On the Unified Field Theory VI

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the operators which correspond to themselves under this automorphism is that the order of G is odd. This is also a necessary and sufficient condition that G is the direct product of K and C. When the order of G is of the form 2m the number of the invariants of K is the same as the number of the invariants of H and at least equal to the number of the invariants of C, and the group generated by H and C contains the square of all the operators of G and is of index 2t under G, where i represents the number of the invariants of C. Hence the order of the cross-cut of C and H is 2'. When either H or K is cyclic then the three subgroups H, K and C must be cyclic and their common cross-cut is of order 2. If this condition is satisfied and G is also decomposable into H and K there are m/2 - 1 or (m - 1)/2 such groups as mn is even or odd. G. A. Miller, Trans. Amer. Math. Soc. 10, 472, (1909).

ON THE UNIFIED FIELD THEORY. VI By TRAcY YERKES THOMAS DEPARTMENT OF MATHEMATICS, PRINCETON UNIVERSITY

Communicated April 2, 1931

Let us suppose that the functions hI in group Go have the values 6 throughout a finite region R of the four dimensional continuum; these special values of the hI can obviously be imposed, at least throughout a sufficiently small region of the continuum, by a suitable coordinate transformation. A coordinate system for which the h' have the above special values will be referred to as a cannonical co6rdinate system. Interpreting the coordinate xl as the time t let us now assume that a disturbance is produced at a time t = 0 throughout a closed region Ro of space.' After a time t = p(X2, X3, X4), the effect of this disturbance will be felt at a point (X2, X3, X4) outside the region X?. For a definite value of t the equation Sp(X2, X3, X4) = t will represent the surface of the space (x2, X3, X4) which, at the instant t, separates the region affected from the region unaffected by the disturbance. If we assume that the values of the h, and their derivatives vary in a continuous manner when we pass from one region to the other, then it follows from the result in sect. 8 of Note V under the hypothesis of canonical coordinates that the surface t -~p(X2, X3, X4) = 0 must be a characteristic surface of the field equations. Wave surfaces are thus identified with the characteristic surfaces. Now assume the expression for the element of distance in the region R in the form

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MA THEMA TICS: T. Y. THOMAS 4

ds2 = dt2

-

PROC. N. A. S.

4

E ya a=2 p= 2 E

dxa dx

(a)

where we have put t = xl and -y,Xo = - g,, in terms of the notation used in previous notes. We shall refer for the moment to the co6rdinates for which the form (a) of the element of distance is valid as Gaussian coordinates in conformity with the terminology used by Hilbert.2 It will be assumed likewise that the co6rdinates of a system of Gaussian coordinates have the significance of coordinates of time and space as implied by the above expression for the element of distance. Use has been made of Gaussian coordinates in this sense in recent work on cosomology.3 Owing to the special form of the element of distance (a) in Gaussian coordinates the function s must, therefore, satisfy the equation 4 4 a aS p aS a=2 P=2 be Z5 over the surface (p = t where ya0 is the cofactor of the element -ya, in the determinant divided by this determinant. The equation

(p(X2, X3, X4) = const.

(b)

therefore represents a family of parallel surfaces, and hence the successive positions of the wave front form a family of parallel surfaces which are propagated with unit velocity. The hypersurface formed from the totality of bicharacteristics issuing from a point P is a characteristic surface (see Sect. 4 of Note IV). Let us denote the sheet of this hypersurface corresponding to IV (4.11) by S; let us also denote the surface of intersection of S with the hypersurface t = const. by S*. It can be shown that from the given position Ro of the wave at time t = 0, the position of the wave (b) at any later time t can be obtained by taking the envelope of the surfaces S* corresponding to the different points of Ro.4 In other words Huygen's principle can be applied for the construction of the wave fronts. A discussion of wave surfaces in general from this point of view has lately been made on the basis of contact transformations by L. P. Eisenhart.5, In particular if the surface Ro reduces to a point P the corresponding wave surface (b) assumes the form (w2)2 + (w3)2 + (w4)2 = const. (c) in a system of metric local coordinates with origin at P. The invariance of (c) under orthogonal rotations of the fundamental vectors he, i.e. Lorentz transformations of the metric local coordinates, has its interpretation in the experimental fact that the velocity of light is independent of the motion of the source or the observer.

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The orthogonal trajectories of the family of wave surfaces (b) are light rays in the usual sense; these curves, however, are geodesics of zero length6 when considered with respect to the Gaussian ccordinates (t, x2, x3, x4). This establishes the geodesics of zero length or bicharacteristics of the field equations as the light tracks in the unified field theory. A treatment of the problems of optics can be made on the basis of the above theory of gravitational and electromagnetic waves. In fact, a step in this direction has recently been taken by Synge and McConnell7 who have deduced formulae for the angle of reflection and refraction of light from a moving surface by a method adaptable to the point of view of the present theory. Other interesting problems in geometrical and physical optics await investigation from this standpoint. 1. We shall add to this note a brief discussion of the problem of the determination of the structure of the continuum by the assignment of data in a system of local coordinates. Let us define a three-dimensional variety S3 and a two-dimensional variety S2 in a system of affine local coordinates zl by

S3:4.(Z',

...,Z4)

=O,2

4)(Z1l ... Z4) 'l(1 J . .,4 *(ZI, ,z4)

= 0 =0

where the functions 4) and ' are assumed to be analytic in the neighborhood of the origin. We shall suppose that 4) and I are independent as functions of z' and z2; this causes no loss of generality as it involves at most a relettering of the coordinate axes. Now put (1.) ¢D(Z), X2 = '(z), X3 = Z3, X4 = Z4. This defines an analytic transformation of coordinates which possesses a unique inverse in the neighborhood of the origin of the local system. We shall say that the origin of the local coordinate system is a characteristic point with respect to the hypersurface S3 if the expression

XI

=

(aq,)2 --2

VaZ1)

()2 -

2

VYZ2) V5Z3) V5Z4)

vanishes at the origin of these co6rdinates. Our discussion will be divided into two parts: (A) the case for which the origin is a characteristic point, and (B) the case for which the origin is not a characteristic point with respect to the hypersurface S3. We shall refer to these two cases as hypothesis (A) and hypothesis (B), respectively. Now suppose that the quantities Kll defined in Note III are given over S3 and the K72 over S2; this can, in fact, be accomplished by taking the K1, to be analytic functions of the coordinates (x2, X3, X4) and the K72 as analytic functions of the coordinates (X3, X4), i.e., analytic in the neighbor-

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PROC. N. A. S.

hood of the values xo corresponding to the origin of the local system. Now observe that the coefficients H of the power series expansions I (2.10) are uniquely determined on account of (1.1). Hence, if we know the derivatives at x' of the h. and Klm up to the order r inclusive, the derivatives of the h' of order r + 1 at x4 will be determined by II (4.2). Differentiation of III (3.3) under hypothesis (A) leads to the determination at x' of the derivatives of the Klm of order r + 1, etc. By this process we obtain uniquely determined convergent power series expansions8 of the hX (x) in the neighborhood of the point x4; transformation by means of (1.1) gives the functions h' in the affine local coordinate system. As a matter of fact the functions h, are likewise uniquely determined in the system of metric local coordinates having its origin coincident with the origin of the affine local coordinates and we, therefore, arrive at the

following EXISTENCE THEOREM. The specification of the functions K71 and Ki, over the varieties S3 and S2, respectively, as analytic functions of the surface coordinates determines uniquely a set of analytic integrals h, of the field equations in a system of affine (or metric) local coordinates having its origin at a point P on S2 not characteristic uwth respect to S3. In particular we can take 4 = z' and T = Z2. We then arrive at a theorem which on the point of view of our previous notes can be stated roughly as follows: The specification at a particular instant t = 0 of the absolute electromagnetic forces K7m throughout a small region Ro determines the structure of the space-time continuum in the neighborhood of Ro. 2. A modification of the above existence theorem is necessary when the hypothesis (B) is adopted. We can suppose that the functions 4 and T satisfy the normal conditions C1 and C2 at a point P as explained in Sect. 1 of Note V. Taking the point P as the origin of a system of affine local coordinates z' we shall then have

az1 az1

6Z2 OJz2

6ZI 6Z2

OZ2 OZ2

OZ3 CZ3 6Z 6Za4

6z4 6z4

0Z2

/

(2.1)

6Z4 OZ3J

at the origin of these coordinates. At the point x' of the (x) coordinate system defined by (1.1) the inequalities V (1.7) will, therefore, be satisfied. Call S* the surface defined by T = 0 and consider the functions U and V defined in Sect. 4 of Note V. If we specify the functions U over the hypersurfaces S3 and S3 and the functions V over the surface S2 as analytic functions of the surface coordinates, we can uniquely determine the successive coefficients of the power series expansions8 of the components

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h, about the point x4 by applying the method of the preceding section to the equations V (4.1), V (4.2) and V (4.3). This gives the following EXISTENCE THEOREM. The specification of the functions U over the varieties S3 and S* and the functions V over the variety S2 as analytic functions of the surface co6rdinates determines uniquely a set of analytic integrals h, of the field equations in a system of affine (or metric) local co6rdinates having its origin at a point P on S2 characteristic with respect to S3 and such that at P the inequalities (2.1) are satisfied. As mentioned in Note V the normal conditions (2.1) place no restriction on the hypersurface S3. 1 Cp. E. Goursat, Cours d'analyse mathematique, 3, 94 (1927). 2 D. Hilbert, "Die Grundlagen der Physik," Math. Ann., 92, 1-32.(1924). 3See, for example, H. P. Robertson, "On the Foundations of Relativistic Cosmology," Proc. Nat. Acad. Sci., 15 (1929), 822-829, where other references are also given; in particular see footnote 4 to this paper. 4 Cp. J. Hadamard, Lecons sur la propagation des ondes, Hermann, 290 (1903). 5 L. P. Eisenhart, "Contact Transformations," Ann. of Math., 30, 211-249 (1929).

6 Proof of this statement can be based on the discussion in Sect. 2 of Note IV. 7J. L. Synge and A. J. McConnell, "Riemannian Null Geometry," Phil. Mag., 5,

7th Series, 241-263 (1928). 9 The proof of convergence involved here will possibly be given later in a comprehensive exposition of the present theory.