JOURNAL OF MATHEMATICAL PHYSICS 46, 012501 (2005)
Weak field reduction in teleparallel coframe gravity: Vacuum case Yakov Itina) Institute of Mathematics, Hebrew University of Jerusalem, Givat Ram, Jerusalem 91904, Israel (Received 11 March 2004; accepted 11 September 2004; published online 17 December 2004)
The teleparallel coframe gravity may be viewed as a generalization of the standard GR. A coframe (a field of four independent 1-forms) is considered, in this approach, to be a basic dynamical variable. The metric tensor is treated as a secondary structure. The general Lagrangian, quadratic in the first order derivatives of the coframe field is not unique. It involves three dimensionless free parameters. We consider a weak field approximation of the general coframe teleparallel model. In the linear approximation, the field variable, the coframe, is covariantly reduced to the superposition of the symmetric and antisymmetric field. We require this reduction to be preserved on the levels of the Lagrangian, of the field equations, and of the conserved currents. This occurs if and only if the pure Yang–Mills-type term is removed from the Lagrangian. The absence of this term is known to be necessary and sufficient for the existence of the viable (Schwarzschild) spherical-symmetric solution. Moreover, the same condition guarantees the absence of ghosts and tachyons in particle content of the theory. The condition above is shown recently to be necessary for a well-defined Hamiltonian formulation of the model. Here we derive the same condition in the Lagrangian formulation by means of the weak field reduction. © 2005 American Institute of Physics. [DOI: 10.1063/1.1819523]
I. INTRODUCTION
Einstein’s general relativity (GR) is very successful in describing the long distance (macroscopic) gravity phenomena. This theory, however, encounters serious difficulties on microscopic distances. So far essential problems appear in all attempts to quantize the standard GR (for recent review, see, e.g., Ref. 1). Also, the Lagrangian structure of GR differs, in principle, from the ordinary microscopic gauge theories. In particular, a covariant conserved energy-momentum tensor for the gravitational field cannot be constructed in the framework of GR. Consequently, the study of alternative models of gravity is justified from the physical as well as from the mathematical point of view. Even in the case when GR is unique true theory of gravity, consideration of close alternative models can shed light on the properties of GR itself. Among various alternative constructions, the Poincaré gauge theory of gravity, see Refs. 2–11, is of a special interest. This theory proposes a natural bridge between gauge and geometrical theories. Moreover, it has a straightforward generalization to the metric-affine theory of gravity,5 which involves a wide spectra of space–time geometries. However, it was elucidated recently that even the restriction of the Poincaré gauge theory to the teleparallel model provides a reasonable alternative to GR.
a)
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012501-2
J. Math. Phys. 46, 012501 (2005)
Yakov Itin
A. Coframe (teleparallel) gravity—basic facts and notations
We start with a brief account of the coframe (teleparallel) model of gravity and establish the notations used in this paper. Details, different approaches, and additional references can be found in Refs. 12–28. Let a four-dimensional (4D) differential manifold M be endowed with two smooth fields: a frame field ea and a coframe field a. In a local coordinate chart, ea = ea共x兲 / x,
a = a 共x兲dx,
a, = 0,1,2,3.
共1.1兲
These fields allow to compare two vectors (more generally, two tensors) attached to different points of the manifold. It is referred to as the teleparallel structure on M. The two basic fields are assumed to fulfill the dual relation: eacb = ␦ba. We denote by c the interior product operator X ⫻ ⌳ p → ⌳ p−1 that, for an arbitrary vector field X 苸 X and a p-form field w 苸 ⌳ p, Xcw : = w共X , . . . 兲. So only one of the fields, ea or a, is independent. Thus, two alternative (but, principle, equivalent) representations of the teleparallel geometry are possible. The frame representation is based on a complex 兵M , ea其 and applies the tensorial calculus as the main mathematical tool similar to the Einstein tensorial representation of GR. The coframe representation, which deals with a complex 兵M , a其, applies the exterior form technique. In the present paper, we use this approach and call it the coframe gravity, in contrast to the metric gravity of GR. In a wider context, the coframe field appears as one of the basic dynamical variables in the Poincaré gauge gravity and in the metric-affine gravity. To extract the pure coframe sector, in these theories, one must require vanishing of the curvature. Here, we treat the coframe field as a self-consistent dynamical variable with its own covariant operators: wedge product, Hodge map and exterior derivative. These two approaches (one with a trivial connection and the other without explicit exhibition of a connection) are principally equivalent. The indices in (1.1) are basically different. The greek indices refer to the coordinate space and describe the behavior of tensors under the group of diffeomorphisms of the manifold M. The italic indices denote different 1-forms of the coframe. The corresponding group of transformations, SO共1 , 3兲, comes together with its natural invariant ab = diag共1 , −1 , −1 , −1兲. The metric tensor on M is expressed via the coframe as g = aba 丢 b ,
共1.2兲
i.e., the coframe is postulated to be pseudo-orthonormal. The coframe field and all the objects constructed from it are assumed to be global (rigid) covariant. In other words, all the constructions are required to be covariant under the global transformations a → Aabb with a constant matrix Aab 苸 SO共1 , 3兲. The metric tensor (1.2) is invariant under a wider group of transformation: local (pointwise) transformations of the coframe with Aab = Aab共x兲. Consider a Lagrangian density, which is (i) diffeomorphism invariant, (ii) invariant under global SO共1 , 3兲 transformations of the coframe, and (iii) quadratic in the exterior derivatives of the coframe. The most general Lagrangian of this form is a linear combination,23,26 3
1 L= i共i兲L, 2 i=1
兺
共1.3兲
where 1 , 2 , 3 are free dimensionless parameters. The linear independent 4-forms appearing here are expressed via the coframe field strength, Ca : = da, 共1兲
共2兲
L = Ca ∧ * Ca ,
共1.4兲
L = 共Ca ∧ a兲 ∧ * 共Cb ∧ b兲,
共1.5兲
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012501-3
J. Math. Phys. 46, 012501 (2005)
Weak field reduction in teleparallel gravity 共3兲
L = 共Ca ∧ b兲 ∧ * 共Cb ∧ a兲.
共1.6兲
The Hodge dual operator * is defined by the pseudo-orthonormal coframe a or, equivalently, by the metric (1.2). One may try to include in the Lagrangian some invariant expressions of the second order (similarly to the Hilbert–Einstein Lagrangian). Such terms, however, are reduced to total derivatives and do not affect the field equations and the Noether conserved currents. So (1.3) is the most general Lagrangian that generates the field equations of the second order. Let us introduce the notion of the field strength Fa: = 共1兲Fa + 共2兲Fa + 共3兲Fa ,
共1.7兲
with 共1兲
Fa: = 共1 + 3兲Ca ,
共1.8兲
Fa: = 2bea c 共m ∧ Cm兲,
共1.9兲
Fa: = − 3a ∧ 共bem c Cm兲.
共1.10兲
共2兲
共3兲
Such separation of the strength Fa involves two scalar-valued forms m ∧ Cm and em c Cm. So some calculations are simplified. For irreducible decomposition of Fa, see Refs. 5 and 23. In the notations (1.8)–(1.10), the coframe Lagrangian (1.3) takes a form similar to the Maxwell Lagrangian, L = 21 Ca ∧ * Fa .
共1.11兲
The free variation of (1.11) relative to the coframe a must take into account also the variation of the Hodge dual operator, which implicitly depends on the coframe. It yields the field equation of the form23 d * Fa = Ta ,
共1.12兲
where the 3-form Ta is the energy-momentum current of the coframe field Ta = 共bea c Cm兲 ∧ * Fm − bea c L.
共1.13兲
The conservation law for this 3-form: dTa = 0 is a straightforward consequence of (1.12). B. Viable models—a problem of physical motivation
A general quadratic coframe model, which is global SO共1 , 3兲 invariant, involves three parameters,
1,
2,
3 — free.
共1.14兲
The ordinary GR is extracted from this family by requiring of the local SO共1 , 3兲 invariance, which is realized by the following restrictions of the parameters:
1 = 0,
22 + 3 = 0.
共1.15兲
The analysis of exact solutions28 to the field equation (1.12) shows that the Schwarzschild solution appears even for a wider set of parameters (viable set),
1 = 0,
2,
3 — free.
共1.16兲
Moreover, for 1 ⫽ 0, spherical-symmetric static solutions to (1.12) do not have the Newtonian behavior at infinity.28
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012501-4
J. Math. Phys. 46, 012501 (2005)
Yakov Itin
So a problem arises: Which physical motivated requirement extracts the viable set of parameters? The quantum-theory solution to this problem is known for a long time. In Refs. 29–33 it was shown that the requirement (1.16) is necessary and sufficient for the absence of ghosts and tachyons in particle content of the theory. Another motivation for (1.16) comes from the requirement that the theory must have a well-defined Hamiltonian formulation (Ref. 34). In this paper we look for a motivation of (1.16) on a classical Lagrangian level. We deal with linear approximation of the general coframe model. The coframe variable can be treated, in this approximation, as a regular 4 ⫻ 4 matrix. Consequently, it reduced to a composition of two independent variables: the symmetric and the antisymmetric fields. Our main result is as follows: Only for (1.16), the coframe model is reduced to two independent models, every one with its own Lagrangian, field equation, and conserved current. In other words, the viable model is exactly this one that approaches the free-field limit, i.e., any interaction between the approximately independent fields appears only in higher orders. Linear approximation of coframe models was usually applied for studying the deviation of teleparallel gravity from the standard GR, and for comparison with the observation data, see Refs. 3, 4, 30, and 31. In our approach the reduction of the lower order terms is used as a theoretical device. We show that this condition is enough to distinguish the set of viable models. The relation between these two approaches requires a further consideration. II. WEAK FIELD REDUCTION A. Linear approximations
To study the approximate solutions to (1.12), we start with a trivial exact solution, a holonomic coframe, for which, da = 0.
共2.1兲
Consequently, Fa = Ca = 0, so both sides of Eq. (1.12) vanish. By Poincaré’s lemma, the solution of ˜ a共x兲, where ˜xa共x兲 is a set of four smooth functions defined (2.1) can be locally expressed as a = dx in some neighborhood U of a point x 苸 M. The functions ˜xa共x兲, being treated as the components of a coordinate map ˜xa : U → R4, generate a local coordinate system on U. The metric tensor (1.2) ˜ a 丢 dx ˜ b. Thus the holoreduces, in this coordinate chart, to the flat Minkowskian metric g = abdx nomic coframe plays, in the teleparallel background, the same role as the Minkowskian metric in the (pseudo-)Riemannian geometry. Moreover, a manifold endowed with a (pseudo-)orthonormal holonomic coframe is flat. The weak perturbations of the basic solution a = dxa are
a = dxa + ha = 共␦ab + hab兲dxb .
共2.2兲
“Weak” means 储hab储 = ⑀ = o共1兲,
储hab,c储 = O共⑀兲,
储hab,c,d储 = O共⑀兲,
共2.3兲
where 储 ¯ 储 denotes the maximal tensor norm. We accept that the coframe and the holonomic coframe dxa have the same physical dimension of [length]. Thus, the components of the matrix hab and the parameter ⑀ are dimensionless. Consequently, the approximation conditions (2.3) are invariant under rescaling of the coordinates. In this paper we will take into account only the first order approximation in the perturbations hab and in their derivatives (i.e., in the parameter ⑀). Note that, in this approximation, the difference between coframe and coordinate indices completely disappears. This justifies our choice, in (2.2) and in the sequel, of the same notation for these (basically different) indices. In accordance with (2.3), only weak coordinate transformations are considered. Under a shift a
xa 哫 xa + a共x兲,
共2.4兲
the components of the coframe are transformed as
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012501-5
J. Math. Phys. 46, 012501 (2005)
Weak field reduction in teleparallel gravity
hab 哫 hab − a,b .
共2.5兲
Thus, in order to preserve the weakness of the fluctuation, it is necessary to require a,b = O共储hab储兲. We will use the term approximately covariant35 for the expressions which are covariant only to the first order of the perturbations. Observe that this assumption restricts only the amplitudes of the perturbations and of their derivatives. It does not restrict, however, the local freedom to transform the coordinates. An appropriative coordinate system can still be chosen in a small neighborhood of the identity transformation in order to simplify the (local) field equations. Similarly, in order to be in agreement with the approximation condition (2.3), the global SO共1 , 3兲 transformations of the coframe field, a 哫 Aabb, must also be restricted. It is enough to require the transformations to be in a small neighborhood of the identity Aab = ␦ab + ␣ab,
储␣ab储 = o共1兲.
共2.6兲
B. Reduction of the field
In (2.2), hab is a perturbation of the flat coframe. Thus we have the following. (i)
To the first order, the holonomic coframe is expressed by the unholonomic one as dxa = 共␦ab − hab兲b .
(ii)
The indices in h
a b
can be lowered and raised by the Minkowskian metric, hab: = amhmb,
(iii)
hab: = bmham .
wab: = h关ab兴 = 21 共hab − hba兲
and
共2.9兲
as well as the trace : = hmm = mm are covariant to the first order. The components of the metric tensor, in the linear approximation, involve only the symmetric combination of the coframe perturbations, gab = ab + 2ab .
(v)
共2.8兲
The first operation is exact 共covariant to all orders of approximations兲, while the second is covariant only to the first order, when gab ⬇ ab. The symmetric and the antisymmetric combinations of the perturbations,
ab: = h共ab兲 = 21 共hab + hba兲 (iv)
共2.7兲
共2.10兲
Under the transformations (2.4), two covariant pieces of the fluctuation change as
ab 哫 ab − 共a,b兲
and
wab 哫 wab − 关a,b兴 .
共2.11兲
Thus the approximately covariant irreducible decomposition of the dynamical variable hab = ab + wab
共2.12兲
is obtained. Thus, instead of one field hab, we have, in this approximation, two independent fields: a symmetric field ab and an antisymmetric field wab. C. Gauge conditions
The actual values of the components of the fields ab and wab depend on a choice of a coordinate system. Thus four arbitrary relations between the components (equal to the number of coordinates) may be imposed. We require these relations to be Lorentz invariant, i.e., covariant in the first order approximation. Thus the most general form of constraints (gauge conditions) that involve the first order derivatives is
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012501-6
J. Math. Phys. 46, 012501 (2005)
Yakov Itin
␣am,m + ,a + ␥wam,m = 0,
共2.13兲
where ␣ ,  , ␥ are dimensionless parameters. Certainly, for some special values of the parameters, these conditions cannot be realized. Indeed, under the coordinate transformations (2.4), Eq. (2.13) changes, in the lowest order, to ˜ ,m +  ˜ + ␥w ˜ am,m = 共␣共a,m兲 + m,a + ␥关a,m兴兲,m . ␣ ,a am
共2.14兲
Thus the conditions (2.13) can be realized, by the coordinate transformations (2.4), if and only if the system of PDE (2.14) has a solution 共x兲 for a given left-hand side (LHS). Let us check the integrability of this system. Equation (2.14) results in ,m ˜ ˜ ˜ am,b,m . 共␣共a,m兲,b + m,a,b + ␥关a,m兴,b兲,m = ␣ am,b + ,a,b + ␥w
共2.15兲
Commuting the indices a and b, we obtain 共␣ + ␥兲 䊐 关a,b兴 = 2共␣m关a,b兴 − ␥wm关a,b兴兲,m .
共2.16兲
Thus, the gauge condition (2.13) with ␣ = −␥ ⫽ 0 cannot be realized by any change of the coordinate system. Now, take the trace of (2.15), 共␣ + 兲 䊐 m,m = ␣mn,m,n +  䊐 .
共2.17兲
Thus ␣ = − ⫽ 0 is also forbidden. We will apply, in the sequel, two separate gauge conditions: for the symmetric field
am,m − 21 ,a = 0,
共2.18兲
wam,m = 0.
共2.19兲
and for the antisymmetric field
Observe, that (2.18) and (2.19) cannot be realized simultaneously by the same coordinate transformation. Indeed, for this, the coordinate functions must satisfy 䊐 a = 2am,m − ,a
and
䊐 a − 共m,m兲,a = wam,m .
共2.20兲
The integrability conditions for these equations yield 䊐 关a,b兴 = 2m关a,b兴,m = − wm关a,b兴,m .
共2.21兲
For arbitrary independent fields ab and wab, these conditions are not satisfied. Certainly, the conditions (2.18) and (2.19) can be realized, separately, by transformation of the coordinates. D. Reduction of the field strengths
By (2.3), let us decompose the field strengths (1.8)–(1.10). The 2-form Ca is approximated by Ca = hab,c dxc ∧ dxb = − ha关b,c兴b ∧ c = − 共a关b,c兴 + wa关b,c兴兲b ∧ c .
共2.22兲
Consequently, the first part of the field strength (1.8), takes the form 共1兲
Fa = − 共1 + 3兲共a关b,c兴 + wa关b,c兴兲b ∧ c .
共2.23兲
As for the second part (1.9), it involves only the antisymmetric field, 共2兲
Fa = − 32w关ab,c兴 b ∧ c .
共2.24兲
The third part (1.10), takes the form
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012501-7
J. Math. Phys. 46, 012501 (2005)
Weak field reduction in teleparallel gravity 共3兲
Fa = 3ac共hmb,m − h,b兲 b ∧ c = 3ac共bm,m − ,b − wbm,m兲b ∧ c .
共2.25兲
Therefore, the field strength is reduced to the sum of two independent strengths—one defined by the symmetric field ab and the second one defined by the antisymmetric field wab, Fa共mn,wmn兲=共sym兲Fa共mn兲+共ant兲Fa共wmn兲,
共2.26兲
where 共sym兲
Fa = − 关共1 + 3兲a关b,c兴 + 3a关b兴关c兴m,m − 3a关b兴,关c兴兴b ∧ c
共2.27兲
Fa = − 关共1 + 3兲wa关b,c兴 + 32w关ab,c兴 − 3a关b兴w关c兴m,m兴b ∧ c .
共2.28兲
and 共ant兲
Hence, for arbitrary values of the parameters i, the field strengths are independent. E. Reduction of the field equations
The field equation (1.12) includes the second order derivatives of the perturbations on its LHS and the squares of the first order derivatives on both sides. In the linear approximation (2.3), the quadratic terms can be neglected. Thus, (1.12) is approximated by d * Fa = 0.
共2.29兲
The covector valued 2-form Fa can be expressed in the unholonomic basis as Fa = Fabcb ∧ c / 2. Accordingly, we derive d * Fa = 21 Fabc,m dxm ∧ * 共b ∧ c兲 = − 21 Fabc,m * 关bem c 共b ∧ c兲 c = 21 Fa关bc兴,c * b . Consequently, Eq. (2.29) reads 共2.30兲
Fa关bc兴,c = 0.
Applying the antisymmetrization of the corresponding indices to the expression (2.26) we derive the linearized field equation 共1 + 3兲共䊐 ab − am,b,m兲 + 3共− ab 䊐 − mb,m,a + ,a,b + abmn,m,n兲 + 共1 + 22 + 3兲共䊐wab − wam,b,m兲 + 共22 + 3兲wbm,a,m = 0.
共2.31兲
Proposition 1: For the case 1 = 0, the linearized coframe field equation (2.31), in arbitrary coordinates, splits into two independent systems, 共sym兲
E共ab兲共mn兲 = 0
and
共ant兲
E关ab兴共wmn兲 = 0.
If 1 ⫽ 0, Eq. (2.31) does not split in any coordinate system. Proof: The equation (2.31) is tensorial to the first order. Thus, by applying symmetrization and antisymmetrization operations, it is reduced covariantly to a system of two independent tensorial (to the first order) equations. The symmetrization yields a system of 10 independent equations, 䊐关共1 + 3兲ab − 3ab兴 − 共1 + 23兲m共a,b兲,m + 3共,a,b + abmn,m,n兲 + 1wm共a,b兲,m = 0. 共2.32兲 The antisymmetrization yields a system of six independent equations,
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012501-8
J. Math. Phys. 46, 012501 (2005)
Yakov Itin
共1 + 22 + 3兲 䊐 wab + 共1 + 42 + 23兲wm关a,b兴,m − 1m关a,b兴,m = 0.
共2.33兲
Evidently, the condition 1 = 0 removes the “mixed terms” and yields the separation of the system. Such splitting holds in arbitrary system of coordinates. Suppose now 1 ⫽ 0. Thus, the “mixed terms” remain in both equations—the w term in (2.32) and the term in (2.33). Let us try to remove these terms by an appropriative choice of a coordinate system. For this we must require the equations
m关a,b兴,m = 0
and
wm共a,b兲,m = 0
to hold simultaneously. These equations can be satisfied only if
ma,m = 0
and
wma,m = 0.
共2.34兲
The actual values of the variables ab and wab depend on a choice of a coordinate system. Recall that the approximation conditions (2.3) do not restrict the freedom to choose the local coordinate transformations. Therefore, by (2.4), four additional conditions (equal to the number of coordinates), can still be applied to the perturbations in order to satisfy (2.34). We need, however, to eliminate eight independent expressions wma,m and ma,m. This cannot be done by four independent functions of the coordinates. Indeed, under the transformations (2.4),
ma,m 哫 ma,m − 共m,a兲,m ,
共2.35兲
wma,m 哫 wma,m − 关m,a兴,m .
共2.36兲
Hence the coordinate transformations must satisfy
共m,a兲,m = ma,m
and
关m,a兴,m = wma,m
共2.37兲
simultaneously. Therefore,
m,a,m = hma,m .
共2.38兲
The consistency condition for (2.38) is hma,b,m = hmb,a,m , which it is not satisfied in general. 䊏 Consequently, for 1 = 0 and generic values of the parameters 2 , 3, the field equation of the coframe field is reduced to two independent field equations for independent field variables. (i)
The symmetric field ab of 10 independent variables satisfies the system of 10 independent equations, 共sym兲
E共ab兲共mn兲: = 3关䊐共ab − ab兲 − m共a,b兲,m + ,a,b + abmn,m,n兴 = 0.
共2.39兲
We rewrite it as 䊐共ab − ab兲 − 共am,m − 21 ,a兲,b − 共bm,m − 21 ,b兲,a + abmn,m,n = 0.
共2.40兲
Substituting here the condition 共2.18兲 and its consequence
mn,m,n = 21 䊐
共2.41兲
䊐共ab − 21 ab兲 = 0.
共2.42兲
we obtain
Equation 共2.42兲 results in 䊐 = 0. Then it is equivalent to
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012501-9
Weak field reduction in teleparallel gravity
J. Math. Phys. 46, 012501 (2005)
䊐 ab = 0.
(ii)
共2.43兲
Consequently, in the coordinates associated with 共2.18兲, the symmetric field satisfied the wave equation. The antisymmetric system of six independent equations for six independent variables, 共ant兲
E关ab兴共wmn兲: = 共22 + 3兲共䊐wab + 2wm关a,b兴,m兲 = 0.
共2.44兲
In the coordinates associated with 共2.19兲 it is reduced to the wave equation, 䊐wab = 0.
共2.45兲
F. Reduction of the Lagrangian
In the sequel of this paper, we consider the models with parameter 1 = 0. Let us examine now the reduction of the Lagrangian (1.3). Proposition 2: For 1 = 0, the Lagrangian of the coframe field is reduced, up to a total derivative term, to the sum of two independent Lagrangians, L共ab,wab兲 = 共sym兲L共ab兲 + 共ant兲L共wab兲.
共2.46兲
Proof: With 1 = 0 the term 共1兲L does not appear in the Lagrangian. Calculate in the linear approximation (we use the abbreviation ab¯ = a ∧ b ∧ ¯), 共2兲
L = 共da ∧ a兲 ∧ * 共db ∧ b兲 = ham,nhbp,qnma ∧ * qpb .
共2.47兲
Applying the formula ⬘兴 b⬘ 关c⬘兴 abc ∧ * a⬘b⬘c⬘ = 6␦关a a ␦b ␦c * 1
共2.48兲
we derive 共2兲
L = 2wab,c共wab,c + wca,b + wbc,a兲 * 1.
共2.49兲
So 共2兲L depends only on the antisymmetric field. Consider now the linear approximation to the term 共3兲L, 共3兲
L = 共da ∧ b兲 ∧ * 共db ∧ a兲 = ham,nhbp,qnmb ∧ * qpa .
共2.50兲
Use (2.48) to get 共3兲
L = 关hab,c共hab,c − hac,b兲 − hab,ahcb,c + ,a共2hba,b − ,a兲兴 * 1.
共2.51兲
Insert here the splitting (2.12). It follows that the Lagrangian (2.51) is reduced to the sum 共3兲
L = 共3兲L共兲 + 共3兲L共w兲 + 共3兲L共,w兲,
共2.52兲
L共兲 = 关ab,c共ab,c − ac,b兲 − ab,acb,c + ,a共2ba,b − ,a兲兴 * 1,
共2.53兲
where 共3兲
共3兲
共3兲
L共w兲 = 关wab,c共wab,c − wac,b兲 − wab,awcb,c兴 * 1,
共2.54兲
L共,w兲 = 2关− ab,cwac,b + ,awba,b − ab,awcb,c兴 * 1.
共2.55兲
Extracting the total derivatives in the mixed term (2.55) we obtain
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012501-10
Yakov Itin
J. Math. Phys. 46, 012501 (2005)
共3兲
共2.56兲
L共,w兲 = 共ab共wac,b − wbc,a兲,c − wba,a,b兲 * 1 + exact terms.
The terms in the square brackets vanish identically as a product of symmetric and antisymmetric tensors. Thus the mixed term 共3兲L共 , w兲 is a total derivative. Consequently, desired reduction of the Lagrangian is obtained. 䊏 The Lagrangian of the symmetric field 共sym兲L = 共3兲L共兲 may be rewritten in a more compact form. Observing the identity
ab,acb,c = ab,ccb,a + exact terms,
共2.57兲
and extracting the total derivatives, we obtain 共sym兲
L = 21 3关ab,c共ab,c − 2ac,b兲 + ,a共2ba,b − ,a兲兴 * 1.
共2.58兲
This form of the Lagrangian is acceptable in arbitrary coordinates. In the coordinates associated with the condition (2.18), the last parentheses in (2.58) vanish. In the first parentheses, we extract the total derivatives and use (2.18) to derive (symbol ⬇ used here for equality up to total derivatives)
ab,cac,b = 共abac,b兲,c − abac,b,c ⬇ − 21 ab,a,b ⬇ 21 ab,b,a ⬇ 41 ,a,a . Consequently the symmetric field Lagrangian (2.53) is reduced to 共sym兲
L = 21 共ab,cab,c − 21 ,a,a兲 * 1.
Analogously, for the Lagrangian of the antisymmetric field identity
共ant兲
共2.59兲 L = 共2兲L + 共3兲L共w兲, we use the
wab,awcb,c = wab,cwac,b + exact terms
共2.60兲
and rewrite it, in an arbitrary system of coordinates, as 共ant兲
L = 21 共22 + 3兲关wab,c共wab,c − 2wac,b兲兴 * 1,
共2.61兲
or, equivalently, as L共w兲 = 21 共22 + 3兲共wab,c共wab,c − wac,b兲 − wab,awcb,c兲 * 1. The gauge condition (2.19) removes the last term while the second term is rewritten as wab,cwac,b ⬇ − wabwac,b,c ⬇ 0. Thus, the Lagrangian of the antisymmetric field is ˜ 共w兲 = 1 共2 + 兲w wab,c * 1. L 2 3 ab,c 2
共2.62兲
G. Reduction of the energy-momentum current
The Lagrangian of the coframe field is decomposed, in the first order approximation, to a sum of two independent Lagrangians for two independent fields. The Noether current expression, being derivable from the Lagrangian, must have the same splitting. Proposition 3: The coframe energy-momentum current is reduced, on shell, in the first order approximation, as Ta共mn,wmn兲 = 共sym兲Ta共mn兲 + 共ant兲Ta共wmn兲,
共2.63兲
up to a total derivative. Proof: The coframe energy-momentum current is of the form
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012501-11
J. Math. Phys. 46, 012501 (2005)
Weak field reduction in teleparallel gravity
Ta = 共bea c Cm兲 ∧ * Fm − bea c L.
共2.64兲
Due to Proposition 2, the second term, in the first order approximation, does not contain the mixed terms ⬘ · w⬘. Hence, it already has the reduced form. To treat the first term, we write the strengths in the component Cm = Cm关bc兴b ∧ c,
Fm = Fm关pq兴 p ∧ q .
共2.65兲
Thus, the first term of (2.64) is approximated by 共bea c Cm兲 ∧ * Fm = Cm关bc兴Fm关pq兴共bea c bc兲 ∧ * pq = 4Cm关an兴Fm关bn兴 * b = 4hm关a,n兴Fm关bn兴 * b . 共2.66兲 The 3-form *b, in the lowest order approximation, is an exact form. Thus, it is enough to show that the scalar factor, on the right-hand side (RHS) of (2.66), has the desired splitting. This expression is a sum of two terms. The first one is proportional to hma,nFm关bn兴 = − hmaFm关bn兴,n + total derivatives, i.e., it is, on shell, an exact form. Now we must show that the second term, which is proportional to hma,nFm关bn兴, does not involve the mixed products of a type · w. The mixed product expression in the latter term is proportional to
mn,a共wmb,n + 2m关n兴w关b兴k,k兲 + wmn,a共mb,n + mbnk,k − mb,n兲.
共2.67兲
By recollection of the terms, we rewrite this expression as 共mn,awmb,n + mn,nwbm,a兲 + 共,awbm,m − ,mwbm,a兲 + 共mb,nwmn,a − bm,awmn,n兲.
共2.68兲
The three brackets above are total derivatives, namely, 关共mn,awmb兲,n + 共mn,nwbm兲,a兴 + 关共wbm,m兲,a − 共wbm,a兲,m兴 + 关共mb,nwmn兲,a − 共bm,awmn兲,n兴. 共2.69兲 Thus, (2.66) and, consequently, (2.64) do not involve the mixed terms. The desired splitting is proved. 䊏 The energy-momentum tensor Tab can be derived from the Noether current Ta by applying the relations T a = T ab * b,
Tab = bebc * Ta .
共2.70兲
Proposition 4: For the field ab in the coordinate system associated with the gauge condition
am,m − 21 ,a = 0,
共2.71兲
Tab = 21 关共mn,amn,b − 41 ablm,nlm,n兲 − 21 共,a,b − 41 ab,m,m兲兴.
共2.72兲
the energy-momentum tensor is
This tensor is symmetric and traceless. Proof: We start with the energy-momentum current for the coframe field Ta = 共bea c Cm兲 ∧ * Fm − bea c L. Due to Proposition 3, in the first order approximation, this current is decomposed to two independent currents. Thus we may assume wab = 0 in order to derive the expression for Ta共兲. In the coordinates associated with the gauge condition (2.71), by (2.59),
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012501-12
J. Math. Phys. 46, 012501 (2005)
Yakov Itin
eaL = 21 3共mn,pmn,p − 21 ,m,m兲 * a . The first term of Ta is derived from (2.64), 共bea c Cm兲 ∧ * Fm = 4m关a,n兴Fm关bn兴 * b = 2共ma,nFm关bn兴 * b − mn,aFm关bn兴 * b兲. Observe that, on shell, up to a total derivative
ma,nFm关bn兴 ⬇ − maFm关bn兴,n = 0. Thus, 共bea c Cm兲 ∧ * Fm = − 2mn,aFm关bn兴 * b . Applying the gauge condition to (2.26) we get Fa = − 3关a关b,c兴 + a关b兴共关c兴m,m − ,关c兴兲bc = − 3共a关b,c兴 − 21 a关b兴,关c兴兲bc . Consequently, 共bea c Cm兲 ∧ * Fm = 23mn,a共m关b,n兴 − 21 m关b兴,关n兴兲 * b . Extracting the total derivatives
mn,amb,n ⬇ mn,nmb,a ⬇ 21 ,mmb,a ⬇ 41 ,a,b , mn,amb,n ⬇ mn,n,a ⬇ 41 ,a,b , it follows that 共bea c Cm兲 ∧ * Fm = 3共− 2mn,amn,b + ,a,b兲 * b . Collecting the terms into Ta and extracting the energy-momentum tensor Tab from the current Ta by Tab = ebc * Ta we get the desired expression. It is clear that energy-momentum tensor is symmetric and traceless. 䊏 In GR, the behavior of small perturbations of the metric tensor is managed by the wave equation. Thus, for a wave propagating in the positive direction of the x axis, only two independent components of the matrix ab remain,
23 = 共兲,
22 = − 33 = 共兲,
where
= t − x.
共2.73兲
The calculation of the energy-momentum tensor for the symmetric field by use of the tensor (2.72) yields Tab = k共,a,b + ,a,b兲.
共2.74兲
2 + 41 共˙ 22 − ˙ 33兲2兲. T01 = − 3共˙ 23
共2.75兲
The energy flux reads
Observe that the expressions (2.74) and (2.75) are the same as the expressions obtained in GR from the energy-momentum pseudotensors. Let us turn now to the antisymmetric field. Proposition 5: In the coordinate system associated with the gauge condition wam,m = 0,
共2.76兲
the energy-momentum tensor of the antisymmetric field is
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012501-13
Weak field reduction in teleparallel gravity
J. Math. Phys. 46, 012501 (2005)
Tab = − 共22 + 3兲共wmn,awmn,b − 41 abwmn,pwmn,p兲.
共2.77兲
This tensor is traceless and symmetric. Proof: The current of the symmetric and of the antisymmetric fields are decoupled. Thus we may assume ab = 0. In the coordinates associated with the gauge condition (2.76), bea c L = 21 共22 + 3兲wab,cwab,c * b . As for the first term of Ta共w兲 we derive from (2.66), 共bea c Cm兲 ∧ * Fm = 4wm关a,n兴Fm关bn兴 * b = 2共wma,nFm关bn兴 − wmn,aFm关bn兴兲 * b . The first term vanishes, on shell, up to a total derivative, wma,nFm关bn兴 ⬇ − wmaFm关bn兴,n = 0. Thus, 共bea c Cm兲 ∧ * Fm = − 2wmn,aFm关bn兴 * b . Inserting the gauge condition (2.76) into (2.26) we derive Fa = − 共3wa关b,c兴 + 32w关ab,c兴兲bc . Hence, 共bea c Cm兲 ∧ * Fm = 2共3wma,nwm关b,n兴 + 32wma,nw关mb,n兴兲 * b . Extract the total derivatives and use the gauge condition to get wmn,awmb,n ⬇ wmn,nwmb,a ⬇ 0, wmn,awbn,m ⬇ wmn,mwbn,a ⬇ 0. Consequently, 共bea c Cm兲 ∧ * Fm = − 共22 + 3兲wmn,awmn,b . The desired expression (2.77) is obtained now by collecting the terms.
䊏
III. THE ROLE OF THE PARAMETERS 1
The case 1 = 0 is extracted in coframe models by existence of a unique spherical symmetric static solution. Since the exact solution yields the Schwarzschild metric this condition generates a viable subclass of gravity coframe models. We have involved an independent criteria. Namely, we have shown that only in the case 1 = 0 the weak perturbations of the coframe reduce to two independent fields with their own Lagrangian dynamics. Consequently the models have a free field limit. This effect is correlated to the recently obtained result 34 concerning the Hamiltonian dynamics behavior. It is interesting to note that in the two-dimensional coframe gravity only one term in the Lagrangian preceded by 1 appears. Thus the corresponding reduction of fields is impossible. 1
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012501-14
Yakov Itin
J. Math. Phys. 46, 012501 (2005)
9
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