly as in standard computation, i.e., approximation by decimal segments at the nth digit. Thus, with our premises, all known properties of the decimals are retained. For a nonterminating decimal x represented by

we have at level n, and the difference dn

= x − nth

0.a1 a2 · · · an an+1 an+2 · · · ,

(2.4)

nth g-term = 0.a1 a2 · · · an

(2.5)

g-term is given by dn = 0.00 · · · 0an+1 an+2 · · · an+k · · ·

which gets smaller in magnitude, with the non-zero digits of the tail an+1 an+2 · · · receding to the right, as n increases to a personal infinity that is equinumerous to a infinite countable set. Thus, as this process continues with n going to any personal infinity and k being finite we get a standard g-sequence of approximations given by 0.a1 , 0.a1 a2 , 0.a1 a2 a3 , . . . , 0.a1 a2 · · · an , . . .

(2.6)

and a corresponding sequence of differences 0.0a2 a3 · · · a2+k · · · , 0.00a3 a4 · · · a3+k · · · , 0. 00 · · · 0 an+1 an+2 · · · an+k · · · .  

(2.7)

n places

Note that each term of the above sequence is a non-terminating decimal. Therefore, we consider a corresponding sequence of differences obtained by taking an approximation for each term of the sequence by truncating the nth term k digits after the first n zeros. Thus, we have, 0.0a2 a3 · · · a2+k , 0.00a3 a4 · · · a3+k , · · · 0. 00 · · · 0 an+1 an+2 · · · an+k , . . . .   n places

This sequence of differences is called a nonstandard d-sequence whose nth term dn is 0.00 · · · 0an+1 an+2 · · · an+k and is not a standard g-term like (2.5). Recall that, in a standard nth g-term of a g-sequence (2.6), the initial digits are repeated in the same order while in the nth term (2.7) of the nonstandard d-sequence in the decimal part, initially there are n zeros and the number of zeros keep increasing as n increases.

18

The Hybrid Grand Unified Theory

Note that as n increases dn gets smaller in magnitude though never reaching zero, since the tail digits are never all zero simultaneously (unless all the ai ’s are zero in the non-terminating decimal x) it becomes indistinguishable from the tail digits of other nonterminating decimals. We call the limit of the nonstandard d-sequence (2.7) the dark number d which is defined and approximated by 0.000 · · · 0an+1 an+2 · · · an+k This term has n zeros at the beginning of its decimal part followed by k more digits, not all zero. In fact, we suppose that an+1 and an+k are not zeros. That is, dn is truncated at the kth nonzero digit in (2.7), with a margin of error (10)−(n+k) . In general, consider the sequence of decimals given by the expression, (δ )n l1 l2 · · · · · · lk ,

n = 1, 2, . . . ,

(2.8)

where δ is any one of the decimals, 0.1, 0.2, 0.3, . . . , 0.9, and l1 , . . . , lk are basic numerals, not all zero simultaneously, n going to any personal infinity and k is finite. For each combination of the choices of δ and the l j s, j = 1, . . . , k, we get a well-defined nonstandard d-sequence whose nth term (which is not a standard g-term) is given by (2.8). This nth term defines and approximates the dark number d which is the limit of the nonstandard d-sequence (2.8). With increasing n this nth term gets smaller but is always greater than 0. If x is a nonterminating decimal, 0 < d < x. We refer to this dark number as the d-limit of (2.8). For example, consider the nonstandard sequence (0.1)n l1 l2 l3

n = 1, 2, 3, . . .

Note that, l1 , l2 , l3 are numerals where l1 , l3 = 0. So, we have the sequence l1 l2 .l3 , l1 .l2 l3 , 0.l1 l2 l3 , 0.0l1 l2 l3 , . . . , with 0.00 · · · 0l1 l2 l3 as the nth term in which there are (n − 3) zeros after the decimal. By varying δ through 0.1, 0.2, . . . , 0.9 and the l j ’s j = 1, 2, . . . , k, we can generate countably infinitely many such nonstandard sequences. In general, for every 0 < r < 1, {rn } generates a nonstandard sequence whose limit is a dark number. Let d ∗ be the set of all dark numbers obtained as limits of nonstandard sequences. To the extent that they are indistinguishable, d ∗ is continuum. Since the elements of d ∗ are indistinguishable, we shall use d ∗ to refer to any one of them in any equation or statement involving dark numbers. Remark 2.1. Note that, in general, dark numbers may be generated by taking any γ and considering the non-standard sequence γ n . Thus, one can see that the set of all dark numbers is uncountable set, that is a continuum. The tail digits of non-terminating decimals join and merge as the continuum d ∗ . We note further that, as in non-terminating decimal, an element of d ∗ is unaltered if a finite number of terms are altered or deleted from its d-sequence. Let us now consider the special case of (2.8) with δ = 0.1 and l1 l2 · · · lk = 1 so that the sequence is {(0.1)n }, n = 1, 2, . . . , and it is identified as the basic or principal nonstandard sequence of d ∗ , and its limit is the the nonstandard basic element of the set d ∗ ; This basic d-sequence is realized when we have the nonterminating decimal x = 0.999 · · · 999 · · ·

Mathematics of Grand Unified Theory

19

consisting of recurring 9’s. The standard g-sequence of approximations given by 0.9, 0.99, 0.999, . . . 0. 99 · · · 9, . . .   n places

The nonstandard sequence of differences dn = x − nth g-term is given by 0.099 · · · 9 · · · , 0.0099 · · · , 0.00099 · · · , . . . . . . , 0. 00 · · 000 99 · · · , . . .  ·  n places

We truncate the terms of the above d-sequence such that we always have k nonzero tail digits. That is, 0.0 99 . . . 99, 0.00 99 . . . 99 · · · , 0.000 99 · · · 99, 0. 00 · · 000 99 · · 99, . . .  ·   · k places

kplaces

k places

n places

k places

The d-limit of this d-sequence as n tends to some personal infinity is a dark number d which is defined and approximated by the nth d-term 0. 00 · · 000 99 · · 99.  ·   · n places

k places

On the other hand, we may also proceed as follows: The nth g-term of the above standard sequence of recurring 9’s can be written as, 0.99 · · · 99 = 1 − 0. 00 · · · 0 1     n places

(n−1) places

which is 1 − 0.99 · · · 99 = 0. 00 · · · 0 1.     (n−1) places

n places

Thus we are lead to the basic d-sequence

{(0.1)n },

n = 0, 1, 2, . . . , which is

0.1, 0.01, . . . , 0. 00 · · · 0 1, . . . .   (n−1) places

The d-limit of the above sequence is the basic element of the set d ∗ . We summarize the above discussion in the following result: Theorem 2.1. The d-limits of the nonstandard d-sequences form the continuum d ∗ . Remark 2.2. It is clear that 0 is the unique lower bound of d ∗ and any sequence of numbers that tends to zero in the standard norm does not really tend to zero in our system but only to d ∗ , that is a dark number. Only, when the impersonal infinity is involved, we get zero but with all other personal infinities we only obtain d ∗ . We introduce the following (linear) order, which is the lexicographic ordering ”<” on R∗ as follows: Two (terminating or nonterminating) decimals are equal if all corresponding digits are equal. That is, N.a1 a2 ... = M.b1 b2 · · · if and only if N = M and ai = bi , for each i = 1, 2, . . . ; Similarly, for N.a1 a2 · · · , and M.b1 b2 · · · ∈ R∗ , we have:

20

The Hybrid Grand Unified Theory

N.a1 a2 · · · < M.b1 b2 · · · if N < M or if N = M, a1 < b1 ; if a1 = b1 , a2 < b2 ; or a1 = b1 , a2 = b2 , a3 < b3 ; etc. Thus, basic d ∗ is adjacent to zero, in the same way as a pair of decimals such as integer N and (N − 1).99 · · · are said to be adjacent. That is there are no terminating decimals between them. Further, if x is a terminating decimal of the form x = N.a1 a2 · · · ak−1 ak then, we know that N.a1 a2 · · · (ak −1)99 · · · and x are adjacent. Since 1 − 0.99999 · · · = d ∗ we may write 1 − d ∗ = 0.99 · · · . Thus, we may write 1 + 0.99 · · · = 1 + (1 − d ∗ ) = 2 − d ∗ = 1.99 · · · More generally, N = (N − 1).99 · · · + d ∗ . We call the pairs (N, (N − 1).99 · · · ) as twin integers. The difference between the integer N and (N − 1).99 · · · , is d ∗ . In the lexicographic ordering the smaller of the pair of adjacent decimals is called the predecessor and the larger the successor. The average between them is the predecessor. Thus, the average between 1 and 0.99 · · · is 0.99 · · · since (1.99 · · · )/2 = 0.99 · · · ; this is true of any recurring 9, say, 34.5799 · · · whose successor is 34.58. Conversely, the g-limit of the iterated or successive averages between a fixed decimal and another decimal of the same integral part is the predecessor or successor depending on their magnitude. We thus, have the following property of d ∗ . (i) For any x = N.99 · · · ,

x + d ∗ = N + 1.

(ii) If x = N.a1 a2 · · · ak , x = N.a1 a2 · · · (ak − 1)99 · · · + d ∗ . On the other hand, if x is any nonzero decimal, terminating or nonterminating, both (0.1)n and x(0.1)n become indistinguishably small as n increases and hence both define and approximate the same dark number. Thus, (iii) xd ∗ = d ∗ . The product of an integer N and 0.99.. may be now understood as N(0.99...) = N(1 − d ∗ ) = N − Nd ∗ = N − d ∗ = (N − 1).99 · · · . We can now turn our attention to algebraic operations involving terminating and nonterminating decimals. Let K be an integer, M.99 · · · and N.99 · · · be decimal integers. Then (1) K + M.99 · · · = (K + M).99 · · · (2) K(M.99 · · · ) = K(M + 0.99 · · · ) = KM + K(0.99 · · · ) = KM + (K − 1).99 · · · (3) 0.99 · · · + 0.99 · · · = 2(0.99 · · · ) = 1.99 · · · , (0.999 · · · )2 = 0.99 · · · . (4) Sum of two decimal integers: M.99 · · · + N.99 · · · = M + N + 0.99 · · · + 0.99 · · · = M + N + 1.99 · · · = (M + N + 1).99 · · ·

Mathematics of Grand Unified Theory

21

(5) Product of two decimal integers (M.99 · · · )(N.99 · · · ) = (M + 0.99 · · · )(N + 0.99 · · · ) = MN + M(0.99 · · · ) + N(0.99 · · · ) + (0.99 · · · )2 = MN + (M − 1).999 · · · + (N − 1).99 · · · + 0.99 · · · = MN + (M + N) + 0.99 · · · = (MN + M + N).99 · · · We call the non-terminating decimals of the form N.99 · · · as decimal integers. Let us define a mapping between integers and decimal integers as follows: For a positive integer N, let f (N) = (N − 1).99 · · · . Such a mapping f is an isomorphism, between the set of all positive integers and the set of all positive decimal integers. For, f (N + M) = (N + M − 1).99 · · · = N + M − 1 + 0.99 · · · = N − 1 + M − 1 + 1.99 · · · = N − 1 + 0.99 · · · + M − 1 + 0.99 · · · = (N − 1).99 · · · + (M − 1).99 · · · = f (N) + f (M). This means that addition of integers is the same as addition of decimal integers and they have identical properties. Next, we show that multiplication is also the same in either set. f (NM) = (NM − 1).999 · · · = NM − 1 + (0.99 · · · ) = NM − N − M + 1 + N − 1 + M + −1 + (0.99 · · · ) = NM − N − M + 1 + (N − 1).99 · · · + (M − 1).99 · · · + (−1)(0.99 · · · ) = NM − N − M + 1 + N(0.99 · · · ) + (−1)(0.99 · · · ) +M(0.99 · · · ) + (−1)(0.99 · · · ) + (0.99 · · · ) = (N − 1)(M − 1) + (N − 1)(0.99 · · · ) + (M − 1)(0.99 · · · ) + (0.99 · · · )2 = ((N − 1) + (0.99 · · · ))((M − 1) + 0.99 · · · ) = ((N − 1).99 · · · )((M − 1).99 · · · ) = ( f (N))( f (M)). The above isomorphism can also be easily extended to terminating decimals, since any terminating decimal can be represented in the form (10)−i N for some i and N is an integer. The extension to negative integers and negative terminating decimals is also analogous. Further, we extend the isomorphism to include d ∗ by defining, f (0) = d ∗ . So, we have (i) (d ∗ )n = d ∗ (ii) (0.99 · · · )n = 0.99 · · ·

n = 1, 2, · · · , and,

(iii) if x = 0, x(0.99 · · · ) = x.

2.2.3 The Structure of R∗ We now, define certain numbers u∗ that are large numbers and are dual to the dark numbers d ∗ . These large numbers may be identified as the upper bounds of divergent sequences of terminating decimals.

22

The Hybrid Grand Unified Theory

Thus, the large numbers may be represented as: ∞p

k

i=0

i=1

∞p

∞p

i=0

i=1

∑ ai (10)i + ∑ bi (10)−i

or

∑ ai (10)i + ∑ bi (10)−i

So, these large numbers include numbers with large integral part and terminating or non-terminating decimal part. We denote by u∗ all large numbers that are dual to dark numbers d ∗ with the property 1 1 ∗ ∗ ∗ ∗ d ∗ = u and u∗ = d . The properties of d and u are solely determined by their non-standard equences. Keeping the set of terminating decimals R as our reference, we define the hybrid real number system R∗ as the g- closure of R. This includes (i) the terminating and non-terminating decimals, (ii) the dark numbers d ∗ , (iii) the large numbers u∗ that are dual to the dark numbers. We also include in R∗ , all the numbers that arise out of algebraic operations between the above numbers. We list below, some of the algebraic operations of d ∗ and u∗ and their properties: (1) For all x, x + u∗ = u∗ ; for x = 0, xu∗ = u∗ . (2) If x is any decimal we have, 0 < d ∗ < x < u∗ . (3)

1 d∗

= u∗ , u1∗ = d ∗ . (ii) 0d ∗ = 0, d0∗ = 0, 0u∗ = 0, u0∗ = 0 ∗

∗

(4) Numbers like u∗ − u∗ , dd ∗ and uu∗ are still indeterminate but indeterminacy is overcome by computation with their nth d-terms. (5) The largest and smallest elements in the open interval (0, 1) are 0.99 · · · and d ∗ , respectively. (6) The largest and smallest numbers in the interval (1, ∞I ) are u∗ and 1 + d ∗ respectively. Since, all the numbers in our hybrid real number system are defined, it is clear that the trichotomy axiom follows from lexicographic ordering. Theorem 2.2. In the lexicographic ordering R∗ consists of adjacent predecessor-successor pairs of decimals (each joined by d ∗ ) so that the closure in the g-norm is a continuum.(see [32]-[49]). To summarize, we have revisited and clarified the construction of hybrid real number system R∗ , which consists of terminating and nonterminating decimal numbers, dark numbers and their duals (large numbers), utilizing the notion of personal infinities and the impersonal infinity. Basic algebraic properties of such numbers are discussed. The significance of the hybrid real number system to physics and other disciplines can be found in [32]-[49]. We conclude this section with the following two comments. Remark 2.3. Recall the following properties associated with the basic digit 9. (1) String of 9’s differes from the nearest power of 10 by, 1. For example, 10100 − 99 · · · 9 = 1 (a string of 100 nines). (2) If N is an integer then (0.99 · · · )N = 0.99 · · · and naturally both sides of the equation have the same g- sequence. Therefore, for any integer N, ((0.99 · · · )10)N = (.99 · · · )10N .

Mathematics of Grand Unified Theory

23

(3) (d ∗ )N = d ∗ ; ((0.99 · · · )10)N + d ∗ = 10N , N = 1, 2, . . . . Now, in the setting of R∗ , it is clear that the triples (x, y, z) = ((0.99 · · · )10T , d ∗ , 10T ), T = 1, 2, . . . , satisfy the relation xn + yn = zn , (of the Fermat’s Last Theorem) for n = NT > 2. Moreover, for k = 1, 2, . . . (kx, ky, kz) also satify the above relation [18, 23].

2.2.4 More Properties of Nonstandard Numbers We have mentioned earlier that d ∗ is a continuum, and therefore that all dark numbers are indistiguishable from each other. We would like to define the notion of “order” for dark numbers, a notion that is useful in avoiding indeterminate forms besides providing some algebra of dark numbers. We begin with the following definition. Definition 2.1. Let γ be a terminating decimal such that 0 < γ < 1 and dγ = g- limn→∞ p γ n , where ∞ p is some personal infinity. A dual dark number uλ of order λ > 1 is defined as the upper bound of the sequence λ n as n → ∞ p . We list below some properties of special classes of nonstandard numbers that can be checked by using their g-sequences. (1) Since γ < 1, we see that, 0 < dγ < x for any decimal γ . (2) xdγ = dγ for any decimal x. (3)

x dγ

= u 1 , for any decimal x. γ

(4) If dγ1 and dγ2 are dark numbers of order γ1 , γ2 respectively, where γ1  γ2 then (a) dγ1 + dγ2 = dγ2 ; (b) dγ1 − dγ2 = dγ1 ; (c)

dγ1 dγ2

= d γ1 and γ2

dγ2 dγ1

= u γ2 . γ1

Since the dark numbers dλ and the large numbers uλ are dual, the above properties hold when we interchange the dark numbers with the corresponding dual numbers. Recall that R∗ is the closure of the terminating decimals in the g-norm. Hereafter, we use the following notation : For any decimal x, ξn (x) denotes the nth g-term of the g- (standard or non-standard) sequence. A decimal is degerate if its g-sequence is finite. Any algebraic operation involving the non-terminating decimals can only be performed through their nth g-terms. Thus, it is important to remember that the margin of error at each step in the computation should be consistent since the result of the computation cannot be any more accurate than the accuracy of the terms involved in the computation. With this in mind, we introduce the following notation: For x = N.a1 a2 · · · an · · · and y = M.b1 b2 · · · bn · · · then we have, for ξn (x) = N.a1 a2 · · · an , ξn (y) = M.b1 b2 · · · bn : (1) ξn (x ± y) = ξn (x) ± ξn (y) (2) ξn (xy) = ξn (x)ξn (y) (3) ξn ( xy ) =

ξn (x) , ξn (y)

provided ξn (y) = 0. 1

For example, to compute f (5) where f (x) = x 3 . We find the 3rd g-term of f (5) as follows: N = 1 is the largest integer such that N 3  5. Now, consider the partition {0, 0.1, 0.2, . . . , 0.9, 1} of the interval, [0, 1]. We find that, for 0.a1 = 0.7 we have (1.a1 )3 = 4.913  5. Thus, the first term of the g-sequence is 1.7. Now, considering the partition {0, 0.01, 0.02, . . . , 0.09} of the interval [0, 0.1],

24

The Hybrid Grand Unified Theory

we find that 0.a2 = 0.0 is the the only point from the partition for which (1.70)3  5 leading to 1.70 as the second g-term. Similarly, we can find the third g-term is 1.709. Similarly, we extend this process to functions defined on (R∗ )k :

ξn ( f (x1 , x2 , . . . , xk )) = f (ξn (x1 ), . . . , ξn (xk )). For a composite function f (g1 (x1 , . . . , xt ), . . . , gs (y1 , . . . , yu )) we have

ξn ( f (g1 (x1 , . . . , xt ), . . . , gs (y1 , . . . , yu ))) = f (ξn (g1 (ξn (x1 ), . . . , ξn (xt ))), . . . , ξn (gs (ξn (y1 )), . . . , ξn (yu ))).

2.2.5 g-Limit and the Limit We adopt the standard topological notion of a limit point in the standard norm and call it limit. An element p of R∗ is the limit of the g-sequence of a non-terminating decimal, if every open interval of p contains an element of the sequence. This limit is attained only when the limiting process invovles an impersonal infinity. From our earlier discussions, it is clear that the limit and g-limit of a g-sequence corresponding to a non-terminating decimal are adjacent. For example, for the g-sequence, 0.9, 0.99, . . . the g-limit is the non-terminating decimal 0.99 · · · and the limit is 1 where as the the g-limit of the g-sequence 4.52, 4.529, 4.5299, . . . is the non-terminating decimal 4.5299 · · · where as the limit is the terminating decimal 4.53. We call the numbers, d ∗ (dark number), u∗ (large number) and any sum of a (terminating) decimal and the dark number d ∗ , to be non-standard numbers. Thus, terms of the g-sequence of a nonstandard number may be written as a sum of decimal and non-standard nth d-terms of d ∗ . Thus, in the computation of the values of a function, F(x), via the nth g-terms, will consist of two components, one with its digits remaining fixed, and the other the nth d-terms of d ∗ . Thus, in computation, we may write, the nth g-term of F(x) as: ξn (F(x)) = H(x) + δ (x), where H(x) and δ (x) are called the major and minor parts, respectively. In our computation, in the light of the notion of order of dark and large numbers, and their algebra, inderterminate forms do not arise [24]. The g-limit of a function, F(x) may thus be computed as follows: Suppose we want to compute the g-limit of F(x) as x → s. We start with a monotone increasing sequence of decimals x0 , x1 , . . . to the left of s such that xi → s. We refine the sequence by inserting the succession of averages of s and xi ’s. Relabeling the terms of the resulting decimals as sequence of si ’s each time, we arrive at some kth refinement. We now compute the values of the function F(x) along the decimals si ’s obtained. Once we find that, ξn (F(si )) is expressible in the form H(si ) + ξn (δ (si ) where H(si ) has fixed digits upto nth term and δ (si ) has terms receeding to the right as n increases. We treat H(si ) as the nth g-term of the g-limit. IF s is nonterminating, we obviously need to truncate s to the desired accuracy of computation. This procedure works for well defined functions. For example, polynomials, exponential functions, logarithmic function, circular functions and their sum, product , quotient and compositions away from discontinuities are well defined functions. Remark 2.4. If we need to perform computations involving non-standard functions, we proceed as follows: For F(x) = G(x) + η (x) where F(x) and η (x) are the major and minor parts of F. Then, g-lim F(x) =g-lim G(x) as x → s, since η (x) → 0 as x → s.

Mathematics of Grand Unified Theory

25

Remark 2.5. The g-limits may be set valued. For example, consider the sequence a3k+1 = (0.123)k , a3k+1 = (0.312)k , a3k+2 = (0.231)k . To find its limits we split it into three component sequences, namely, {a3k }, {a3k+1 }, {a3k+2 }. The g-limit of the original sequence is three valued, consisting of dark numbers of different orders, which are special elements of d ∗ .

2.3 Special Functions and the Generalized Curves 2.3.1 Some Mischievous Functions We consider mischievous functions, mischievous only in relation to the well-behaved functions because they render certain established methods ineffective. However, once tamed they are useful. Examples of mischievous functions are the infinitesimal zigzag and the wild oscillation, sinm 1x , (sinn 1x )(cosm 1x ), where m and n are integers.

2.3.2 The Infinitesimal Zigzag We shall generate a sequence of functions representing the polygonal lines: Cn : yn = yn (x), 0  x  1; n = 1, 2, · · · that converges to the curve C0 point-wise (or in the sup norm) as follows: Without loss of generality, take C1 : y1 = y1 (x), 0  x  1, the polygonal line joining A and B formed by sides AD and DB of triangle ABD with vertices at A(0, 0), B(1, 0) and D(1/2, 1/2). For the second term in the sequence, join the midpoint P of AD to the midpoint Q of AB and point Q to the midpoint R of DB to form the polygonal line APQRB from A to B. We denote this function by C2 : y2 = y2 (x), 0  x  1. We continue similar construction on the polygonal APQRB. From the geometry of the figure, the slope of C at any point in [0, 1] is +1 or −1 except at corner points where it is simultaneously +1 and -1 (derivative √ does not exist in this set of measure 0). Also, |C2 |, the length of C2 , = |C1 | = length of C1 = 2. Continuing similar construction on the finer polygonal lines we obtain a sequence of polygonal lines Cn : yn = yn (x), n = 1, 2, . . . , 0  x  1, which has the following properties: (1) |Cn | = y n

 Δ

1

((1 + (y )2 ) 2 dx +



1

Σ

((1 + (y )2 )) 2 dx =

√ 2 = |C1 |,

yn ; Δ = {x|y n (x) =

where is the derivative of +1} and Σ = {x|y n (x) = −1} (the set at which y n is simultaneously +1 and -1 has measure 0). (2) The sequence Cn : n = 1, 2, · · · is uniformly convergent point-wise and since each Cn is continuous, the limit denoted by C0 : y0 = 0, 0  x  1, is continuous. In fact, y0 coincides with [0, 1] whose equation is y = 0 which is absolutely continuous. Hence, y0 is also absolutely continuous. (3) What about the derivative of y0 ? Does it exist? If it does, what is it? We cannot have y 0 = 0. For, if that were so, it would violate the dominated convergence theorem applied to the integrals in (a) since this would imply 

1

[0,1]

((1 + (y 0 )2 )) 2 dx = 1 =

 √ 2 = lim |Cn | = lim

[0,1]

1

((1 + (y n )2 )) 2 dx,

as n → ∞. In fact, the derivative of y0 does not exist in the ordinary sense because the sequence y n , n = 1, 2, . . . or −1, +1, −1, +1, . . . , does not converge to a single point since its set is {−1, . + 1}, i.e., y 0 is set-valued.

26

The Hybrid Grand Unified Theory

This function C0 (y0 , y 0 )) : y0 = 0, y 0 = ±1, 0  x  1 is an unambiguous counterexample to a theorem in [96] that says, an absolutely continuous function is differentiable, almost everywhere. This function is absolutely continuous but nowhere differentiable. It raises some important points which are the source of this particular contradiction: (1) Inadequacy of the present notion of function; this was pointed out in [125, 126] 70 years ago and, again in [128]. A function defined by its values alone cannot distinguish the function C : y = 0 from C0 : y0 = 0 which are distinct in at least two ways: one is differentiable and the other is not and they also have different lengths. (2) Inadequacy of the notion derivative; that the derivative of a function cannot be adequately expressed by its values because derivative is a property belonging to a larger space (extension of n-space in the general case) whose restriction to the space of real-valued functions contradicts some of its properties (e.g., properties of absolutely continuous function). Therefore, there is a need to extend the notion of function to include those with set-valued derivative. Also, the present defect in the notion of limit is passed on to other notions defined by limits including the derivative. (3) Existence of new kind of function called infinitesimal zigzag: The function C0 : y0 = 0, y 0 = ±1, 0  x  1, belongs to a wider class of curves called generalized curves. It is different from the ordinary curve C : y = 0, 0  x  1. Yet their values coincide point-wise. In fact, there are countably infinite number of functions of this kind. One can see that although the sequence of functions Cn , n = 1, 2, . . . converges to the segment AB point-wise, its standard limit, say, in the sup norm, is something else: the infinitesimal zigzag, C0 : y0 = 0, y 0 = ±1. This example raises two very important points: (a) Fallacious proof of existence of a mathematical object by approximation or convergence as well as the erroneous use of numerical and algorithmic methods without existence theory (in fact, this is a variant of vacuous statement). (b) The inadequacy of the values of a function in characterizing its derivative; thus, the present notion of derivative is inadequate to capture its complexity as a property of a function. These inadequacies of the notion of function and derivative as well as numerical method without rigorous justification have far-reaching significance for all of analysis and beyond. In particular, any theorem on derivative inherits this problem. That is drawn out by mischievous functions that we shall deal with later. The infinitesimal zigzag is our first example of a mischievous function. It serves as counterexample to a number of well-established theorems.

2.3.3 Significance of the Infinitesimal Zigzag This example tells us that the sup norm and the metric induced by point-wise convergence are not natural for purposes of optimization, especially, optimal control; moreover, the properties of a curve are fully accounted for by neither its values nor its parametric representation alone. We must take into account its behavior in terms of its derivative and the behavior of the latter as well so that if f (t),t ∈ [0, 1], is the parametric representation of a curve C we represent it by the pair C : ( f , g) where g is the derivative of f . Then the natural metric for purposes of optimization is the Young measure or curvilinear integral of some objective function along it (which can be the cost function) [125, 126, 131]. If we represent that measure by the integral, I(C) =

 [0,1]

( f (t), g(t)) dt.

Mathematics of Grand Unified Theory

27

Then I(C) is the Young measure of the curve C. When the integrand is 1, I(C) is called the length of the curve. Thus, a curve is a linear functional and curves of the same Young measure belong to the same equivalence class representing that linear functional. This makes functional analysis available to optimal control theory. In an optimal control problem the derivative g is the control parameter so that it is independent of f ; in other words the system is controlled by finite set of values of the derivative. Another case of optimization where the ”obvious” curve is not the optimal solution is this example: find the minimum of the integral,  [0,1]

((1 + x2 )(1 + ((x 2 − 1)2 )100 ) dt

(where x is derivative) among admissible functions x(t) subject to x(0) = x(1) = 0. The obvious optimal curve among conventional curves is x = 0, subject to x(0) = x(1) = 0 and the minimum is 2100 . However, by admitting infinitesimal zigzag, which is like the ordinary curve x = 0 but whose derivative is set-valued and concurrently takes the values +1 and -1, and attaching a probability weight 1 2 to each of these values, we obtain a minimum of 1. Thus, the conventional theory of curves yields incorrect solution of this variational problem. Here, the infinitesimal zigzag is a generalized curve or, to be precise, this generalized curve is the equivalence class of curves of the same length (with set-valued derivative). Incidentally, all four types of cosmic waves and the superstring are generalized curves because they have one thing in common: set-valued derivatives; so is the path of a primum (elementary particle [131]; also see Chapter 3).

2.3.4 The Wild Oscillation sin 1x Our next mischievous function is a simple one, the wild oscillation, F(x) = sin 1x . This is a special case of the more general mischievous function sinm x1k , where k, m are positive integers. It reveals a flaw in the Lebesgue theorem on the Riemann integral that says: A bounded function is Riemann integrable if and only if its set of discontinuity has measure zero [96]. The bounded function F(x) = sin 1x whose only discontinuity is at x = 0 is not Riemann integrable in any neighborhood of the origin. Known proof of integrability of sin 1x involves construction of a Riemann integral outside an ε -neighborhood of x = 0, where ε > 0, which exists, and taking a sequence of such integral as ε → 0, which converges. The limit of such a sequence, however, is not necessarily Riemann integrable, certainly, not sin 1x because no Riemann sum of this function can be formed in any neighborhood of 0. The best we can say here is that one can construct a convergent sequence of Riemann integrals with some relation to the function sin 1x in the same ε -neighborhood of x = 0 but its limit is something else. This is, in fact, a form of the Perron paradox on the use of necessary conditions without an existence theory. It also illustrates the same fallacy mentioned earlier in the proof of existence of some mathematical object by approximation or convergence as well as the use of an algorithmic solution of a problem without rigorous justification. In the development of the Henstock integral the function sin x12 plays a central role. However, the theory is flawed by the present inadequate notion of derivative. While this function can be shrunk to zero by the factor x2 its derivative cannot because as we shall see later derivative of a function belongs to a higher space and is independent of the function. The function considered in [64] is x2 sin x12 , 0  x  1. It is asserted that its derivative F (x) exists at x = 0 and F (x) = 0 because at that point its one-sided derivative can be trivially computed since, using the ordinary definition of

28

The Hybrid Grand Unified Theory

derivative, we have,

|ΔF| |x2 |  = |x|, |Δx| |x|

so that lim ΔF |x| , as x → 0+, exists. The above inequality follows from the fact that F(x) is bounded by its envelope y = ±x2 . Note that F(x) is continuously differentiable outside x = 0. In fact, we have, at x = 0, F (x) = 2x sin

1 2 1 − cos 2 . x2 x x

The term 2x cos x12 oscillates rapidly along all values in the interval (−∞, ∞) as x → 0+ and does not converge. This is a particular kind of discontinuity, an example of what we shall call chaos. Our final mischievous function is a function of the type, 1

ez 1 1 (sinm 2 + cosn 2 ) x x xk or



1

ez xk



sinn

1 x2

,

(2.9)

where z = x2 , k, m, n are positive integers. Finding the limits of these functions as x → 0 quickly reveals L’Hospital’s rule breaks down. The reason: these functions do not satisfy its hypothesis, that the function should not be zero in some neighborhood of the function; each of the functions in (2.9) has countably infinite zeros in any neighborhood of the origin . Also by rearranging the factors one gets different limits. The limit of the first term of (2.9) is evaluated in [34] using generalized derivative, i.e., the expectation of set-valued derivative with probability distribution, to generalize L’Hospital’s rule and apply this method to the functions in (2.9).

2.3.5 Rapid Spiral and Oscillation A primum is mathematically modelled by the rapid spiral (in cylindrical coordinates, center at origin), x = t, r(t) = β (sinn π t)(cosm kπ t), t ∈ [−1/k, 1/k], θ = nπ t, n, m, k, integers, n >> k, m even. Its cycle energy is Planck’s constant h = 6.64 × 10−34 joules [133]. Energy conservation and flux compatibility pull the primal cycles together to form a set-valued function that requires the generalized integral (see Chapter 3) to do calculation on it.

2.3.6 Rectification Partial rectification of inadequacies in the notion of functions and derivatives is done in [125, 126] by representing a vector function f with values in R by a pair f , g, where g, called the derivative of f , is a vector function which takes values in a field of vectors belonging to a separate space isometric to Rn . This representation formalizes the independence of the derivative from the function. (For full development of this idea and the requirements on f and g see [125, 126]). Regarding the derivative component of the pair ( f , g) Young went so far as admitting set-valued derivative with the introduction of chattering controls [131]; in the study of convex vector functions the notion of a set of landing hyperplanes at a point is admitted. This corresponds to a set of tangent lines for a real-valued function

Mathematics of Grand Unified Theory

29

The thrust of rectification focuses on the derivative component of the pair ( f , g) in the representation of a function. It is this approach that led to the introduction of generalized curves which, in turn, established an existence theory and resolved the Perron paradox in the calculus of variations, paving the way for the latter’s modern formulation in optimal control theory. However, with the appearance of set-valued functions in the study of fractal and chaos this rectification effort still falls short; we need to allow set-valued function in the pair ( f , g) as a way to capture the complexity of certain notions, particularly, function and derivative. Note that the approach in [125, 126] reflects the enrichment methodology: enriching the space with new elements to achieve an existence theory or convergence. Let { fn (t), gn (t)}, n = 1, 2, . . . , t ∈ [a, b] be a sequence of functions in the new sense, where each fn (t) is continuous and gn (t) measurable and well-defined almost everywhere. We suppose further that the end points [an , bn ] of the domains of definition of fn (t) and gn (t) tend, respectively, to 0 and 1 as n → ∞. For our specific purposes here we require that an  0, and each of fn (t) and gn (t) has common extension to some interval T containing both [0, 1] and the sequence of intervals [an , bn ], n = 1, 2, · · · We define the limit set of { fn (t), gn (t)} as the pair ({ f0,0 (t)}, {g0,0 (t)}), where { f0,0 (t)} = Slim{ fn,k (t)} = the set of limit points of a diagonal element { fn (tk )}, as n → ∞ and tk → t and {g0,0 (t)} = Slim{g0,0 (t)} = the set of limit points of a diagonal element {gn (tk )}, as n → ∞ and tk → t. Since fn is continuous its limit is independent of the sequence {tk }; not so with gn since we only require measurability. Thus, we distinguish the limit set of a sequence of pairs { fn (t), gn (t)} by the particular sequence {tk }, where tk → t. As a special case we may take tk → t for all t ∈ T. Then the closure of such sequence { fn (t), gn (t)} under this convergence is called the space of generalized curves. The complete formulation of the theory of generalized curves as linear functionals is given in [125, 126]; we summarize it below. Essentially, generalized curves are curves with set-valued derivatives developed as linear functionals [125, 126].

2.3.7 Application of the Infinitesimal Zigzag We make references to the superstring although we shall take it up later since we want to introduce its mathematical models. A superstring is a nested fractal sequence of superstrings where the first term is a close helix; it has a flux called toroidal flux through its helical cycle which is a superstring traveling at speed greater than the speed of light; the toroidal flux has a toroidal flux in its helical cycle which is a superstring traveling at speed greater than c, etc. The projection of a helix on the plane through its axis is a sinusoidal or oscillatory curve. Given any oscillatory curve y = sinm bx (not necessarily rapid oscillation), which is rectifiable, we can deform it into some isosceles triangle so that its length is preserved and equal to the sum of the lengths of AD and DB. In turn, we can deform it into a finer oscillatory curve K1 , with length preserved. We iterate this deformation forming an alternate sequence of polygonal lines and oscillatory curves Kn from A to B. Again, the sequence Kn tends towards a generalized curve called infinitesimal oscillation whose function component coincides with the zero function C : y = 0, 0  x  1. Its length is equal to the original length |K| of K and its derivative at any point x ∈ [0, 1] is set-valued and equal to the limit points of the derivatives of the sequence of oscillations at x. Since the segment AB is arbitrary we can prescribe its length to be an arbitrary number ε > 0. Then we have the following: Theorem 2.3. Given an oscillatory curve K, any number ε > 0 and a line segment AB, there exists a continuous deformation of K into a fine oscillatory curve inside an ε -neighborhood of AB that preserves the length of K [18]. Proof. Let A be a given point and B a point in the ε -neighborhood of A and suppose AB = ε2 > 0. There exists a deformation of K, with length preserved, into two sides of an isosceles triangle

30

The Hybrid Grand Unified Theory

ADB where |AD| + |DB| = |K|. There exists a sequence of polygonal curves Cn and corresponding oscillatory curves Kn such that for each n, |Kn | = |Cn | = |AD| + |DB| = |K| and Kn tends to the segment AB. Hence there exists a positive integer N such that whenever n  N, the curve Kn lies inside the ε -neighborhood of A.  Theorem 2.4. Given an oscillatory curve K, there exists a continuous deformation of K, with length preserved, into an arbitrarily small neighborhood of a point [18, 25, 17]. Proof. Since the length of AB is arbitrary, ε > 0 and |AB| = ε2 . The rest of the proof follows from the previous Theorem.  Note that in each case the oscillatory structure is preserved as well as its length. Thus, it is possible to shrink an oscillatory curve of any length into an infinitesimal oscillation at a point. Now, let β > 0, where β is small, and let K be an oscillatory curve of large length K. Let ε = β2 < |K| 2 . As before, we deform K into the two sides of an isosceles triangle ADB with base AB, where |AB| = ε . Let h be the altitude of this triangle, then h is roughly |K| = β2 . By the Archimedean property of the decimals there exists some positive integer n such that |K| |K| |K| < n+1  n . 2 2n+2 2

(2.10)

Therefore, in the sequence of oscillatory curves Ki with |Ki | = |K|, for each i = 1, 2, · · · , which tends towards the line segment AB, there is one whose amplitude satisfies the inequality (2.10). We state this as a theorem. Theorem 2.5. Let K be an oscillatory curve with large length |K| and let ε > 0, ε = β2 < K2 . Then one can continuously deform the oscillatory curve K into an arbitrarily small neighborhood of a point with its length and amplitude prescribed to satisfy, |K| |K| |K| < n+1  ε  n , 2 2n+2 2 for some integer n, [18]. The following theorem [52] is now obvious and follows from the above theorems: Theorem 2.6. The real line is chaos [17]. All of the above four theorems model different aspects of shrinking of a superstring. This discussion has other implications for physics that can explain certain phenomena such as the tremendous but undetected energy in the nucleus of an atom. Tremendous because we can pack an infinitesimal helical loop into an arbitrarily small neighborhood of a point at very high energy level hν (h is Planck’s constant and ν frequency) and that would not be detected by our means of observation such as light because if the wavelength of the latter is sufficiently fine there would be no interference or discordant resonance with this infinitesimal helix due to difference in orders of magnitude of their cycle and wavelength (the helix can be non-agitated superstring [33]). Helix and oscillation (sinusoidal) are universal configurations of matter and they are related: the projection of a helix on a plane through its axis is sinusoidal. Infinitesimal helix and oscillation are both generalized curves because their derivatives are set-valued. The superstring, basic constituent of matter, is an infinitesimal helical loop, a generalized curve [22].

Mathematics of Grand Unified Theory

31

2.3.8 The Generalized Curves We summarize the development of the generalized curves in [125, 126]. The theory was an offshoot of the effort to provide existence theory to resolve the Perron paradox in the Calculus of variations. The paradox stems from the use of necessary conditions without existence theory. We reconstruct the basic formulation to get a sense of the method of enrichment. Consider the parametric equation x(t), t1  t  t2 , of the curve C, where x(t) is absolutely continuous and takes values in Rn . At each t we take a vector y(t) taken from a vector field in a separate space isometric to Rn which we also denote by Rn . We require the vector y(t) to lie in a unit sphere S, i.e., |y(t)| = 1. We also require C to lie in a compact cube in Rn and denote by A the compact Cartesian product of these two sets. Our Lagrangian belongs to the space C0 (A) of continuous functions with compact support A. We further assume that our Lagrangian is homogeneous in y so that it is determined in Rn by its restriction to A since if L is the Langrangian and (x, y) is any point in Rn × Rn then there exists some scalar α  0 such that y = α y , where y ∈ S. From the homogeneity of L in y, L(x, y) = L(x, α y ) = α f (x, y ), where f ∈ C0 (A) is the restriction of L to A. We make a minor adjustment for simplicity of notation by representing C(x(t), y(t)), a  t  b, as a curve defined on the compact set A. Here we have attached a derivative y(t) to stress the independence of derivative from x(t). We also assume the parameter t to be an arc length from the initial point of C to avoid dependence of the curvilinear integral along C on its parametrization but more on intrinsic properties. We consider two curves C1 : (x1 (t), y1 (t)), a  t  b and C2 : (x2 (t), y2 (t)), c  t  d, equivalent if their curvilinear integrals satisfy: I(C1 ) =

 [a,b]

f (x1 (t), y1 (t)) dt =

 [c,d]

f (x2 (t), y2 (t)) dt = I(C2 ),

(2.11)

for all f ∈ C0 (A). Thus, a curve C0 is completely determined by the values of the curvilinear integrals along it for all f ∈ C0 (A), i.e., a linear functional g in the dual space C∗ (A) of C0 (A). We define fine convergence of a sequence gn , n = 1, 2, · · · , of elements of C ∗ (A) and say that gn converges to g0 or gn → g0 if gn f → g0 f for all f ∈ C0 (A). The closure of this space of curves in the new sense is called the space of generalized curves. In this sense a generalized curve is the fine limit of ordinary curve. A generalized curve is also called a generalized flow, where the latter is an element of the positive cone of C∗ (A), i.e., g f  0 for all f ∈ C(A) with f  0. It is the space of generalized curves that provides an existence theory in the calculus of variations and optimal control theory and renders Perron paradox inoperative there, showing again how a contradiction is resolved by some sort of enrichment, embedding or completion. This fine convergence induces a metric on Rn called Young metric. Thus, the distance between two curves C1 and C2 is given by, I((C1 ) − I(C2 )). This norm or metric is really the sup norm in the space of generalized curves defined as sup |g1 f − g2 f | for all f ∈ C0 (A). We require that for all these linear functionals g1 , g2 to be well-defined in the intervals of definitions of their respective duals (ordinary curves) in A have common endpoints. This is the same requirement for fine convergence, that the dual sequence of gn in A must have endpoints tending towards those of its limit g0 . The length of a generalized curve g is the value of the integral in (2.11) when f = 1. This norm is the right one that is consistent with the counterexample

32

The Hybrid Grand Unified Theory

presented earlier, the polygonal line that converges to an infinitesimal zigzag. In that case the function √ 1 f is the arc length ((1 + (y 0 (t))2 ) 2 which is constant and equal to 2 for each Cn . Thus, gn f =

 [0,1]

1

((1 + (y 0 (t))2 ) 2 dt =

√

2.

Of course, the derivative of f , y 0 (t), is set-valued with value {1, −1}, and its dual in C∗ (A), where A[0, 1][−1, 0], is the infinitesimal zigzag. The development of the generalized curves started 70 years ago but it needs to be rediscovered because contemporary studies on fractals, bifurcation and chaos can learn much from it, not to mention its many applications. In fact, a generalized curve such as the infinitesimal zigzag is both chaos and limit set of fractal but the term fractal is of recent origin. In the case of bifurcation (or more generally, multifurcation), which is the transition from chaos to fractal, it can be explained by the fact that even well-behaved functions y = f (x) passing through some point (x0 , y0 ) is only one of the countable infinity of local solutions of some differential equation near (x0 , y0 ) satisfying an initial condition on its derivative there. Put another way, near the point (x0 , y0 ), there is a countable infinity of functions ( f (x), g(x)), which are local solutions of some set-valued differential equation. For each choice of g in a set-valued differential equation of the form x ∈ {g(t, x(t)} we have, for a given probability distribution, a corresponding branch of the solution. This is how multifurcation occurs at every point on the initial function as well as on each branch. This is how chaos ultimately results. This is similar formulation in the development of generalized surfaces that we used for dealing with the undecidable proposition FLT. Later it was extended to relaxed trajectories of control theory [131]. For extension of this methodology to the development of generalized surfaces, see [57, 127, 128, 132]. To summarize, the generalized curve mathematically models the superstring, primum (elementary particle) and spiral path of visible matter falling into and spinning around the eye of cosmological vortices both of which are continuous arcs with set-valued derivatives. Rapid spiral mathematically models a simple primum and rapid oscillation mathematically models the photon and the primum in flight. R∗ mathematically models physical time and distance and non-standard g-sequence of d ∗ the nested fractal superstring and d ∗ the tail end of its toroidal fluxes, a superstring and continuum. The decimals model the metric system and the integers the countably infinite and discrete dark and visible matter; and, as we shall see later, GUT qualitatively models our universe. Note that a physical system may have more than one mathematical model but only one dynamic model within equivalence.

2.4 Generalized Fractal and Chaos 2.4.1 Definitions and Examples Recall that classical fractal construction is iterated affine transformation of a given generator, usually, some geometrical figure; it is mainly computational. Affine transformation is a combination of contraction and translation so that the effect is to generate a sequence of self-similar figures, i.e., each term except for the first is similar to the preceding term, similarity coming from both contraction and translation and at smaller and smaller scale. We generalize this fractal construction to include as well, rotation, taking mirror image, sliding along a curve and replication all of which preserve similarity. The last characteristic is the most important for applications, especially, in biology. We have seen this splitting in the case of the roots and branches of a tree and the veins of its leaf. This also happens in mitosis or cell-division where self-similarity is in terms of the replication of the genes in every offspring cell. Moreover what is replicated need not be geometrical figure but general properties of the terms of the fractal sequence. We call this fractal formation generalized fractal; only qualitative

Mathematics of Grand Unified Theory

33

mathematics is capable of modeling this process especially when there is multiple replication or multifurcation. In this case every branch continues as nested fractal, i.e., each term in the sequence is contained in or a part of the preceding term (in this case at decreasing scale). The essential characteristic here is replication more than self-similarity because the terms may not be geometrical but some characteristics or processes such as the splitting of the roots and branches of a tree and veins of its leaf. Mitosis in a living cell is a special fractal that biologists use to describe its replication in the offspring cell, especially, its genetic content; here decreasing scale does not apply but self-similarity and replication do. Nested fractal is nature’s way of packing huge energy in a physical system or attaining maximum efficiency in a physical process. Therefore, it is an expression of energy conservation and, like oscillation, a universal configuration of nature that extends even to man-made structures such as machines, buildings and bridges due to its optimal properties. In physical biology, encoding of information in the brain is a fractal process [25, 31, 46]. The fractal structure of the roots of a tree allows it to absorb maximum nutrients from the ground and that of its branches as well as the veins of its leaf allows optimal efficiency in the distribution of nutrients to the stomata for fruit production (photosynthesis) and distribution to where humans can pick them up. Chaos is mixture of order none of which is identifiable. For example, in regions where there is much under-ocean volcanic activity the ocean surface heats up, throws the gas molecules above it into motion (kinetic energy) that pushes them apart and creates low pressure. Low pressure sucks gas molecules around it and the initial rush throws the trillions of molecules into collision that makes it impossible to monitor or even predict the path of any single gas molecule. At the same time, every molecule is subject to the laws of nature. This is then a classic case of chaos. But, this is transitional since collision is energy dissipating. Energy conservation induces global order on it and turns it into a cyclone. The formation of tropical cyclone is an example of standard dynamic system. It starts on a calm summer day, which is order, then chaos ensues as a transition to coherence of order called turbulence, in this case, a cyclone which is a vortex in the atmosphere. Why this transition from chaos to turbulence is realized is due to energy conservation. Chaos is energy dissipating due to collision which is distortion of order among the gas molecules; therefore, energy conservation induces its evolution into global order, the cyclone. Then the cyclone vanishes when nothing infuses energy on it, e.g., warm corridors on the ocean, and its energy dissipates into the atmosphere; when a cyclone passes through a warm corridor its power rises because heat or kinetic energy is fed to it. However, when the eye hits and gets plugged in by a mountain its power declines because friction dissipates its energy. It is impossible to model chaos computationally, only qualitative modeling can. Another example is the fundamental chaos called dark matter one of the two fundamental states of matter that we shall consider later in Chapter 3; fundamental in the sense that it is part of the cycle of matter: all comes from it and will ultimately return to it. However, fundamental chaos is not energy dissipating because the superstrings do not interact among themselves; therefore, it is stable and has zero entropy, the only stable physical chaos. (The real line is mathematical chaos [17]; so are infinitesimal zigzag and oscillation [18, 25, 17]; we have more examples below).

2.4.2 The Peano Space-Filling Curve The problem here is to map the unit interval [0, 1] onto a unit square by continuous function. The usual tool is a theorem that says: the limit of a uniformly convergent sequence of continuous func-

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tions is continuous. But all constructions so far are difficult to visualize. We make the geometrical construction of this curve quite simple [25, 17, 31]. Divide the unit square into 9 little squares by the lines, 1 x= , 3

2 x= , 3

1 y= , 3

2 y= . 3

Take the following initial generators: 1 g1,1 : y = x, 0  x  ; 3 1 g−1,−1 : y = x, 0  x  − ; 3

1 g−1,1 : y = −x, 0  x  ; 3 1 g1,−1 : y = −x, 0  x  − . 3

The first term f1 (t) of the fractal sequence and its generators in the construction of the Peano spacefilling curve are given below. These generators are obtained by rotating positively the first initial generator g1,−1 by π2 radians (counterclockwise) one at a time until it gets to g1,1 . The initial function in the iteration process is constructed as follows: f1 (t) : g1,1 (t), g1,−1 (t) + (1/3, 1/3), g1,1 (t) + (2/3, 0), g−1,1 (t) + (1, 1/3), g−1,−1 (t) + (2/3, 2/3), g−1,1 (t) + (1/3, 1/3), g1,1 (t) + (0, 2/3), g1,−1 (t) + (1/3, 1), g1,1 (t) + (2/3, 2/3), 0  t  1/3. The functions that comprise f1 (t) are suitable translations of the initial generators (segments) given above, to form a polygonal line through the diagonals of blocks B1 , B2 , · · · , B9 , in that order. Note that f1 (t) is a uniformly convergent sequence of continuous functions. To construct the function f2 (t) we contract f1 (t) by one third by pushing it along its projection cone (without distortion) towards the origin, so that the vertices of the contracted square lie at (0, 0), (1/3, 0), (1/3, 1/3), (0, 1/3), to obtain the initial generator g2,2 which is flipped over the y-axis, -x-axis, -y-axis one at a time to obtain the following generators for f2 (t): (1) g2,2 (t) = (1/3) f1 (t), 0  t  1/3; this is represented by the polygonal line in B1. (2) g−2,2 (t) is obtained by reflecting g2,2 (t) about the y-axis. (3) To obtain g−2,−2 (t), reflect g−2,2 (t) about the x-axis. (4) To obtain g2,−2 (t), reflect g−2,−2 about the y-axis. We construct f2 (t) as follows: f2 (t) : g2,2 (t), g2,−2 (t) + (1/3, 1/3), g2,2 (t) + (2/3, 0), g−2,2 (t) + (1, 1/3), g2,2 (t) + (2/3, 0), g−2,−2 (t) + (2/3, 2/3), g−2,2 (t) + (1/3, 1/3), g2,2 (t) + (0, 2/3), g2,−2 (t) + (1/3, 1), g2,2 (t) + (2/3, 2/3), 0  t  1/3. We now iterate the process, with orientation preserved, to generate a sequence of functions, f1 (t), f2 (t), . . . , fn (t), with the following properties :

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(1) Each fn (t), n = 1, 2, . . . , is made up of polygonal lines as suitable translations of the generators of fn (t); (2) Self-similarity is obvious from the construction since the orientation of fn (t), for each n, is preserved; (3) Each fn (t) in the sequence is continuous; (4) The sequence fn (t), n = 1, 2, . . . , is uniformly convergent; (5) Therefore, the sequence fn (t), n = 1, 2, . . . , converges to a continuous function f (t).

2.4.3 Filling up the Unit Cube The above construction can be extended to fill up the unit cube by the continuous mapping of the unit interval as follows: (1) Consider the unit cube with vertices at A(0, 0, 0), B(1, 0, 0), C(0, 1, 0), D(0, 0, 1). (2) Subdivide the cube by the planes, x = 1/3, x = 2/3, y = 1/3, y = 2/3, z = 1/3, z = 2/3, into 27 little cubes. (3) As in the construction of the Peano space-filling curve, stretch and deform the unit interval AB to a polygonal line and map each segment in suitable order into the diagonals of the little cubes starting with the one at the origin joining the vertices B000 (0, 0, 0) and B111 (1/3, 1/3, 1/3), · · · , B262626 (2/3, 2/3, 2/3) and B272727 (1, 1, 1), where the last segment is mapped into the diagonal of the little cube opposite the first little cube at the origin. Call this polygonal line f1 . (4) Contract f1 to 1/3 into the first cube (by pushing it through its projection cone with vertex at the origin. This contracted polygonal line becomes the first generator of f2 ; flip f2 to its mirror image about the y-axis, the latter to its mirror image about the -x-axis, and the latter to its mirror image about the -z-axis, etc. until each octant has the contracted little cube with the polygonal line from the origin to the opposite vertex. The eight polygonal lines in the eight little cubes will form the generators of f3 . (5) Take suitable translations of these generators to fill up all 27 cubes with the generators of f2 and contract f2 to 1/3, etc. (6) Iterate this procedure to obtain a uniformly convergent sequence of continuous functions whose limit is a continuous function. We shall call this the Peano cube space-filling curve. Note that this construction can be extended to n-hypercube.

2.4.4 The Infinitesimal Zigzag as Limit Set of Fractal Consider triangle ADB which, without loss of generality and for convenience, we represent by the segment AD which we denote by g1,1 (t). We take as initial generator the segment AD which we denote by g1,1 (t). We rotate g1,1 (t) by −2π to form another basic generator g1,−1 (t). These generators are given by the parametric equations: g1,1 (t) : (x, y) = (t,t), g1,−1 (t) : (x, y) = (t, −t), 0  t 

1 2

To construct function f1 (t), we take g1,1 (t), 0  t  12 as part of f1 (t) and combine it with the translation of g1,−1 (t) given by (x, y) = g1,−1 (t) + (1/2, 1/2), 0  t  12 .

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The Hybrid Grand Unified Theory

To construct f2 (t) in the iteration process, we contract g1,1 (t) by 12 and denote this by g2,2 (t). This is one of the generators. Rotate g2,2 (t) by − π2 to form another generator g2,−2 (t). The generators are given by g2,2 (t) : (x, y) = (t,t), g2,−2 (t) = (t, −t), 0  t  1/4. The equation of f2 (t) is given by the following system of equations, f2 (t) : (x, y) = g2,2 (t), (x, y) = g2,−2 (t) + (1/3, 1/4), (x, y) = g2,2 (t) + (1/2, 0), 1 (x, y) = g2,2 (t) + (3/4, 1/4), 0  t  . 4 We iterate this construction to generate a sequence of functions fn (t) represented by the curve Cn , n = 1, 2, . . . , with the following properties: (1) The sequence fn (t), n = 1, 2, . . . , is uniformly convergent and each fn is continuous, √ (2) For each n, the length |Cn | of Cn satisfies |Cn | = |C1 | = 2, (3) The sequence fn converges to a continuous function fn (t) represented by the curve C0 : y0 (t) = 0 which coincides point-wise with the √ordinary √ constant curve C; x(t) = 0, 0  t  1. But distinct from it since |C0 | = lim |Cn | = lim 2 = 2 = |C|. This curve C0 is the infinitesimal zigzag. One property of the superstring is that, left alone, it shrinks steadily. We model this behavior mathematically using the superposition of a sinusoidal curve over a polygonal line (the sinusoidal curve is the projection of the helix on the plane through its axis). Given any real number r > 1, there exists an isosceles triangle ADB with sides AD and DB having total length AD + DB = r. We form a sequence of polygonal lines whose limit in the sup norm is AB but whose length is r. Thus, AB is a coincidence of countably many curves distinct from AB and from each other. The ordinary segment AB is the visible element of the countably infinite space of generalized fractals. The rest is dark matter. The preceding construction can be generalized to apply to any triangle, ADB, where the slopes of AD and DB are m1 and m2 . respectively, and −∞  m1  ∞, −∞  m2  ∞. Then we can pack an infinity of infinitesimal zigzags into AB whose lengths vary along all values in the interval (|AB|, ∞) each of which has set-valued derivative (m1 , m2 ) at each point on AB.

2.4.5 A More General Fractal Given compact set B in Rn (subspace of the n-Cartesian product) and any real number s, 0  s  1, then the set sB = {sb | b ∈ B} is similar to B in the sense that for any point b ∈ B with component bk , k = 1, 2, . . . , n, sbk /bk = s. i.e., ratios of components are preserved. The set sB is a contraction of B along the projection cone B∗ with vertex at the origin. 

In fact, B∗ can be expressed as, B∗ = 0t1 {tB}, and is obtained by taking the union of B with its projection cone whose vertex is the origin. Note that, since for any real number s, 0 < s < 1, lim sm = 0, the compact set B in Rn can be shrunk to a point at the origin by iterated construction, sB, s2 B, . . . , sm B, . . . Suppose B has diameter δ , where δ = sup[d(p, q) | p, q ∈ B], and d(p, q) is the Euclidean distance between p and q. Let s = 1/k, where k is some integer. We consider a sequence of affine transformations of the compact set B whose initial generator is the set sB which we denote by g1,1 . By rotation and translation, we adjust g1,1 so that a diameter coincides with the first component coordinate axis

Mathematics of Grand Unified Theory

37

with the left endpoint at the origin ). We take k successive translations of g1,1 as follows: G11 = g1,1 G12 = g1,1 + (1/k, 0, . . . , 0) G13 = g1,1 + (2/k, 0, . . . , 0) ...

...

G1k = g1,1 = ((k − 1)/k, 0, . . . , 0). We take the union of G1,m , m = 1, 2, . . . , k − 1, to obtain, G1 =

k−1

m=1 G1,m .

This method can be further generalized by allowing linear combination of different contractions of the generator at each stage in the iteration process. To construct G2 we contract G1 by 1/k to obtain the generator g21 . Then we take successive translations of g21 as follows: G21 = g2,1 G22 = g2,1 + (δ /k, 0, . . . , 0) G23 = g2,1 + (2δ /k, 0, . . . , 0) ...

...

G2k = g2,1 + ((k − 1)δ /k, 0, . . . , 0). Then we have, G2 =

k−1

m=1 G2m .

The second generator is translated and put end to end; this third generator is contracted by 1/3 and again translated to put end as shown. We iterate this construction to obtain a sequence of compact sets G1 , G2 , . . . , Gm , . . . which shrink to the first component axis joining the origin and the point (δ , 0, . . . , 0). Again, obvious generalization can be done by allowing linear combination of different contractions of the generator at each stage in the affine transformation. This fractal mathematics, particularly, infinitesimal oscillation is the key to an understanding of physical singularities such as the black hole and the tremendous but dark or latent energy in the nucleus of an atom. As we shrink n oscillations to a point with its length preserved, its energy hv rises without bounds. However, it is dark (undetectable) with respect to our present means of observation such as light due to difference in orders of magnitude between their frequencies (the same principle that applies to non-resonance of radio or TV reception). Note that the proofs of the above theorem are geometrical and much simpler and suitable for animation. In fact, fractal construction of the Peano space-filling curve has been animated in the presentation of [21]. Fractal is everywhere in nature because it is an equivalent form of energy conservation. It is one of the twenty equivalent forms identified in a law of nature called Energy Conservation Equivalence.

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2.5 The Integrated Pontrjagin Maximum Principle 2.5.1 The Integrated Version This version of the Pontrjagin maximum principle was developed in [128, 131] (with slight improvement in [30]) from the original principle by Pontrjagin and his associates [95], to apply to generalized curves, specifically, relaxed trajectories of optimal control theory. Anchored on a number of existence theorems, it was meant to rid optimal control theory of the Perron paradox, a contradiction arising from the use of necessary condition without existence theory [131]. The principle itself is a necessary condition consisting of three parts but it rests on proof of existence of optimal solution of the optimal control problem among relaxed trajectories. We state this principle here because it calculated the trajectories, positions and velocities of the general gravitational n-body problem once the qualitative solution was provided by GUT. But why do we really need to go through this level of sophistication when GUT has already determined that the n-bodies fall to the cores of their respective cosmological vortices along rotating spiral streamlines? The streamlines are not ordinary curves but generalized curves, i.e., having set-valued derivatives, and for purposes of applications we need the specific equation of the trajectory of each body in the problem. In fact, we can look at the falling body as subjected to two-valued control set one value is the pull of gravity and the other the impact of the centrifugal force of spin. The path of a body is what is called infinitesimal simplicial curve, i.e., piece-wise arcs corresponding to alternating constant values of the control set U consisting of two elements, a generalized curve called relaxed trajectory and superposed on it is another generalized curve due to the micro component of turbulence but it has no visible impact on this macro problem.

2.5.2 Formulation of the Problem We shall first summarize the formulation of the time-optimal control pre-problem for each body in the original setting of naive optimal control theory [95], i.e., without the benefit of existence theory. Then we update the formulation to be able to utilize the Integrated Pontrjagin Maximum Principle. We ask for the minimum of the integral 

f (t, x, u, w) dt,

for trajectories x(t), controls u(t) and constants w, subject to x˙ = g(t, x, u, w) dt, where u(t) range in given set U and w in W and appropriate conditions are specified (note that the dimensionality is at our disposal). We can eliminate the constants w by regarding the pair (x, w) as a point in higher space and adjoining the differential equation dw =0 dt which insures that w is constant along any trajectory. Then we add to our end condition that the initial or final values of the projection w of (x, w) lie in W . Thus, the only effect of the constants is to alter dimensionality and end conditions.

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39

To further simplify the problem we introduce another coordinate x0 subject to x˙0 = f (t, x, u), and include x and g for the pairs (x0 , x) and ( f , g). We also add the end condition x0 at the final end of a trajectory which correspond to the time t = 0; in other words, we reverse time and find the minimum of −x0 for trajectories x(t) and u(t) subject to x˙ = g(t, x, u),

(2.12)

where u(t) ranges in U and x(t) satisfies appropriate end conditions. Thus, in this problem what is being minimized is the function x0 which in applications can be the cost function. Without loss of generality we assume that the pair of endpoints of x(t) belongs to some pre-assigned closed set B of the Cartesian product of x-space with itself (i.e., the initial values of t is not restricted directly, e.g., we can set t + t0 in place of t). We denote by G(t, x) the set of values of the vector g(t, x, u) for fixed (t, x) as u ranges in U. Then we ask for the minimum of −x0 subject to the condition x˙ ∈ G(t, x).

(2.13)

The problem with constraint (2.12) is called the controlled pre-problem; the one with the constraint (2.13) is called the uncontrolled pre-problem. Note that the latter has larger space of trajectories from which to find the minimum. As the space of trajectories becomes larger the better is the chance of existence of minimum. The space of trajectories of either problem is still conventional and we can still improve the solution by enlarging the space beyond conventional trajectories into the space of generalized curves. This is exactly what we need for the gravitational n-body problem because the trajectories we are looking for are the spiral streamlines each of which is the local resultant of the effect of gravity and centrifugal force mathematically modeled by a relaxed trajectory. Instead of the constraint equation, x˙ = u, where the velocity vector is controlled directly and coincides with the control vector u we attach a probability or unit measure to u (normalized probability distribution) so that the actual velocity dx/dt becomes the integral of u with respect to this unit measure. This is called the weighted average or expectation value for this probability measure. Then the control function u(t) that yields a specific trajectory x(t) will now be replaced by the probability measure v(t). Such measure is called chattering control value v, and we say that it reduces to a conventional control u if the measure is totally concentrated at u. We write V for the space of chattering control values v, i.e., the set of unit measures on U.

2.5.3 Existence Theorems We quote some existence theorems on conventional and relaxed trajectories from [131]. (1) Existence of solutions to the general initial value problem for ordinary differential equations. Let f (t, x) be a vector-valued function with values in n-space and suppose in some neighborhood of (t0 , x0 ), f (t, x) is continuous in x for each t, measurable in t for each x, and uniformly bounded in (t, x). Then there exists an absolutely continuous function x(t) defined in some neighborhood of t0 such that x(t0 ) = x0 and, almost everywhere in that neighborhood, x˙ = f (t, x(t)).

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(2) Halfway Principle of McShane and Warfield. Suppose given a continuous map p∗ from Q to P, and a measurable map p∗∗ from R to P such that p∗∗ (R) ⊂ p∗ (Q) ⊂ P. Then there exists a measurable (lifting) map q∗ from R to Q such that p∗∗ = p∗ q∗ .  x) the set of the values of G(t, x, v) when (t, x) is kept fixed and v allowed to We denote by G(t, vary in V . (3) First Corollary. Let x(t) be continuous in the finite time interval T and let z(t) be measurable  x). Then there exists a measurable vector-valued function in T such that z(t) ∈ G(t, x) or z(t) ∈ G(t, conventional or chattering control u(t) or v(t) such that z(t) = g(t, x(t), u(t)) or z(t) = g(t, x(t), v(t)). (4) Second Corollary (the Filippov Lemma). If, in particular, x(t) is an (uncontrolled) conven x) almost everywhere, tional trajectory subject to x˙ ∈ G(t, x) or relaxed trajectory subject to x˙ ∈ G(t, then there exists a measurable conventional or chattering control u(t) or v(t), so that x(t) coincides with the corresponding controlled trajectory satisfying the differential equation dx(t) = g(t, x(t), v(t)), dt

x˙ = g(t, x(t), u(t)) or almost everywhere.

(5) Uniqueness Theorem for the Initial Value Problem of an Ordinary Differential Equation dx/dt = f (t, x). Suppose, in addition to the hypothesis of (1), that for some constant K, the function f (t, x) satisfies, whenever (t, x1 ) and (t, x2 ) lie in some neighborhood N of (t0 , x0 ), the Lipschitz condition, | f (t, x2 ) − f (t, x1 )|  K|x2 − x1 |. Then in some neighborhood of t0 there exists one and only one absolutely continuous function x(t) such that  x(t) = x0 + f (τ , x(τ )) d τ , Δ

where Δ = [t0 ,t]. Let f ∈ C0 (T ×U), the space of continuous functions f (t, u) on T ×U, where U is the set of control values u, and T is some fixed time interval. We write Δ for some variable time interval of U. For such pair ( f , Δ) consider the function w of ( f , Δ) defined by the integral w( f , Δ) =

 Δ

f (t, v(t))dt,

where v(t), t ∈ T is a measurable chattering control (or v is conventional control, i.e., unit measure concentrated at some point u ∈ U). We understand the integrand f (t, v(t)) as shorthand notation for an integral of f , for constant t, with respect to the probability measure v(t) on U. Then we regard w as a measure and identify w( f , Δ) with the integral w( f , Δ) =

 Δ

f (t, v(t)) dt.

(2.14)

Then we write w = v(t)dt.

(2.15)

Bearing in mind that (2.15) is a double integral; every control measure is determined by a chattering control v(t).

Mathematics of Grand Unified Theory

41

The control measure w will be termed simplicial if it is defined by ((2.14) where v(t) reduces to a conventional piece-wise constant control u(t). Then we use the formula (2.14) and reinterpret u(t) as a unit measure on U concentrated at one point u(t). We denote by W the space of all control measures w. We introduce fine convergence in this measure. A sequence of control measures wν , ν = 1, 2, . . . , is termed convergent if, for each f , the values wν ( f , Δ) tend to a limit w( f , Δ) uniformly in Δ; then we say that wν tends to w. (6) Theorem. (i) The space W is sequentially complete. (ii) In order that a real function w of f and Δ be of the form w( f , Δ) where w ∈ W, the following system of conditions is both necessary and sufficient. (1) w( f , Δ) is linear in f and additive in Δ; (2) f (t, u)  0 in Δ ×U implies w( f , Δ)  0; (3) f (t, u) = 1 in Δ ×U implies w( f , Δ) = |Δ|. Theorem 2.7. (i) The space W is sequentially compact. (ii) Simplicial control measures are dense in W . We term bundle of relaxed, conventional or simplicial trajectories the family of trajectories which meets given bounded closed subset of (t, x)-space corresponding to closed time intervals, possibly degenerate ones all contained in a fixed time interval. A sequence of functions xν , t ∈ Tν , t ∈ T0 , ν = 1, 2, . . . , where Tν are closed time intervals all contained in some fixed time interval will be said to converge uniformly to x0 , t ∈ T, if, first, T0 xν , t ∈ T0 , a closed time interval whose extremities are the limits of those of T , and, second, for some choice of a closed time interval T containing T0 and all but a finite number of the Tν , there exist, for large ν , extensions of our functions of the form xν , t ∈ T , which tend uniformly to a corresponding extension to T of x0 (t) (T0 may reduce to a single point). (7) Theorem. A bundle of relaxed trajectories is sequentially compact and complete and the corresponding bundle of simplicial trajectories is dense in it. (8) Corollary. Suppose the set G(t, x) of the values of g(t, x, u) for fixed (t, x) is convex. Then any bundle of conventional trajectories is sequentially complete and compact, and the corresponding bundle of simplicial trajectories is dense in it. (9) Existence Theorem for Relaxed Solutions. Let Q be a bounded closed set of (t, x)-space, P a closed set in the Cartesian product of (t, x)-space with itself and T a closed finite interval of t. We denote by Σ the set of relaxed trajectories x(t) defined on closed subintervals of T which meet Q and passes a pair of extremities situated in P. The function g(t, x, u) which appears in the differential equation x˙ = g(t, x, u) is supposed continuous and subject to the Lipschitz condition in x. Then either Σ is empty or there exists in Σ a relaxed trajectory for which the difference at the endpoints of the coordinates x0 of x assumes its minimum.

2.5.4 The Pontrjagin Maximum Principle For greater generality we suppress dependence on the chattering control v(t) by writing g(t, x) for g(t, x, v(t)). Consider a convex family G of such functions, a family such that every convex combination Σαi gi , of a finite number of members gi , of G with constant coefficients αi  0, where ∑ αi = 1, is itself a member of G (in the chattering case the family G consists of functions g of the form g(t, x, v(t)) and convexity holds in a stronger sense in which the coefficients αi are allowed to be measurable functions of t instead of constants). In addition we require that every function g(t, x)

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in G is continuously differentiable in x for fixed t and measurable in t for fixed x, and also that each g and its partial derivative gx are bounded functions of (t, x) or, more generally, bounded in absolute value by some integrable function of t only. These various requirements are to hold only in some bounded open set O of (t, x) space. In the chattering case all these requirements are satisfied if we make the stipulation that g(t, x, u) is continuously differentiable. We consider the family H = yG of Hamiltonian functions h(t, x, y) = yg(t, x), where y is a variable vector and each g ∈ G gives rise to a corresponding λ ∈ H. we shall be concerned with points (t, x) that lie in a sufficiently fine neighborhood of the set described by a given fixed trajectory C of the form x(t),t1  t  t2 . In terming C a trajectory we imply that the function x(t) is, almost everywhere in its interval, a solution of the differential equation, x˙ = g(t, x(t)),

(2.16)

for some fixed corresponding member g ∈ G; moreover, x(t) is to be absolutely continuous. We term ordinary point of C a point at which (2.16) equation holds; in particular, we say that C has ordinary endpoints if the derivatives x(t ˙ i ) exist and have the values g(ti , x(ti )), i = 1, 2. We suppose the function x(t) continued outside its interval of definition, when convenient, subject to the same differential equation and absolute continuity. In view of the uniqueness theorem any such extension is uniquely determined once we fix the member g ∈ G and an initial condition of the type x(t0 ) = x0 . We write q for the ordered pair of endpoints of C and P for a small neighborhood of q. Thus, p lies in the space of such ordered pairs q, i.e., in the Cartesian product of (t, x)-space with itself or, equivalently, O with itself. We denote by Q the subset of P consisting of ordered pairs (p, q) of endpoints of trajectories in O, sufficiently close to C. Any such trajectory has the form χ (t), τ1  t  τ2 where χ (t) is absolutely continuous and satisfies, for almost all t of its interval of definition, a differential equation similar to (2.16) with g replaced by some member g of G. In P we shall suppose given a smooth manifold M with q as boundary point where M is represented by local coordinates as a smooth one-to-one map of a smooth Euclidean domain with its boundary. We take the interior of M and boundary of M as corresponding images of the interior and boundary of this domain. We suppose the dimension of M to be  1 so that it does not reduce to a point. We suppose, further, that the boundary of M at q has a tangent subspace, which is itself the boundary of a tangent half-space (a tangent half-subspace by taking half-lines tangent to M at q). We suppose that a neighborhood of q in M has a continuous one-one map onto a neighborhood of q in this tangent half-subspace such that q corresponds to itself and that, if q + δ p denotes the image of p in M, we have P = q + δ p + o(p − q) where o(p − q) is small compared with p − q as p → q. In particular, local coordinates can be thought of as coordinates on the tangent half-subspace. Moreover, a vector φ = 0 in the underlying (2n + 2)-dimensional Euclidean space will be termed an inward normal of M at the point q if it is, first, orthogonal at q to the boundary of M, i.e., to a hyperplane through q that contains the tangent subspace of this boundary and, second, directed towards the side of this hyperplane that contains the tangent half-subspace of M. A trajectory C is an M-extremal if it contains no interior point of M; we term conjugate vector along C an absolutely continuous and nowhere vanishing vector-valued function y(t) with values in nspace defined on the same interval as x(t). If h is the Hamiltonian corresponding to the element g ∈ G that enters into the differential equation (2.16) satisfied by x(t), we term corresponding momentum

Mathematics of Grand Unified Theory

43

and denote by η (t) the (n + 1)-dimensional vector derived from y(t) by the adjunction of the initial component h(t, x(t), y(t)) = 0(t). We term corresponding transversality vector for C the (2n + 2)-dimensional vector (η (t1 ), η (t2 )).

(2.17)

2.5.5 The Integrated Form of the Pontrjagin Maximum Principle Let g ∈ G. Let h be the corresponding Hamiltonian function yg(t, x) and let C be an M-extremal of the form x(t),t1  t  t2 , satisfying, almost everywhere, the corresponding differential equation (2.16) which we now write x˙ = ∂∂ hy . If C has ordinary endpoints or M consists of pairs with the same coordinates as q, then there exists a conjugate vector y(t) along C such that the pair (x(t), y(t)) satisfies the following three conditions (a), (b), (c): (a) The canonical Euler equations: x˙ = (b) The Weierstrass condition: As function of λ ∈ H, the quantity

∂h , ∂y

y˙ = −

∂h . ∂y

 Δ

h(t, x(t), y(t)) dt

assumes its maximum when λ = h. (c) The transversality condition: Then the transversality vector (2.17) is an inward normal of M.

Chapter 3

Grand Unified Theory (GUT)

3.1 Introduction The great achievements of mathematical physics, such as providing a broad picture of our universe from the very small to the very large and giving both to technological advances, are the result of mathematical modeling whose principle tools are computation and measurement. Nonetheless, to solve long standing unsolved problems, to resolve unanswered questions and unsettled issues of physics, mathematical modeling is inadequate. We need qualitative modeling, which is the complement of mathematical modeling, as discussed in Chapter 1. In Section 3.2, we proceed to find the basic constituent of matter which is the key to our understanding of nature. Starting from the existence of dark matter which consists of small objects undetected by light and whose properties are known only by its impact on visible matter, we employ qualitative modeling to discover certain natural laws that are sufficient to provide the structure and behavioral properties of dark matter and explain that the basic constituent of matter is the superstring and the primum is the unit of visible matter. Historically, quantum gravity was shaped by the search for graviton, which was supposedly responsible for gravity, both at micro and macro scales. As it turns out, the graviton does not exist even as a theoretical necessity. Section 3.3 proceeds to explain quantum gravity, by qualitative modeling, as the local dynamics of vortex flux of superstrings while Section 3.4 explains macro gravity explicitly as the dynamics of cosmological vortices of superstrings from the super...super galaxy, our universe, through the galaxies, stars, planets, moons and cosmic dust. Finally, Section 3.5 deals with explanation of certain natural phenomena, verifying grand unified theory and Section 3.6 provides an overview of GUT..

3.2 The Search for the Basic Constituent of Matter Given the patterns and regularity in nature that we have observed, we are now ready to upgrade the first law of thermodynamics into what we consider the most fundamental law of nature so that other natural laws that we need to discover must be consistent with it and if some natural phenomenon appears to contradict it we find some natural law that reconciles them. This will be our strategy. Energy Conservation. In any physical system and its interactions, the sum of kinetic (visible) and latent (dark) energy is constant, gain of energy is maximal, loss of energy minimal. The first statement says that matter and energy can only be converted from one form to another and in any conversion the total energy remains constant. The other two statements say that nature V. Lakshmikantham et al., The Hybrid Grand Unified Theory, Atlantis Studies in Mathematics for Engineering and Science 3, DOI 10.1007/978-94-91216-23-7_3, © 2009 Atlantis Press/World Scientific

45

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The Hybrid Grand Unified Theory

optimizes. Our primary agenda now is to discover the basic constituent as the key to our knowledge of nature. Therefore, we discover other natural laws to reach that objective. The next natural law states the various expressions of energy conservation so that we can choose the relevant expression for specific purposes such as describing or explaining the configuration of the basic constituent and any other configuration is energy dissipating. Energy Conservation Equivalence. Energy conservation has other forms: order, symmetry, economy, least action, optimality, efficiency, stability, self-similarity (fractal), coherence, resonance, quantization, smoothness, uniformity, motion-symmetry balance, non-redundancy, evolution to infinitesimal configuration, helical and related configuration such as circular, spiral and sinusoidal and, in biology, genetic encoding of characteristics, reproduction and order in diversity and complexity of functions. This law provides information not achievable by computation, e.g., the shape of the electron and configuration of photon and cosmic wave. Non-redundancy means that nature does not create another physical system when there is already one with the same functions. The so-called third quark in the nucleus outside the proton discovered in 2004 does the same function as the negative quark in the proton, joins two positive quarks. The third quark also joins two positive quarks, one from each of two protons. Therefore, they must be both negative quark. This law explains a lot of other things such as the spiral paths of free-falling bodies observed in young galaxy. The steady formation of matter in the Cosmos combined with energy conservation reveals another law of nature. Existence of Two Fundamental States of Matter. There exist two fundamental states of matter: visible and dark; the former is directly observable, the latter is not. With the existence of dark matter established, this question is no longer vacuous: what does dark matter consist of? The answer, the superstring resolves the 5,000-year-old quest for the basic constituent of matter. For the moment it is just a name but as soon as we discover enough natural laws to give it “flesh”, i.e., structure, behavior and properties, we shall do so. Why is it that we cannot detect the superstring? Our medium of observation is light. To be detected, an object in the Cosmos must emit or reflect light or be silhouetted against a source of light. Obviously, the superstring does not emit light because it is lighter and smaller than a photon of light. When does an object reflect a photon of light? When its rest mass is at least equal to the momentum of a photon of light and obviously that of the superstring is much less. Can it be silhouetted against a light source? Obviously, not since we have not detected it. What can be the reason? For an object to be detected by light it must demolish a wavelength to be detected by the sensor, the spectroscope. It is called interference or discordant resonance. For it to occur its size must be of the same order of magnitude as visible light and the finest visible light is just beyond the blue spectrum and towards but before the ultra-violet which is of order of magnitude 1014 meters. We proceed from this assumption based on the principle of resonance that says, only object of size at least the same order of magnitude of the wave length of light interferes or has discordant resonance with it. They explain why the superstring could not be found in the atom. We know that the superstring exists but we do not know its structure and properties. We think backwards starting from its impact on visible matter. We already know one impact: formation of matter in the Cosmos. Another impact is the stability of our universe that has existed for billions of years. This means that the superstring is indestructible. As a piece of matter, the superstring must have motion. Furthermore, every piece of matter must be reducible to it. These are the parameters

Grand Unified Theory (GUT)

47

within which we endow the superstring with full structure and properties. We need more natural laws to achieve this; we first define some concepts. Flux is motion of matter with identifiable direction at each point, e.g., water current, wave and cyclone. Turbulence is coherence of fluxes, e.g., typhoon as vortex. In any flux there is always resistance by the medium called viscosity. This is mitigated by local vibration called micro component of turbulence so that the resistance is turned to advantage to facilitate passage through resonance. Thus, cosmic waves ride on the synchronized vibration of the superstrings. The latter is part of its latent energy. Riding on vibration is also a form of resonance. Chaos is mixture of order none of which is identifiable. A classic example of chaos is the initial phase of formation of tropical cyclone when trillions of gas molecules rush towards tropical depression and collide among themselves. The motion of each molecule cannot be identified due to the immensity of their number and yet each molecule is subject to the laws of nature. Moreover, since collision is energy dissipating, energy conservation induces this chaos towards global order, a vortex called tropical cyclone. The next law explains cyclone and dark vortex formation. Flux-low-pressure complementarity. Low pressure sucks matter around it and the initial chaotic rush of dark matter towards low pressure stabilizes into local or global coherent flux; conversely, coherent flux induces low pressure around it. The discovery of this law was inspired by a high school experiment in physics many years ago: place two books on the desk with parallel back edges three inches apart; place a thin light paper over the gap and blow under. What will happen to the paper? Intuition says it will fly off but in reality it will sag down and get sucked by the flux of air under it. Now, let us to go back to the superstring and its indestructibility. Destruction occurs only in interaction of objects of comparable order of magnitude of size when there is interference or discordant resonance. What is the possible cause of its destruction? A hammer cannot do it because there is no resonance there, destructive or harmonious. The only possible cause of its destruction is the impact of cosmic wave since it is due to vibration of the dark component of atomic nucleus whose order of magnitude is comparable to that of the superstring. Therefore, there is resonance between their vibrations. The vibrating atomic nuclei generate and propagate basic cosmic waves across dark matter, i.e., the medium for cosmic waves that we referred to earlier is dark matter. The nuclear vibration induces the superstrings around it to vibrate and propagate outward, by resonance, and that is how cosmic wave is generated Now let us think hard. What configuration should the superstring have to be indestructible to the impact of cosmic wave and at the same time have motion in it like the atom? Let us do a little thought experiment first. Imagine an eggshell that contains an eggshell that contains an eggshell, ad infinitum. This is called nested fractal sequence of eggshells. If we hit the outermost eggshell in the sequence by a hammer can we destroy all the shells? No, only at most a finite number of them. Why? The tail eggs must be so small they will hide in the huge spaces of the hammerhead among the atoms and because of their nested fractal configuration the surviving sequence of shell will remain nested fractal sequence of shells and so it survives. Therefore, to be indestructible, the superstring must be nested fractal. How about its shape? How should it look like? From energy conservation and energy conservation equivalence it must have the following qualities: smooth, has maximum symmetry, a loop for motion in it to be sustained and helical to optimize space (like the DNA strand). Let us now take as a law of nature that the superstring is nested fractal and call it the fractal principle. When hit by cosmic wave, a superstring, possibly after shedding off its first few outer layers in the fractal sequence, is imparted with kinetic energy and thrown off into collision with other

48

The Hybrid Grand Unified Theory

superstrings (as one possibility since they have the same order of magnitude of size) bouncing into erratic motion like Brownian motion that forms an erratic flux. Only two things can happen here: (a) the kinetic energy imparted by the cosmic wave is exhausted or dissipated and the superstring comes to rest or (b) it comes near its earlier path and gets sucked by it, by flux-low-pressure complementarity, to form a loop (closed path) with itself as a flux along the loop. Energy conservation induces this loop to evolve into an optimal configuration, namely, helical with the original superstring as its flux, called toroidal flux along its cycles (we shall understand later why this name is used). This is now a new superstring. By the fractal principle its toroidal flux is a superstring whose toroidal flux is a superstring, etc.; then its nested fractal structure becomes clear. The first term of the newly formed superstring is what has been referred to as the superstring because its toroidal flux was not known then. This new superstring is semi-agitated, i.e., its cycle length (CL) lies between 10−16 and 10−14 meters; a superstring is non-agitated if CL < 10−16 and agitated if CL > 10−14 m. Semi-agitated and non-agitated superstrings are dark and agitated superstring is visible. Left alone, a superstring steadily shrinks, by energy conservation, since the motion of the toroidal flux is more energy dissipating over a longer distance and nature optimizes. Both semi- and non-agitated superstrings have the configuration of a hollow torus, hence, the term toroidal flux. The g-sequence of a decimal models the nested fractal superstring and d* models its tail end, a continuum. We can now introduce the black hole as massive concentration of non-agitated superstrings that accumulates in the eye of cosmological vortex such as star or galaxy because the eye is a region of calm and de-agitates matter at its boundary. We consider details later. There is another possibility that can occur: the kinetic energy of the latter is absorbed by its first term as nested fractal by expanding (i.e., raising the latent energy of the toroidal flux) to a semiagitated superstring. This is not new but converted to semi-agitated superstring. Note that the toroidal flux has just enough energy to suck itself and form a loop that evolves to semi-agitated superstring, a form of quantization principle of quantum physics as a law of nature. Consequently, just like an inert atom, the dark superstring does not interact with anything else. Therefore, it is absolutely stable and this is true of all superstrings that provides absolute stability to dark matter. Right here we have an absolute frame of reference for our universe. Since cosmic waves come from everywhere in the Cosmos at all times there is steady formation of semi-agitated superstrings as well as conversion from non-agitated to semi-agitated superstrings (the reverse process occurs at the eye of every cosmological vortex where there is calm and de-agitation).

3.2.1 The Primum What happens when a semi-agitated superstring is struck by cosmic wave? (a) It breaks and its toroidal flux remains non-agitated and at rest in dark matter, the path of the latter terminated by the impact, or (b) the impacted segment expands and becomes agitated, i.e., visible; it is called primum, a unit of visible matter. This completes the requirements on the superstring as basic constituent of matter, dark or visible. Among the familiar simple prima are the electron, positron, and the positive and negative quarks. The neutrino, proton and neutron are coupled prima. Since a simple primum forms by the expansion of the helical cycles of a segment of semi-agitated superstring it remains a helix whose lateral section has sinusoidal envelope (by energy conservation equivalence). In other words, its envelope is obtained by the rotation of a full arc of a sinusoidal curve of even power (that preserves symmetry). “The more energized a primum the thinner its discular cross-section” is due to centrifugal force of its toroidal

Grand Unified Theory (GUT)

49

flux and the greater the exponent of the generating sinusoidal arc the flatter its tip. The plane through its rim is called equatorial plane, its extreme cycle at its rim the equator. The primum is a different ballgame, so to speak, because unlike the dark superstring it is interactive. Hit by cosmic waves from all directions, its toroidal flux is thrown into erratic motion (analogous to Brownian motion) called spike in the neighborhood of its helical path but continues to travel along its helical cycles, by flux-low-pressure comlementarity, at great speed (we shall confirm later that its speed is greater than that of light). Consequently, it pulls the superstrings around the primum into a vortex flux (induced by its toroidal flux) with cylindrical eye along its axis. Again, this flux is motion of matter and is, therefore, energy called charge. This makes the primum a magnet. When its vortex flux is clockwise viewed from the N-pole it has negative charge (positive if counterclockwise, by convention). The electron has negative charge, −1, the unit of charge. Its energy is 1.6 × 1019 coulombs (q). Its mirror image with respect to a plane normal to its equatorial plane beyond its vortex flux is the positron, its anti-matter [39]. Its vortex flux is counterclockwise, its charge +1. The positive and negative quarks have charges +2/3 and −1/3, respectively [58]. They are the basic simple prima because they make up all atoms. They are produced in large quantities by the second. In fact, they make up the tissues of living organisms from the embryos onwards. The genes convert dark matter to the tissues of living organisms in the cellular membrane [45, 78, 87] by their radiation (brain waves). Of course, the basic prima are also produced in staggering amount in the Cosmos; they make up the cosmic dust that congeal into galaxies, stars, planets and moons.

3.2.2 Primal Polarity The properties of a primum as magnet follow from flux compatibility and primal interaction is governed by it along with flux-low-pressure complementarity. Flux compatibility. Two prima of opposite toroidal flux spins attract at their equators but repel at their poles; otherwise, they repel at their equators but attract at their poles. Two prima of same toroidal flux spin connect equatorially only through a primum of opposite toroidal flux spin between them called connector. Note that the negative quark is connector between positive quarks just as the valence electron is connector between two atoms of a molecule. Note that every simple primum has charge. Therefore, a primum that is neutral is necessarily coupled, e.g., neutron and neutrino. The charge of such primum is neutralized by the opposite charges of its component prima. A primum has spin, i. e., rotation as unit of matter different from its toroidal flux spin. For example, the electron has spin 1/2, meaning it rotates about its axis in the direction of its toroidal flux which is clockwise. To be able to discuss primal polarity we first make the following assumptions to be confirmed later: (1) Earths gravity (as well as gravity of any cosmological vortex) is part of the dynamics of its vortex flux of superstrings called gravitational flux. Its direction is from West to East and the Earth being collected mass around its eye is pulled by it into a spin of 24-hour cycle. (2) There is a lag in the Earths gravitational flux from the equator to either Pole, fastest at the Equator and zero flux at either Pole. Since non-agitated and semi-agitated superstrings have infinitesimal induced flux around their cycles, they do not interact with other superstrings and prima and are, therefore, randomly oriented.

50

The Hybrid Grand Unified Theory

When a semi-agitated superstring is agitated and pops out of dark matter as free primum and its equatorial plane is oblique to the direction of the gravitational flux, it rotates counterclockwise in the Northern Hemisphere (clockwise in the Southern Hemisphere) in view of (2) and aligns its equatorial plane in the direction of the gravitational flux. This is the optimal energy-conserving alignment of a free primum. By flux compatibility, a positive primum is pushed up so that there is abundance of free protons and positrons in the upper atmosphere (confirmed by shower of fragments of protons smashed by energetic cosmic waves that fall on Earth [98]). Free neutral prima are oriented randomly. Free positive ions are counterclockwise eddies in the Earths gravitational flux and are also pushed upwards, by flux compatibility. However, being heavy, they remain in the clouds in the lower atmosphere. The electron as clockwise eddy in the Earths gravitational flux is pushed downwards, by flux compatibility. Thus, there is abundance of free electrons on the ground. Other free negative prima including the negative quarks should be abundant on the ground but we do not know where they are; this needs investigation. When the voltage between the positive ions in the lower atmosphere and the electrons on the ground reaches critical level they rush towards each other, collide and explode as lightning (a bolt of lightning has the energy of one megaton of TNT). A moving charge is electric current and the electron is known to be carrier of electric current. It is not clear if it is the only carrier. At any rate, this separation between positive and negative prima contributes to the stability of cosmological vortices like Earth. Outside the Earth’s gravitational flux, primal orientation is determined by the dominant gravitational flux there; between planetary gravitational fluxes it is the Suns gravitational flux that prevails. Outside the solar system, the Milky Ways, etc. It is clear that the basic prima form steadily, especially, in living things. Since they are trapped in the cells of living organisms they are not polarized and separated by the Earths gravitation flux and form their atoms, molecules and tissues. In the vast neutral regions of the Cosmos they form atoms and molecules that get trapped in micro vortices and collect at their cores as cosmic dust. They, in turn, get entangled in macro vortices and collect and form cosmological bodies such as galaxy, star, planet and moon.

3.2.3 Primum, Photon and Wave-Particle Duality Basic cosmic wave travels at the speed of light. When it scoops up a primum the latter flattens in the direction of flight due to dark viscosity and becomes rapid oscillation. When it breaks away from its loop it becomes a photon and remains stable when its forward flux speed is equal to the speed of its carrier wave, i.e., the speed of light, c = 105 km/sec in vacuum. Otherwise, it disintegrates, the toroidal flux remaining at rest in dark matter. This occurs, for instance, when the carrier wave passes through opaque barrier that absorbs the photon. That is why one cannot find a photon at rest; its so-called rest mass is calculated from its kinetic energy. The fact that the forward flux speed of a photon equals the speed of light implies that the toroidal flux speed along the fine oscillating arc and also the primums helical cycles it comes from is greater than the speed of light (7 × 1022 cm/sec according to [3]) which confirms our previous statement. (For details see [41]). The wave-particle duality in the behavior of a photon comes from the fact that it is, indeed, a particle, a piece of matter in the form of rapid oscillation of its toroidal flux embedded between arcs of adjacent parallel basic cosmic waves of opposite crests as its envelope (follows from energy conservation equivalence). Since this carrier envelope is sinusoidal, i.e., a wave, it exhibits wave

Grand Unified Theory (GUT)

51

characteristic. Even simple primum that rides on basic cosmic wave flattens into rapid oscillation with the loop remaining intact and exhibits wave-particle characteristics. With its loop intact it is stable unless destroyed by collision or energetic cosmic wave. It is this wave characteristic of the photon that deflects it inward when it passes the gravitational flux of a cosmological body like the Sun, as predicted by Einstein, and allows us to see stars at the back of the Sun during solar eclipse that blocks the Suns intense light. If it were an asteroid instead of a photon it would have been deflected away, fortunately, by flux compatibility (thus, the Earths gravitational flux has shielding effect against asteroids). We model below the photon embedded between its pair of carrier basic cosmic waves by rapid oscillation and the primum in flight also by rapid oscillation [18, 41, 35, 31]. Energy conservation and flux compatibility pull the primal cycles together to form a wild oscillation that requires the generalized integral to do calculation on it [18, 35].

3.2.4 Matter- Anti-Matter Interaction Two simple prima are anti-matter to each other if each is mirror image of the other with respect to a normal to their common equatorial plane between their equators. Having opposite toroidal flux spins, anti-matter attracts each other at the equators. When they get close, the momentum of their attraction forces their cycles to overlap and their fluxes to collide leading to explosion that throws them as photons into opposite directions. The positron is the anti-matter of electron. The logic of GUT says that every simple primum has anti- matter. However primal polarity and the Earths gravitational flux separate them from each other. In a controlled environment positive anti-matter can be produced on the ground and interact with its anti-matter to mutual destruction.

3.2.5 Qualitative and Computational Models of the Primum and Photon The primum at rest is modelled computationally (in cylindrical coordinates) by the helix, x = t, 1 y(t) = β (sin nπ t)(cosm kπ t), θ = nπ t, t ∈ [ −1 k , k ], n, m, k, integers, n k, m even. Its cycle energy −34 is Plancks constant h = 6.64 × 10 joules (cylindrical coordinates). Scooped up by basic cosmic wave, the primal cycles flatten to rapid oscillation, x = t, y(t) = β (sin nπ t)(cosm kπ t), z = 0, in the direction of flight due to dark viscosity. When it breaks off from loop it becomes photon, y(t) = β (sin nπ t)(cosm kπ t), z = 0. The toroidal flux speed of a primum or photon is: 7 × 1022 cm/sec [3] which is much faster that the speed of light of 3(1010 ) cm/sec. The primum is stable when its forward toroidal flux speed equals the speed of light. The primum is stable in flight since its loop is intact. In the case of the photon, when the forward speed of its toroidal flux is not equal to the speed of light, i.e., the speed of its carrier wave (e.g., when blocked by opaque barrier), it disintegrates, its toroidal flux remains in dark matter; thus, the photon is never at rest except when it disintegrates and energy conservation holds. From the energy of the photon one can, theoretically, calculate the number n of its full arcs since each has kinetic energy of h. This is also the same number of cycles of the primum it comes from. However, due to the uncertainty of small number, in this case, the Plancks constant the computation cannot be done and computation on the primum requires integration of set-valued functions [18].

3.3 Quantum Gravity Historically, quantum gravity as a field was shaped by the search for the graviton that is supposedly responsible for gravity both at the micro (e.g., charge) and macro scales. Like the search for the basic

52

The Hybrid Grand Unified Theory

constituent of matter, it did not materialize. In the case of the superstring scientists were looking for it in the wrong place, the atom, because they did not realize one essential requirement: indestructibility, and the atom is not the place to find it since everything there appears destructible. As it turns out the graviton does not exist. It is not even a theoretical necessity since primal and atomic attraction is taken care of by flux compatibility that explains electromagnetism and unifies it with quantum gravity.. Earlier, scientists were also looking for the gluon that supposedly binds two protons together in the narrow confines of the nucleus that, as someone calculated, would require 27 tons of force. This staggering force is derived from electrodynamics that says two particles of charges q1 , q2 that are s cm apart has repulsive force, F = q1 q2 /s2 dynes, which is unbounded as s → 0. This force does not exist also for it takes only one neutron of low kinetic energy (0.25 calories) to penetrate the uranium 235 nucleus and split it. The quark discovered in Chicago (1990) joins two components of the proton which are positive quarks; therefore, we give it the name negative quark. The gluon discovered in 2005 also joins two positive quarks, one from each of two protons in the nucleus. By the law of non-redundancy [38] this is the same negative quark discovered in 1990 only that it functions outside the proton. Thus, the momentary excitement that followed its re-discovery fizzled out quickly. Quantum gravity is the local dynamics of vortex flux of superstrings, induced or pulled by primal toroidal flux into a vortex. Since the nucleus is coupled primum, quantum gravity extends to atomic and molecular interactions. Quantum algebra calculates flux intensity of the primum, in this case, the nucleus, measured as charge. For example, the atomic number of an atom gives the number of orbital electrons and, by flux compatibility, also the number of protons in the nucleus which is proportional to the intensity of nuclear charge since the neutrons do not contribute to it being neutral. We have applications of quantum algebra (below) including calculation of the mass of the neutrino which was unknown previously.

3.3.1 Primal Coupling and Interaction The proton consists of two positive quarks joined equatorially by a negative quark. Its charge is 2/3 − 1/3 + 2/3 = 1 and there is net counterclockwise vortex flux around it with the vortex flux of the negative quark an eddy in it. The neutrino being neutral must be coupled pair of prima of opposite but equal charge joined at their equators so that if the charge of one is q that of the other is −q and q + (−q) = 0, i.e., neutral. The neutron consists of a proton, an electron and a neutrino. We apply flux compatibility and flux-low-pressure complementarity to compose the neutron; this is part of quantum algebra. By flux compatibility, the electron may attach itself equatorially at any point on either positive quark of the proton. However, by energy conservation, the strongest and, hence, most stable coupling is to attach itself to both positive quarks beside the negative quark. Naturally, it pushes the negative quark a bit, by flux compatibility, the four primal centers forming a quadrilateral. Then there arises four coherent vortex fluxes around the center of the quadrilateral: counterclockwise around each of the positive quarks and clockwise around each of the electron and negative quark, making this area in the interior of the quadrilateral around its center low-pressure region that, by flux-low-pressure complementarity, sucks matter around it. It cannot suck charged primum, however, because the charged prima already in the cluster repel it. Therefore, it sucks only neutral prima and the neutrino is the only available one, light enough to fit in and so it squeezes itself in. This completes construction of the neutron. Then its charge is 2/3 − 1/3 + 2/3 − 1 = 0, i.e., neutral, and has no net vortex flux around it.

Grand Unified Theory (GUT)

53

The rest masses of the neutron, proton and electron are known [135]: Neutron: 1.674 × 10−27 kg Proton: 1.672 × 10−27 kg Electron: 9.6107 × 10− 31 kg.

(3.1)

Converting to atomic mass unit (amu) we obtain their masses as follows: Neutron: 1.0087 amu Proton: 1.0073 amu Electron: 5.4860 × 10−4 amu.

(3.2)

Taking the difference between the mass of the neutron and the combined mass of the proton and electron, we find the mass of the neutrino [47]:

η = 8.5 × 10−4 amu or 1.55 times the electron mass.

(3.3)

In primal interaction the toroidal fluxes do not touch for that would violate energy conservation; it is governed by flux compatibility and flux-low-pressure complementarity. (For computational models of prima, see [65]-[68]; ours is qualitative model)

3.3.2 Do Coupled Prima have Anti-Matter? Every simple primal component of coupled primum has anti-primum and can be mutually destroyed by its anti-primum. However, whether the coupled primum can be mutually destroyed by its coupled anti-primum, depends on their configuration. In the proton the positive and negative quarks have their respective anti-prima. Let us call them anti+quark and anti-quark, respectively. Then the protons anti-proton consists of two anti+quark (charge -1 each) joined by an anti-quark (charge +1/3). Their axes are also coplanar; therefore, their attractive momentum when they get close forces destructive coupling. The most likely outcome is: two photons are thrown in opposite directions normal to their equatorial planes and the middle antiquark collides with and is destroyed by the toroidal fluxes of the two

3.3.3 Genesis of the Atom and Formation of Heavy Isotope The first thing that forms in an atom is the nucleus consisting of protons alone. How many atoms form initially depends on the boundary conditions. In the very early universe, after the Cosmic Burst, only the lightest hydrogen atom and a few other light elements formed. Today, the cooler planets like Earth allow formation of complex atoms, e.g., uranium. The massive planets like Jupiter have powerful spin and hot cores that only allow formation of light and simple atoms like gases. The Sun consists mainly of hydrogen although there are helium and boron also which are similarly light. Given the right conditions, e.g., temperature, distribution of quarks, the number of protons in the initial formation of a nucleus is stochastically determined. In the nucleus, protons are joined by negative quarks through their positive quarks and the initial protons remain in the cylindrical eye of their combined vortex flux around the eye. This eye sucks neutral prima; charged prima are repelled by either the positive or negative quark in it. It sucks neutron to form heavy isotope. The suction energy that keeps the neutrons there is, of course, proportional to the number of protons. The periodic table shows that the nucleus holds more neutrons than protons. The neutrino may be sucked also but it simply whizzes by at great speed. Dark superstrings continually accumulate in the nucleus like a mini-black hole, by flux-low-pressure complementarity. They are converted to kinetic

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energy (prima and photons) in nuclear explosion. Electrons are attracted by the vortex flux around the nucleus (provided by the protons) but being quite light they are swept into orbit. This completes the formation of atom; it is neutral when the number of orbital electrons equals the number of protons, by flux compatibility, ion otherwise. The nucleus is in the eye of its vortex flux; its spin throws the protons and neutrons into circular arrangement along its equator, by centrifugal force. flux compatibility establishes the following coupling: When the nucleus is large, circular arrangement is unstable. Therefore, they form discular layers of equatorially coupled neutrons, hollow around the axis due to centrifugal force. This arrangement conforms to energy conservation equivalence. The sucked neutrons cluster evenly, as much as possible along the nuclear equator on both sides of the equatorial plane, also by energy conservation and centrifugal force. This is the most stable nuclear coupling. Since the nucleus consists of prima, it is a coupled primum. Energy conservation requires that in a big nucleus, the closed helices are arranged side by side along its cylindrical boundary which should conform to some quantization principle analogous to the Pauli exclusion principle in terms of the number of particles in a ring. Energy conservation requires compactness of coupling as much as compact as possible. Thus, the nucleus is so compact that, unlike the electron, cannot even be detected by the electron microscope. What happens to the superstrings absorbed steadily by the nucleus? Since they are dark (weightless) they are unaffected by centrifugal force and remain in the center of the nucleus, just like the black hole in a cosmological vortex (another example of quanum-macro gravity duality). The energy released in nuclear explosion comes from conversion of these superstrings to visible matter and energy such as alpha and beta particles and radiation (photons). Neutrinos may be trapped (e.g., due to loss of energy just as they are trapped in a pool of water 2 miles underground; they are detected by the bubbles they create). They insert themselves between adjacent rings. In molecular formation the valence electron serves as connector between atoms, by flux compatibility. Some atoms have weak ionization energy that the outer or valence electrons are easily knocked off by indexcosmic waves. This is true of malleable materials like metals that are endowed with free electrons making them good electrical conductor. Among gases the most energy conserving arrangement is diatomic, i.e., coupled pair of atoms of the same gas. Ions are unstable because they interact with other atoms. It follows that the most stable primal clusters are neutral.

3.3.4 Superconductivity and the Bose-Einstein Condensate In electrical conduction, resistance is due mainly to collision with atoms and other electrons. It is a source of energy dissipation. It results in heating up the conductor or producing light. However, even without electrons, the coherent flux (of superstrings) it rides on, being moving charge, is electric current. In fact, it is this coherent toroidal flux that powers the magnetic train. The difference is: there is no resistance. This phenomenon is called superconductivity. This occurs when the conductor is reduced to 110◦ K which can be achieved by soaking it in some liquefied gas, e.g., helium and hydrogen. It appears that 110◦ K is another constant of nature, the threshold for superconductivity. Can GUT explain it? Conductor like metal has free electrons. When coherent flux of superstrings pushes the electrons through it in electrical circuit (due to voltage different between terminals, a measure of the energy

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of the flux on which the electrons ride in an electrical conductor) they get smashed or converted to photons or vibrate and heat up the conductor. Electric current is sustained when there is continuous availability or conversion of electrons. At normal temperature, valence electrons are knocked off by the flux to carry electricity. These too can be exhausted. However, the flux itself involves collisions and agitation in view of the micro component of turbulence. This is the principal source of electrons and electricity whenever voltage is maintained between two terminals. When the temperature is sufficiently low the residual kinetic energy of any material is too low that even the agitation by the micro component of turbulence is not sufficient to convert superstrings to electrons. Thus, there is flux, therefore, electricity, but no resistance. This is called superconductivity. Is there any potential industrial application for it? Hardly; if the idea is to transmit electricity cheaply, it is a long shot. It will be prohibitively expensive to reduce the temperature of a conductor to 110◦ K. Moreover, there is no place on Earth at this temperature that can make materials superconductive. Another phenomenon related to superconductivity is Bose-Einstein condensate (BEC) which has been verified by experiments in the US. When the temperature of a material is sufficiently close to absolute zero, temperature of 1◦ K has been achieved by slowing down the mechanical motion of a material sufficiently, e.g., soaking the molecules in suitably viscous liquid (analogous to the natural cooling of the quasars in the early universe due to dark viscosity [20]). Then new phenomena occur that, otherwise, do not occur at normal temperature: (1) atomic boundaries are blurred, (2) atoms pass through each other without deflection but sometimes rebounding as if something hard inside collided, (3) superfluidity (zero viscosity), (4) formation of large vortices in superfluidity and (5) atoms merge forming a single large ellipsoidal globule; this ellipsoid absorbs other atoms in ones and twos and by the dozen and, with a startling suddenness, remains a huge motionless blimp. All of these phenomena are explained by quantum gravity. At the general level, cooling close to absolute zero temperature is de-agitational and converts kinetic to latent energy; then flux intensity and repulsion also decline. There is also greater flux resonance and coherence that may knock off orbital electrons so that the nuclei may get close together and form larger clusters and vortices. At the same time, helical coupling (one primum going inside another) occurs [68]; moreover, the weakening of fluxes allows atoms to pass through each other just as semi- and non-agitated superstrings pass through each other with minimal chance of collision. This is analogous to the almost zero chance of stellar collision when two galaxies collide despite the relatively high density of stars at their metropolis, i.e., the region of greatest star density around the eye, because they are widely dispersed even there (another case of dual phenomena between quantum and macro gravity). The occasional rebounding mentioned in (2) occurs when nuclei collide. In a stable cluster there is balance between flux repulsion and flux attraction. However, atoms in gaseous condensate, experience a small mutual repulsion or attraction depending on their species. Thus, atoms of sodium, rubidium 87 and hydrogen repel their own kind. Lithium 7 and rubidium 85 atoms attract. By introducing magnetic trap, cluster of atoms can be held together even if there is slight net repulsion between them. Large clusters are formed this way. In fact, rubidium 87 and sodium can be routinely condensed to millions of atoms, as much as 20 times larger than they would be in the absence of the repulsion.

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Attractive atoms like lithium and rubidium 85 exhibit this surprising phenomenon. While this attraction allows formation of large cluster, with magnetic trap to enhance stability, there is a limit of 1,500 atoms to its clustering; beyond that, total attraction is so great the cluster collapses, contracts, becomes too dense and triggers explosion that spills atoms out of the trap. Superconductivity and BEC have conventional models but quantum gravity is their only qualitative model. Naturally, conventional models do not provide physical explanation. Another natural law that has application even to social science is the next one. Law of Uneven Development. In any process or interaction development proceeds unevenly and perfect balance and uniformity is unstable. This implies that perfect balance is unstable and its instability is usually mitigated by the universality of oscillation. In the above example of lithium and rubidium there is a temporary stable balance of their clusters between repulsion and attraction until they become so dense that it triggers explosion. There is, again, a dual phenomenon to it in macro gravity. For example, the balance between centrifugal force and force of gravity in planetary orbit oscillates across the point of balance in the radius resulting in elliptical orbit. In this case oscillation stabilizes the unstable balance between gravity and centrifugal force. Binary stars are rare because their initial even power in attracting and collecting mass is transitory as one may eventually gain more power over the other when more mass falls into the spinning matter at the core of one. One will generally collect more and raise, by momentum conservation, the level of spin and suction over the other so that the balance will tilt in favor of it and the other will become a minor vortex. Among countries uneven development is the rule rather than exception.

3.3.5 Electric Current, Generation and Conduction The earliest generation of electric current is direct current using a battery. By using a chemical electrolyte that ionizes a metallic electrode soaked in it, electrons are freed that get attracted to a semi-metal like lead, the other electrode that has minimal free electrons. When a conductor connects this electrode to the metallic electrode, the repulsion among the electrons combined with the attraction to the positive ions in the metallic electrode, a continuous flow of electrons is sustained and since electrons are charged, it produces electric current. But battery has limited capacity for electric generation; it also encounters much resistance in transmission due to electronic collision with the atoms of the conductor. Later, electric generators and power plants were invented for massive generation of electricity. How do they work? Let us look at a magnet. Recall that the atomic nucleus is a magnet. When they are polarly joined Npole to S-pole into a string they produce coherent flux of a magnet called magnetic field. However, it is too weak to be detected. But when strings are joined equatorially (by suitable connectors) they form a bundle whose flux aligns the nuclei beyond the N-pole to form a bundle, by resonance, that join up to its corresponding S-pole on the other end of the magnet. The maximum number of strings in a bundle depends on the material, by the quantization principle of quantum physics. When a closed circuit conductor cuts across a bundle, its coherent vortex flux induces a coherent flux along the conductor as electric current. At temperature above 110◦ K electrons flow through it and encounter resistance along the conductor mainly due to primal collision with the atoms. The computational mathematics of electricity and magnetism are given in elementary textbooks on the subject. We introduce its qualitative foundation.

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The cross-section of a bundle is similar to that of the nucleus of an atom except for the huge number of equatorial cycles of the former. As in the nucleus there may be minor eddies in the interior but the more important feature of this configuration is the induced coherent flux around it whose intensity is the algebraic sum of the charges of the aligned nuclei in the cross-section that contribute to it. In an ordinary magnet only a small percentage of the nuclei are polarly coupled and aligned. Magnetic intensity depends on the proportion of aligned polarly coupled nuclei. The rotation of the coil of the armature of the electric generator causes rapid reversals of the induced fluxes that yield alternating electric current. It also causes slicing of opposite fluxes that is agitational and converts superstrings to prima that are carriers of charge. This is the essential mechanism for generating alternating current. Wiggling the wire or increasing the coils around the armature produces higher voltage, i.e., greater intensity of flux and electric current. The action is intensified by the rapid rotation of the armature. Sufficiently intense flux may knock off electrons from the atoms and ride on the flux or, being agitational, may convert superstrings to electrons that ride on the coherent flux and encounter resistance from collision with other prima and the atoms of the conductor to generate heat. In running an electric motor, it is the coherent flux that is important and not the flux of electrons. In fact, the latter may heat up the motor and contribute to inefficiency. Therefore, for heating and lighting purposes the electron flux is needed but it is the flux intensity that runs the magnetic train and electric motor. Since flux is moving charge, it is the primary factor in the generation of electric current. Thus a good conductor must have the atoms suitably distanced to serve as medium for electric current as coherent flux. Porous material is not suitable since the atoms are far apart and do not connect their fluxes to serve as medium for electric transmission. How can one make a magnet? Wind a conductor around the ferrous material and let a direct current through it. Wind another coil over the first coil and let an alternating current through it. The alternating current shakes the atoms and the direct current aligns them, by flux compatibility.

3.3.6 Thermonuclear Reaction There is current misunderstanding of this phenomenon; that is the reason there is no breakthrough in plasma research in the last 60 years. It is the belief that when hydrogen atoms are suitably compressed (e.g., forcing them through narrow tunnel mechanically) their nuclei merge, shed their energy in the form of heat and form heavier nuclei like helium. In the hydrogen bomb, compression is supposed to be effected by the explosion of the trigger atom bomb that presses the hydrogen atoms against the bomb shell and among themselves. This understanding is at odds with energy conservation. There is no natural law that supports this is the reason fusion research or plasma physics has had no breakthrough. Even in the formation of heavy isotope, once the protons form a nucleus no proton can be added because all charged prima will be repelled by the positive or negative quarks already in the nucleus. To merge two nuclei would mean introducing a proton into the nucleus of one of them. This amounts to reverse alchemy unsupported by any law of nature. The repulsion between two protons alone in the narrow confine of the nucleus has been estimated to be 27 tons. The most likely scenario is as follows: The explosion of the trigger atom bomb agitates the superstrings in the hydrogen nuclei and converts them to simple prima (alpha and beta particles) and photons (radiation), the energy released in thermonuclear explosion. Other simple prima converted by the agitation form neutrino and light nuclei like heliums. The trigger in hydrogen bomb cannot be duplicated in a reactor and, therefore, the chance of a breakthrough in fusion research is nil.

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3.3.7 Further Verification of Quantum Gravity (1) The following verifies dark-to-visible-matter conversion by shock wave agitation: (a) Laboratory results at Pavlov Institute of Physiology, State Technical University, St. Petersburg, show that brain waves produce molecules in the neural membrane [78, 87]. (b) Generation of balls of fire and earth lights by seismic waves coming from volcanic activity and lightning; balls of fire are produced around geological faults and volcanoes while lightning in lower atmosphere produces earthlights in the mesosphere [90]. (2) GUT explains spin of quantum physics as rotation of primal vortex as a unit distinguished from the vortex flux spin induced by the toroidal flux that gives rise to charge. Charge is measure of induced vortex flux intensity, nothing to do with the mass of the primum. (3) It is known that the charge of an electron is fixed; this follows from the quantization principle; the charge of any primum is its signature; moreover, only free prima are oriented by the Earths gravitational flux; those attached to primal cluster or riding in a coherent flux such as electric current are not. (4) The gluon is now proven to be a left quark, by energy conservation in the form of nonredundancy, the left quark must be the connector that holds the protons in the nucleus together. (5) That the atomic nucleus is miniature black hole (i.e., massive concentration of superstrings) is verified by the release of great energy in nuclear and thermonuclear explosion. (6) Technology is verification of theory and the most remarkable verification of quantum gravity in biology is the great breakthrough by the Genome Project which offers great possibilities for genetic engineering, particularly, remedy for gene-induced diseases like cystic fibrosis, systemic lupos erythematosus, cancer, diabetes and muscular dystrophy. The project essentially maps the genes responsible for every physical characteristic and there are, on the average, 25,000 30,000 of them in the human body. Experts in this field predict that in a few years one can walk into a diagnostic clinic and have his genes examined for possible defect that can be neutralized by genetic alteration and modification like the way cystic fibrosis gene is neutralized. What is missing in this great project is the explanation of how the gene does its many tasks such as producing the tissues and chemicals of the human body. This missing link is what GUT provides: the gene emits radiation that converts dark to visible matter where it is needed and some special gene neutralizes harmful radiation emitted by defective gene. (7) We have considered implicitly that the superstring has mass since it is the fundamental constituent of matter. We now take it explicitly as basis of two important predictions of the theory: (a) the neutrino has mass since it is a pair of equatorially coupled left and right prima; this has been verified by experiment; its mass has been computed from the known masses of the proton, neutron and electron and found to be 1.55 times electron mass; (b) since a photon has toroidal flux, it has mass. Therefore, like ordinary matter it is pulled by gravity; this has been verified by the deflection of a ray of light that passes through the gravitational field of the massive Sun. This was a remarkable prediction made by Einstein but he had no explanation for gravity. Now we know that the Einsteins curvature of space formed around a massive body is its gravitational flux, a vortex flux of superstrings. The superstrings are like viscous liquid that form concentric rings around the eye of a spinning vortex core (just like what happens when one steers an egg). (8) Waves of stars by the trillions in concentric rings generated by and propagated outward from the point of collision between one of the pair of small galaxies in one direction and a large one in another oblique to it [110]. Since stars are sparsely dispersed even in the metropolis of a galaxy

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star collision is quite unlikely. Therefore, the agitation that converted dark to visible matter in this instance was mainly due to the slicing of the colliding galaxies dark halo. As in macro gravity which is dependent mainly on qualitative mathematics due to great masses, forces and distances involved, much of the verification of quantum gravity requires qualitative mathematics for the opposite reason: it deals with very small sizes and masses and faint interactions that are not even measurable with present technology. Some of the interactions are impossible to model other than qualitatively and by simulation.

3.3.8 Updates The following is an update from the Gale Encyclopedia of Science. Very long radio waves: 108 − 104 m; long waves: 104 − 10 m; short waves: 10 − 10−3 m; infrared rays: 10−3 − 10−6 m; ordinary light: 10−6 m; ultra-violet rays: 10−6 − 10−9 m; r¨ontgen rays: 10−9 − 10−11 m; gamma rays < 10−11 m. (The finest laser (directed light) has wavelength 10−9 m). Furthermore, [3] gives some numerical characteristics of the proton: circular velocity 7 × 10−22 cm/sec (speed of light: 3 × 10−10 cm/sec), mass: 1.67 × 10−27 kg, energy: 8 × 1014 joules= 5 × 1026 MeV, angular velocity: 4 × 1036 per sec, relaxation time: 1022 sec. We classify under visible light all known or detectable electromagnetic waves down to the finest at gamma-ray wavelength which is less than 10−11 m. Ordinary light is visible. In GUT the boundary between visible and dark wavelength is 10−14 m. This is a reasonable assumption. At any rate, experimental physics will have to make refinements in the future. Whatever the right figures might be the three sub-states of the superstring will remain, agitated, semi-agitated and non-agitated, and the analysis provided by GUT remains valid.

3.4 Macro Gravity Our setting for macro gravity is the timeless and boundless Universe of dark matter. The steady shrinking of the superstrings, by energy conservation, combined with the law of uneven development creates nested fractal sequences of low pressure regions or cosmic depression, i.e., regions of depression in a depression, containing regions of depression, etc. By flux-low-pressure complementarity they evolve to nested fractal sequences of cosmological vortices. They are called ordinary or usual. Again, by the law of uneven development, the most probable formation is one main cosmological vortex containing nested fractal sequences of galaxy clusters, galaxies, stars, planets, moons and cosmic dust. Vortices around the main eye are called minor vortices; they are also nested fractal sequences of vortices. A cosmological vortex is initially dark and isolated. Its kinetic energy of spin raises its temperature that agitates and converts superstrings to prima. Conversion is augmented by seismic waves from the micro-component of turbulence at the core. Converted prima gains instant momentum that raises the spin of the core (collected mass around the eye). As spin rises, so does dark-to-visible matter conversion. Then dark viscosity extends the influence of the core outward. By gravity (suction by the eye) it draws in weaker cosmological vortices around as minor vortices and pulls them into rotating spiral streamlines of visible matter falling into the core further augmenting its spin and dark-to-visible matter conversion. More powerful cosmological vortex around can pull the vortex towards and turn it into its minor vortex. We take the galaxy as unit cosmological vortex. All galaxies are usual, i.e., formed by the

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natural shrinking of superstrings. However, we consider special galaxies, including our universe as supersuper galaxy that got their initial visible matter from some cosmic burst [31, 41, 38]. Thus, the predecessor of any usual galaxy is nested fractal regions of depression that evolved to nested fractal sequences of cosmological vortices. This dynamics is also true of its minor cosmological vortices. Beyond the deep interior or inner and outer core, visible matter is generated by micro component of turbulence and the seismic waves it generates. Every vortex has eye due to energy conservation; there would be much friction, collision and dissipation of energy without it. In quantum gravity, the energy of the primal induced vortex flux of superstrings is charge; in macro gravity it is gravity, a feature of quantum-macro gravity duality. Vortex spin is most powerful near the eye. Not only do the staggering spin and micro component of turbulence of the core agitate and convert the superstrings to prima, core spin also pulls visible matter in its vicinity into rotation around it due to dark viscosity and imparts centrifugal force on it. The more vigorous the spin the bigger the eye since centrifugal force pushes its boundary outward and the stronger its suction. Thus, any vortex in the Cosmos has eye, a rarefied region of calm and de-agitation. Therefore, by flux-low-pressure complementarity, it sucks and accumulates matter around it. Thus, two opposite forces are at work on any mass around the eye, from its immediate vicinity to the rim of the cosmological vortex: suction by the eye and centrifugal force on it directed outward. Calculation of inward flux pressure is provided by Newtons gravitation law. Around the eye is balance of vigorous spin, hence, strong centrifugal force against strong inward flux pressure so that minor vortices caught at this balance take elliptical orbits (elliptical due to radial oscillation). As spin increases, this balance extends outward and the vortex expands. At the height of its power, peripheral stars may be catapulted out of its influence. There are such stars in the cosmos [135]. However, there is countervailing dynamics: as gravitational flux thins out, viscosity declines and the impact of core spin declines as well. Their limit of balance marks the summit or maximum power of the vortex. Then it declines, leaving minor vortices free and the core isolated. By the quantization principle there is optimal spread of nested fractal cluster of regions of depression that forms nested fractal sequences of cosmological vortices, not a super...super... galaxy like our universe but usual cosmological vortex. In the ascendancy of any cosmological vortex towards its summit, the more massive it is (i.e., having powerful core spin generating great kinetic energy and high temperature) the more gaseous (e.g., Jupiter, Uranus, Saturn, Neptune). It has solid compact deep core, however, e.g., the Suns deep core, with specific gravity 150, due to its composition of compactly clustered pure prima and light nucleons; it is also compressed by gravity but as secondary factor. The Earths inner cores specific gravity is also 150 verified during earthquake by seismic waves that travel faster through denser material. Due to its high temperature 6, 000◦ C [111], it cannot have complex atoms there; mainly pure prima which are compact and account for its great density. In the early phase of a cosmological vortex, suction by the eye is greater than the centrifugal force on the minor vortices and they fall smoothly along spiral streamlines that wind around the eye. This is seen in young galaxies called spiral nebulae. As matter falls into and around the eye, energy conservation raises the spin and, by dark viscosity, catapults the minor vortices into orbits where gravity balances the centrifugal force. Stars with opposite spins and comparable mass clash and explode as supernova if they get close together; a rare occurrence, since they are thinly dispersed even at the nucleus of galaxy. In the evolution of a galaxy the minor vortices that remain in elliptical orbit around the core are few exceptions. Most minor vortices are sucked by the eye. Only the few that gets into elliptical orbits away from the accumulated mass around the main eye, escape being devoured. Still fewer are pieces of debris from minor vortex that just missed collision with the main core and catapulted back into elongated elliptical orbits called comets.

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We see here the genesis not only of the galaxy but also all its component minor vortices including stellar and planetary systems. A stellar system apart from the solar system has been discovered with a large star at its core and smaller stars orbiting around it [135]. Our solar system is not quite as massive as this one because its minor vortices collect only planets around their eyes. At any rate, the solar system consists of nested fractal sequences of cosmological vortices with the Sun as common first term followed by the planets as minor vortices and, further on into the fractal sequence, the moons and cosmic dust. Then we append the nested fractal sequences through the atoms and superstrings and we have the full stretch from the supersuper galaxy, our universe, through the superstrings. We summarize our discussion by the next natural law. Formation of a Macro Vortex. The steady shrinking of superstrings, by energy conservation combined with the law of uneven development, induces formation of nested fractal sequences of depression that evolve into nested fractal sequences of cosmological vortices. In a typical galaxy the main vortex contains nested fractal minor vortices consisting of stars and each of the latter contains nested fractal sequences of vortices consisting of planets, etc., all the way to the moons and cosmic dust. In the opposite direction, the galaxies belong to galactic clusters, super galaxy clusters and on to the super...super galaxy, our universe. The fractal-reverse-fractal locator [52] traces the vortices in our fractal universe starting from any cosmological body including cosmic dust particle where one can trace a fractal sequence up into the macro scale (reverse-fractal) and end up in the super...super galaxy; or go down the sequence at the micro scale and end up at cosmic dust. As example of this reverse fractal clustering, Andromeda is actually a galactic cluster having 22 galaxies as minor vortices; so is Milky Way with 11, one of which a dead galaxy, the Sagittarius [85], and both of them belong to the larger galactic cluster called the constellation Virgo. A galaxy is usual universe by itself but a supsup galaxy is formed in special way.

3.4.1 The Cosmology of our Universe Our universe is quite special because it was started by a Big Bang, explosion of a black hole, a very rare phenomenon that could have been triggered by suitable sequences of cosmic wave hits on that primordial black hole, the destiny of the core of a previous universe [20, 47]. In traditional cosmology, the Big Bang is assumed to have occurred spontaneously with neither rhyme nor reason as if our universe started with staggering violation of energy conservation. In fact, it was a natural phenomenon subject to the laws of nature. We state the fundamental premise of macro gravity. The Big Bang. A black hole exploded 8 billion years ago (t = 0), formed our Cosmic Sphere and agitated dark matter between its inner and outer boundaries and interior and near exterior. The primordial black hole was immersed in and at resonance with dark matter having tremendous latent energy; suitable agitation by sequence of cosmic waves triggered chain reaction that converted it into a great burst of kinetic energy called Cosmic Burst also referred to as second big bang [102]. Anything that explodes, destroys whatever order there is in it so that it is followed by initial chaos that, in view of energy conservation, evolves into a new order. On this basis we can now account for the formation of our universe. The Big Bang created two physical systems: a super...super depression and an expanding spherical wave front, the Cosmic Sphere. During this period, 0 < t < 1.5, the Cosmic Sphere was compressed layer of dark matter trapped and pressed between the force of explosion and suction by its interior (by flux-low-pressure complementarity) and pounded and agitated by the less energetic

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shock waves (concentrated cosmic wave with enhanced latent energy) bouncing between its inner and outer boundaries. This agitation endowed the Cosmic Sphere and the superstrings trapped in it with enormous latent energy. Compression of trapped dark matter prevented conversion to prima, only conversion to semi-agitated superstrings. The more energetic shock waves pierced the Cosmic Sphere and converted dark to visible matter in its exterior. The expanding Cosmic Sphere weakened and, combined with outward pressure from the compressed and semi-agitated superstrings, burst at t = 1.5 billion years. The Cosmic Burst was much more powerful than the Big Bang because of the accumulated infusion of latent energy from the agitation. The Cosmic Burst released the semi-agitated superstrings into the fractal sequences of regions of low pressure in the vicinity of the once Cosmic Sphere as simple prima at first due to high temperature and kinetic energy of our early universe. They formed the bright and radioactive clusters called quasars that peaked at t = 2.5 billion years from the start of the Big Bang [107]. Then they got entangled into the evolving fractal sequences of usual cosmological vortices in their neighborhood. Dark viscosity slowed down their motion, reduced kinetic energy and temperature and allowed formation of coupled prima such as proton, neutron and light elements like hydrogen, helium and boron and, combined with further decline of temperature, the quasars evolved in the early galaxies. The Cosmic Burst added to the breadth and depth of this super...super depression and, at the same time, drew in matter into it that formed the transitory phase of chaos and, by flux-low-pressure complementarity, evolved into a super...super galaxy, our universe. As our universe increased its spin it imparted greater centrifugal force on the galaxies but balanced by the suction by the eye that induced elliptical orbits around it (elliptical due to radial effect of the laws of uneven development and universality of oscillation). As its power rose further, centrifugal force surpassed gravitational suction and accelerated its radial expansion now described by Hubbles law. We do not know if this acceleration is rising or slowing down but our universe is expanding very rapidly [41, 38, 102]. The energy imparted by the Big Bang plus the accumulated energy of the Cosmic Sphere due to 1.5 billion years of agitation by shock waves from the Big Bang made our universe a super...super galaxy 1010 light years across [85]. There are evidences of the existence of some universe elsewhere, perhaps, with even more powerfully accelerated expansion beyond a critical level than our universes as shown by galaxy clusters catapulted by it now traversing our universe [106]. Such a universe could not have been formed the usual way. The collision of galaxies coming from different directions is evidence [110]. Galaxies of our universe have only one direction radial and outward. They move away from each other and cannot collide. There are also evidences that the Milky Way was far from the Big Bang. One is: we can see our universe when it was only 3% of its present age. Another is it is the oldest galaxy in its neighborhood, its spiral of falling minor vortices faint, most of them already sucked by gravity suggesting old age. The discovery of stars in it, older than the Big Bang [103], reveals that the milky way formed prior to the Big Bang and was drawn to our universe as it expanded to a super...super galaxy. One question remains: can a super...super galaxy like our universe form the usual way? It is unlikely, since there is optimum spread for nested fractal sequences of depression to evolve into cosmological vortex, being deprived of great infusion of energy by a big bang and cosmic burst. There is another dynamics: as visible matter falls into the core, resonance with its dark component pulls dark matter with it and thins out the gravitational flux reducing dark viscosity and hence suction by the eye on the minor vortices. Suction reaches a peak and declines resulting in an isolated vortex once again (an example of this state is the Sagittarius debris of stars as remnant of its past as a galaxy [85]). Formation of galaxies continues, however, and two baby galaxies have been discovered

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since 2004. Stars 200 million times as massive as our Sun have been discovered recently. They are cores of weak galaxies. What is the destiny of our universe? Being a galaxy, albeit a super...super galaxy, it has the same destiny as any galaxy and we shall find out later.

3.4.2 Early Cosmological History of the Earth Like any cosmological vortex formed the usual way, the Earth started as dark vortex, a term in the nested fractal sequences of cosmological vortices of the Sun. Now and then its core consists mainly of simple prima that could not cluster into more complex physical systems due to high temperature and kinetic energy that sets them into high intensity vibration. As the core accumulates, mainly through dark-to-visible matter conversion by its heat and generation of seismic waves by the micro component of turbulence in the core, superstrings around it convert to visible matter, the latter shields the rest of the outer layers from the hot core allowing formation of more complex systems such as atoms and molecules and even biological species close to the Earths surface. Ingredients of many biological species have their origin in magma oozing out of the Earths crust from the Earth’s interior [111]. This explains why volcanic islands such as the Galapagos, west of Ecuador, and Hawaii have the most diverse collection of animal biological species. At the same time, the core’s simple prima couple into compact clusters, by energy conservation, making the core solid. Agitation by the hot spinning core converts dark to visible matter within it and the latter acquires instant mass and momentum that augments momentum and spin. The outer layers from the mantel to the crust are mainly outcome of the core’s expansion but are augmented by dark-to-visible matter conversion by seismic waves generated by the micro component of turbulence at the core [52]. Its intensity and contribution to dark-to-visible matter conversion rises with the enhancement of the mass of the core. The outer layers are further augmented by falling minor vortices and debris due to suction by the eye that, in turn, enhances its spin, by momentum conservation, and broadens its influence on the surrounding cosmological vortices due to dark viscosity [47]. However, the increased angular momentum and spin of the core is enhanced principally by dark-to-visible matter conversion within the hot spinning core that gains instant mass and momentum upon conversion and augments its momentum and spin. The same is true of dark matter sucked by the eye, by fluxlow-pressure complementarity], upon their conversion to visible matter by the hot spinning core. The Earth’s minor vortices were formed in the same way at smaller scale. Its only surviving minor vortex is the Moon. At the same time, the increased spin of the Earth broadens its eye due to centrifugal force which, in turn, increases its gravity. Some scientists conjecture that the demise of the dinosaur was due to the deteriorating inadequacy of its anatomy relative to its increasing weight caused by the increasing gravity of the Earth long before its total disappearance 65 million years ago. Dark-to-visible matter conversion in the core and mantel expands the Earth’s interior and creates outward pressure from Earths mantel that results in magma oozing out of the crust that builds mountain ranges along both sides of constructive tectonic plate boundaries under the oceans and become part of the crust. Some broke out of the ocean surface and formed islands. Most of the islands of Hawaii and the Galapagos were formed this way. Constructive plate boundaries, called the Pacific Ring of Fire, surround the Pacific Ocean along most of its rim where much volcanic activity occurs some under the ocean, others find cavities and create volcanoes inland and feed them with lava. The Pacific Ocean seabed is studded with numerous under-ocean volcanoes responsible for much of weather turbulence there including tropical cyclones. There are also cracks and holes on the plates that form hotspots where magma oozes out and creates volcanoes and feeds them with lava. It appears that most biological species form from magma

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as it cools to surface temperature. The Galapagos has the most diverse biological species in the world; it is here where Darwin spent the years 1831-1836 gathering data for his theory of evolution of biological species. In any cosmological vortex, the Earth included, the eye is region of calm; over time, it de-agitates the accumulated mass around it during its degeneration phase and transitions towards the destiny of its core, black hole, massive concentration of non-agitated superstrings. In an active cosmological vortex like a star or Earth, matter at the eyes boundary is de-agitated; in particular, the prima lose kinetic energy, their cycles collapse converting them to semi-agitated superstrings, then to non-agitated superstrings that settle at the center of the eye as black hole, at the center since it is weightless and unaffected by centrifugal force. This will be the future destiny of the Earth in the far future. At this time, the Earth is at its ascendancy phase of its cycle as shown by the receding Moon which means that the power of its spin is still growing and the Earth itself is still developing to higher order, e.g., new biological species continue to form the existing ones well on their evolutionary advance. Why should the spin of the Earth continue to rise when there are no longer minor vortices falling on it to enhance its rotational momentum (the impact of meteors and asteroids is negligible)? As long as the eye is still there, it continues to suck dark matter that gets agitated and converted to visible matter by the hot spinning core and enhances the power of the spin. Only when the core has been sufficiently converted to dark matter will the suction vanish; then the black hole in the eye becomes naked. There are many such naked black holes in the Cosmos, marked by absence of matter and catalogued by astronomers.

3.4.3 Cosmological Vortex Interaction and the Trek Back Home to Dark Matter flux compatibility and flux-low-pressure complementarity laws have direct bearing on vortex interaction. However, energy conservation and energy conservation equivalence laws are the most fundamental ones in any interaction of matter. Other natural laws are their consequences and they provide insights in understanding natural phenomena. To this category belong the oscillation universality, fractal, uneven development and resonance laws. Spin determines interaction between cosmological vortices which is mediated by their gravitational flux by virtue of flux compatibility: two vortices of opposite spins are attractive through the common coherent induced flux at their rims; they are repulsive otherwise. If they have the same spin and their masses have the same order of magnitude, they evolve into binary vortices each revolving around the other and mutually riding on each others spiral flux; centrifugal force prevents them from falling into each other. If they have the same spin, regardless of their relative masses, they have mutual repulsion unless one is a giant compared to the other in which case the more massive one may gobble up the less massive. However, if one is large compared to the other and has opposite spin, the latter rides as minor vortex or an eddy on the gravitational flux towards and merges smoothly with the core of the former unless the centrifugal force on the smaller vortex balances the main gravitational flux pressure in which case it takes elliptical orbit around the main eye. Otherwise, if suitably light, it gets catapulted off the vortexs influence. Elliptical orbit, being due to radial oscillation, is the most probable orbital configuration since perfect balance, which yields circular orbit, is unstable in view of the law of uneven development. A minor vortex along the core spiral streamline with spin opposite that of the core vortex either forms elliptical orbit around it as an eddy or gets sucked into and is crushed by the core. As an eddy a vortex has relative autonomy. Two contiguous vortices of comparable masses with the same spin do not crash into each other due to mutual repulsion of opposite fluxes, by flux compatibility. As in a game of chance, an even game is unlikely over a period of time. While a pair of vortices

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may have initially the same mass and vortex power, once one vortex gains advantage, by the law of uneven development, it builds up over time until it is more massive than the other. Then one becomes a minor vortex of the other. Thus, the most likely configuration of fractal sets of vortices is one with single main core vortex and many minor vortices of diverse masses along its flux spirals and those of minor vortices. Binary stars form when they have even mass at the waning phase of their power. While calm and de-agitation in the vortex eye nurtures a black hole in its eye, what happens to the accumulated matter around the event horizon? In a galaxy it evolves into huge star as transitional phase towards black hole. The minor vortices evolve into stars or planets or moons. When the accumulated mass at main eye is greater than two-thirds the mass of our Sun, vortex spin is so powerful that it remains gaseous, as any star is; otherwise, it cools down and congeals into a solid planet (e.g., Earth) or planetoid. The more massive a planet is (i.e., having powerful core spin that generates great kinetic energy and high temperature) the more gaseous it is (e.g., Jupiter, Uranus, Saturn, Neptune). Massive cosmological vortices are gaseous but have solid and compact collected deep interior masses. An old star has weakened kinetic energy and is at the transitional phase towards its destiny, a black hole. Its gravitational flux has declined, spin around the eye has ground to a halt and matter there is well on its way to joining the black hole in the eye. Primal bonding has weakened but the separated prima stick to the eye by its suction. Over time they collapse to semi- and then non-agitated superstrings and join the black hole in its eye and the latter will become naked and nothing is left but a void. Among the intriguing questions raised by this theory is the possibility of tampering natural object to break global flux coherence and quash its capability to exert gravitational pull on other objects. (Local flux coherence cannot be eliminated because the atoms are local turbulence as vortices). Moreover, by flux-low-pressure complementarity, such tampering cannot shield objects from the gravitational pull of another. However, like the stealth bomber that breaks coherence of reflected radar beams to evade detection, a sufficiently tampered body, e.g., debris like asteroids, may lose global coherent fluxes that, while acted upon by gravity, may no longer exert gravitational pull or push on other bodies. They are bodies that have lost cosmological history. To verify, we utilize some natural laboratory: the asteroid belt between the orbital corridors of Jupiter and Neptune [97, 111]. They are ideal laboratories for this purpose. The irregular shape of the asteroids and the objects that form the planetary rings reveals lack of cosmological history, meaning, lack of gravitational vortex flux; they are debris rather than matter formed at vortex cores. They do not form gravitational clusters either, that is, they do not exert gravitational pull among themselves and yet they have masses. This is the first major verification of this prediction. They are also counterexamples to Newtons law of gravitation. Remember Galileo’s amazement about his own discovery that the rate of acceleration of a freefalling body above Earth is constant regardless of mass [36]. For insight consider a water vortex, say, a sink full of water with objects of different weights floating on it, release the water through an orifice at the bottom-center of the sink. A vortex will form and the floats will be accelerated at the same rate along spirals towards the bottom- center. In Galileos experiments the bodies were floating along the Earths vortex flux spirals through their dark component. The same dynamics applies to the “cannibalistic” activity of Milky Way on smaller galaxy with opposite spin that enters its gravitational influence [85]. The Milky Way can cannibalize only a smaller galaxy of opposite spin. If they are of the same order of magnitude of mass they mutually explode as supernova. A lighter galaxy of same spin is repelled and cannot be cannibalized. Recent study reveals that cosmic dust particles are oblong, confirming they have cosmological history, i.e., like a planet, a piece of cosmic dust is accumulated mass around the eye of a micro vortex.

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While its axis of rotation wobbles like the summer and winter solstices, the principal determinant of its motion is galactic gravitation of outside stellar gravitational fluxes. Like the Earth it has crust and mantle. It is estimated that interstellar dust constitutes one thousandth of the Milky Ways mass and hundreds of times more than the mass of the galaxys planets [4]. Therefore, its contribution to planetary mass is considerable. Cosmic dust continues to form and congeal into stars. While the first term of our universe as nested fractal sequences is a super...super galaxy, the last terms of its cosmological vortices are the cosmic dust. Then, we append the fractal superstrings to the cosmic dust and we have the full stretch of our fractal universe all the way from the supersuper galaxy through dark matter. The fractal-reverse-fractal locator [52] traces any vortex in our fractal universe starting from any cosmological body including cosmic dust particle up to the macro scale (reverse-fractal) and end up in the super...super galaxy or go down the sequence at the micro scale and end up at cosmic dust. Conventional science takes the view that these dust clouds formed during the last 1.5 billion years. However, formation of cosmic dust occurs whenever there is agitation by indexcosmic waves, e.g., cosmic ripples and γ -ray bursts and GUT provides physical explanation of its existence and origin as part of the formation of visible matter by seismic waves at galactic cores. Since the eye is region of calm and de-agitation, the superstrings at its boundary are de-agitated, evolve toward infinitesimal tori and cluster into black hole, huge concentration of mass inside the eye. Thus, every vortex is incubator of black hole; it is also its own graveyard, both of which have been verified in recent years among galaxies including the Milky Way [109, 107]. By energy conservation this is the scenario that happens in every galaxy: As falling visible matter declines spin momentum declines as well and, by energy conservation, the prima lose power and energy and, hence, charge. In the case of a star it becomes a neutron star, a misnomer since there is no such thing. Since energy is no longer infused into the primal toroidal fluxes the centrifugal force on the primal helical cycles and the prima that comprise the core collapse to semi-agitated superstrings that lose charge and evolve to non-agitated superstrings that join the black hole in the eye. Then the black hole becomes naked and that cosmological vortex has reached its own graveyard, a black hole back in dark matter. It is clear that black hole, being dark, does not suck. It is the eye of the vortex that nurtures it that does. When it becomes naked there is no longer suction but absence of visible matter that was sucked previously by the vortex that nurtured it. There are many such “voids” in the sky catalogued by astronomers. Each superstring completes a cycle: non-agitated → semi-agitated → primum (agitated superstring) → semi-agitated → nonagitated belonging to a black hole back in dark matter. The cycle may be cut short at any point and the superstring returns to dark matter as non-agitated superstring. It is clear that the destiny of a galaxy or any cosmological vortex, including our universe, as its power wanes, is a set of black holes, the most massive one in its main eye and the minor ones in the eyes of minor vortices. It is now well verified that there is a black hole at the eye of every galaxy [109, 107]. Moreover, the accumulated mass around the event horizon of a galaxy forms a giant star as transition phase towards black hole. This was verified in 1997 with the discovery of a giant star, observed through the Hubble, 10 million times the mass of our Sun; more massive ones, as much as 200 million times the Suns mass have been discovered since then. This evolutionary formation of a galaxy applies also to formation of a universe the usual way. This dynamics of a galaxy including its genesis, evolution and trek back home to dark matter applies to all galaxies. All universes, special or ordinary, share common evolutionary history, they differ only in the circumstances of birth.

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3.4.4 Cosmic Waves We conclude this section with the prime mover of our universe from the Big Bang through the formation and development of living organisms and human thought: indexcosmic waves. They trigger thought and sustain consciousness. We build on our earlier discussion of cosmic waves. There are three kinds of indexcosmic waves. Type I or basic cosmic or electromagnetic waves are generated by the normal vibration of atomic nuclei that possess the vibration characteristics of individual nucleus and induces the vibration of dark matter around it by resonance that propagates along spherical wave front and serves as medium for their propagation, also by resonance. Basic cosmic wave is carrier of primum in flight, e.g., neutrino and photon, a primum that has broken away from its loop. All cosmic waves travel at the speed of light in vacuum so that their speed is a constant of nature. Since the nucleus is nested fractal consisting of agitated, semi-agitated and non-agitated superstrings the cosmic waves it generates is also nested fractal, i.e., its envelope is a sinusoidal curve, its arc is wiggled into finer sinusoidal curve, each fine arc wiggled into finer sinusoidal curve, etc. This endows electromagnetic waves tremendous latent energy. That is why radio signals encoded on electromagnetic waves can penetrate barriers and one can tune on the radio inside the room. Electromagnetic wave is not only primum and photon carrier but also medium of communication for the brain. For instance, neural vibration that carries thought has superposed vibration characteristics of electromagnetic wave beyond normal vibration characteristics. This type of encoded electromagnetic wave is called brain wave. It is the same brain wave that carries the information from the sense organs to the brain or command by the brain to the arms or legs in an athletic game. In short, brain waves are the medium of communication from the brain to other parts of the body and back. Some are conscious transfer of information and others executed by the secondary nervous system. Type II cosmic or seismic waves are generated at the interface of turbulence such as at destructive boundary between tectonic plates or between two lava slabs, in either case pressing against each other with staggering force that causes the atoms and molecules to vibrate at huge frequency. This is called micro component of turbulence. The effect is for the dark component of each interface to wiggle the basic cosmic waves into two-layered nested fractal sequences of basic cosmic wave with macro envelope. Seismic waves are generated and propagated during earthquake and its macro envelope is seen as wave motion on the ground. Even without earthquake the compression between destructive tectonic plate boundaries generate seismic waves that produce balls of fire around geological faults. The flow of compressed lava underground also generates seismic waves that produce balls of fire around volcanoes; so does lightning in the lower atmosphere that produces earthlights in the mesosphere [90]. Seismic waves are propagated in all directions with layer boundary parallel to the interfacing turbulence. Seismic waves are also generated by the turbulence at the core of cosmological vortices; they are partly responsible for the generation of visible matter at the core as well as cosmic dust around it. Nuclear explosion and supernova also generate seismic waves (type III) but they are propagated in all directions from the explosion with layer boundary normal to the radius of propagation.

3.5 More Verification of GUT Verification of physical theory lies in its ability to make predictions and guide experiments to test them, explain natural phenomena, provide basis for gathering information on its subject matter and guide development of technology based on it. Verification of GUT in this Section involves mainly explanation of natural phenomena.

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We first note that although the distribution of dark halo does not affect suction by the eye of a vortex, it has an effect on vortex interaction. For example, flux compatibility is not operative in the direction normal to the equatorial plane. Fortunately, asteroidal paths lie in the solar equatorial plane (SEP) and planets that have their equatorial planes parallel to it have greater protection against asteroidal collision while those whose equatorial planes are much tilted away from SEP have lesser protection, e.g., Uranus. The Earths equatorial plane is tilted about 15◦ from the SEP and is protected by its thick gravitational flux around it. This accounts for rare hit on Earth by asteroids despite the millions of them whizzing by its vicinity every year since we are so close to the asteroid belt along the orbital corridor of Jupiter [97]. This prediction can be tested on planets with widely tilted equatorial planes, e.g., Uranus. We note further that the dark halo of any cosmological vortex and its gravitational flux extend much farther than its visible halo. For instance, in the solar system it takes several centuries for some comets to negotiate its path along the solar gravitational flux. This tells us that its gravitational flux goes much farther than the farthest planet from the Sun.

3.5.1 Issues, Facts and Internal Dynamics of our Universe We first summarize some information about our universe based on GUT that it is a super...super galaxy and derive some information about it. (1) The radius of our universe has order of magnitude 1010 billion light years and its core is a tightlypacked cocoon-shaped galaxy cluster 650 million light years across (discovered by French astronomers in 1994. Its estimated age is 8 billion years. Of course, without benefit of theory this estimate is subject to further refinement but good enough to start with; refinement will come later. (2) A great surprise of the last century that haunts relativists was the discovery of the staggering rapid expansion of our universe at accelerated rate [102]. Based on extensive direct measurement of the separation of galaxies from our vantage point Edwin Hubble formulated his law that expresses the rate of separation of a galaxy from us at distance s from Earth: ds = ρ s, dt

(3.4)

where ρ = 1.7 × 10−2 /km distance of the receding galaxy from Earth. For convenience, we measure distance S along a great circle in the spherical dark halo of our universe. Then, dS = ρ S. dt

(3.5)

Since the discovery of the rapid expansion of our universe, estimate of its age has varied from the original 8 billion to the present 14.7 billion and talk of raising it to 20 billion is abuzz. Each time an older star is discovered the estimate is adjusted to accommodate it. This star-chasing game is based on the wrong premise that only our universe exists. In fact, there are others as we have seen. Therefore, we stick to the original estimate of 8 billion to solve (3.5) and find the radius r as function of t. Since dS/dt = 2π dr/dt and (3.5) is independent of the distance between us and the other galaxy it holds when S = r (i.e., when the distance from Earth is r). Then, dr dr ρ 2π = ρ r or = dt. (3.6) dt r 2π

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Solving r, reckoning time from the Big Bang and taking light year and 1 billion years as units, r(t) = 1010 e(ρ /2π )(t8) light years, r (t) = (ρ /2π )1010 e(ρ /2π )(t8) light years/billion years, r (t) = (ρ /2π )2 1010 e(ρ /2π )(t8) light years /(billion years)2 Using standard units we have, at t = 8, r(8) = 3.2 × 1022 km, r (8) = 840 km/sec, r (8) = 7 × 10−2 km/secsec. Since r > 0, our universe is on the young phase of its cycle, its power of spin still rising. The value of ρ is based on direct observation and analysis of Doppler effect on spectrum of light coming from receding source. Now Encarta Premium has this figure: ρ = 260, 000 km/hr/3.3 million light years, i.e., the receding galaxy is moving away from Earth faster by 260,000 km/hr for every 3.3 million light years distance away from us. Does it make sense? Converting to standard units and simplifying we get ρ = 3 × 10−19 /km; inserting this value in (3.6) we obtain, r (t) = 5 × 10−14 km/sec, the supposed rate of radial expansion of our universe and acceleration of 3 × 10−32 km/secsec. They point to a static universe that does not match present observation and measurement. Moreover, if this figure were correct we would have been roasted by intense heat due to the steady formation of stars in the Cosmos, one per minute [4, 107], and emergence of two baby galaxies discovered since 2004. On the contrary; the average temperature of the Cosmos remains steady at 4◦ C. (3) Contrary to popular belief, the Big Bang is not a theory but an event that occurred some 8 billion years ago. The Big Bang alone cannot explain the spin of cosmological vortices like galaxy, star and planet and, certainly, not gravity and a lot more. The Big Bang destroyed whatever order existed in that primordial black hole that exploded and converted its latent energy to heat, radiation and punching force in the form of shock waves. Therefore, our universe came neither from that primordial black hole nor from the Big Bang. Rather, the Big Bang started a chain of events subject to the laws of nature that led to the formation of our universe as a super...super galaxy. The Cosmic Burst (also referred to as the 2nd Big Bang) marks the birth of our universe. Its aftermath gave rise to the prima and the first atom [103]. (4) The current understanding of supernova as explosion of star, that it is part of its evolution, is incorrect since cosmological vortices are stable. Moreover, a supernova is a rare phenomenon and does not match the fact that there are trillions upon trillions of stars in the Cosmos. Left alone, a star evolves towards higher order, a black hole in its eye. Therefore, the only plausible explanation is collision of two stars of opposite spins in their common equatorial plane. If they have the same spin they avoid each other, by flux compatibility. With opposite spins they attract each other and, by its momentum, the fluxes between their eyes merge smoothly at first until the rim of one goes past the eye of the other and their fluxes there, being opposite, collide resulting in double explosion. Then the flux barrier between their eyes breaks and causes huge depression that violently sucks matter around causing more powerful third explosion. This phenomenon is analogous to its quantum dual: primum-antiprimum mutual destruction. Supernova is quite rare since stars are sparsely distributed even in the metropolis of a galaxy. (Photographs of supernova show the three rings of visible matter on expanding shock waves [111]). (5) Gravity is dynamics of cosmological vortices; it acts on visible matter via resonance with its dark component. Macro gravity is the dynamics of global vortices of superstring arising from the natural steady shrinking of the superstrings and quantum gravity is the local dynamics of vortex flux of superstrings induced by primal toroidal flux. At both dynamics gravity is attractive or repulsive, such interaction governed primarily by flux compatibility and flux-low-pressure

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complementarity. For example, when both the Sun and the Moon are overhead, it is low tide, contrary to common belief. This is due to the fact that the Earth, Moon and Sun have the same spin so that the Suns and Moons gravitational fluxes interacting with Earths combine and reinforce each other and repel the Earths gravitational flux, by flux compatibility, that pushes the ocean down to a low tide. Gravity as we know it is due to the suction by the Earths eye, the Earth being a vortex. The other side to it is the gravitational flux that also interacts with other gravitational fluxes such as the Suns and the Moons. Quantum gravity includes primal, atomic and molecular interactions as well as biological processes. Note that quantum gravity can also be attractive or repulsive. Primum-anti-primum mutual destructive interaction is dual to the supernova, destructive collision of cosmological vortices of opposite spins both dynamics subject to flux compatibility. (6) GUT now explains the puzzle of the existence of stars older than the Big Bang. They formed prior to the Bang; since they belong to the Milky Way, it also proves that the Milky Way is older than the Big Bang.

3.5.2 The Milky Way and Andromeda and their “Cannibalistic” Activity Since we cannot see much of the Milky Way due to dust cloud and cannot view it from the outside either, the much younger Andromeda, the nearest giant spiral galaxy 2.2 million light years away has caught the attention of astronomers for it is believed it has common features with the Milky Way. Andromeda is the farthest object that can be seen from Earth by the naked eye. It is also the brightest galaxy in the neighborhood. Its visible discular halo is 200 million light years across, its mass equivalent to 3,500 billion times the mass of the Sun. It has 22 minor galaxies two of which are at opposite sides of and near the rim of its visible discular halo and appear headed for gravitational gobbling by its eye [85]. Andromeda has diameter of 200,000 million light years. It has a double nucleus near the center and is moving towards Milky Way at the rate of 50 km/sec (again, this departure from Hubbles law is further evidence that the Milky Way formed independently of our universe). The Milky Way is a medium giant in the Cosmos containing 400 billion stars including our Sun. Its visible discular halo along its galactic equatorial plane is 100 million light years across, its core or metropolis 10 million light years thick. Its concentrated discular gravitational flux within its spherical dark halo along the equatorial plane extends far beyond its visible halo. Its dark halo has greater concentration in the visible discular halo due to flux-low-pressure complementarity, resonance with the dark component of visible matter in the visible halo and centrifugal force on the latter. The Milky Way has 11 satellite galaxies including a dead one, the Sagittarius, now a cloud of stars that has been cannibalized by it, some stars already gobbled and others falling into its core [85]. They are distinguished from Milky Ways stars by their trajectories which appear to be directed straight towards the core. This is the effect of the saw-tooth action at Milky Ways gravitational flux rim as it slices the Sagittarius cloud of stars and throws them into a sector between the tangent and normal to the rim.

3.5.3 Point of No Return There are three speculations about our universe (a) steady state, that it will remain as it is forever, (b) pulsating universe, that it will eventually reverse its present expansion and head toward a big crunch that can be viewed as another big bang and can start a new universe and (c) forever expanding universe.

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Obviously (a) does not describe its present state because it is steadily changing; in fact, every second it is 840 km bigger in radius (from Hubbles law) and expansion is still accelerating [102]. Item (b) violates energy conservation; it would require staggering amount of force to reverse the huge momentum created by billions of years of acceleration due to its centrifugal force of spin. The only other force in the direction of reversal is gravity which can only diminish with the thinning of dark halo. Moreover, there is no evidence among the billions of galaxies that it can happen. The most that can happen is weakening suction due to the thinning of the dark halo or gravitational flux in which case the galaxies will be on their own momentum. This is exemplified at a smaller scale by the status of the Sagittarius cloud of stars, the remnant of a dead galaxy. With respect to (c) while the galaxies’ momentum may sustain their outward flight, dark viscosity will catch up with and constrain them to a halt as they trek back to their destiny, black holes in dark matter. At the same time, except for the lucky few that get caught in elliptical orbits, the minor vortices will steadily fall to its graveyard, the eye, enter the transitional phase of de-agitation and ultimately join the black hole in the graveyard, the eye. This means that (c) is impossible. What will happen then? Our universe will continue to expand, reach its peak, the core will lose influence on the minor vortices as dark halo thins out so that the minor vortices are left on their own, each vortex will approach its graveyard, its eye, and join the black hole there as dark matter.

3.5.4 Stable Universe GUT has qualitative model of our universe that upholds energy conservation from the Big Bang through its destiny in dark matter. In our qualitative model, the Big Bang was explosion of a black hole, the destiny of the core of a previous universe in the Universe. Unlike other models GUT has stable qualitative model that does not need the unsettling anthropic principle [76]. Dark matter is the stable frame of reference that resolves Einsteins twin paradox. In fact, our model leaves neither a problem nor a paradox in physics, astronomy and cosmology unresolved.

3.5.5 A Paradox No More Originally referring to our universe, Olbers paradox [76] says that it cannot be infinite, otherwise, accumulated light coming from all directions would have fried us in intense light by now which has not happened. The average temperature of our universe is steady at 4◦ C despite the rapid formation of stars at the rate of one per minute plus formation of at least two baby galaxies discovered recently. Their appearance is offset by the rapid expansion of our universe. This paradox is now moot, of course, since our universe is limited and visible matter in it is finite since the prima, atoms and molecules have finite measures. However, we can extend this issue to the unbounded Universe. Can visible matter be infinite? Since dark matter is unbounded there is no reason to rule out unbounded visible matter. There are two possibilities regarding accumulation of light. (a) visible matter is suitably dispersed that light reaching our universe does not accumulate since light from far off source dissipates energy before reaching us or (b) light reaching the vicinity of our universe is deflected by its gravitational flux (in the same way that asteroids missing the narrow injection angle are deflected by the Earths gravitational flux or light deflected away from the Sun). Either case has no bearing on our universe’s temperature or its distribution. In the case of asteroids and other bodies deflection is outward. How come light that passes through the gravitational flux of the Sun is deflected inward? The difference is the photon of light has wave envelope that gives it

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wave characteristics such as refraction.

3.5.6 The Transitory Natural Laws The natural laws of our universe are revealed by natural phenomena, i.e., visible physical systems. Before the Big Bang there was none in this region of the Universe. After the Big Bang, especially, after the Cosmic Burst, natural phenomena appeared with increasing complexity. Naturally, they revealed laws of nature that needed to be discovered and articulated. At this time, we have new phenomena that did not exist previously, e.g., biological phenomena. They are governed by biological laws. All these laws, however, will become empty as they lose natural phenomena that reveal them or are explained by them and our universe collects into black holes in the eyes of its cosmological vortices.

3.5.7 Ultra-Energetic Cosmic Waves Cosmic rays of energy level as much as 1021 eV have been reported recently [98, 105]. Traditional theories require them to be heavy elementary particles, possibly protons, coming from outside a 100million-light-year radius from Earth. The estimate of distance of origin is based on supposed absence of possible source within that radius from the perspective of traditional theories. Acceleration of material object to great speed e.g., proton, is possible through centrifugal force imparted by the powerful spin of some galaxy. However, as charged particle proton encounters much resistance and is not a likely candidate for such great speed. A neutron has a better chance, being neutral, but it must ride on some cosmic wave of relatively large envelope; that cannot be sustained at great speed by the natural vibration of dark matter. These energetic cosmic rays are not necessarily particles but cosmic waves that pack huge amount of latent energy through their fractal configuration. They may or may not carry matter but they impart kinetic energy on objects upon impact, e.g., prima or fragments of atoms. Basic cosmic waves may carry light neutral primum or photon at great speed, e.g., neutrino. These energetic cosmic rays are known to smash protons in the mesosphere [90] that, incidentally, confirms our prediction that positive prima are pushed high up by the Earths gravitational flux. Such cosmic waves could have come from the cores of powerful galaxies. Then there are energetic gammaray bursts coming from distant regions of the Cosmos and some scientists theorize that they are due to black hole explosion.

3.5.8 Celestial Spectacle Many interesting dynamics are displayed by galaxies and stars. Among the spectacular dynamics in the Cosmos is stellar and galactic jet outflow of hot gas (actually, pure prima) exhibited by nascent stars and galaxies [14]. Jet outflow was the first known case of matter speeding several times faster than light. How do we explain it? The eye of a vortex is cylindrical and normal to the plane of its discular halo along the equatorial plane, much like that of the tornado or typhoon as well as primum. The rapid flux of matter, visible or dark, into and around the event horizon builds up tremendous kinetic energy and accumulation of hot gas that must find a soft spot to escape through and that soft spot is the eye itself. Thus, jet outflow of hot gas pops out of the eye of a young galaxy or star in opposite directions at great speed several times the speed of light [14]. As the accumulated mass at the core cools down, the extremities of the eye sucks the masses at the poles inward leaving a spherical mass that is slightly flattened at the poles. This is quite evident in Earths flattened polar region. (At Lookout Restaurant between Sydney and Wollongong, Australia, one can see the curved rim

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of this flattened ocean horizon)

3.5.9 More Verification in the Cosmos (1) Emergence of quasars (pure prima) right after the Cosmic Burst when the temperature was still too high to allow formation of complex cluster. They were among the predecessor of the present galaxies that obtained visible matter from the cosmic burst; their number peaked at t = 2.5 billion years from the start of Big Bang [107]; dark viscosity reduced them to ordinary galaxies. (2) The resolution of Einsteins twin paradox by GUT that establishes dark matter as absolute frame of reference. (3) It follows from flux-low-pressure complementarity that the Universe cannot have a boundary; therefore, it is timeless and unbounded.

3.5.10 Verification in the Solar System Below is a sweeping verification of GUT in the solar system: (1) The shielding effect of the Earths gravitational flux, by virtue of flux compatibility and momentum conservation accounts for the rare hit by asteroids (only twice known) despite the millions of them that whiz by annually. Even if an approaching asteroid is headed for either Pole of the Earth away from the Earths equatorial plane, it will be drawn to it, by flux-low-pressure complementarity and begins to be deflected as it enters the rim of the Earths gravitational flux along its equatorial plane. If it misses the narrow injection angle of about 2 degrees used by astronauts to return to Earth in the direction of the Earths gravitational flux it will be tossed by its own momentum past Earth and miss it completely. If it approaches the gravitational flux of the Earth on the other side at its normal speed of at least 25,000 mph against the direction of the Earths gravitational flux then, by flux compatibility, it gets deflected away. There is a threshold of gravitational flux strength beyond which the shielding effect reverses to attraction. This is shown by the high frequency of asteroid hits on such powerful planets as Jupiter. In the 1990s asteroids from the tail of a dying comet landed on Jupiter that caused powerful earthquakes detected by seismographs on Earth. There is also a threshold in the other direction. When the gravitational flux is too weak the cosmological vortex loses its shielding effect. This is verified by the much poke-marked surface of the Moon. At the same time, light objects have less momentum and greater sensitivity to gravity which explains the meteor shower coming from tails of comets that hit Earth frequently. (2) The nested fractal structure of the Sun and its planets and the moons, confirms universality of fractal as natural law of GUT; it is conceivable that some large moons may have their own moons or, perhaps, revolving asteroids that got caught by their gravitational fluxes. (3) Gaseous composition of large planets due to powerful spin (hence kinetic energy), e.g., Jupiter, Saturn, Uranus and Neptune. (4) The diverse tilts of the planets relative to the Suns equatorial plane point to the relative autonomy and independent bearing of the minor vortices as eddies; another evidence is the opposite direction of four moons of Jupiter relative to the other 12 moons embedded in Jupiters gravitational flux; they belong to two different eddies in Jupiters gravitational flux, one inside the other (such double-layered vortex happens in kitchen sink and toilet bowl). (5) A comet belongs to a different category. It is more than an asteroid because it has regularity. It is a planetoid that plunged into the Sun but missed it and gets catapulted back by the Suns

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gravity into an elongated orbit. Its tail of debris is pushed by solar radiation away from the Sun as it travels around it. (6) The wobbling of the Earth called summer and Winter Solstices has oscillatory pattern consistent with the oscillation universality principle; the same applies to the reversal of the polarity of the Earths magnetic field every 25,000 years. (7) Gravity wave in the ocean is the counter-part of solar turbulence like sunspot. (8) The discular shape of the visible solar halo along SEP where the planetary orbits lie and planetary dark halo traced by the thin rings of Saturn, Uranus, Neptune and Jupiter [88]; it is quite likely that these rings were formed by debris (due to asteroid hits) that flew off the large planets by centrifugal force as a result of their powerful spins or tails of comets caught by the planets gravity. The perihelion shift of Mercury is due to its location on the thick region near the core of the solar concentrated discular gravitational flux. (9) The tidal cycle reveals an error in both Relativistic and Newtonian physics. Both theories predict that it would be high tide when the Sun and the Moon are overhead presumably due to their combined gravitational pull on the ocean. This is not borne out by observation. In fact, it is low tide at this relative position of the Sun, Earth and Moon since they have the same spin and, therefore, the Earths equatorial gravitational flux has opposite direction to that of the combined fluxes of the Suns and Moons fluxes on this side of the Moon. Consequently, they are repulsive and the Suns and Moons gravitational fluxes push the ocean down to a low tide. Incidentally rural fishermen can predict the occurrence of low tide during the lunar cycle based on the position of the Moon. They cannot be wrong here because their livelihood is linked to the abundance of fish caught in ponds and springs on the ocean bed during low tide. Their prediction is consistent with GUT. (10) Vortex spins in sinks and pools: counterclockwise in the North and clockwise in the South both of which are oriented by the Earths gravitational flux spin; this also applies to spins of typhoons and tornadoes as well as the hair spin around the cowlick on ones head. (11) Like water eddy the hair spin at the cowlick on ones head is determined by the gravitational flux lag from the Equator to the Poles. Since the hair sticks out of the scalp at the cowlick it is the tail end that spins around. In the North there is greater counterclockwise spin than clockwise; it is the reverse in the South. An informal survey reveals that in St. Petersburg 75% of the hairs tail end spin counterclockwise around the cowlick; it is about the same percentage that spins clockwise in Sydney. (12) Similar informal survey reveals 100% counterclockwise and clockwise spins at kitchen sink and toilet bowl in the two cities, respectively. When there are two kitchen sinks with bottom orifices that join into a single outlet pipe they form vortices of opposite spins in accordance with flux compatibility. It can be expected, however, the manner by which the outlet pipes are angled with respect to the vertical can conceivably affect the direction of vortex spin.

3.6 An Overview of GUT In order to present an overview of the Grand Unified Theory (GUT), it is appropriate to first recall the brief history of its development. GUT was envisioned by Albert Einstein in the 20’s to unify gravity and electromagnetism. The field equations of the theory of relativity and Maxwell’s equations of electromagnetism provided the necessary mathematical descriptions of these two forces. But the identification of physical processes or interactions that account for these forces remained obscure. Later quantum physicists joined in and widened the search for grand unification to include the weak

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and strong forces of nuclear physics. The weak force is supposed to account for decay of elementary particles and the strong force is the binding force that keeps the protons together in the narrow confines of the nucleus despite the enormous force of repulsion between them. They attributed the strong force to an elementary particle called gluon that glues the protons together and postulated another elementary, graviton, to account for the weak force. After nearly 3 decades of this unification pursuit by various physicists, in 60’s Steven Weinberg and Abdus Salam succeeded in unifying electromagnetism with the weak force through their gauge symmetry theory by attributing electromagnetism to interaction by exchange of photons and the weak force to the exchange of W and Z intermediate particles called bosons. This still left the gravity and the strong force out. The primary quest of knowing the basic constituent of matter led to vigorous research and smashing atom by more and more powerful particle accelerators, which only revealed that most elementary particles that last only for split second did not mean the important requirement of nondestructability. In 1950’s Paul Dirac succeeded in developing the first string theory that describes particles as loops of strings vibrating in higher dimensions, as much as 26 dimensions. However, strings are not directly observable and since they are supposed to have zero mass, do not occupy space and yet exert a force, it is an obvious violation of energy conservation. Though several string theories developed since Dirac, there is no significant improvement over his initial discovery and the quest has continued on. After realizing that descriptions of reality and computation alone are inadequate for the purposes of grand unification, we have to turn to a new methodology of qualitative modeling that explains nature through its laws and using qualitative mathematics necessary for the discovery of the laws of nature. In this section, we like to summarize the ten most important natural laws that facilitate the grand unification pursuit and briefly survey the mathematics related to GUT’s development and applications

3.6.1 The Mathematics of GUT Classical mathematics directly and indirectly involved with GUT and applications include: Youngs generalized curves and surfaces, rapid oscillation, Hubbles law, and the integrated Pontrjagin maximum principle. In order to answer some of the questions of physics raised in Section 1.2, it was necessary to reassess and update the underlying fields of Fermat’s Last Theorem which generated considerable mathematics central to GUT. They are: rectified foundations, generalized fractal and integral, set-valued functions and derivatives, the new real number system, new real line, new nonstandard analysis, solution of the gravitational n-body problem, quantum algebra and qualitative mathematics. Qualitative mathematics is the crucial factor for the development of GUT as physical theory, i.e., synthesis, summation and integration of the human collective experience and accumulated information anchored on the laws of nature. The new reals R∗ , free from ambiguity and contradictions, has three simple axioms: (1) R∗ contains the basic integers 0, 1, . . . , 9, and (2) the addition and (3) multiplication tables of elementary arithmetic that well-define the terminating decimals first. Then a nonterminating decimal is well-defined as the Cauchy limit of its standard Cauchy sequence. A new element of R∗ which is not a real is the nonstandard number, d ∗ = 10.99 . . . , called dark num-

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ber, the Cauchy limit of the equivalence class of its nonstandard Cauchy sequences represented by its principal nonstandard Cauchy sequence, 0.1, 0.01, . . . ; d ∗ is set-valued and lies between two adjacent decimals, e.g., 4.3500 . . . and 4.3499 . . . . It is the well-defined counterpart of the infinitesimal of the calculus. The nonstandard number u∗ is the upper bound of the equivalence class of divergent sequences, the counterpart of infinity of calculus. Since no decimal exists between two adjacent decimals, d ∗ is a continuum. The lexicographic (natural) ordering of the new reals puts adjacent decimals side by side “glued” together by d ∗ . R∗ contains countable union of continua each consisting of a recurring 9 paired with its predecessor adjacent to it. The recurring 9s including the new integers are isomorphic to their predecessors. R∗ has natural ordering, is non-Hausdorff and non-Archimedean but the decimals are; moreover, the decimals are countable, hence, discrete, and inherits its natural ordering R∗ has countably infinite counterexamples to FLT and proof of Goldbachs conjecture. The next two items improve calculation considerably: (1) The nth term of the Cauchy sequence of a nonterminating decimal approximates it at margin of error 10−n ; its value or magnitude in the standard norm is adjacent to it. (2) Cauchy convergence induces Cauchy norm that has advantages over the standard metric: (a) avoids indeterminate forms, (b) Cauchy limit is adjacent to its standard norm (limit point of Cauchy sequence), therefore, calculating it yields the latter as well, (c) tames the chaos in R∗ and (d) calculates the value or limit of function directly as decimal, digit by digit. The most recent result is the following: Theorem 3.1. R∗ − {u∗ } is a continuum, non-Archimedean and non-Hausdorff but the subspace of decimals is countably infinite, discrete, Archimedean and Hausdorff in the Cnorm. R∗ × R∗ × R∗ models the timeless unbounded physical Universe; R∗ physical time and distance, the non-standard Cauchy sequence of d ∗ the nested fractal superstring and d* the tail end of its toroidal fluxes, a superstring and continuum. The decimals model the metric system and the integers the countably infinite and discrete dark matter. Our universe is finite, so is its dark and visible matter. Work on GUT did not unfold until this question was posed: why are long-standing problems in mathematics and physics remain unsolved, e.g., FLT and Laplace problem? This question initialized reassessment and updating of their underlying fields: mathematical foundations, the reals and physics. Updating of foundations led to development of the new reals. In physics, inadequacy of mathematical modeling required qualitative modeling to solve long-standing unsolved problems and resolve unanswered questions and unsettled issues. The gravitational n-body problem posed by Simon Marquiz de Laplace at the turn of the 18th Century says: Given n bodies, b1 , b2 , . . . , bn , in the Cosmos at time T , with masses, m1 , m2 , . . . , mn , at points, x1 , x2 . . . , xn , and velocities, v1 , v2 . . . , vn , subject to their gravitational attraction, find their positions, velocities and paths at later time t. The solution could not be found because it is vaguely formulated since mass and gravity are ill-defined and the 5,000-year-old search for the basic constituent of matter unresolved. Newtons mathematical model of gravitation simply describes motion of bodies subject to gravity and says that force F of attraction between two masses m1 , m2 at distance s apart is F = Gm1 m2 /s2 (in suitable units), G = 6.7 × 10−11 m3 /kg/sec2 .

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The 10 natural laws that solve the problem qualitatively also anchored GUT and the integrated Pontrjagin maximum principle provided the other details asked by the problem. We shall now present the 10 natural laws that are central to GUT.

3.6.2 The Laws of Nature With the first law of thermodynamics that we consider most fundamental we proceed to find others consistent with it. However, this law needs modification as follows: Natural Law I (Energy Conservation) In any physical system and its interaction, the sum of kinetic and latent energy is constant, gain of energy maximal and loss of energy minimal. The Hubble reveals that matter forms steadily in the supposedly empty Cosmos at the rate of one star per minute. A nascent galaxy discovered recently makes this estimate conservative. How do we reconcile it with energy conservation? The answer is the next natural law. Natural Law II (Existence of Two Fundamental States of Matter) Two fundamental states of matter exist: visible and dark; the former is directly observable, the latter is not. Dark matter consists of small objects undetected by light. Its properties are known by its impact on visible matter. Its energy density according to De Broglie is 1026 joules/cu. ft. or the equivalent of 1018 kg/cu. meter or 8, 8 × 108 volts/cm. Natural Law III (flux-low-pressure complementarity) Low pressure sucks matter around it and initial chaotic rush of dark matter (flux) towards low pressure stabilizes into local or global coherent flux; conversely, coherent flux induces low pressure around it. This law explains vortices, e.g., typhoon, tornado, quantum and cosmological vortices. Natural Law IV (Energy Conservation Equivalence) Energy conservation has other forms: order, symmetry, economy, least action, optimality, efficiency, stability, self-similarity (fractal), coherence, resonance, quantization, smoothness, uniformity, motion-symmetry balance, non-redundancy, evolution to infinitesimal configuration, helical and related configuration such as circular, helical, spiral and sinusoidal and, in biology, genetic encoding of characteristics, reproduction and order in diversity and complexity of functions, configuration and capability. This law provides information not achievable by computation, e.g., shape of electron and configuration of photon and cosmic wave. Non-redundancy means that nature does not create distinct physical systems with the same functions. The so-called third quark in the nucleus outside the proton discovered in 2004 does the same function as the negative quark in the proton joins two positive quarks; the third quark joins two positive quarks, one from each of two protons. Therefore, it must be a negative quark. Now we ask what dark matter consists of? The answer is superstring. The existence of the superstring is assured by Natural Law II. Dark matter is made up of superstrings at various phases of their cycles from non-agitated through semi-agitated, agitated and back to non-agitated in a black hole. Cosmic wave traverses dark matter from all directions. When it hits a non-agitated superstring the latter is thrown off and bounces against other superstrings. When it loses imparted energy, it grinds to a halt as non-agitated superstring. However, when it gets near its previous path it is sucked, by flux-low-pressure complementarity, to form a loop. By Natural Laws III and IV, it evolves into its most stable form: helical semi-agitated superstring loop, the original superstring becoming its toroidal flux, a non-agitated superstring endowed with its latent energy. Repeated application of Natural Law IV yields:

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Natural Law V (The Basic Constituent of Matter and its Nested Fractal Structure) The basic constituent of dark matter, superstring, is a loop and nested fractal sequence of superstrings or toroidal fluxes, with itself as first term; each toroidal flux in the sequence is a superstring having toroidal flux, a superstring, traveling at speed beyond that of light along its cycles, etc.; each superstring except the first, is contained in and self-similar to the preceding term in structure, behavior and properties. Natural Law VI (Semi-Agitated Superstring Formation) When suitable cosmic wave hits (agitates) a non-agitated superstring one of the following occurs: (a) its first term as nested fractal sequence expands and becomes a semi-agitated superstring with the rest of the sequence as its toroidal flux; (b) it is projected into the first term of a new superstring with itself as the toroidal flux or simply loses the energy imparted on it and remains non-agitated. Dark matter consists of superstrings that fill up the timeless unbounded Universe, by Natural Law III. A superstring is dark if its cycle length (CL) is less than 10−14 m, non-agitated if CL < 10−16 m, semi-agitated if 10−16 < CL < 10−14 m and agitated and visible if a segment has CL > 10−14 m. By energy conservation, a superstring shrinks steadily; its tail end of toroidal fluxes is a superstring, a continuum. Natural Law VII (Dark-to-Visible-Matter Conversion) When shock wave hits semi-agitated superstring the following occurs: (a) the outer superstring breaks, its flux torus remaining non-agitated superstring or (b) a segment bulges as primum, unit of visible matter. Dynamic-mathematical model of primum (cylindrical coordinates) is: r = β sin nπ x(cosm kπ x), x ∈ [−1/k, 1/k], where n, k, m are integers, n k, m is even and n odd, a surface generated by rotating one period of the oscillation, g(x) = cosm kπ x about the x-axis, the cycles winding around it. The energy of a cycle is h = 6.64 × 10−34 joules, Plancks constant. Since toroidal flux of primum hit by comic waves from all directions is thrown into erratic motion (spike) in its infinitesimal neighborhood, it pulls the superstrings around it into a vortex flux of superstrings, a magnet. Moreover, when basic cosmic wave scoops up a primum and the latter breaks off its loop, it flattens in direction of flight (due to dark viscosity) as photon (rapid oscillation of toroidal flux)); it is stable when forward flux speed equals speed of carrier, i.e., speed of light); thus, speed of toroidal flux of primum is greater than that of light. Plancks constant h is energy of one full sinusoidal arc of photon. Its dynamic mathematical model is: p(x) = β (sin nπ x)(cosm kπ x), x ∈ [−1/k, 1/k]. Natural Law VIII (Flux Compatibility) Prima of opposite toroidal flux spins attract at their equators, repel at their poles; otherwise, repel at their equators, attract at their poles. A primum equatorially connects two prima of opposite toroidal flux spin as connector. Thus, a negative primum connects two positive prima equatorially and a positive primum connects two negative prima equatorially. This is the essence of primal interaction. Quantum gravity is dynamics of vortex fluxes of superstrings induced by primal toroidal flux spin. Its mathematics is called quantum algebra. The primum is a magnet and its properties follow from natural laws III and VIII. Its charge is due to induced vortex flux that also creates magnetic field. Using right-hand rule, a primum has positive charge if its induced flux, viewed from its north pole, is counterclockwise, negative otherwise. The electron’s charge of −1(1.6 × 10−19 coulombs (q)) is unit of charge. Other common simple basic prima: positive and negative quarks, charges +2/3 and -1/3. When primum pops out of dark matter Earth’s gravitational flux aligns its axis normal to it. Then by flux compatibility positive prima and

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ions go up and negative ones go down. Positive ions, being heavy, stay in lower atmosphere but light positive prima go into the mesosphere or beyond. When voltage between ground and cloud reaches critical level protons and electrons rush towards each other, crash, collide and explode as lightning. Charge is due to stationary primal vortex flux (static electricity); when it travels along conductor it is electricity. Every primum has charge but some coupled prima are neutral. The neutrino is neutral; therefore, it is equatorially coupled pair of prima of equal but opposite charges; their sum is 0, i.e., no net charge around it as they neutralize each other. Proton is pair of positive quarks joined equatorially by negative quark as connector; its charge: +2/3 − 1/3 + 2/3 = +1. Thus, proton has counterclockwise net induced flux, negative quark an eddy in it. Energy conservation requires axis of its component quarks be coplanar. Proton, electron and neutrino comprise neutron. Electron may attach to positive quark (flux compatibility) but energy conservation requires it attaches between two positive quarks of proton beside negative quark. flux compatibility pushes negative quark so that all four prima form quadrilateral. The coherent fluxes around center, formed by component prima, two clockwise and two counterclockwise, creates low pressure that sucks matter around it; cant suck charged primum, therefore, neutrino fits in being light, neutral. Thus, neutrons charge: +2/3 − 1/3 + 2/3 − 1 + 0 = 0; has no net flux around it. Since neutrons, protons and electrons masses are known we can calculate neutrino mass η : 1.0087 − (1.0073 + 5.4860 × 10−4 ) = 8.5 × 10−4 amu, i.e., η = 8.5 × 10−4 amu or 1.55 times the electron’s mass (roughly 1/6). First thing that forms in atom is nucleus consisting of protons equatorially joined by negative quarks or layers of such; initial number of protons in it is stochastically determined depending on temperature. Their vortex fluxes add up to form coherent vortex flux around nucleus as its eye. Then the eye sucks neutral prima since charged prima are repelled by quarks. In particular, it sucks neutrons forming heavy isotope, the number of neutrons sucked depending on number of protons. Electrons are attracted by vortex flux but, being light, pulled into orbit by it. Atom is neutral when number of orbital electrons equals number of protons in nucleus; otherwise, its an ion. Natural Law IX (Internal-External Factor Dichotomy) Interaction and dynamics of physical system are shaped by internal and external factors; in general internal is principal over external and the latter works through the former. Principle X (Law of Uneven Development) In any process or interaction development proceeds unevenly and perfect balance and uniformity is unstable. Macro gravity is dynamics of cosmological vortices of superstrings from the supersuper galaxy, our universe, through the galaxies, stars, planets, moons and cosmic dust forming nested fractal sets of vortices. Pull of gravity is an aspect of that dynamics, specifically, suction by eye of cosmological vortex.

3.6.3 Formation of a Universe The Universe, consisting of dark and visible matter, is unbounded, by Natural Law III, and there is no evidence of beginning or end either. However, local universes including ours continue to form, their destiny back in dark matter. By energy conservation, the superstrings shrink steadily and, by law of uneven development, form nested fractal sets of low pressure regions that, by flux-low-pressure complementarity, evolve into nested fractal sets of cosmological vortices, the usual way of forming a universe. By law of uneven development, the most probable formation has one core vortex, the rest nested fractal sets of minor cosmological vortices around main eye. Less probable is formation of binary cosmological vortices of comparable masses, each revolving around the other, there being a

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stable balance between their mutual suction and centrifugal force on each. Their collected masses form binary stars. Even much less probable is formation of trinary stars (interacting equatorially among themselves). Trinary stars are stable only if they have same spin and their mutual gravitational attraction is balanced by their equatorial repulsion due to flux compatibility and centrifugal forces. Every vortex has an eye bounded by spinning matter around it. The eye is region of low pressure and calm and, by flux-low-pressure complementarity, sucks matter around it. At the same time, dark viscosity and spinning collected mass at eyes boundary (event horizon) pulls and catapults dark matter into a spin around it. Resultant of centrifugal force and suction by eye is spiral streamline path along which matter falls towards eye. Streamlines cover entire region around eye within the reach of suction. By momentum conservation falling matter along spirals raises spin of collected mass around eye and, by dark viscosity, extends spiral streamlines and suction outward. Two forces act on any visible matter along spiral: suction by eye and centrifugal force of spin. When these two forces are balanced it takes an orbit around the eye. By oscillation universality of uneven development, the point of balance oscillates along its orbital radius forming elliptical orbit around eye. How does visible matter form in a cosmological vortex? The spinning collected mass around the eye is turbulence that agitates and converts dark matter to hot prima and gases. The micro component of turbulence generates and sends seismic waves (type II cosmic waves) outward that convert dark to visible matter first as prima, then light atoms that get entangled in micro vortices and collect at their micro eyes as cosmic dust. Cosmic dust gets entangled into cosmological vortices and collects at their eyes to form masses like stars and their planets and moons. This process forms masses in the Cosmos at the rate of one star per minute. As macro vortex sucks matter into its core it thins out its dark halo (gravitational flux of superstrings) around the eye. Viscosity declines and so does the reach of suction and spiral streamlines covering around it leaving the outer minor vortices free. This starts disintegration of the macro vortex. However, in the thinning of gravitational flux the remaining minor vortices at balance between suction and centrifugal force escape falling into core and remain in orbit. The lucky survivors in the solar system are its planets. Each follows same pattern of development. The Earth has only one surviving minor vortex, the Moon. Jupiter has 16 and Saturn twenty. There is evidence that Earth is at its younger phase of development, e.g., Moon is receding one inch per century, i.e., Earths gravitational spin still rises. Moreover, there is evidence that 65 million years ago Earths mass was 67% of the present. Suitable measurements in the solar system may determine if the Sun is on the younger phase of its development. When spin of cosmological vortex rises and exceeds centrifugal force on minor vortex, the latter gets catapulted out of it. This is verified by independent stars traveling along straight paths. The eye of every cosmological vortex is region of low pressure and calm that de-agitates prima at inner boundary of eye to become, ultimately, non-agitated superstrings, by energy conservation. Then they form massive concentration of non-agitated superstrings inside eye called black hole. Thus, the eye of cosmological vortex is its own graveyard and the destiny of the mass it absorbs by gravity is black hole in its eye. Consequently, the destiny of a universe including ours is a cluster of black holes back in dark matter. Our universe, as super...super galaxy, is the first term of its nested fractal sequences of galaxy clusters, galaxies, stars, etc. Each galaxy has its nested fractal sequences of cosmological vortices with itself as common first term and then all the way down through the stars, moons, cosmic dust, atoms, prima, superstrings, a superstring being itself nested fractal sequence. Thus, these nested fractal sequences stretch from macro scale through dark matter. Technology based on GUT is called new technology. One example is the magnetic train that runs on and is controlled by vortex fluxes of superstrings of opposite equatorial directions based

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on flux compatibility. This is a case of invention ahead of theory just as the steam engine was ahead of thermodynamics. There is now prototype of a 15-volt 700-watt free energy converter (using magnets) that converts dark matter to electricity. It can be up-scaled to build electric power plants of any megawatts of capacity. Another research challenge is to produce metallic alloy for building foundations and concrete reinforcement resistant to softening or melting when impacted by high-intensity seismic waves or concrete composites that do not crack or pulverize under impact of high-intensity seismic waves (during earthquake each case). These alloy and composite provide remedy for the problem of the space shuttle Colombia. Unfortunately, the program has been stopped because the problem that caused the previous disaster persisted in succeeding flights. Still another is to make an instrument that can monitor propagation of seismic waves from geological faults and tectonic plate boundaries to predict earthquake or monitor seismic waves from volcanic activity to predict eruption. A new technology that can terminate an approaching tornado has been conceptualized. It can be installed below ground surface at suitable side of city or farm for protection. Main natural law involved is flux-low-pressure complementarity. Another new technological possibility is suitably placing magnets at crossroad underpasses and coils on buses so that when busses pass through them direct current is generated and stored in suitable batteries inside to keep them going until the next underpass chargers. The flying saucer if it exists works on the same principle as magnetic levitation, a well known phenomenon. Ordinary gyroscope can serve as steering and control. A fabric of ferromagnetic material that scatters electromagnetic radiation and blocks magnetic field serve as protective shield against radiation. New technology based on magnetic levitation can revolutionize travel within the Earths gravitational flux that extends far beyond the Moon. Also Kerlian photography is well known and belongs to the category of new technology. It has practical applications in crime investigation. GUT offers myriad of new technological possibilities that call for active research collaboration and technology generation that can ultimately elevate the human condition and serve mankind. We have merely scratched the tip of the iceberg, so to speak. At the same time, new research possibilities have opened up in both mathematics and physics for graduate students, mathematicians and physicists.

Chapter 4

Theoretical Applications

4.1 Introduction This chapter is mainly theoretical application of GUT to create physical theories in the other disciplines of the natural and behavioral sciences with one exception, the solution of the gravitational n-body problem of physics. The theoretical articles are: The Amazing Brain (physical psychology) in Section 4.4, Turbulence: theory, verification and applications (atmospheric and geologic sciences) in Section 4.3, and only the computational component of the solution of the gravitational n-body problem is treated here in Section 4.2 as the qualitative component is covered by Chapter 3, The Grand Unified Theory. However, this problem has particular historical significance because it was the catalyst that launched this journey into physics. The applications of GUT to engineering and medicine are treated in [40,41].

4.2 The Solution of the Gravitational n-Body Problem 4.2.1 Preliminaries Posed by Simon Marquis de Laplace at the turn of the 18th Century in his book, Celestial Mechanics, the gravitational n-body problem actually sparked the development of GUT. Its inclusion among the applications, therefore, reverses its important place as the first milestones in the historical and theoretical development of GUT. Around 1992 this question was posed: why are there long-standing unsolved problems of mathematics and physics that defied resolution for so long? For both mathematics and physics the answer is: inadequacy of their underlying fields. For physics the principal underlying field is mathematical physics and the inadequacy lies in its methodology of mathematical modeling that describes nature mathematically and statistically. Let us state the problem to see why. Given n bodies in the Cosmos at time T and positions, x, . . . , xn , velocities v1 , . . . , vn , and of masses m1 , . . . , mn , subject to their gravitational interaction, find their positions, velocities and paths or trajectories at later time. Even the solution of the simplest case, n = 2, where the two bodies are the Sun and Earth does not match the behavior of the Earth relative to the Sun. One solution presented at the 2nd International Conference on Dynamic Systems and Applications, May 1995, Atlanta, says that the Earth orbits around the Sun along a rotating plane whose axis of rotation coincides with the Sun’s polar axis which is not true. Something is missing in this problem: Neither “body” nor “gravity” is known. Therefore, we do not know the behavior of this physical system and cannot predict its future course required by V. Lakshmikantham et al., The Hybrid Grand Unified Theory, Atlantis Studies in Mathematics for Engineering and Science 3, DOI 10.1007/978-94-91216-23-7_4, © 2009 Atlantis Press/World Scientific

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the problem. It requires a physical theory, specifically, some laws of nature, to define the relevant concepts involved and solve the problem. Newton’s so called law of gravitation is not a theory but a mathematical model, i.e., description of motion of bodies under the influence of that unknown force called gravity. Even “body” is not defined properly. To know what a body is it is necessary to answer what the Greeks and the Chinese and, inevitably, most cultures, sought thousands of years ago: the basic constituent of matter. Physicists have been smashing the atom for over 50 years to find it. They failed because they have been looking for it in the wrong place. Both their methodology and tools are inadequate, the latter mainly computation and measurement. The characterization of undecidable propositions in the course of the resolution of FLT revealed the inadequacy of mathematical modeling: mathematical and physical spaces are distinct and, therefore, independent; the former being man-made and defined by its axioms, the latter objective and subject to and defined by the laws of nature. Therefore, a problem in one space cannot be solved by the other; one can only use reasoning by analogy; sometimes it works but not all the time. This is the context by which we can look at the inability to solve the gravitational n-body problem for two hundred years. In the actual solution of this problem [48] 10 laws of nature were needed; they were the initial laws of nature of GUT now stated and used in its development along with other natural laws. They were required to discover and define the basic constituent of matter, the superstring which is dark, and its conversion to visible matter. They also define gravity in its full manifestation as natural phenomenon, i.e., as dynamics of a cosmological vortex. Laplace posed the problem to prove the stability of the solar system which was then thought to be our universe. Therefore, in his formulation he expected, as solution, that the n bodies would eventually arrange themselves into a planetary system like Jupiter and its moons where one body, Jupiter, is at the center and the others orbiting around it or, like the planets, would find their respective orbits around the Sun. It is not quite that simple now because the bodies may belong to or be under the gravitational influence of different cosmological vortices. Let us first classify the familiar bodies in the solar system; then the general case will become obvious: (a) Cosmological body, i.e., the core or collected mass around and at the event horizon of the eye of a cosmological vortex to which category belong the stars, planets, planetoids and moons. (b) Comet, damaged cosmological body, most likely planetoid, due to near hit on the Sun but damaged and catapulted into an elongated orbit by the Sun’s gravity (we have to confine our definition to the solar system because we do not know if a comet has counterpart in a galaxy). (c) Debris, e.g., asteroid, comes mainly from comets tail or planetary collision with planetoid or collision among wayward planetoids; they cluster around huge planets like Jupiter and Neptune because their strong gravity serves as trap for collision except those that get suspended in neutral regions (i.e., between planetary vortices and under the Suns dominant gravitational flux) influence. (d) Meteors, debris from tail of comet light enough to be attracted by or plunge into weaker planets like Earth. (e) Cosmic dust, converted prima from dark matter that gets entangled into micro cosmological vortices. Technically, cosmic dust consists of cosmological vortices but so light they cannot move much as their dark component cannot overcome dark viscosity (their motion is analogous to Brownian movement but confined to narrow neighborhood). One thousandth of Milky Ways mass is cosmic dust [4]. Of course, the bulk of its mass is generated by the turbulence within it. Items (b) to (e) were already discussed earlier but we shall summarize the discussion here and explain their behavior. Moreover, in the case of (a), the qualitative solution of the n-body problem

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was also done under macro gravity. Thats exactly what the 10 initial laws of nature did: provide the qualitative solution. The bodies lie along spiral streamlines and the latter are rotating around the core; for some, when centrifugal force equals pull of gravity, they take elliptical orbits. Then the integrated Pontrjagin maximum principle provided the computational component of the solution that we shall undertake here. This is really the role of computation in physics: to corroborate, concretize and provide details of the qualitative solution. The computational tool we use is the Pontrjagin maximum principle, specifically, to find the equations of the trajectories of the n-bodies. We shall take the general case were the n bodies do not all belong to the same cosmological vortex. In view of the fractal nature of our universe with itself as the common first term of the fractal sequences [85] there is a minimal cosmological vortex that contains all of them. We put the origin of our global coordinate system at its eye. Then with the fractal-reverse-fractal locator [52] we locate the positions of the n bodies and using suitable composite functions find their equations relative to the global coordinate system.

4.2.2 The Solution of the Gravitational n-Body Problem We solve the problem one at a time as a time-optimal problem on a cosmological body. We formulate the problem in the equatorial plane of the core vortex and the problem is to bring the body (a minor cosmological body) to the surface of the collected mass at the main eye where the origin of the local coordinate system is located at its center. This is an uncontrolled problem in the context of the Pontrjagin maximum principle because the controls are fixed: the pull of gravity and the centrifugal force on the body. This makes the solution a relaxed trajectory, a special case of generalized curve, having set-valued derivative. The control set consists of the values of the derivative constrained by some differential equation. More set-values of the derivative of a relaxed trajectory come from the micro component of turbulence along the streamlines as interfaces of turbulence. We consider the family H = yG

(4.1)

of Hamiltonian functions h(t, x, y) = yg(t, x), where y is a variable vector and each g ∈ G gives rise to a corresponding h ∈ H. We shall be concerned with points (t, x) that lie in a sufficiently fine neighborhood of the set described by a given fixed trajectory C of the form x(t), t1  t  t2 . In terming C a trajectory we imply that the function x(t) is, almost everywhere in its interval, a solution of the differential equation, .

X = g(t, x(t)),

(4.2)

for some fixed corresponding member g ∈ G; moreover, x(t) is to be absolutely continuous. Here G is a convex family of functions, a family such that every convex combination,

∑ αi gi ,

(4.3)

of a finite number of members gi , of G with constant coefficients αi  0, where ∑ αi = 1, is itself a member of G. In addition we require that every function g(t, x) in G is continuously differentiable in x for fixed t and measurable in t for fixed x, and also that each g and its partial derivative gx are bounded functions of (t, x) or, more generally, bounded in absolute value by some integrable function of t only. These various requirements are to hold only in some bounded open set O of (t, x) space. In the chattering case all these requirements are satisfied if we make the stipulation that g(t, x, u) is continuously differentiable and strengthen the convexity condition by requiring that the αi s are measurable functions of t.

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(In the formulation of the Pontrjagin maximum principle x is a vector in n-space; so that the optimal solution of (4.2) would be an absolutely continuous vector function in n-space. Here the solution will be an absolutely continuous parametric function in the plane in the parameter t.) The appropriate case for this problem is the matrix A having eigenvalues with negative real parts. Then trajectories x(t) approach the origin for large t, i.e., the system is stable. By appropriate choice of unit of time (change of scale) we assume the eigenvalues to be λ ± i, or λ , λ − 1, where λ < 0. By affine transformation of x and translation of u we suppose that B is unit matrix and take the appropriate case where the matrix A is given by,  A=

λ −1 1 −λ

 (4.4)

To find the optimal control function we use Young’s theorem [131]. Theorem 4.1. Let u(t) be a suspected optimal control which transfers our particle from the initial position x0 at time t0 to the origin at time t = 0 and suppose that the corresponding function x(t), which defines the trajectory described by the particle is not constant in any time interval terminating at t = 0, i.e., that the final arc of the trajectory does not reduce to a single point, consisting of the origin. Then u(t) is an optimal control, and the only one to make this transfer. The control parameter u belongs to a convex set U, a parallelogram none of whose edges is parallel to a coordinate axis. The Pontrjagin maximum principle asserts the existence of nonvanishing solution y = ψ (t) of the adjoint equation, dy = −A∗ y, dt

(4.5)

where A∗ is the transpose of (4.4), such that the Hamiltonian (scalar product), H(x, y, u) = (y, Ax + Bu)

(4.6)

attains, when x = x(t), y = ψ (t), its maximum in u = u(t). The initial condition at time t0 is x(t0 ) = x0 and at final time at the origin both x = 0 and transversality condition in t which states that H > 0 for the argument (x, y, u) = (0, ψ (0), u(0)) holds. [129] shows that the condition H > 0 holds on suspected solution and is constant there and hence can be verified at any time t along that trajectory. The control u(t) is piece-wise constant. We term switching time for suspected solution one for which u(t) is discontinuous in t. For fixed t the maximum of H(x(t), ψ (t)) is attained when the scalar product (ψ (t), Bu) is maximum and, being linear, attained at a vertex of U. It occurs there only unless there is one edge with direction vector w on which the scalar product is constant so that by subtracting w it satisfies, (ψ (t), Bw) = 0. The solution of the adjoint equation is, y = ce−λ t η , where η = ei(t+α ) ,

(4.7)

where α and c are integration constants and c > 0. Note that the η -curve is a rotating unit circle in the counterclockwise direction so that the product with the radial curve y = ce−λ t tends towards the origin as t increases, i.e., the spirals tend towards

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the origin. We can interpret this as the path of a falling body in a cosmological vortex with counterclockwise spin. This family of spiral curves covers the whole vortex. Of course, the mirror image of this family relative to the y-axis is a family of spirals in the clockwise direction and corresponds to the paths of falling bodies in a cosmological vortex with clockwise spin. If there was another body in the same cosmological vortex then there is a unique spiral that connects it with the origin. It does not really matter if the body is in orbit or not, the bodys trajectory will be a rotating spiral. For each body in this cosmological vortex we have all the information the problem asks for: position, trajectory and velocity.

4.2.3 General Case of the Problem We have solved the uncontrolled problem for the case of one body in a cosmological vortex. When there are more than one then each body lies in some spiral and we simply find their equations one at a time. When the n bodies do not all belong to the same cosmological vortex then, in view of the fractal nature of our universe with itself as the common first term in its nested fractal sequences, there exists a (minimal) cosmological vortex that contains the cosmological vortices of the other bodies of the problem [86]. We place the global coordinate system in that vortex with the origin at its center (it is possible that the common cosmological vortex of the n bodies is our universe itself). To find the locations of the other bodies we apply the fractal-reverse-fractal vortex locator on each [52]. Then we find their equations with respect to the global coordinate system. We do it one at a time for each body. Without loss of generality we take the Moon as one of the n bodies. Then using (4.7) we can find its rotating spiral trajectory about the Earth. The Earth may be one of the bodies, it does not matter. At any rate, we can find the Earths rotating trajectory in the local coordinate system with origin at the center of the Sun. Then we can find the appropriate composite function that gives the equation of the rotating trajectory of the Moon in terms of the local coordinate system with origin at the center of the Sun. Moving backwards, we can find in exactly the same manner the equation of the rotating trajectory of the Sun about the center of the Milky Way in terms of the local coordinate system with origin at the center of the Milky Way and the appropriate composite function that would yield the equation of the trajectory of the Moon. We can do the same thing for all the intervening cosmological vortices from the Earth through the common cosmological vortex and give the equation of the Moon in terms of the global coordinate system located there. Then we can replicate this scheme for all the other bodies of problem.

4.2.4 Solution for the Other Categories of Bodies in the Cosmos Massive piece of debris in the cosmos, e.g., asteroid, comet, has greater viscosity relative to the gravitational flux it is under the influence of. Long before an asteroid reaches the vicinity of a cosmological vortex, say Earth, it gets deflected away from it by the gravitational flux. The Earths gravitational flux reaches far beyond the Moon and an asteroid starts getting deflected there. Only when it enters the narrow injection angle of 2◦ (used as guide by astronauts in coming back to Earth) will it crash into Earth. Even when it intersects the spiral streamline in the same direction, its momentum is too strong to be sucked by the Earths gravitational pull towards the eye along that streamline. If the asteroid

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approaches Earth at the opposite edge of its gravitational flux, it will be deflected away just the same by flux compatibility. One might wonder why asteroids do not approach Earth from the poles where the injection angle is much wider. The answer is: asteroids come from tails of comets or debris from planetary collision with planetoids or collision among wayward planetoids that lie in the Suns dark halo along SEP. They are attracted towards massive planets by gravity. Some get suspended in neutral regions (i.e., dominated by the Suns gravitational flux). How can a planetoid go wayward? When a comet passes nearby it pulls the planetoids away from their orbits around the Sun, by flux-low pressure complementary. In the case of Jupiter with powerful gravitational flux, while it has also a natural shield against the massive asteroids or wayward planetoids, its powerful gravity overcomes their momentum to escape suction along the spiral streamlines and so they fall into the planet. We have seen this spectacle in the 1990s when large asteroids collided with it and caused powerful earthquakes detected on Earth. We already know what happens to cosmic dust and so we will go to such level of detail...

4.2.5 Extension Efficiency is improved by varying controls and this is where we need other families of spiral curves from the family of curves by shifting origins in a systematic manner to compose a simplicial trajectory corresponding to piecewise constant controls. In general a convex polygon in the same n-space will do for the control region. It can be dials switched from one position to another. In our case we have taken a parallelogram U with no side parallel to a coordinate axis. We determine the directions of y orthogonal to sides of U and the values of t at which u(t) switches from one vertex to another. These are at corresponding intersections of curves η (t), ∞ > t > 0, with lines l1 , l2 through the origin perpendicular to the sides of parallelogram U (note that we have shifted origins relative to the vertices). The η -curves are counterclockwise arcs of unit circle and the intersections occur at four sets of periodic values of t of period 2π . One of these families is a trajectory corresponding to constant controls determined by the origin; they represent streamlines. They are visible in the galaxies with visible matter, specifically, minor vortices as tracer. A streamline is the most efficient path of a particle (uncontrolled) to reach the origin. To find the suspected optimal control we apply the maximum principle and the above Young’s theorem [99] to the solution of (4.2): y = ce−λ t ζ ,

ζ = aei(t+λ ) ,

(4.8)

where c and a are constants of integration and a > 0. We determine the directions of y orthogonal to sides of U and the values of t at which u(t) switches from one vertex to another. These are at corresponding intersections of curves ζ (t), with lines l1 , l2 through the origin, perpendicular to sides of parallelogram U. The ζ -curves are counterclockwise arcs of unit circle and the intersections occur at four sets of periodic values of t of period 2π . Without loss of generality we suppose l1 , l2 to lie in the first and third quadrants, respectively, and denote by αk (k = 1, . . . , 4) angular sectors on which l1 , l2 divide the plane and by uk (k = 1, . . . , 4) the vertices of U. The indices indicate orientation so that as we proceed counterclockwise around the origin starting from the real axis we pass from α1 to α2 by crossing l1 , then from α2 to α3 by crossing l2 and so on. For each k, the vertex uk of U is in sector αk and when lk is in sector αk the control u(t) along suspected optimal trajectory takes the value uk . We term sojourn time at uk the time that elapses between consecutive switches for which u(t) = uk , i.e., ζ (t) is in angular sector

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ak , and the sojourn time is its measure which does not depend on the time and place at which the previous switches occur. To avoid confusion due to the shifting of coordinates we now refer to the curves in (4.8) as the ζ - and φ -curves, respectively. To determine the trajectories, we denote by xk (k = 1, . . . , 4) vertices −A−1 uk of the parallelogram derived from U by affine transformation u → x given by Ax +u = 0 or x = −A−1 for which uk satisfies d (xxk ) = A(xxk ), dt obtained by translating the origin to point xk from corresponding φ -curve solution of dx/dt = Ax: x = aeλ t ζ (t). Here, a and ζ are constants of integration and a > 0 except for the trivial solution φ = 0. The origin is reached for t = 0. The ζ - and φ -curves coincide and the latter is derived by multiplying by the factor aeλ t which deforms them towards the origin. What remains is fitting together suitable translations of the φ -curves for which the origin has been shifted, in turn, to the appropriate vertex xk to form the suspected optimal trajectory which, by the above Youngs theorem is uniquely determined by the set U, is necessarily optimal. Also, by the Pontrjagin Maximum Principle, the terminal segment hits the target, boundary of a half-sphere with center at origin, orthogonally. The sequence of convex polygons, edges approaching zero length, as control sets generate the sequence of optimal trajectories converging weakly to the optimal generalized curve (absolutely continuous with set-valued derivative). Controls can be dials on control panel. As matter accumulates at the core the spin increases due to momentum conservation and, with dark viscosity, extends its impact on wider region. The particle outside the core takes increasing centrifugal force which, combined with gravitational flux pressure, yields a resultant that serves as control with reverse effect: delays the transfer of particle to the core until the balance is attained between the centrifugal force and the gravitational pressure. Stability ensues with radial oscillation yielding elliptical orbit. If the centrifugal force critically exceeds the gravitational pressure, the particle takes a parabolic orbit and escapes. This physical solution can be mathematically modeled by a control system where the sequence of control sets U consists of convex polygons having arbitrarily fine edges and the objective function is suitably modified to generate the sequence of piece-wise constant controls u(t) and the corresponding trajectories weakly converging to a generalized elliptical trajectory (see figures). The conjugate curves representing the streamlines from which the sequence of optimal controlled trajectories is formed with controls induced by the resultant of gravitational flux pressure and centrifugal force at any point on the plane. However, the computational method (e.g., symbolic computation) applied to the solution of the pre-problem offers more efficient solution, the same method used in space exploration. Nevertheless, both physical and computational mathematics apply here only because the pre-problem has been solved and the solution is anchored on qualitative mathematics. Here we see the precise complementary relationship between computation and its complement, qualitative mathematics and, in terms of methodology, conventional and dynamic modeling. Qualitative modeling and its mathematical component, qualitative mathematics sets the physical basis for the problem, that gravity is dynamics of vortex flux of superstrings. Then computation finds the trajectory and locates the body along it.

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Both the solutions of equation (4.2) and the adjoint equation (4.5) are parametric equations of two families of trajectories, one family spiraling counterclockwise towards and winding around the target circle with center at the origin and the other spiraling clockwise towards and winding around the same target circle. Each trajectory in either case corresponds to constant control, fixed parameter. Thus, control is applied by varying the value of the parameter, in effect, the derivative. The solution we got is not the optimal solution but an improvement over one with constant control, i.e., along a fixed spiral. A much improved trajectory is one with rapid switching between u1 and u2 that generates a simplicial curve analogous to the zigzag except that instead of mixture of line segments we have here mixture of spiral segments. The limit of a sequence of such simplicial curves, as the switching becomes more rapid indefinitely, is an infinitesimal simplicial curve, a generalized curve, along spiral trajectory. That limit is an optimal trajectory. Any body B among the rest of the n bodies can be located by fractal transformation using the collected mass of the core vortex as the fractal generator. Body B takes an elliptical orbit when the gravitational pressure at B balances the centrifugal force on B combined with usual radial oscillation at the point of balance. The main and secondary streamlines on the corresponding spins can be mapped. The main spin generally retains residual secondary spin (e.g., the Earth’s oscillatory wobble known as SummerWinter Solstice) or takes minor spiral spin. The Earth’s main spin is the 24-hour rotation. Around the polar axis at the opposite poles are conical spiral streamlines whose cross sections normal to the axis form concentric circles covering the plane with center at the axis of major spin (only Mercury is in one of those polar streamlines which accounts for its perihelion shift). All such spirals are dark streamlines towards the event horizon of the vortex eye. As the accumulated mass at the core cools off pressure at the eye reverses and exerts sucking action at the poles resulting in flattened polar caps.

4.2.6 Concluding Remark This solution is direct application of qualitative modeling. To solve a problem one devises a theory that provides the solution. Neither Newtons laws of motion nor Einsteins field equations can solve this problem. Since they are not qualitative but mathematical models they can only describe the behavior of a physical system but cannot explain their behavior. Therefore, they have no predictive capability on the n bodies. Relativity has the basic ingredients of a theory but there are problems with the postulates. GUT facilitates the search for verification or counterexamples to other theories. For example, the asteroids found between the orbits of Mars and Jupiter and the larger ones around the corridor between the orbits of Neptune and Pluto are verification of its prediction that bodies in the Cosmos that have lost gravitational flux coherence (e.g., debris) also lose gravitational interaction between them. Since the asteroids have mass they are counterexamples to Newtons law of gravitation. We note at this point that the gravitational n-body problem has different solutions when the n bodies are not collected mass at cosmological vortices (e.g., asteroids and comets). Let us consider an example of the extended solution of the n-body problem.. Suppose the problem is to land a rocket on the Moon. In this case the entire trajectory is mixture of segments of trajectories from the Earth to the Moon as follows: (a) trajectory from launch site to a point of tangency P with an orbit around Earth; (b) then the space vehicle goes around the Earth in this orbit a few times to prepare for “break-off” at a certain point Q in this orbit so that we have now a mixture of paths from launch site to point Q.

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(c) Then the space vehicle fires its rocket at point Q for the trajectory towards a point of tangency S at a lunar orbit; the space vehicle may go around this lunar orbit to survey the terrain at landing site. (d) Then at some point T on this lunar orbit the space vehicle launches the Lunar Lander towards, stops and hovers at some point Z, say, six feet above target and drops it on the target (in this case the orthogonality condition is satisfied).

4.3 Theory of Turbulence Chaos, turbulence and fractal are generally lumped up together in the scientific literature; unfortunately, they are not equivalent dynamics but distinct phases of the standard dynamics: (1) evolution from order to chaos, (2) transitional phase of chaos and (3) evolution from chaos to order called turbulence. Then turbulence may or may not fade away depending on whether the conditions for its existence continue to exist. For example, Jupiter has a cyclone that has been there for 300 years. The usual examples of standard dynamics are weather and financial market dynamics [61, 111]. The standard turbulence problem is: find mathematical model of the standard dynamics that has predictive capability on its future state. While there is great effort at mathematical modeling of weather turbulence and the stock market the frequent embarrassment of the weatherman and occasional ruin of the stock broker confirms this problem is unsolved. The irony is: the solution of the turbulence problem in the stock market will also wipe out the stock market for good as market fluctuation grinds to a halt. The common thread of this dynamics is uncertainty local and infinitesimal at all phases and also global at phase (2) which places them under the category of ambiguous physical system (involving large set and numbers). Rapid infinitesimal local motion induces uncertainty at phases (1) and (3) (e.g., Brownian motion of gas molecules) similar to uncertainty at limit set of fractal. We focus on earthly turbulence: earthquake, volcanic activity, typhoon, tornado and lightning. We recall that turbulence is motion of matter with identifiable direction at each point. Wave of any kind is turbulence and even stationary object is turbulence and when it interfaces with other turbulence such as compression or a rocket whizzing through the stationary atmosphere then all the dynamics of turbulence comes into play.

4.3.1 Qualitative Modeling Weather turbulence like cyclones and tornadoes is one of the oldest problems of physics. We broaden the problem beyond computational modeling and include these questions: what are the necessary conditions that trigger turbulence? Do individual properties of the molecules have some bearing on these conditions? What actually happens at the onset of turbulence or when it is already in progress? Of course, the more practical question is: how can we control or terminate it? Although sophisticated monitoring may provide leads to the answers they are not enough. In the US they have extensive empirical data especially on tornadoes but they have no answers. Obviously, computational modeling is inadequate here. Therefore, we again employ qualitative modeling. With it the problem is easy to resolve because we have already a theory to start with. In fact, all we need is, apply GUT to the problem of weather and geological turbulence using some natural laws particularly

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flux-low-pressure complementarity, the existence of micro component of turbulence and threshold for the generation of seismic waves [51] and flux compatibility.

4.3.2 Geological Turbulence Our universe is turbulence from the super...super galaxy itself all the way through the cosmological vortices, atoms, prima and superstrings. But, we focus on Earthly turbulence and on only five of them that fall under the standard dynamics: earthquake, volcanic activity, typhoon, tornado and lightning.

4.3.3 Earthquake Compression, grinding and lateral tension at geological faults and tectonic plate boundaries involving millions of tons of force causes interpenetration (interfacing part inducing changes on the other), collision and energetic vibration of atoms and molecules induced by the micro component of turbulence. Since this happens at interface of turbulence the dark components of the interfacing turbulence compose or generate and propagate seismic waves. Present seismograph detects only their visible high-frequency wave components and its envelope is visible to the naked eye (e.g., the wave motion of ground surface during earthquake). Seismic wave is actually a nested fractal sequence of waves from the visible or macro through the micro and then dark component that ends up at the interface, each of the interfacing parts containing its respective fractal sequence of waves. The waves we see on the ground during vertical or lateral earthquake are the visible envelopes of seismic waves. They are actually nested fractal sequences of waves that end up as dark (energetic) cosmic waves at the interface that soften or melt metal and crack or pulverize brittle material. The micro component of seismic waves irritate the brains of animals causing erratic behavior: horses and water buffalos jumping and running around erratically; dogs howling in distress; ants and termites coming out of their mounds and hives; roaches flying like bees to escape irritation at wall crevices and ceiling; schools of whales and jellyfish leaving the ocean depths for shallow waters; they are signs of impending earthquake. The high intensity of seismic waves just before an earthquake convert superstrings to prima that form atoms and molecules, e.g., earthlights and volatile gases like radon that shoot off flames from the ground, combined with change in water level at open wells and widening of geological faults; they are danger signs the Chinese are adept at interpreting to predict an impending earthquake. Previously labeled UFO, earthlights of varied colors, intensity and motion have been sighted in California (along Andreas fault), Colorado, South Wales and Mexico. Geologists associate them with earthquakes but have no theory to explain them. Earthlights were sighted over Mexico City two years before the devastating earthquake of 1985. Physical characteristics of earthlights have predictive value. Bluish earthlight being energetic indicates high compression and tension at plate boundaries and faults and reddish one (less energetic) indicates low compression and tension. Just like the guitar string, greater tension on it produces energetic vibration and higher pitch. Then the impending earthquake can be assessed for its intensity. Analyzing shock waves from earthquake provides information about the interior of the Earth. Since seismic waves travel faster on denser materials scientists are able to measure the specific gravity of the Earths inner and outer core. Its core is extremely hot due to its staggering spin. Therefore, only prima and light elements like hydrogen exist there. They are compact and this accounts for the great density of the core. Compression by gravity is a secondary factor. Present measurement says that the Earths core has specific gravity of 150 [111]. The present guess is that it consists of molten iron but iron is a complex atom and cannot form at such high temperature, kinetic energy and vibration.

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Just like the Sun the Earths core (that contains the eye) may also contain hydrogen and helium and, certainly, it is also accumulating its own mini-black hole. Along geological fault or conservative plate boundary, two kinds of dynamics stemming from independent motion of the interfacing parts and the law of uneven development occur. They are due to the turbulent motion of the Earths mantle under the Earths crust. In both compression and lateral tension there is grinding, interpenetration (each part inducing changes on the other) and energetic vibration of the atoms and molecules that propagate seismic waves. This dynamics cannot be reproduced in man-made laboratory because of the huge forces involved. Verification requires simulation, observation at faults and plate boundaries and gathering and consolidating data from the literature, US Geological Survey databases and even features by Sky Cables Discovery (e.g., Hostallen Project) and National Geographic. Lateral tension also involves compression due to uneven interfaces at faults (usually macro sinusoidal). Therefore, it generates seismic waves which can be monitored and level of compression and tension measured. Opposite infinitesimal lateral movement along faults is verified by macro geological motion and formation. For example, the two parts of Andreas Fault in California are moving in opposite directions that a mountain near Lake Tahoe now used to be in Mexico millions of years ago. The two parts of the Alpine Fault in New Zealand has now slid past each other by 300 km [111]. The wavy fault interface adds to resistance and, therefore, power buildup; earthquake occurs when these obstacles including huge rocks between sliding parts snap and break, sending powerful shock waves (visible envelope of seismic waves) in all directions on the ground. However, their dark components are thrust in all directions. They convert dark matter to earthlights and balls of fire above the ground. Between earthquakes, compression, grinding and lateral tension generate shock waves converting superstrings to visible matter, e.g., earth lights. They can be analyzed to study fault and plate boundary dynamics that may have important practical applications. Engineers thought that shock waves and jolting movement caused by earthquake damage highrise structures and towers and in the 60s they started putting rollers and springs at their base to absorb the shocks and jolting action. It did not work. The great devastation in Tokyo and Taiwan caused by earthquakes in the 80s and 90s showed that such precaution has no significant effect. Form pictures of the devastation, especially, in Taiwan, the softening of metallic attachment (malleable material) at foundations and cracking and pulverization of concrete (brittle material) were quite evident. The remedy is non-existent yet: alloy that withstands the softening effect of seismic waves and composite resistant to cracking and pulverization by seismic waves.

4.3.4 Volcanic Activity In the interfaces of compressed lava, laminas or slabs moving unevenly along the Earths crust crevices, compression and grinding also occur that generate seismic waves. They convert superstrings to prima that form atoms and molecules that produce earthlights and balls of fire (earthlights are high up in the atmosphere, in the mesosphere, while balls of fire hover just above ground or ocean surface). Their physical characteristics shed light on accumulation of lava long before they reach ground level. They can be used for calculating power of impending eruption and predicting its occurrence. Like visible components of shock waves from seismic activity they are detected as high-frequency waves by seismographs. In fact, huge movement of lava causes slight tremors also that may generate seismic waves around the crevices due to the motion of huge rocks and crust. The fractal structure of seismic wave is reflected in the structure of the lava outflow. Some geologists study it to get underground information. Technology can be generated to measure and monitor lava accumulation and movement from tectonic plate boundary through the Earths surface and gather in-

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formation revealed by lava. For instance, most living organisms come from volcanic lava. That is why the volcanic Galapagos Islands off the western Coast of South America have the most complete collection of animal species. Darwin spent much time there in his study of the evolution of species. Perhaps, the volcanic Hawaiian Islands come second to Galapagos in diversity of animal species. Ocean species also abound in the boiling waters of the deep around under-ocean volcanoes. (This information tells us that ingredients of living organisms also originate in the interior of the Earth). Seismic waves (fractal seismic waves from macro to dark frequency) from volcanic activity are generated in three ways (1) grinding of lava slabs under extreme temperature, (2) motion of boulders that give way to lava flow and accumulation that induce compression, lateral tension and grinding at faults nearby and (3) effect of lava flow as it passes across plate boundaries and crevices. Most lava flow out of constructive boundary (constructive because the lava outflow pile up mountains on both sides of the boundary); and it induces separation and, therefore, intensifies compression at the opposite subductive boundary of the plate where earthquake occurs. Thus, there is a close link at different levels between volcanic and seismic activity. Its impact on visible matter is verified in the Philippines by earthlights and vigorous cloud motion over seemingly dormant volcanoes. Moreover, the mysterious under-ocean bright lights in the Southern Philippines a few years ago may have link to the strong tremors that followed (they could have been under-ocean earthlights). Areas of intense volcanic activity lie along constructive and subductive plate boundaries; constructive because they are open and allow lava outflow that deposit huge piles of lava. The Pacific Ring of Fire (string of volcanoes) along the Pacific Rim consisting of constructive, subductive and conservative plate boundaries generates intense volcanic activity. They have direct link with el nio and, ultimately, cyclones. There is minimal volcanic activity at conservative boundary, only when there is subduction. Along constructive plate boundaries massive outflow of lava heats up the ocean surface to form el nio but also accumulate along plate margins that materially affects atmospheric behavior and volcanic activity.

4.3.5 Atmospheric Turbulence We first remark that vortex fluxes, visible or dark, is subject to the same two laws of nature, flux compatibility and flux-low-pressure complementarity. The reason is: pressure is a collective phenomenon and independent of the individual properties of structures of the physical systems that form the flux. Thus, the spin of typhoon or tornado is the same as the spin of its dark component and subject to the same laws that govern the behavior of water eddy. In the northern hemisphere, for instance, the lag in the Earths gravitational flux from the Equator to the North Pole causes the dark component of the typhoon or tornado, as dark eddy, to spin counterclockwise. This is the reason for of the counterclockwise spin or typhoon or tornado in the Northern hemisphere. Typhoons form over shallow constructive plate boundaries along the Pacific Ring of Fire, cracks in oceanic plates that form under-ocean volcanoes and hot spots. The Pacific Ring of Fire has branches consisting of plate boundaries most of which are constructive. Constructive plate boundaries too deep in the ocean such as the Marianas Trench and the Philippine Trench in the bottom of the Philippine Deep do not contribute to the formation of typhoons but influence their paths. The el ni˜o it forms are not contiguous but broken so that they give rise not to coherent vortex flux of air but

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incoherent and chaotic air wind movements that hamper the normal evaporation of water that cause rainfall. Instead of typhoon they cause drought in the region. The region east of the Marianas Trench all the way through the Central Pacific and on to the Coast of Central America down to the Peruvian Coast is studded with volcanoes and, in the Central Pacific and around Galapagos Islands, also hot spots. Typhoons originate here. The region supplies contiguous el ni˜os that create and power typhoons. First, there is calm over these regions, say, in the Northern Hemisphere, the initial phase of the standard dynamics. Then huge solid el ni˜o pocket forms, sometimes as broad as the Canadian landscape and as thick as 650 meters deep. This warms up the air and raises kinetic energy of the air molecules, pushing each other away to thin out the lower atmosphere, induce low pressure and cause air to rush out through thin air above and form atmospheric depression. By flux-low-pressure complementarity, depression sucks denser air around and causes initially chaotic inward rush of air due to collision; this is the second phase or transitional phase of chaos. Chaos is energy dissipating and, by energy conservation, stabilizes into coherent flux called turbulence, specifically, a vortex called cyclone (typhoon or hurricane), third phase of the standard dynamics. As we have seen earlier a vortex spins counterclockwise in the North; this is due to the counterclockwise twitch caused by gravitational flux lag towards the North Pole that resonates with the dark component of the rising column of air (exactly like the way water eddy is formed when there is uneven water flow or current; the tangential component of the gravitational flux is most intense at the Equator and goes to zero at the Poles). This results in helical streamlines (paths of air molecules) from the ocean surface through the rising column around the vertical cylindrical eye with micro local oscillation as local component of turbulence. As it leaves the eye the helical path is thrust outward by centrifugal force but its vertical profile reaches its peak along parabolic arc and goes downward also along parabolic arc, in accordance with Newtons mathematical model of an object thrown into the atmosphere subject to the pull of gravity, with vertex somewhere midway between the boundary of the eye and the rim. Then the helical path reaches the bottom and winds back to (and is sucked by) the eye, winds around upward again and the cycle repeats, etc. The vortex is donut-shaped but flattens at the bottom as it grates the ocean. The vortex is covered by these helical streamlines; its top having parabolic profile but almost flat due to centrifugal force on the molecules imparted by vortex spin. The vortex is almost solid; almost, because there is a secondary horizontal circular cylindrical eye at the middle of the donut along its axis and around the main vertical eye. This secondary eye is verified during typhoon by the circular motion of tree branches as this eye passes by. Thus, a typhoon is donut-shaped vortex with two eyes the main one being the hole of the donut and the other called axial eye a hollow cylindrical eye along the donuts axis. Here, the typhoon is an eddy in air current and its dark component an eddy in the Earths gravitational flux. As long as there is upward flux the counterclockwise twitch sustains the spin and centrifugal force and momentum imparted on the helical flow of molecules. This initially increases the typhoons power until it reaches its peak. Then the vortex begins to ease the depression and erode its own power. Then the depression fades out as soon as the vortex resolves the pressure gradient with the surrounding atmosphere through interpenetration. The eye minimizes collision and friction, i.e., dissipation of energy. Eye formation is forced by centrifugal force of spiraling molecules that push the other molecules outward. Thus, the greater the spin, the bigger the eye is. When typhoons main eye is plagued off by mountain along its path the cyclone loses steam. This explains weakened state of typhoon over Manila that passed first over the mountain ranges Banahaw east of the city. Only a few times when a typhoon detours around the mountain ranges of Quezon and Laguna Provinces into the sea then turns northward to Manila hits the city hard. The typhoon has the effect of leveling up depression and weakening its own power. As it weakens, centrifugal force on spinning air molecules declines, the main eye shrinks and the typhoon

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reduces its own size and fades away; or, the vortex may simply break and become incoherent wind movements followed by calm. There is substantial verification of typhoon dynamics aside from spin. Rural folks know when typhoon approaches and distinguishes it from ordinary monsoon wind: in an approaching typhoon broken tree branches shoot downwards and pierce the ground because that is the direction of the streamlines; in ordinary monsoon wind branches just fly off. They also distinguish the calm when the typhoons eye is passing by from the calm when it is over as the wind direction reverses after the eye has gone by; they know where the path of the eye is relative to their location by the manner by which wind direction shifts. In the western Pacific typhoons normally follow the contours of the Pacific Wind Cycle. However, frequent heating up over the ocean surface above the Trench at the bottom of the Philippine Deep that yields broken el ni˜o, while not sufficient to form tropical depression, combined with similar temperature variation around Mayon Volcano affects the path of an approaching typhoon: it tends towards hot corridor where atmospheric pressure is less, pulling southward instead of following a contour of the Pacific Wind Cycle. Then its path can be predicted by monitoring departure of its path from normal course along the Pacific Wind Cycle as it crosses into the Philippines area of responsibility west of the Marianas. It is theoretically possible to reroute a typhoon but that would be impractical. It is more advantageous to cope with it by utilizing its energy, e.g., floods. The flood waters can be stored and recycled for purposes of hydroelectric generation, irrigation and treatment for potable water. [111] provides data on the relationship between under-ocean activity and hot spots and el nio and, therefore, topical depression, cyclones and tornadoes. Using this theory we can now explain the difference in impact of el ni˜o between the Eastern, Central and Near Western Pacific regions (up to the Marianas Trench), on the one hand, and Western and Southwestern Pacific, on the other. Volcanic activity abounds in Central Pacific from the Hawaiian Island group through Marshall Islands and the Marianas where most tropical depression forms. There is only moderate volcanic activity along trenches around the Philippine Plate, having mainly destructive boundary there; the 15% conservative plate boundary there hardly contributes to volcanic activity. Moderate volcanic activity along the trench in the Philippine Deep does not produce solid el nio but only broken little pieces that induce chaotic rush of air molecules that stabilize as local incoherent fluxes and interfere with normal evaporation of water but does not stabilize into cyclones. They do not sufficiently heat up and moisten air above the ocean. This explains frequent drought in this region. However, they reduce pressure sufficiently to extend the typhoon corridor toward the China Sea. This theory provides cheap and accurate means of predicting a typhoons course without radars that get knocked off by advanced strong winds anyway. On the other face of Earth are numerous under-ocean volcanoes in the Atlantic just East of the US and North of Cuba and across the Caribbean aside from the constructive plate boundary across the Equator from North to South between the Continental Americas and Europe generators of devastating hurricanes in that region. There is also constructive plate boundary just west of the US from Oregon through Washington State and Canada and another one west of Central and South America, aside from the group of volcanoes and hot spots around Galapagos Islands that extend far towards Central Pacific; they are also generators of huge el nio and devastating hurricanes

4.3.6 Tornado Tornado occurs at sharp interface of hot and cold air pressing against each other. In the US the Tornado Belt where most tornadoes hit is the 1000- by 650-km corridor between Oklahoma and

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Florida and between the Midwest and Texas. This is where the Pacific and Atlantic Wind Cycles interface and press against each other, by flux compatibility, the former bringing cold air from the North, by flux-low-pressure complementarity, the latter warm air from the Equator. Since the Pacific Wind Cycle, which is roughly elliptical, is much wider than the Atlantic Cycle, the interface over the Tornado Belt goes from North to South but slightly tilted towards the East in the North. It oscillates between Oklahoma and Florida. The cold air from the North merges with the Pacific Wind Cycle in this interface. The largest temperature gap occurs in Spring and early Summer, but the gap is greatest in early Summer when strong tornadoes occur. The strongest known tornadoes in recent years occurred in Oklahoma in 1995 where the spin close to the eye was 510 km/hr (intensity F-5). The whole funnel was 1,000 km thick and 1.5 km long (upward) and at that height the funnel spreads outward and tapers like an umbrella. The eye itself was 90 meters wide with five little funnels spinning inside. By the time the tornado broke off it cleaned up a 6-km long strip of land 1 km wide going northward. Another tornado of the same power and characteristics occurred in Texas in 1998. We consider tornado formation in the Tornado Belt. At the interface, warm air from the Equator pushes against the cold air from the North and goes up, being lighter, while the denser cold air goes down. This action creates a little vortex with horizontal axis going in the general direction from South to North. The gravitational flux twitch combined with the greater force of suction on denser air tilts the north end of the vortex downwards. Since the denser air comes from the bottom end of the vertical eye it pulls the vortex down, by Newtons action-reaction law, and swoops towards and touches down on and grips the ground by its suction. At this time we have a full-blown tornado with a counterclockwise spin of its funnel. The suction or grip is firm and stable on flat ground but on rugged surface or incline it is jumpy, does not stabilize, lifts off and breaks up, the reason powerful tornadoes are only stable on the plains. Forested areas also weaken its power. The counterclockwise spinning funnel is like a wheel. Since the denser air offers more friction than lighter air the spinning funnel is pushed northward with easterly tilt greater in the northern section of the Belt due to the thicker Pacific Wind Cycle. Thus, a tornado in this Belt travels northward (with easterly tilt) at about 30/kph. As the tornado moves forward several things happen: (a) it flattens structures and grates the ground; (b) in the interior of the funnel the debris is sucked by the eye, pushed up and scattered outward by as much as 210 km, farthest thrown being light material like paper and card; (c) near the edge of the funnel on the ground the micro component of turbulence scoops up material, e.g., log, post, lumber from debris, and, by the combination of centrifugal force and forward push of the funnel, throws them as missiles (this is called torpedo effect) at structures along its forward path and weakens them before the funnel flattens them. The funnel remains strong as long as there is suction and sufficient temperature gradient between the two interfaces. Tornado touchdown is generally of short duration due to interpenetration that wipes out the sharp interface of warm and cold air. Its duration ranges from split second to a couple of hours. The tornado of longest known duration lasted for over a couple of hours and traveled in the northeasterly direction from Monroe City, Wisconsin (typical in the North end of the Belt). Sharp interface of warm and cold air can happen anywhere because of the irregular course of wind flow but that is rare. However, when it happens at sea it forms waterspout that scoops up water and dumps it elsewhere, sometimes on land. This can create a deep hole on the ground.

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Finally, this chapter has solved the turbulence problem for these four types and opens the door wide open for their control (partially for typhoon, earthquake and volcanic activity but full for tornado); a new technology (based on GUT) has been conceptualized for monitoring potential tornado occurrence and aborting, deflecting and terminating or breaking a tornado in progress to protect cities and farms; it needs collaboration between the theoretician and structural engineer and funding for patenting.

4.3.7 Lightning Lightning occurs when there is accumulation of positive ions in the lower atmosphere and the electrical potential between the ions and the ground reaches critical level. The electrical potential depends on the amount of positive ions since there is abundance of electrons on the ground due to the push by the Earths gravitational flux. When critical potential is reached, the positive ions rush down and the negative ions rush up. They meet and collide resulting in explosion. Some estimates put the energy of a bolt of lightning at one megaton of thermonuclear explosion. Since lightning is an explosion it generates and propagates seismic waves that convert dark matter to earthlights in the mesosphere, some 50 to 90 km above the Earths surface. They are familiar spectacle in the skies for airline pilots. How do positive ions accumulate in the lower atmosphere? One source is the gravitational push by the Earths gravitational flux, by flux compatibility. Another is turbulence that knocks off valence electrons of gas and turns them into positive ions. This occurs, for instance when there is forming cyclone and tornado in the clouds and even when the tornado has touched down the spinning funnel continues to ionize the air close to the ground in which case the critical electrical potential is low. Thus, a tornado in progress has its funnel wrapped in little sparks of lightning close to the ground. This also occurs, especially, during the chaotic phase in the formation of typhoon. However, lightning stops when the electrical potential has been wiped out by the release of energy.

4.4 The Physics of the Mind 4.4.1 The Mind We shall focus on thought or intelligence, which is part of the mind; the latter includes emotions centered in the heart, as the capability of the brain to engage in what we call mental activity such as cognition, learning, memory, recollection, reasoning (drawing out conclusions or making inferences), creativity and building mathematical systems and physical theories. This capability is acquired from training, experience and study. We use mind in place of thought, however, since thought cannot be separated from it. Learning and creativity allow the mind to imbibe values including critical thinking and inductive and deductive reasoning. Invention of concepts and construction of mathematical spaces are creative mental activity. Technological inventiveness and athletic skill are both mental and physical activity and the former is creative as well. However, it usually takes a creative move for an athlete to win a game and there is mental activity involved there. This is also true of theoretical research which requires creativity and those who bring in creative strategy or tactic to excel in it. Mathematical concepts and principles which are man-made and have no physical referents are either learned (acquired) from training and study or from the creative activity of the mind. Physical concepts and principles (laws of nature) both of which have physical referents, are also created by the mind but induced by experience, direct or indirect, and study. Visible matter such as atom and star are directly observable but dark matter (consisting of semi-

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and non-agitated superstrings) is only indirectly observable through its impact on visible matter. Its existence and structure were assured and provided by the discovery of the initial 10 laws of nature in 1996 [48] that anchored the flux theory of gravitation, now GUT. Mathematical concepts, e.g., set, distance, angle and speed, are also creations of the mind to express relationship between physical objects, in the case of the first three concepts, and describe some natural phenomenon by the fourth concept. While psychology describes how the mind works it requires physics to explain the underlying physical processes involved. This is our focus and our principal tool is qualitative mathematics. Aside from mental activity the brain or central nervous system, along with its extension to the secondary nervous system consisting of the spinal column and the network of nerves, controls all the normal functions of the human body. The nerves serve as communication channel or conductor of signals to and from the nervous system to other parts of the body. They transmit information from the different parts of the body to the nervous system and bring back interpretation or command from the latter. The center of thought and all conscious activity (e.g., body command to execute an athletic game) is the cortex consisting of the creative-integrative (CIR) and sensation regions. The rest of the nervous system controls all the organs and automatic functions of the human body such as reproduction, immunity, instincts and reflexes. The non-mental activity of the nervous system will be treated later.

4.4.2 The Human Brain The human brain is about 2.2 pounds in weight (double that of the chimpanzees). On top of its upper lobe and around it is the cortex, 1/4 inch thick and consisting of two billion neurons. It is convoluted which gives it effective surface area of 16 square ft where dendrite tips touch the surface. Neurons in the cortex have dendrites and axons sticking out of their nuclei. A sequence of brain waves that carries signal or information from one point to another or from a sense organ to the cortex and vice versa passes from one neuron to another by jumping over the gap between dendrites accompanied by spark, like that emitted by a radio transmitter, propagates the information and establishes connection or communication channel between them. As signal passes through, both the neurons and their dendrites and the connection between them are activated so that they let through only signals with the same characteristic vibration that encodes the information via resonance. From behind the forehead through the top of the cortex is the CIR; around it on the left, back and right of the upper lobe are the sensation regions. They serve as storage of information; sensation occurs in the CIR. A concept, physical or mathematical, is modeled in the CIR as suitably activated (vibrating) network of neural clusters. During sleep a concepts components are dispersed and encoded and stored in the appropriate sensation regions (sight, hearing, taste, smell and touch) as corresponding suitably activated neural network for storage (this occurs always but most intense during sleep at 90-minute cycle according to psychologists). By energy conservation, the concepts neural network in the CIR is de-activated as its components are dispersed to the appropriate sensation regions and activated for storage. The CIR recalls and recomposes them into the same neural network of the same concept when needed for mental activity through its ability to focus and direct cosmic waves to vibrate their stored components of their neural regions. By energy conservation, the neural network components in the sensation regions are de-agitated when recall and energy transfers to the CIR are completed. It is like liquid and gas flow to regions of low pressure. These neural dynamics of memory and recall have been modeled mathematically in [93–95]. Psychologists have considerable knowledge of the functions of the brain. For example, it is known that what the mind learns during the day is transferred to some parts of the brain and stored as

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long-term memory in due course, especially, during sleep. But, what physical processes occur in the brain during this transfer or when the mind is engaged in thought is unknown. This is what we fill in by bringing in GUT. The brain is the most active organ of the body, 1/4 of the energy the body gets from food goes to it. Most of that energy is spent in mental activity such as concentrating and directing cosmic agitation to the sensation regions in search of concept components for recall to draw conclusions or make inferences in both deductive and inductive reasoning as well as give commands to the body to execute conscious activity. All these processes involve transfer of energy from agitated to nonagitated neural regions, by energy conservation.

4.4.3 Brain Waves cosmic waves are the prime mover of our universe. Every piece of matter vibrates due to the impact of cosmic waves coming from all directions and their vibration characteristics are principally determined by their internal physical structure in accordance with the internal-external dichotomy law [47]. This is the reason every piece of matter has temperature above absolute zero, i.e., it has kinetic energy. Moreover, thought is triggered by cosmic waves and consciousness is due to the normal or residual vibration of the entire neural network in the cortex. There are three types of cosmic waves. But, for our present purposes, basic or electromagnetic waves are of interest. Basic cosmic waves are generated by the normal vibration of atomic nuclei and propagated outward in all directions through dark matter from each nucleus. Since the nucleus is fractal, the basic cosmic waves it generates are also fractal and, therefore, energetic. Photon rides on basic cosmic wave and information is encoded in it also, e.g., wiggled by neural vibration or convertor of a sense organ. Basic cosmic wave with rider or encoded with information is called type I. Thus, brain waves are type I basic cosmic waves encoded with information by neural vibration. Electromagnetic waves are energetic or high-frequency waves (kinetic energy of a wave is given by E = h f , where h is Plancks constant and f is frequency). Its fractal structure gives it tremendous dark or latent energy. Radio and TV signals are encoded on them and being energetic they penetrate great barriers. For example, one can tune on the radio inside a closed room. However, they are distorted or deflected by metal, e.g., TV antenna and steel reinforcement in concrete. They can be scattered and effectively blocked also by fabric made of randomly arranged thin pieces of ferromagnetic material. Telekinesis (bending metal by brain waves) shows the energy of brain waves but it is neither understood nor explained why only very few individuals have this capability (the explanation is in [45]).

4.4.4 Resonance An important law of nature governing wave interaction is the principle of resonance that says: maximum resonance between waves or vibrations occurs when they have the same wave characteristics but its principal determinant is frequency or wave length (one is inversely proportion to the other). Resonance regulates radio reception in terms of oscillation frequency or kilohertz. When a tuning fork of certain pitch or frequency is struck its vibration resonates with and vibrates a tuning fork of the same frequency nearby and produces sound of the same pitch. When resonance between waves or vibrations is instantaneous it is called interference or discordant resonance. This happens, for instance, in a cycle between interacting waves of different wavelengths or two waves of the same wavelength but one is shifted by a fraction of a phase. Naturally, resonance between waves reinforces each other since their energies add up just as fluxes of same direction do when their speeds have the same order of magnitude in which case their micro

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components of turbulence also have the same order of magnitude. Since the frequencies of the micro components of interfacing turbulence are proportional to their speeds there is greater resonance between them the closer their speeds are in orders of magnitude. During the disastrous final flight of the Columbia space shuttle on February 1, 2002 which was flying in the Northern hemisphere it was revealed that more insulation panels burned on the North side of the shuttle than on the left. It meant greater resonance on the North side between the micro components of turbulence of the insulation panels and the gravitational flux that slows down to zero towards the North Pole.

4.4.5 Sensation and Concepts Any sense organ has two components, receptor of visible signals and encoder on and convertor of information (signals) to brain waves for transmission to the CIR through the nerves. In some sense organs the receptor and encoder-convertor are one, in others separate. Consider the organ of taste, the tongue; its receptor consists of separate groups of taste buds (for sweet, sour, bitter and salty) along its edge differentiated by their visible vibration characteristics that make them resonate with visible waves generated by the food molecules of corresponding tastes. The most accurate medium for molecular vibration is liquid, saliva in this case. Take sugar molecules. They must be soaked in saliva so that their molecular vibration is transmitted accurately and the taste buds for sweetness which are attached to the nerves at their base resonate with and are vibrated by it. Their nuclear vibration wiggles the basic cosmic waves they generate and superpose or encodes the vibration characteristics for sweetness on them and turns them into brain waves which are transmitted to the CIR through bundles of linear chains of nerve cells. Each unit characteristic resonates with a component of the taste buds characteristics and is encoded by its vibration on the basic cosmic waves its nuclei generates and the more intense a characteristic the longer the chain of brain waves that carry it (by energy conservation). A change in characteristic corresponds to a branch of that chain with new characteristic so that the chain of the same characteristics goes through the cortex along a single neural chain, starts a nodal (separation) point in the CIR and vibrates and activates a neural network issuing from the same nodal point, each characteristic encoded in separate branch, i.e., the different characteristics split at the nodal point. When the signals are simply repeated no new neural activation or connection occurs, the encoded neural network simply continues to vibrate. Signals from the other taste buds will activate their respective neural network and, altogether, their nodal points are interconnected to form a single nodal region. As long as this neural network is being vibrated by the same signals from the taste buds the same sensation continuous. The characteristics in this case come from the different flavors of the sugar sample. For instance, chocolate candy has its specific flavor with its component characteristics. Of course, an event, e.g., eating candy, has different components of taste, smell, texture, shape and color each of which encoded in the CIR and their nodal regions are interconnected. When the signals stop the entire network encoded by the event ceases vibration and at some point the CIR sends the various components to the respective sensation regions for storage by activating corresponding neural network there with the same characteristics. When it is over the encoded neural network reverts back to its normal vibration. The concept formed by the event is now modeled by its neural network joined together by a common neural region. When the CIR needs that concept for mental activity it vibrates its nodal region and recalls and recomposes the components around it. When the same event is repeated it reactivates the same neural network. This understanding of taste sensation has provided some useful technology for the control of functional disorder like diabetes. Insulin is needed by the body to bring sugar into the cell for nourishment. When pancreatic secretion of insulin is inadequate, a condition suffered by diabetics, blood

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sugar level rises. Sugar is viscous and high blood viscosity can damage the fine blood vessels of the kidney and the eye, raise blood pressure and cause other complications. Since taste is due to the physical structure of the food molecule that determines its vibration characteristics, why not simulate it by using non-viscose materials? In fact, this is what synthetic foods are. Synthetic sugar “Equal” consists of suitably designed carbon molecules with the vibration characteristics of sugar molecules; it is 550% sweeter than ordinary sugar of the same concentration but without the viscosity that harms diabetics. We now have a number of synthetic foods such as “beef” and crab “meat”, from organic materials like vegetables. They have the taste but not the bad cholesterol. However, there is another factor that maintains intensity of signals from the sense organs. It is well-known that in the membrane of the neural cell, ionization of sodium atom creates electrical potential followed by its jump into lower potential accompanied by electrical impulse. What causes this ionization? It occurs when an outer orbital electron is expelled from the atom. The only force that does it comes from energetic brain wave passing through. The sodium jump vibrates the brain waves at visible frequency so that it does not distort the encoded message (similar to the effect of the radio signal oscillator at the station), it only maintains the energy of the brain waves and, hence, intensity of signals when they reach the cortex. The entire neural network activated by the event reverts back to its normal vibration as soon as it is over and the concept induced by the event is now encoded. During the time of activation and encoding there is sensation while, at the same time, the dendrite tips spark and propagate brain waves across dark matter. Energy conservation requires that when an encoded neural network in the CIR is deactivated its components are transferred to and activate and encode similar neural network with their characteristics in the corresponding sensation regions. As soon as the transfer is completed the network reverts back to normal vibration and the components are now stored as long-term memory. This transfer usually occurs during sleep. When a nodal region is hit by suitable basic cosmic waves directly from the Cosmos or indirectly through the CIRs ability to direct and focus cosmic waves, i.e., by vibrating its neural region, the encoded neural network there vibrates with exactly the same characteristics as when it was activated and encoded and if the components are in the sensation regions they are recalled to the CIR and recomposed at their nodal region. This is consistent with the psychologists prescription for recalling non-physical concepts by association. For instance, the concept of time is mathematical and manmade, the result of the CIR’s creative ability. It is not an event in the real world and is not associated with sensation; nor does it have physical referent. Therefore, we can determine specific time only in association with contiguous events there that have interconnected neural regions. It means activating their neural network so that they get recalled and recomposed at their interconnected neural regions. This also works for recalling contiguous physical concepts because vibration of one nodal region vibrates others connected to it. For instance, seeing an old friend whose name one forgets recalls the name (not all the time, the brain goofs sometimes). How does one recall the year when a song became a hit? This requires association with the right contiguous events that recalls physical concepts such as ones place of work, friends and associates, etc., at the time. Their recollection (activation of their neural network) fixes the year of the event. An event may have several components. For example, a fire cracker explosion has light, sound, smell and touch components. After an event its components are sent out to their respective sensation regions in due course automatically, by energy conservation, but available for recall by the CIR when needed for recomposition as concept. When the nodal region of an encoded neural network is activated directly by cosmic waves the recomposition and recollection is spontaneous and simultaneous with sparks and propagation of cosmic waves at dendrite tips. When concepts are recalled, the dendrite tips of their neural network

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also spark and send out brain waves with vibration characteristics corresponding to the information they are encoded with. In general, since mental activity involves passage of vibration characteristics, it lights up dendrite tips and propagates brain waves. Therefore, recall of concept components to their nodal point in the CIR or dispersal of concept components to their respective sensation regions for storage involves passage of energy from activated to non-activated network and quite analogous to passage of gas or liquid from high to low pressure. While generation and propagation of brain waves are well understood and explained by the laws of nature reception is not fully understood. One such unexplained phenomenon quite well-known in parapsychology is extra-sensory perception (ESP) where the individual receives information from external source without involvement of any of the sense organs. There is no known scientific confirmation of this phenomenon in humans but there is ample confirmation by scientific studies on dogs. For instance, a dog knows when the master is within 10 km from home and starts getting excited waiting at the door or may even go out to meet him.

Recall that in the sensation of taste the taste bud is both receptor and converter- encoder. This is not so with hearing. Its receptor of visible or ordinary sound waves is the eardrum. The outer ear catches and directs the sound waves through the outer canal to the eardrum which is vibrated by resonance. On the other side of the eardrum in the inner ear is the cochlea, a spiral canal filled with liquid. Near the eardrum immersed in the fluid are the three tiniest bones in the human body, malleus (hammer), incus (anvil) and stapes (stirrup), in triangular arrangement that detects the direction the sound comes from.

Lining the interior of the cochlea are strands of hair separated according to vibration characteristics. They are the converter- encoder of ordinary sound waves to brain waves which are then transmitted to the CIR for composition as concepts. Sound waves from the outside vibrate the eardrum. The handle of the malleus is attached to the eardrum. As the eardrum vibrates, the hammer hits and vibrates the incus. As the incus vibrates it vibrates the stapes that, in turn, stirs the cochleal fluid and amplifies and propagates the vibration and the latter resonates with and vibrates the corresponding strands of hair that line the inner wall of the cochlea. Each strand resonates with specific vibration characteristics. Their atomic nuclei encode the vibration characteristics on the basic cosmic waves (now, brain waves) that carry and transmit the information to the CIR that converts it to suitable neural network. Thus, the hair strands are the converter- encoder in this case. In due course the signal components are encoded and stored in the appropriate sensation regions. Energy conservation insures that when a concepts components are stored in the sensation regions its neural network in the CIR deactivates and restores normal vibration and vice versa. (See [50] on the functions of the five sense organs) There is one more missing link on sensation: how does the feeble encoded vibration of the receptors get transmitted to the CIR and corresponding sensation regions? The neural bundle from a sense organ through the corresponding sensation region has the same vibration frequency; the CIR has all the frequencies of the neural bundles from the sense organs. Therefore, the neural network in both the CIR and the sensation regions resonate with and are composed by brain waves from the corresponding sense organs by resonance. As brain waves (encoded with signals from a sense organ) pass through the nerve bundles they ionize and raise concentration of positive ions, e.g., sodium ions, in neural membrane creating electrical potential between membrane and cell interior. This is resolved by sodium jump that vibrates the cells and produces electrical impulses at visible frequency so that they do not distort the encoded information they only energize the passing brain waves.

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4.4.6 Value, Perception and Cognition The CIR can focus and direct cosmic waves to the sensation regions through the nodal regions and activate the neural network there encoded with concept components that, in turn, are recalled and recomposed to form appropriate concepts at their respective nodal regions. According to psychology, ones perception of an event comes mainly from memory, only 20% from the event itself in the external world. Psychologists use this knowledge to assess ones personality by asking the subject to tell what he perceives from a set of objects. It reveals the contours of his lifes experience that shapes his personality. Values and ones way of drawing out conclusions or logic, which we include among values, are learned in accordance with Pavlovs theory of learning or conditioning. Values consist of a system of “correct” choices, decisions, conclusions, etc., corresponding to the right neural connections to their respective concepts in the CIR. They are formed through training, peer pressure, experience and study. In this sense, ones logic or way of thinking belong to the category of values. Correct choices, etc., are imbibed by approval and incorrect ones by rejection first by the parents, later by the teachers, peers and experience. More values may be learned from experience and training. Correct choices, decisions, etc., are encoded as neural connections to their premises or boundary conditions (modeled by neural network) in the CIR. Incorrect choices do not establish neural connection. A discipline of knowledge (mathematical space or physical or social theory) consists of a system of concepts and values (including logic) encoded as suitable neural network whose nodal regions are connected to the central neural region of the discipline in the CIR at which its axioms or natural laws (of a physical theory) or social principles (of social science) are recomposed. Different disciplines of knowledge may be autonomous but connected through their central nodal regions so that the person can switch focus from one system to another. In rare cases distinct and disconnected systems of values are involved and the individual suffers from multipersonality where one “may not know” the others. A normal individual may have several connected systems of knowledge including practical knowledge that allows him to function day-to-day. This is the case, for instance, with an expert in different fields. Information from any discipline may be recalled, recomposed at their neural regions and combined with new information to form new knowledge. The CIR takes an active role here because it may happen that the various disciplines involved may have different systems of values. It composes them into knew knowledge and system of values. The fact that the individual can connect crisscrossing information from different disciplines suggests that the nodal regions are not just linearly connected but fully integrated, i.e., every nodal region in one discipline is connected to all the nodal regions of the others. However, for an individual to have full control of his knowledge there has to be top-central nodal region for the totality of individual knowledge. To avoid the mischief of language (e.g., self reference) there must be pyramidal hierarchy of knowledge so that nodal region of a discipline of knowledge should be the central nodal region of all its branches. For example since quantum physics is a branch of physics its central nodal region must be a nodal region of physics. Ones knowledge is in the mind, its representation as language is in the visible world. Scientific knowledge expressed in suitable language is built, stored in databases and applied as physical theory anchored on some laws of nature. That is the relationship between thought and knowledge. The same thing happens in other endeavor. During, say, a game of bowling the player models the lane in the CIR (at the central nodal region of this game) by appropriate neural network and commands the body to move accordingly to play the game on the basis of his perception of the lane, knowledge of the game and previous experience. A skillful bowler accurately models the lane on the basis of which the CIR commands execution of the game based on accumulated experience and knowledge of the game.

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In a game or any physical endeavor it is thought that gives the command through brain waves that activate the appropriate parts of the body to execute it. Sometimes, thought triggers automatic execution. For instance, when one thinks about lemons the energetic brain waves it generates convert the superstrings in the salivary gland to salivate and this involves dark-to-visible matter conversion [45]. Now, thought can move the PC curser and even operate the motor that moves prosthetic arm. In theoretical research the mind utilizes observation and experimental data, training, social or physical theory, if any, and mathematical tools to discover and articulate (create) some laws of nature upon which to build a physical theory. Although the laws of nature refer to physical concepts their statements are mathematical principles because they have no physical referents, i.e., one does not find them in the visible world. They are created by the mind based on synthesis or analysis of known information and experimental data which, at the same time, verify them and the theory built on them. For the other function of the brain, as control center of all processes in the body, stress dulls its ability to keep the auto-immune system in top shape that it is unable to do its normal function of destroying unwanted systems in the body. Ordinarily, the body has normal level of cancer cells (due to background radiation, chemical irritation and natural mutation) and the auto-immune system adequately destroys and maintains them at tolerable level. However, under stress, it may fail and be overcome by the replication of cancer cells; then the person becomes sick. That is why healthy body and mental health go together. In some cases extreme stress or emotional trauma may cause physiological imbalance such as raised level of brain cell regulators, e.g., serotonin. This is what happens in depression where the neural network vibrates even when they are not supposed to and the individual hears voices or sees things (audio or visual hallucination) which are non-existent or makes unusual decision, e.g., attempting suicide. Such imbalance may be due to genetics or emotional trauma. In general, strong emotion is accompanied by physiological changes, e.g., raised adrenalin secretion induced by anger. Pain sensation may also cause physiological changes, e.g., extreme pain produces molecule in the neural membrane [78, 87]. When the cortex does not have the ability to direct or focus cosmic waves or concentrate on a particular neural activity it is a mental condition called autism. In one form the person is simultaneously aware of everything around him and cannot respond in an orderly manner. Naturally, he has short attention span since other sensations are competing for attention; then he becomes hyperactive. This condition can now be controlled by therapy.

Chapter 5

Hybrid Unified Theory

5.1 Introduction If we assume nothing, no theory can be developed. Therefore, we accept the Vedic declaration “That which exists is ONE, scientists call it by various names.” Ancient scientists, known as Rishis, described it using thousands of names. We shall be content in utilizing the following: namely, the Impersonal Being, the Absolute Reality, That ONE, and the Selftron, and show that our assumption is not vacuous. That part of Veda Samhitas known as Vedanta is the gist of the Vedas and contained in Upanishads which are the final parts of the Vedas. Concerning our visible universe, the Isopanishad proclaims the following: “The invisible (the Impersonal Being) is infinite; the visible universe too is infinite. From the invisible Impersonal Being, the visible universe of infinite extension has manifested. The infinite Impersonal Being remains the same, even though the infinite universe has manifested out of it.” It should be stressed that this does not imply that the visible universe and Impersonal Being have become separate entities, rather the universe is recognizable, yet it is still connected or a part of the Impersonal Being. It is naturally harder for us, in general, to understand the idea of Impersonal, for we are always clinging to the Personal. Impersonality includes all personalities, is the sum total of everything in the universe, and infinitely more besides. Let us mention that the true knowledge is defined by the ancient scientists as that which remains unchanged in the past, present and the future. We know that an attempt to explain all concepts can hardly be called scientific. Some concepts require the acceptance without explanation. Whatever we the scientists say is expressed in terms of limited and approximate descriptions, which are improved in successive steps. We progress from truth to truth, from lesser truth to higher truth, but never from error to truth. We view the truth, get as much of it as the circumstances permit, color it with our own feelings, understand it with our intellect, and grasp with our own minds. This makes the difference between human beings and sometimes generates contradictory ideas. Nonetheless, we all belong to the same universal truth. New knowledge comes from inspiration and revelation and not from just rational thought. Different scientists describe their creative ability by various equivalent means using different language. Newton and Ramanujan, for example, credit their creative ability to the divine intervention. What we call the most correct, systematic, mathematical language of the present time, and the hazy, mystical, mythological language of the ancients differs in clarity. Clearly, the two approaches of thought, namely, ancient and modern, have their roots in different parts of human culture and tradition. V. Lakshmikantham et al., The Hybrid Grand Unified Theory, Atlantis Studies in Mathematics for Engineering and Science 3, DOI 10.1007/978-94-91216-23-7_5, © 2009 Atlantis Press/World Scientific

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It is important to realize that the root sounds of the mantric language employed in the Vedas cannot be put into words in any final manner. As an example, consider one of the words which we utilize later, namely, Agni, which is fire in the sense we normally employ. Agni is also translated as the God of Fire which is a personification of all fire. Agni is the energy of transformation that is essentially the energy of pure consciousness. Agni is whatever burns, penetrates, perceives, labors, creates, envisions, wills, aspires and ascends with force. Agni is symbolically a fire, for example, the way we speak of the fire of genius. Agni is fire but not simply a fire, even as seen by a fertile but primitive imagination. As the principle of light and energy in general, Agni is the sun by day, the moon by night, the fire in home, the stars in the sky, and the wind in the atmosphere or lightening in the clouds. Whatever manifests as power, strength and spirit is Agni. Thus Agni represents heroic man, the swift stallion, the strong bull, or the soaring eagle. In terms of our faculties, Agni is wakeful awareness and mindfulness, the will to truth, consciousness, the state of seeing, the mind, the intelligence, listening, chanting and the voice. In short, Agni is whatever in us that manifest light, power and effulgence of the Impersonal Being. To render the word Agni only as fire is like calling the Milky Way, not a galaxy but a stream of milk [82]. There were many different schools of thought in ancient times as in modern times. Also, several ancient scientists were satisfied with an approximate understanding of Nature, (Prakriti in Sanskrit), described selected groups of pheonomena, neglecting other phenomena, which in their opinion, are less important. The theory of the school of Mimamsakas, for example, concludes that the very performance of an act carries its result as action is always followed by reaction. The universe has been in existence in time, without a beginning. There was no time at which it did not exist. Hence, do not worry about who created it. There is no necessity to give credit to the creator or God [82]. Like modern scientist, ancient scientist’s solution of the mystery of the Universe, from the analysis of the external world was as satisfactory as it could be. There have been attempts to catch a glimpse of the beyond, through which one can understand the Imperson Being, by studying the external world. This has been a failure because the more we study the material world, even a little true knowledge we possess vanishes. The greatest of the ancient scientists are known as Rishis. According to Max Planck, “Rishis were human beings in whom Nature’s Grace Absolute has chased away the eternal darkness of the mind, destroyed the natural perversity of the intellect, and driven away the last vestige of egoism and falsehood” [92]. They realized that the knowledge to the highest or beyond must come only through internal search. They, therefore, were concerned with absolute knowledge involving an understanding of the totality of existence. They were interested, as Mundaka Upanishad questions, “What is it, which having been known, everything else becomes known?” They first perceived and realized, and then written down in Upanishads [123]. We must remember that until the present interpretation of the new physics even the word transcendence, for example, was seldom mentioned in the vocabulary of physics. The term was considered heretical. In modern physics, the universe is experienced as a dynamic inseparable whole, which always includes the observer in an essential manner. In this experience, the traditional concepts of time and space of isolated objects, and of cause and effect, loose their meaning. Such an experience is very similar to that of ancient scientists. The only difference being they repeatedly insist that the ultimate reality can never be an object of reasoning or demonstrable knowledge. This is exactly what our scientist Max Planck observed [92], namely, “science cannot solve the ultimate mystery of Nature, and that is because in the last analysis, we ourselves are part of Nature and therefore, part of the mystery we are trying to solve.” In view of the ideas of modern physics and the recent fruitful research in several disciplines, it appears that the time is almost ripe to have at least a hybrid interaction between the theories of modern

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and ancient scientists. We feel that this hybrid interplay discussed in this paper and in [59, 80, 83] has shed some light on the existing theories and improved the approximations. Fortunately, the theories of ancient scientists have begun to interest a significant number of people and the terms yoga, meditation, reincarnation, consciousness and karma, no longer are viewed with ridicule or suspicion, even within the scientific community. For example, David Bohm asserts “mind and matter are mutually enfolding projections of a higher reality, which is neither matter nor consciousness” [8].

5.2 Mind and Consciousness The mind is limited in the sense that it cannot go beyond certain limits, of time, space and causation. The mind is intimately connected with the brain, which changes every time the body changes. The body is not the real human being nor the mind, for the mind waxes and wanes. The body and the mind are continuously changing and are, in fact, only names of a series of changing phenomena, like the rivers whose waters are in a constant state of flux, yet present the appearance of unbroken streams. With every sense we have, there is first external instrument in the physical body and behind that there is the organ, and yet these are not sufficient. When the mind detaches itself from the organ, the organ may bring any news to it, but the mind will not receive it. When it attaches itself to the organ, then alone it is possible for the mind to receive the news. Even that does not explain the perception. One more factor is essential. There must be a reaction within and with this reaction comes the knowledge. The state of mind that reacts is known as intellect, called Buddhi in Sanskrit. But this does not complete the process either and will not be completed unless there is something permanent in the background, on which all the different impressions are projected. This something on which the mind is painting all these pictures, this something on which our sensations carried by the mind and intellect, are placed, grouped and formed into a unity, is what is known as Selftron, the Atma in Sanskrit [80, 121]. When we move a finger, that apparently simple motion is the culmination of millions of unutterably complex chemical and electrical interactions, occurring within milliseconds in a neatly ordered sequence in our brain. The entire process of moving a finger is called the firing mechanism which starts in a region of the brain called supplementary motor area. If the subject of the experiment did not actually move his finger at all, but merely thought about moving it, detectors in the experiment indicated that his supplementary motor area was firing, although the motor cortex of the brain, which controls the movement of the muscles themselves was not. Hence, Sir John Eccles, the Nobel Laureate [15], triumphantly said that the supplementary motor area is fired by intention. The mind is working on the brain and thought does cause brain cells to fire. The physiology of movement proves that we have freedom of will, that something outside of a purely mechanical process is involved in our actions. We have the mental ability to decide to act. If we do it at this elementary level, moving a finger, it follows that we can do it on more complex levels of human action and interaction. The external instruments are in the gross physical body of human being but the mind and intellect are not. They are in the fine body, or the subtle body, known as Sukshma Sarira in Sanskrit, made up of fine particles. Beyond that is the Selftron, that is behind the mind. We know that there is something behind the mind because knowledge is self-illuminating and the basis of intelligence cannot belong to dull, dead matter. It is the intelligence that illuminates matter. Neither the body, not the mind, not even the fine body is self-luminous. That which is self-luminous cannot decay. The luminosity of that which shines through the borrowed light comes and goes. The moon, for example, waxes and wanes since it shines through the borrowed light of the sun. Thus we see that all the powers of the gross

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body are borrowed from the mind, and the mind, the fine body, borrows its powers and luminosity from the Selftron. In reality, there is only one existence designated as the Impersonal Being, the Absolute Reality. Through the cosmic illusion known as Maya, the apparent multiplicity is created and the Impersonal Being appears as the individual Selftron and the Visible Universe. Maya is not a theory for the explanation of the Visible Universe but it is simply a statement of facts, as they exist. If we think that the shapes and structures, things and events, around us are realities of nature, instead of realizing that they are concepts of our measuring and categorizing minds, then this point of view of ours is known as Maya. It is the cosmic illusion of taking these concepts for reality, of confusing the map with the territory. As long as we confuse the myriad forms of the creative activity of the Impersonal Being, with reality, without perceiving the unity of the Impersonal Being underlying all these forms, we are under the spell of Maya [80, 121]. All objects are fictions, chimers of the mind. It is our left-brain hemisphere that tricks us into seeing sheep, trees, human beings and all the rest of our neatly compartmentalized world. We seek out stability with our reasoning faculty, and ignore flux. Through this classifying and simplifying approach we make sections through the stream of change, and we call these sections things. And yet a sheep is not a sheep. It is a temporary aggregation of subatomic particles which were once scattered across an interstellar cloud, and each of which remains within the process that is the sheep for only a brief period of time. That is actual, irrefutable case. We slip so easily into the habit of assuming what we see and feel in our minds is what is actually going on outside ourselves, beyond the portal of the senses. Although we experience the world as a series of sensory objects, what actually comes to our senses is energy in the form of vibrations of different frequencies: very low frequencies for hearing and touch, higher frequencies for warmth, and still higher for vision. The radiations we pick up trigger neural codes that are made by the brain into a model of external world. Then this model is given subjective value and, by the trick of the brain, projected outward to form the subjective world. That inner experience is what we habitually equate with external objectivity. But it is not objective. All of the perceived reality is a fiction [12]. Nature (Prakriti) is eight fold, namely earth (Bhumi), water (Aapa), fire (Agni), air (Vayu), space (Akasa), mind (Manas), intellect ( Buddhi) and ego (Ahankara). The first five elements represent in the microcosm, the five sense organs by which the individual comes to experience and live in the world of sense objects. This list is nothing but the fine body and its vehicles of expression, namely, sense organs. These are the channels through which the world of stimuli reaches within. The inner point of focus of the five sense organs is called the mind. The impulses received by the mind are rationally classified and systematized into the knowledge of their perception by the intellect. In all these three levels, namely, sense reception, mental perception and intellectual assimilation, there is a continuous sense of I-ness, which is called the ego. This is the lower nature of the Selftron. The higher nature is the pure consciousness, the Absolute Reality. It is this entity that makes it possible for the body, mind and intellect, made up of mere materials to act as if they were in themselves so vitally sentient. It is the Selftron that maintains, nourishes and sustains all the possibilities in the manifestations of the world of plurality [7, 80]. The Selftron is the only reality in human body and is not material. Because it is immaterial, it cannot be a compound, and therefore, it does not obey the law of cause and effect. That which is immortal can have no beginning because everything with a beginning must have an end. It also follows that it must be formless, since there cannot be any form without matter. It must be all pervading because that which is formless cannot be confined in space. When the Selftron acts through the phenomenal universe, its action is visible and is called Liftron, known as Prana in Sanskrit. We need to explain the word Prana. The ancient scientists refer to the anu (atom) and to the paramanu (finer electronic energies) but also prana which is the creative liftronic force.

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Modern scientists have identified the thought process with molecular changes in the brain and have been happy with this approximation. They did not like for a long time, the existence of consciousness behind the brain. However, because of the recent findings that human beings who have suffered removal of large parts of the brain retain their memory and are able to live intellectually proves that consciousness cannot be centered in the brain. The consciousness is now an active field of study. According to Schr¨odinger “consciousness affects all life, that the direction of evolution is governed by consciousness, but, most importantly, that there is only one consciousness” [89]. The school of Buddhists also denies the existence of the Selftron. They say that the world is itself sufficient and we need not ask for any background at all. There is no use of thinking of something as a support for this universe. The school of dualists claims that there is no such thing as unchangeability in the universe; it is all change and nothing but change. They admit that the Selftron exists but assert that the body, mind and the Selftron are three separate existences. We see therefore that not only we have different schools of thought in modern times but also we had in ancient times. Buddhists are perfectly right to say that the whole Universe is a mass of change. But as long as we are separate from the universe, as long as we stand back and look at something before us, as long as there are two things, the lookers on and the thing looked upon, it will appear always that the universe is one of change. However, the truth is that there is both change and changelessness in this universe. Those who see only motion never see absolute calm, and those who see absolute calm, for them motion has vanished. The dualists are also right in finding something behind all as a background which does not change. This is the Selftron. But the analysis that it is neither the body nor the mind, something separate from them both, but underlying them is wrong. It is not at the same time all these. We cannot observe change without there being something unchangeable. We can only conceive something changeable by knowing something which is not changeable, and this must also appear more changeable in comparison with something else which is less changeable and so on, until we are bound to admit that there must be something which never changes at all. The whole of this manifestation must have been in a state of unmanifestation, calm and silent, being the balance of opposing forces. This universe is ever hurrying on to return to that state of equilibrium again [121, 123].

5.3 Inanimate and Animate Matter The boundary between inanimate and animate matter or living and non-living matter cannot be definitely fixed, although we are used to compartmentalize them as different. The question is where do we draw the line between living and non-living matter. No one would question the statement that the human body is a living entity, consisting of countless cells, each of which is alive or that these cells join forces to form organs, which are also alive. Modern scientists agree that all living systems reproduce and use energy; they take in nutrients, process them, extract energy, and excrete waste products. One can easily observe this process at work in our fellow living beings. But is it also going on at the cellular level and at the planetary level? The conversion of energy or substances from one form to another in order to maintain the functioning of the organism is it’s metabolism. The earth absorbs sunlight and heat is radiated out into space as a waste product. Therefore, the earth has a metabolism and it is alive. Similarly, the universe is alive too. Picture an astronomical phenomenon occurring on more human time scale, and the idea of a living universe becomes easier to envision and accept. Let us look at the evolution of a star by assuming one billion years equal to one minute. We see a cloud of hydrogen gas sucked into a compact core and then transformed into heavier atom, cooked by the nuclear blaze at the stars center. The heavier atoms are excreted in the form of stellar winds or a violent stellar explosion. Similar

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to other living beings, stars also reproduce; their waste products are fed into other regions of space, where they become part of new contracting clouds destined to become stars, within which energy will be exchanged and still more waste products are excreted. Our speeded-up view of what happens in space reveals constant evolution and movement. These little life centers, these cells we call stars, are part of larger living organisms, the galaxies. The nucleus of a galaxy can be likened to a heart. We know that it pumps plasma, that is, hydrogen gas with some impurities, the nutrients, out into the surrounding veins; we call the spiral arms, streamers of intergalactic hydrogen that reach out and touch neighboring galaxies. Since our universe is made up of living galaxies, it is certainly alive. To mitochondria and bacteria, the organism that is their host is as vast and mysterious as the universe to us. Like the organelles, we may be part of some as yet incomprehensible living thing made up of organisms on all scales such as galaxies, gas clouds, star clusters, stars, planets, animals, cells and microorganisms. Thus from the biological view point, the earth and the universe can rightly be considered as living beings and therefore we need to relocate the line between living and non-living [119]. More than a century ago, in 1899, Sir Jagadis Chandra Bose began a comparative study of the curves of molecular reaction in inorganic substance and those in the living animal tissue. To his awe and surprise, the curves produced by slightly warmed magnetic oxide of iron showed striking resemblance to those of animal muscles. In both, response and recovery diminished with exertion, and the consequent fatigue could be removed by gentle massage or by exposure to a bath of warm water. Other metal components also reacted in similar ways. Bose who synthesized the teaching of his forefathers with the revelations of modern scientific research finds that every fiber in a green leaf, apparently sluggish mass of foliage, is infused with sensibility. Flowers and plants cease to be merely a few clustered petals, a few green leaves growing from a woody stem. They are man’s organic kin. Thus Bose’s research confirms not only vedantic teachings, but the deep, world wide philosophic conviction that beneath the chaotic, bewildering diversity of nature, there is underlying unity. At the close of one of his Royal Society addresses, after he had shown the complete similarity between the response of apparently dead metals, plants and animal muscles, Bose poetically uttered the conclusion at which he has arrived: “It was when I came upon the mute witness of these self made records and perceived in them one phase of a pervading unity that bears within it all things; the mote that quivers in ripples of light, the teeming life upon our earth, and the radiant suns that shine above us; it was then that I understood for the first time a little of that message proclaimed by my ancestors on the banks of the Ganges thirty centuries ago. They who see but ONE in all the changing manifestations of this universe, unto them belongs Eternal Truth-unto none else, unto none else” [116]. According to ancient scientists, the five senses of perception are the means to take cognizance of the five gross elements, namely, space, air, fire, water, and earth (Akasa, Vayu, Agni, Aapa, and Bhumi, respectively in Sanskrit). The ear perceives the sound which is characteristic of space. The skin all over the body is endowed with the sense of touch which is peculiar to the air. The eye cognizes form which is revealed by the light of fire. The tongue experiences taste of things dissolved in water. But for the aid and agency of water nothing can be tasted. The nose can detect smell produced by the earth. The knowledge of the five instruments are thus recognized as the revealers of the five gross elements of which the universe is constituted. In human beings and animals, the liftron works through these five senses. In lower forms, there is involution of senses from thought(sound), magnetic sensation(touch), photonic sensation (sight),

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electrical sensation (taste) in fluid medium, and molecular sensation or movement (smell), in that order. As the beings take lower and lower forms in the evolutionary scale, the sensations come down the scale, till at last the so called inert objects sense purely by smell or molecular movement. All this sensing in whatever levels of evolutionary ladder the being may be stationed, is done by the reflection of liftron. The liftron is the birth right of all forms including the space itself that we normally see as the etheric nothing, which also determines its life span. Without liftron, they would be unintelligent and chaotic. We are told that between atomic disintegration and atomic birth, it takes 216,000 years and that is why we only see atomic disintegration and not its rebirth [7, 123].

5.4 Evolution and Involution From the time the human beings appeared on this planet earth, they have been enjoying all the aweinspiring, sublime and beautiful things including the celestial bodies. The whole mass of existence which we call Nature, Prakriti in Sankskrit, has been acting on our minds and thoughts all the time. The same questions have been raised several times, namely, what is this visible universe? Who projected it? Does it have a beginning or an end? What is the place of human beings in it? Etc. Many times various answers have been presented by several ancient and modern scientists. Looking around us we observe a continuous change. We put a seed in the ground and later find a seedling peep out, lifts itself slowly above the ground, and grows till it becomes a tree. Then it dies, leaving only a seed. The seed does not immediately become a tree, but has a period of very fine unmanifested state. It has to work for some time beneath the soil. It breaks into pieces, degenerates and regeneration comes out of that degeneration. The bird springs from the egg, lives its life, and then dies, having produced eggs which are seeds of future birds. The child is born, grows into a human being, departs from the world through death. However, the subtle body made up of fine particles, lives on in an unmanifested state for some time and then takes the next embodiment, depending upon the resultant tendencies produced by its actions [83]. We know that the rotating clouds of hydrogen contract to form stars, heating up in the process until they become burning forces in the sky. When most of hydrogen fuel is used up, stars expand and then contract again in the final gravitational collapse. This collapse turns the stars into invisible neutron stars. Then living sometime in that invisible state, the stars are formed again. Thus we see that the fine forms slowly come out and become grosser, until they reach their limit, and then go back to finer forms. This is what is known as evolution. It is an old idea, as old as human society [121]. Everyone is aware of the ape-origin of humans which is the Darwin’s theory of evolution. He maintained that man had risen, though by slow and interrupted steps, from some primordial cell through the fish, the amphibians and the mammals to an old-world simian stem. From the point on, he held the development of upright posture and large brain could bring enough modifications to produce the modern man. A Chinese scientist, Tsou Tse, in the sixth century B.C., has the following on the evolution. All organizations originate from a simple species. This single species had undergone many gradual and continuous changes and then gave rise to all organisms of different forms. Such organisms were not differentiated immediately, but on the contrary, they acquired their differences through gradual change, generation after generation [10]. These theories of evolution are only partial truths. Darwin’s theory, for example, is not conclusive because a special aspect of the peculiarities of nature has been elevated to the level of a theory. It is a descriptive statement of that aspect, which was taken as a final truth. Recent discoveries in many branches of research are dispelling long held scientific opinion that

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the upward evolution of life and intelligence that produced human beings was an accidental process. The very existence of living matter is leading many scientists to acknowledge an inherent design in creation. A careful analysis suggests that even mildly impressive living molecule is quite unlikely to form randomly. One intriguing observation that has bubbled up from physics is that the universe seems calibrated for life’s existence. If the force of gravity were pushed upward a bit, stars would burn out faster, leaving little time for life to evolve on the planet circling them. If the relative masses of protons and neutrons were changed by a hair, stars might never be born, since the hydrogen they consume would not exist. If, at the Big Bang, the initial conditions had been juggled, matter and energy would never have coagulated into galaxies, stars, planets or any other platforms stable enough for life as we know it. Many biologists have long believed that the coming of highly intelligent life was close to inevitable. The blind evolutionary process of natural selection’ and survival of the fittest’ that fashioned complex living creatures from the earth’s raw material cannot persist without organization of life [118]. The question is from what does this evolution come? Here is where the ancient scientists discovered one more thing, which we the modern scientists have yet to perceive clearly and that is involution. Every evolution pre-supposes an involution, because nothing can be evolved which is not already there, the sum total of energy in the universe remaining the same throughout. Evolution does not come from nothing but comes out of the previous involution. The child is the human being involved and the human being is the child evolved. The seed is the tree involved and the tree is the seed evolved. All the possibilities of life are in the germ. From the lowest protoplasm to the most perfect human being, is really just one life, moving from one stage of evolution to another. In the beginning the whole of this universe has to work similarly for a period, in that subtle form, unseen and unmanifested and out of that comes new projection. The entire period of one manifestation of this universe, its going back to the finer form, remaining there for some time, and coming out again, is a grand cycle [121]. “What is it by knowing which this whole universe will be known?” was one search by some ancient scientists of Vedanta school. They did not care for particular but always searched for the general, rather universal. They postulate that this phenomenal universe was resolved into material called Spactron, and force called Liftron (Akasa and Prana respectively in Sanscrit). All the things which are around us, and which we see, feel, touch, taste, are simply different manifestations of the Spactron. It is all pervading and finer substance. All that what we call solids, liquids, or gases, figures, forms or bodies, the earth, sun, moon, planets, stars and galaxies, all of these are composed of the Spactron. All forms of energy in the universe, all motion, attraction and even thoughts are different manifestations of Liftron. This Liftron, acting on the Spactron, creates the whole of universe. The next step is to resolve the Spactron and the Liftron into still higher entity called Mahat, in Sanskrit, which is the Selftron, devoid of all formation but full of potentialities for infinite enfoldment. This Selftron manifested, changed, evolved itself into Spactron and Liftron. By the combination of these two, the entire universe has been released. At the beginning of a cycle, Liftron sleeps figuratively speaking (or lay latent), in the infinite Spactron, which is motionless. Then by the action of Liftron, motion arises in Spactron and the Liftron begins to vibrate and out of this vibration come various celestial bodies, stars and solar systems with earth, plants, animals and human beings, and the manifestations of all the various forces and phenomena. Thus every manifestation of energy is Liftron and every manifestation of matter is Spactron. When the cycle ends, all matter transforms successively into finer forms and more uniform vibrations, and finally all merge back into original Spactron. All attraction, repulsion and motion will slowly resolve into original Liftron. Then this Liftron will lay latent for a period to evolve again as before. This process of creation and dissolution goes on through eternity, that is, this entire process

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is both static and dynamic periodically, in cycles [121].

5.5 Big Bang and Big Crunch Through the powerful telescopes, we observe our Universe in a ceaseless motion. One of the most important discoveries in modern astronomy is the fact that the Universe as a whole, with its millions of galaxies, is expanding. The analysis of the light received from distant galaxies has shown that the entire set of galaxies expands in a well orchestrated manner and recession velocity of any galaxy is proportional to the galaxy’s distance. When we talk about an expanding Universe, it is in the framework of general relativity. To explain the expansion of the Universe, the microwave background radiation, and the cosmic abundance of helium, the Big Bang theory was developed. From the relation between the distance of a galaxy and its recession velocity, one can calculate the starting point of the expansion, which is the age of the Universe. Assuming that there has been no change in the rate of expansion, which is by no means certain, one arrives at an age of the order of 10,000 million years. The cosmologists believe that the Universe came into being as a highly dynamic event between 10,000 and 20,000 million years ago, when its total mass exploded out of a small primeval fireball. According to the Big Bang model, the moment of the Big Bang marked the beginning of the universe and the starting of space and time. Stephen Hawking [11] states that the Big Bang, like the Black Hole, must have a singularity, which is a point in space-time at which the space-time curvature becomes infinite, namely, a condition of infinite density and zero volume. The laws of science breakdown at a singularity. It was at such a point or state that the Universe has originated. Thus the Big Bang singularity was the beginning of the universe as well as the beginning of time. When it began it had zero size and infinite heat. A second later, it had grown exponentially and its temperature had dropped to about ten thousand degrees centigrade or about a thousand times the temperature at the center of our sun but no more than the temperature in a hydrogen bomb explosion. For a few million years the Universe cooled and expanded and so on, the theory goes. This singularity is supposed to have ended with Big Bang after which the normal laws of science would apply. If the whole Universe re-collapsed, there must be another state of infinite density in the future, the Big Crunch, which would be an end of time. Astronomers do not know what preceded the Big Bang and what is at the end of it either. The so-called Big Crunch is yet to come, if at all. In the computation of the age of the Universe and the corresponding Big Bang theory, it is assumed that there has been no change in the rate of expansion of galaxies and that the outermost galaxies are traveling more than the speed of light. However, as observed very recently, the universe is clumsy, irregular relative to the distribution of galaxies, which is diametrically opposite to earlier assumptions. Furthermore, we are not clear about the known and not yet known phenomenal Universe. The notion of dark matter is questionable because at the time of observation, the previously effulgent celestial bodies of the Universe could be in the invisible form. For example, our sun and the entire solar system was in invisible state two billion years ago. Anyone observing from outside our little world would find it as dark matter. Similarly, we may be observing the galaxies as they used to be in their earlier birth or in their death, so to speak. Moreover, the Big Bang model is not of much help in the explanation of the so-called Big Crunch either. What was before the Big Bang and after the Big Crunch is unknown. The theory of Big Bang and its conclusions are therefore partially true. Since there is no matter that we can see or know, hidden away in the black holes or in hot invisible gas between galaxies, the universe will hold gravitationally and partake of a succession of cycles, (Yugas in Sanskrit), expansion followed by contraction, universe up on universe, cosmos without an end.

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According to ancient astronomers, the present Universe has been in existence a little more than 155.52 × 1012 . Long before the start of the Big Bang, the Selftron remained in equilibrium, in which everything lay latent. Then there was an impulse of desire to release the phenomenal universe. This existential urge created an outburst of heat from which Mahat was evolved. Recall that Mahat is the same Selftron devoid of all formation but full of potentialities for infinite enfoldment. The Spactron and the Liftron were released from Mahat, which resulted in an explosive outpouring of great heat and looked like an abrupt leap of Big Bang. A concise description of what possibly must have happened after the Big Bang, can be seen in Sagan [20]. At the end of the cycle, all matter transforms into vibrations and condenses into the Spactron. Similarly, all forces merge into Liftron and the two dissolve into Mahat, which becomes enfolded into the Selftron. In this process, a great amount of cold is generated and the entire phenomena may be considered as the Big Crunch. The Big Bang and the Big Crunch seem to appear sudden events, only in the sense that the lift of a seedling and the fall of a ripe fruit appear sudden, although they are continuous processes. A simple example may illustrate the total process. Consider a spider which has its web hidden in its stomach. Without any force, it brings out the web with its own power, for its own sake. At some stage, it swallows its web. Though the processes are seen separately, they are one. Thus the spider is always one like the Selftron. In order to know more details of the theory of astronomical cycles relative to the time scales of our Universe, we need to consider our baby universe in which we live. This is discussed in the next section.

5.6 Sristi and Laya (Creation and Dissolution) Sristi is the word employed for what we call evolution or creation by the ancient scientists. Sristi means to release and does not imply to create something out of nothing as the word creation connotes. Thus Sristi means bringing into existence what has been latent in the Selftron (the Impersonal Being). This phenomenal universe is released by a continuous process and is a crystallization of the unmanifest dormant names, forms, and qualities into their manifest forms of existence. That which is called manifest is available either for the perception or for the understanding of the intellect. That which is not available for any one of these instruments of cognition, feeling or understanding is considered unmanifest. This visible universe emanated from and recedes into the Selftron in an everlasting process and lives through a time period of 311.04×1012 mean solar years known as grand cycle and then dissolves into unmanifest state, in which it remains in equilibrium the same period and then manifests into the phenomenal universe again. We learn that this process of Sristi and Laya have followed one another since endless time. Laya means the process of involution. The present Universe has been in existence for 155.52 × 1012 plus 1.972949109 × 109 solar years up to 2007. To learn how the time period of the universe is arrived by the ancient astronomers, we need to learn about our little baby universe where we live. Let us therefore concern ourselves next with present formation of our little baby universe or world, known in Sanskrit as Brahmanda, which is a huge egg-shaped ellipsoid containing our solar system. Geocentrically speaking, our earth is in the center of our Brahmanda, the distance east to west through the equator to the boundary of the Brahmanda is about fifteen billion miles, whereas the distance from north to south is approximately twenty one billion miles. Our Brahmanda with its solar system is located in the stellar system called Milky Way, or simply the galaxy and makes one revolution around the galaxy in 306.72 × 106 mean solar years. There are 21 × 1063 Brahmandas in the universe showing immense vastness of the universe. Each Brahmanda has its own time scale

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or life period. Our Brahmanda’s life is 4.32 × 109 years. By the end of this time, the sun becomes an invisible star but still the entire Brahmanda goes around the galaxy. This period of 4.32 billion years is known as Kalpa, in Sanskrit, a cycle, which is the minimum time when all planets make an integral number of revolutions, together with the apogees or aphelia and orbital nodes. They all will be in conjunction at the zero degree of Aries. Since by this time, most of hydrogen fuel is used up by the sun, the gravitational pull makes it absorb entire planetary system with a gigantic explosion and become an invisible star. After 4.32 billion years of this invisible state, the collected rotating clouds of hydrogen gas by the invisible sun, contracts, heats up in the process until it becomes a burning fire. It still continues to rotate ejecting material which spirals outward and condenses into planets, which circle around the sun. The smallest cycle, known as Kali Era, is 432,000 years at the start of which all the planets including the sun and the moon make integral number of revolutions and are in conjunction at zero degree Aries. Presently we are in Kali Era which started in 3102 BC, February 20th at 2 hours, 27 minutes and 30 seconds with the conjunction and thus 5108 years are completed from the start of Kali Era. The French Astronomer, La Kalie did verify this astronomical fact. In fact, from the present Kalpa our Brahmanda has completed 1.972949109 × 109 years and our Universe is 155.52 × 1012 plus 1.972949109 × 109 years old, as already mentioned. The foregoing figures are relative to our solar time scale on Earth. If, for example, humans are living above earth’s North Pole, then for them our six months will be one day time or equivalently our one year is one night and day. As a result, if we measure in this time scale, above figures need to be divided by 360. As another example, if humans live outside of our Brahmanda in the Universe, our Kalpa, that is, 4.32 billion years will be one day time for them since our solar system will exist that much time. Hence a day for them will be 8.64 billion years and with respect to such time measure, the age of the universe would be one hundred years and presently its age would be fifty years and less than six hours in the first day. In fact, all manifested forms and shapes of existence in the Universe have birth and death, so to speak, at allocated times, including the entire Universe. We also learn that it took 360,000 years after our Brahmanda was formed, for the vegetable life to start, which needed 12,960,000 years to develop fully. It then took 3,744,000 years for the actual creation of human life. Thus a total of 17,064,000 years were required before the existing type of life began on our earth after our baby universe was formed [82, 113, 120].

5.7 Infinite Void and Infinite Light The physicists begin their inquiry into the essential nature of things by studying the material world. Penetrating into an ever deeper realm of matter, they have become aware of the essential unity of all things and events. Furthermore, they have also learned that they themselves and their consciousness are an integral part of this unity. They arrive at this conclusion starting from the external world but their observations take place in realms of atomic and subatomic world that are inaccessible to the ordinary senses. They have developed a four dimensional space-time formalism which implies concepts and observations belonging to different categories in the ordinary three dimensional world. The multidimensional experiences transcend the sensory world and are therefore almost impossible to express in ordinary language. The modern physics shows us, at the macroscopic level, that material objects are not distinct entities, but are inseparably linked to their environment. Their properties can only be understood in terms of their interaction with the rest of the world. According to Mach’s principle, this interaction reaches out to the universe at large, to the distant stars and galaxies. The basic unity of the cosmos manifests itself, not only in the world of very small but also in the world of very large.

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The subatomic particles have no meaning as isolated entities, but can only be understood as interconnections between preparation of an experiment and the subsequent measurements. Quantum theory thus reveals a basic oneness of the universe. It shows that we cannot decompose the world into independently existing smallest units. As we penetrate into matter, nature does not show us any isolated basic building blocks, but rather appears as a complicated web of relations between various parts. These relations always include the observer in an essential way. Physicists experience the four dimensional space-time world through the abstract mathematical formalism of their theories, but their visual imagination, like everybody else, is limited to the three dimensional world of senses. Since our language and thought patterns have evolved in the three dimensional world, we find it extremely hard to deal with the four dimensional reality of relativistic physics. In modern physics, mass is no longer associated with a material substance, and therefore, particles are not seen as consisting of any basic stuff, but as bundles of energy. Since energy is associated with activity, with processes, the implication is that the nature of subatomic particles is intrinsically dynamic. These particles can only be conceived in relativistic terms, that is, in terms of a framework where space and time are fused into a four dimensional continuum. Subatomic particles are dynamic patterns which have a space aspect and a time aspect. Their space aspect makes them appear as objects of certain mass, their time aspect as processes involving the equivalent energy. In quantum theory, we do not speak about a particle’s trajectory when we say that the particle is also a wave. What we mean is that the wave pattern as a whole is a manifestation of the particle. The introduction of probability waves, resolves the paradox of particles being waves by putting in a totally new context. But that leads to another pair of fundamentally opposite concepts of existence and nonexistence. This pair of opposites is transcended by the atomic reality. Being a probability pattern, the particle has tendencies to exist in various places and thus manifest a strange kind of physical reality between existence and nonexistence. We cannot describe, therefore, the state of the particle in terms of fixed concepts. The particle is not present at a definite place, nor is it absent. It does not change its position, not does it remain at rest. The concept of quantum field, that is a field which can take the form of quanta or particles, is a new concept, which has been extended to describe all subatomic particles and their interactions, each type of particle corresponding to a different field. The quantum field is seen as the fundamental physical entity, a continuous medium that is present everywhere in space. Particles are merely local condensations of the field; concentrations of energy which come and go, thereby losing their individual character and dissolving into the underlying field. This underlying activity is the only reality and all its phenomenal manifestations are seen as transitory. Thus the quantum field theory is a well defined concept which only accounts for some of the physical phenomena, namely, inert matter and hence it is only a partial truth. When physicists began to study quantum theory of fields, they discovered that a vacuum or void is not inert and featureless, but alive with throbbing energy and vitality. A real particle such as electron must always be viewed against this background. The virtual particles can come into being spontaneously out of the void and vanish again into the void. According to field theory, events of this kind happen all the time. Thus the void or physical vacuum is far from empty. It contains unlimited numbers of particles which come into being and vanish without end. It contains the potentiality of all forms of the particle world. These forms are not independent physical entities but merely transient manifestations of the underlying infinite void. Thus the infinite void is a truly living void, pulsating in endless rhythms of creation and destruction [9]. The reality of the quantum field theory cannot be identified with the Absolute Reality of the ancient scientists. The difficulty is that it is, as pointed out earlier, an incomplete generalization since

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we have taken only one of the facts of nature and that is inert matter. We have left out the other side, namely the living matter. If we penetrate into the living matter, we will find infinite light or effulgence. The infinite void and infinite light cannot be conceived as mutually exclusive opposites, but only two aspects of the same reality, which coexist. However, the scientific exploration of inert and living matter requires different approaches. The perception of Absolute Reality depends on awareness of oneness of all animate and inanimate life, the interdependence of their multiple manifestations, and their cycles of change and transformation. The silencing of the mind that is, shifting the awareness from the rational to the intuitive mode is achieved by concentrating one’s attention to a single item. The human mind can be trained and disciplined to focus its powers enough to exercise control over the sensory perceptions. One can go even further in silencing or controlling the mind, intelligence and ego, in addition to the senses thereby realizing the universe in higher dimensional spaces to get a glimpse of infinite effulgence. Ancient scientists suggest that it is possible for some individuals to go to infinite dimensions and merge with the Impersonal Being or equivalently the Selftron. Our immediate reaction may be that it is impossible. However, recently the definition of space has been modified. There is a growing acknowledgement among some physicists world wide, including several Nobel Laureates, that the universe may actually exist in higher dimensional space. Light, in fact, can be explained as vibrations in the fifth dimension. Higher dimensional space, instead of being empty, actually becomes the central actor in the drama of nature [75]. The question, whether the mind is of higher dimension is not considered by many physicists. However, in complex biological and computational systems, mind is a fundamental process in its own right, as widespread and deeply embedded in nature as light or electricity. Mind is, in a word, elemental, and it interacts with matter at an equally elemental level, at the level of the emergence of individual quantum events. What mathematics seems to say is that, between observations, the world exists not as a solid actuality but only as shimmering waves of possibility. Whenever it is looked at, the atom stops vibrating and objectifies one of its many possibilities. Whenever some one chooses to look at it, the atom ceases its fuzzy dance and seems to freeze into a tiny object with definite attributes, only to dissolve once more into a quivering pool of possibilities as soon as the observer withdraws his attention from it. This apparent observer-induced change in an atom’s mode of existence is called the collapse of the wave function or simply the quantum jump [71]. John Von Neumann addressed the problem that something new must be added to collapse of the wave function, something that is capable of turning quantum possibilities into definite actualities. Searching his mind for an appropriate actually existing nonphysical entity that could collapse the wave function, Von Neumann reluctantly concluded that the only known entity fit for this was “consciousness”. In his interpretation, the world remains everywhere in a state of pure possibilities except where some conscious mind decides to promote a portion of the world from its actual state of indefiniteness into a condition of actual existence. By itself the physical world is not fully real but takes shape only as a result of the acts of numerous centers of consciousness. Ironically this conclusion comes not from an ancient mystic but from one of the world’s most practical mathematicians deducing the logical consequences of a highly successful and purely materialistic model of the world. It is therefore clear that the use of our own nervous system as instruments of experiments allows us to interact with not only the material level of existence but also non-material (living matter) levels. This should be a part of the scientific training of the future [116]. Through meditation, one can also get the important habit-altering brain change. Over the years, we develop circuits and channels of thought in our brain. These are physical pathways which control the way we think, and habits become so fixed that they turn into rigid wiring. In other words, the circuits or channels become so deeply ingrained it is almost impossible to transform them. Scientific research has shown that electrical activity between left and right sides of the brain becomes coordinated during certain kinds of meditation or prayer. Through these processes, the mind definitely becomes more capable of being altered and having its capabilities maximized. When we are in a

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state of enhanced left-right hemispheric communication, plasticity of cognition occurs, in which we actually change the way we view the world. When we change our patterns of thinking and acting, the brain cells begin to establish additional connections or new wirings. These new connections then communicate in fresh ways with other cells, and before long, the pathways that kept the phobias or other habits are replaced or altered and therefore changed actions and a changed life will follow. The implications are exciting and even staggering. It is interesting to know that approximately one hundred billion nerve cells in the brain have a total number of possible connections of the order 25 × 1030 [6]. The physical benefits of meditation have recently been well documented by medical researchers. It tends to lower, for example, or normalize blood pressure, pulse rate, and the levels of stress hormones in the blood. In short, meditation reduces the wear and tear on both body and mind, helping people to live longer and better. It has also been found recently that no technique of extreme physiological treatment may be expected to provide a cure for chronic pain unless the patient commits to a systematic inner change in thoughts and behavior. As a result of this discovery, pain specialists are beginning to recognize the great practical utility of the science of yoga and meditation in dealing with the physical and mental causes of chronic pain. To learn self-control requires will power and training in right thinking, right attitudes, right activity, the very things one learns from the disciplines of yoga [112]. Yoga is the ability to direct the mind exclusively towards a single object and sustain that direction without distractions. The mind serves the dual purpose. It serves the perceiver (the Selftron) by presenting the external to it. It also reflects or presents the perceiver to itself for its own enlightenment. When the mind is free from the clouds that prevent perception, all is known and there is nothing more to be known. When the highest purpose of life is achieved the three basic qualities, behavior, attitude, and expression, do not excite responses in the mind. That is freedom. In other words, the perceiver is no longer colored by the mind or the mind has complete identity with the Selftron [89].

5.8 Phenomenon and Noumenon There are two principles of knowledge. The first one is that we can know by referring the particular to the general and the general to the universal. The second is that anything, of which the explanation is sought, is to be explained, as far as possible, from its own nature. Taking up the first principle we see that all our knowledge really consists of classifications leading to higher levels of refinement. When something happens once, for example, we are dissatisfied but when the same thing happens again and again, we call it a natural law. This implies that from the particular we deduce the general. The generalization of the personal God is another example. The personal God was called the sum total of all consciousness, which is an incomplete generalization. This is because we take only one side of the facts of nature, the consciousness, and generalize it as personal God. But we left out the other side of nature, namely, the inert nature and hence it is a defective generalization. There is another deficiency about the personal God, which relates to the second principle. If God is the creator of the universe, who has nothing to do with nature, and if nature is said to be the outcome of the command of that God and to have been produced from nothing, then it is a very unscientific theory. In fact, the theory of personal God implies that he is endowed with the qualities of human beings enhanced very much, who by his will created the universe out of nothing and yet is separate from it. The discussion about substance and qualities is very old. On a higher plane, the fight about substance and qualities, takes the form of the fight about phenomenon and noumenon. There is the phenomenal world, the universe of continuous change, and there is something beyond which does

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not change, and the duality of existence, noumenon and phenomenon, some scientists hold to be real, while other scientists claim that we have no right to admit the two, for what all we see, feel, and think is only phenomenon. We have no right, some schools of thought, like Buddhist school, to assert there is anything beyond the phenomenon, and apparently there has been no answer to this. According to the school of Vedanta or Advaita, one thing alone exists, and that one thing is either phenomenon or noumenon. It is not true that there are two, something changing, and, in and through that, something which does not change. But it is one and the same thing which appears as changing and is in reality unchangeable. We have for example, come to think of the body, mind and Selftron as separate, but really there is only one and that one appears through these various forms. Take the well known example, the rope appearing as the snake. Some people, in the dark or for some other reason, mistake a rope for a snake, but when knowledge comes, the snake vanishes and it is found to be a rope. By this illustration, we see that when the snake exists in the mind, the rope has vanished, and when the rope exists, the snake has gone. When we see only the phenomenon around us, the noumenon is vanished, but when we see the noumenon, the unchangeable, it naturally follows that the phenomenon has vanished. We can now understand the position of the realist and the idealist. The realist sees the phenomenon only, and the idealist looks at the noumenon. For the genuine idealists, who have truly acquired that power of perception, whereby they can get away from the ideas of change, the changeful universe has vanished for them, and they have right to say that it is all delusion and there is no change. The realists, on the other hand, look at the changeful phenomenon and for them the unchangeable does not exist, and they have right to conclude that the phenomenon alone is real. Out of this discussion, we find that the personal God is not sufficient; we have to reach something higher, that is, to the idea of the Impersonal. It is the only logical step we can take, since the Impersonal, is a much higher generalization. Only the Impersonal can be infinite, the personal is limited. This generalization does not imply that the Personal God is destroyed, nor the personal human being is lost. The Advaita idea is not the destruction of the individual or the Personal God, but their vindication. We can not prove the existence of the individual, for example, except by referring to the universal and proving that the individual is really the universal. The thought of the individual as separate from everything else is not tenable. When we apply the second principle, we are led to a still bolder idea which is more difficult to understand. It is nothing less than the Impersonal Being, the highest generalization is in ourselves and we are That. The Chandogya Upanished declares by saying “That which is the subtle cause of all these things, of it are all the things that are made, that is All; that is the Truth, Thou art That.” We are the Impersonal Being. That God for whom we searched all over the universe has been all the time within ourselves. The human beings we now know, are manifestations of that Impersonal Reality. To understand the personal we have to refer to the Impersonal. The particular must be referred to the general and the Impersonal is the true Selftron. As manifested beings we appear to be separate but our reality is ONE and the less we think of ourselves as separate from That ONE, the better for us. The more we think of ourselves as separate from the whole, the more miserable we become. Just as there are millions of people who believe in the personal Creator, there have also been thousands of brightest minds in this world who have felt that such ideas were not sufficient for them and wanted something higher and remained outside of such theories. This implies we need a theory large enough to take in all various views. There must be some independent authority, and that cannot be any one book, but something that is universal and what is more universal than intuitive reasoning? What we want is progress, development and realization. The only power is in realization and that lies in ourselves and comes from thinking. The glory of the human beings is that they are thinking beings.

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We have to understand the Impersonal for it is in and through That alone that the other concepts can be explained. As an example, let us take the Personal God, who is the highest reading of the Impersonal that can be reached by the human intellect; and what is the universe but various readings of the Impersonal Being or the Absolute? This universe in itself is the Absolute, the Unchangeable, the noumenon and the reading there of constitutes the phenomenon. We find that all phenomena are finite, and every phenomenon that we can see, feel or think is finite, limited by our knowledge. The Personal God, as we conceive of Him, is in fact a phenomenon. Whatever is real in the universe is that Impersonal being, and the forms and names are given by our intellects. We as personalized, differentiated human beings, forget our reality and the teaching of the Advaita is not that we must give up these differentiations, but that we must learn to understand what they are. We are in reality that Infinite Being and our personalities represent so many channels through which this Infinite Reality is manifesting itself, and the whole mass of changes that we call evolution is brought about by the Selftron trying to manifest more and more of its infinite energy. We cannot stop anywhere on this side of the Infinite and our power and wisdom cannot but grow into the Infinite. We have not yet realized the Infinite power and wisdom. All the knowledge that we have in this world is within us and nothing is outside. The gigantic intellect and the infinite energy lie coiled up in the protoplasmic cell, the energy is in the cell, potentially no doubt, but still there. It may seem like a paradox but it is true. The whole is the Absolute and within it every particle is in a constant state of flux and change. It is unchangeable and changeable at the same time, impersonal and personal in one. Thus we see that instead of doing away with the Personal and the relative, the Impersonal, explains it to the full satisfaction of our own intuitive reason. When we shall get rid of our minds, out little personalities, we shall become one with it. This is what is meant by “Thou art That” [121, 123].

5.9 Conclusion The infinite power of love existed in the hearts of those great ancient scientists, Rishis. They knew that all human beings must have their own paths, but the paths are not the goal. All paths are only steps towards the Truth (That ONE), and are not the Truth itself. However, they were aware that the human society should grow and went on applying their techniques by leading them upwards, step by step. Such were the writers of the Upanishads [122]. The sound is defined as vibration. At the level of subatomic world, all matter is the same. But objects appear differently to the eyes because energy produces vibrations of different frequencies at various points. Vibrations create sound and conversely, if sound is to result, vibrations should be created. Since vibrations of different frequencies occur in the flood-stream of energy, the explanation of modern scientists, for the creation of the universe, and the pronouncement of ancient scientists that the creation resulted from the life-breath of Impersonal Being, are mutually in agreement. The passage of breath through the various pulse-nerve centers creates vibrations which are responsible for the health of beings or lack of it. The breathing is nothing but the regulation of the vibrations within us. All matter, whether animate or inanimate, emanate from the Impersonal Being, multiply themselves and manifest in various forms, and get transformed or disappear. The different vibrations necessary for such mutations should naturally be caused in the substance called Impersonal Being. If the origin of phenomenal existence is traceable to vibrations and sound, then it stands to reason that the same vibrations and sound can correct the erring forces of nature and cleanse the mind of improper thoughts. The mind is capable of having two states based on two distinct tendencies. These are distraction

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and attention. However, at any moment only one state prevails and this state influences the individual’s behavior, attitude and expressions. The silencing or restraint of senses occurs when the mind is able to remain in the chosen direction and the senses disregard different objects around them and faithfully follow the direction of the mind. Then the senses are mastered or silenced and the mental activities form an uninterrupted flow only in relation to this object. This state is known as Samadhi. Soon nothing except comprehension of the object is evident, which is as if the individual has lost his own identity. This is the complete integration with the object of understanding and nothing is beyond reach. This state is more intricate than Samadhi according to Patanjali Yogasutra [89]. Mikhael Aivanhov, a Bulgarian, who lived in France, expresses a similar idea in a modern way. When we live in a state of harmony our inner forces react and reject impurities. Harmony should be our first thought or intention and our last. Harmony is the opposite of stress, hectic activity, restlessness, competitiveness, unease and the myriad other conditions that characterize the modern psyche and culture. We cannot be loving and compassionate, for instance without being attuned to the higher reality. In harmony, we transcend the ego, and our body-mind becomes a conduit for the higher reality. Harmonize everything within ourselves, and we will become capable of acting with such wisdom, such depth, such intelligence, that we will wonder “How did we do it?” [54]. Another approach of describing a similar situation is as follows. Our deeper understanding leads to another kind of power, a power that loves life in every form as it appears, a power that does not judge what it encounters, a power that perceives meaningfulness and purpose in the smallest details upon the earth. This is authentic power. When we align our thoughts, emotions, and actions with the highest part of ourselves, we are filled with enthusiasm, purpose and meaning. Life is rich and full. No matter how many different ways we describe, we have only one method of acquiring true knowledge. From the lowest mortal to the highest Rishis, all have to employ the same method, the method of concentration. In doing anything, the stronger the power of concentration, the better the outcome. This is the one key that opens the gates of nature, and lets out the floods of light. That is why Patanjali defines yoga as the “cessation of thought waves of the mind” and provides the processes of several levels to attain concentration suitable for our various temperaments and purposes [89]. Ancient scientists rightly attached great importance to the various temperaments or inheritance that constitutes the natural inequality of individuals. These inequalities are due to the eternal recurrence of rhythmic character of the world process, which is explained by the theory of reincarnation. Accordingly, they explained their theories in several ways in order to reach the human beings of different temperaments. Thus we see, for example, the process of creation (Sristi) and dissolution (Laya) has been described in many ways [82]. The following is an approximate translation of Rig Veda’s description of the situation before the present creation (Sristi) of the Universe, which loses a good deal of poetic beauty in translation. “Existence or nonexistence was not then. That ONE breathed in the infinite vacuum by itself. Other than that there was nothing. In the beginning there was gloom hidden in gloom. All that existed then was infinite void and formless. From that voidness emerged through the might of the heat of austerity, the desire, the first seed of mind arose in That. There was a bond of existence in nonexistence. Who knows the truth? Who will pronounce it, whence this birth, whence this Sristi arose? Whether it created itself or whether, it is not? That ONE, who looks upon it, surely knows or may be knows not” (Rig Veda, X, 129, 1-7) [122].

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Index

n-body problem, vi, 5, 39, 76, 84, 90

dual numbers, 23

adjacent decimals, 20, 76 Anti-Matter, 51, 53 anti-matter, 49, 51 anti-primum, 53 axiom of choice, 3

earth lights, 58, 93 energy conservation, 8, 10, 33, 37, 46–48, 51–54, 57–61, 63, 64, 66, 71, 75, 77–80, 95, 99–102 energy conservation equivalence, 47, 48, 50, 54, 64

Banach-Tarski Paradox, 3 Big Bang, 1, 6, 61, 62, 67, 69–73, 114–116 black hole, v, 6, 37, 48, 53, 54, 58, 61, 64–66, 69, 71, 72, 77, 80, 93, 115 Bose-Einstein Condensate, 54

Fermat’s Last Theorem, 23, 75 flux, 9, 11, 29, 47–52, 54–58, 60, 62, 64–66, 68–74, 77–81, 84, 87–90, 94–101, 109, 110 Flux Compatibility, 78 flux compatibility, 28, 49–54, 57, 64, 68–70, 73, 74, 78–81, 88, 92, 94, 97, 98 flux-low-pressure complementarity, 47–49, 52, 53, 59–65, 70, 73, 77, 79–81, 92, 94, 95, 97 Fractal, 10, 35, 36 fractal, vii, 7, 9–11, 29, 32–34, 37, 46–48, 60–62, 64–67, 72, 73, 76, 77, 79, 85, 87, 90–94, 100

chaos, vi, 7, 13, 28–30, 32, 33, 47, 61, 62, 76, 91, 95 consciousness, vii, 67, 100, 108–111, 117, 119, 120 cosmic burst, 60, 62, 73 cosmic dust, 5, 11, 45, 49, 50, 59, 61, 65–67, 79, 80, 84, 88 Cosmic Sphere, 61 Cosmic Waves, 67 cosmic waves, vi, 6, 9, 11, 27, 47–51, 61, 67, 72, 80, 92, 99–105 cosmological vortices, 5, 32, 38, 45, 50, 59–67, 69, 70, 72, 77, 79, 80, 84, 87, 90, 92 curvilinear integral, 26, 31

Generalized Curves, 31 generalized curves, vi, 13, 26, 27, 29–32, 38, 39, 75 generalized fractal, vi, 1, 11, 13, 32, 36, 75 generating sequence, 16 Goldbach’s conjecture, 2 Grand Unified Theory, 74, 83

Dark Matter, 64 dark matter, vi, vii, 1, 5, 7, 13, 14, 33, 36, 45–51, 59, 61–64, 66, 67, 71–73, 76–81, 84, 93, 98, 100, 102, 115 dark number, vi, vii, 13, 18–25, 76 decimal integers, 20, 21

Hybrid Real Number System, 14 hybrid real number system, 13, 14, 16, 22 Hybrid Unified Theory, 107 impersonal infinity, vii, 14, 19, 22, 24 131

132

infinite light, 119 infinite void, 118, 119, 123 Infinitesimal Zigzag, 25, 26, 29, 35 infinitesimal zigzag, 25–27, 32, 33, 36 internal-external dichotomy, 100 involution, vii, 112, 114, 116 Kerlian photography, 6, 81 Laya, 116, 123 lexicographic ordering, 19, 20, 22 Liftron, 110, 114, 116 macro gravity, vi, 45, 54–56, 59–61, 85 macro vortex, 11, 80 Mischievous Function, 25 mischievous function, 25–28 nested fractal sequence, 9, 11, 29, 47, 59–63, 66, 67, 78, 80, 87, 92 nonstandard d-sequence, 17–19 Nonstandard Numbers, 23 nonstandard numbers, 23 nonterminating decimals, 13, 16–18, 20 noumenon, 120–122 Perron paradox, 5, 27, 29, 31, 38 personal infinity, 2, 15–19, 23 phenomenon, 7, 45, 54–57, 61, 69, 81, 84, 94, 99, 103, 111, 120–122 Planck’s constant, 9, 28, 30 Pontrjagin’s Maximum Principle, 41, 43 Pontrjagin’s maximum principle, vi, 38, 75, 77, 85, 86, 89, 132 Pontrjagin’s maximum principle, vi Primum, 48, 51 primum, 27, 28, 32, 45, 48–55, 58, 66, 67, 69, 72, 78, 79 qualitative mathematics, vii, 5, 7, 8, 33, 59, 75, 89, 99 qualitative modeling, vi, 7, 8, 33, 45, 75, 76, 90, 91 quantum field, 118 Quantum Gravity, 58 quantum gravity, vi, 45, 51, 52, 55, 56, 58–60, 69, 70 quark, 46, 48–50, 52, 53, 57, 58, 77–79 radio waves, 59

The Hybrid Grand Unified Theory

rapid spiral, 28 Russell Paradox, 2 seismic waves, 9–11, 58–60, 63, 66, 67, 80, 81, 92–94, 98 Selftron, 107, 109–111, 114, 116, 119–122 Sristi, 116, 123 Structure of R∗ , 21 super...super galaxy, 11, 45, 61–63, 66, 69, 80 superconductivity, 6, 54, 55 superstring, vi, vii, 5, 11, 27, 29, 30, 32, 33, 36, 45–50, 52–55, 57–67, 69, 76–80, 84, 89, 92, 93, 99, 105 toroidal flux, 29, 32, 48–54, 58, 66, 69, 76–78 trichotomy axiom, 2–4, 22 Turbulence, 92, 94 turbulence, vi, 6, 7, 33, 38, 47, 55, 59, 60, 63, 65, 67, 74, 80, 84, 85, 91, 92, 95, 97, 98, 101 twin integers, 20 Ultra-Energetic Cosmic Waves, 72 Vedas, 13, 107, 108 Vedasamhitas, 13 vortex flux, 45, 49, 52–54, 56, 58, 60, 65, 69, 78–80, 89, 94 Wave-Particle Duality, 50 wave-particle duality, 50 Wild Oscillation, 27 wild oscillation, 25, 27, 51 yoga, 109, 120, 123 Young measure, 26, 27