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Advances in Mathematical Physics
Volume 2016, Article ID 3645129, 15 pages
http://dx.doi.org/10.1155/2016/3645129
Research Article

On the Splitting of the Einstein Field Equations with respect to General Threading of Spacetime

Department of Mathematics, Kuwait University, P.O. Box 5969, 13060 Safat, Kuwait

Received 5 July 2016; Accepted 3 October 2016

Academic Editor: Eugen Radu

Copyright © 2016 Aurel Bejancu and Hani Reda Farran. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Based on general threading of the spacetime , we obtain a new and simple splitting of both the Einstein field equations (EFE) and the conservation laws in . As an application, we obtain the splitting of EFE in an almost FLRW universe with energy-momentum tensor of a perfect fluid. In particular, we state the perturbation Friedmann equations in an almost FLRW universe.

1. Introduction

The present paper is a continuation of [1], wherein an new approach on the threading of spacetime with respect to an arbitrary timelike vector field has been developed. The study in [1] refers to Lorentz metrics given by (2a), (2b), and (2c) and subject to the condition that is independent of the time coordinate. We remove this condition, and therefore the results are valid for any Lorentz metric of a spacetime. Another important issue of the present paper is that the whole study is developed in the general setting of a spacetime with a spatial distribution that is not necessarily integrable. The threading frames and coframes, the spatial tensor fields, and the Riemannian spatial connection are the main tools used throughout the paper. These geometric objects enable us to obtain new and simple splitting of EFE and to apply it to the structure of an almost FLRW universe. The new approach developed in this paper can be extended to threading of higher-dimensional universes. In this respect, we mention [2], wherein threading of a universe has been developed.

Now, we outline the content of the paper. In Section 2 we present the main geometric objects which constitute the foundation of a general threading formalism of a spacetime with respect to an arbitrary timelike vector field. We close this section with local expressions of the Levi-Civita connection in terms of spatial tensor field and of the local coefficients of the Riemannian connection (cf. (23a), (23b), (23c), and (23d)). In Section 3 we state, for the first time in the literature, the spatial Bianchi identities in the general case when the spatial distribution is not necessarily integrable (cf. (37), (38), and (39)). The structure equations on induced by the threading formalism are presented in Section 4 (cf. (47a), (47b), (47c), (47d), (48a), (48b), (48c), and (48d)). They play an important role in the next sections, wherein we relate tensor fields on with spatial tensor fields. In Section 5 we obtain simple expressions for the local components of the Ricci tensor of with respect to the threading frame field and for the scalar curvature (cf. (63a), (63b), (63c), (64a), (64b), (64c), (66a), and (69)). The splittings of both the Einstein gravitational tensor field and the energy-momentum tensor field are stated in Section 6 (cf. (73a), (73b), (73c), (81), (84), and (86)). In Section 7 we obtain the spatial, mixed, and temporal EFE (cf. (89), (90a), (90b), and (91)). Also, we state (93) which, in the particular case when the threading is taken with respect to a unit vector field, becomes the well-known Raychaudhuri-Ehlers equation. A new splitting of conservation laws with respect to general threading of spacetime is given in Section 8 (cf. (98), and (99)). Also, we compare our results with what is known in the literature on this matter. Finally, in Section 9 we apply the general theory developed in the paper to the threading of an almost FLRW universe. We close the paper with Conclusions and Appendices A, B, and C.

2. The Threading Formalism with respect to a Nonnormalized Timelike Vector Field

Recently, a new approach on the threading of spacetime with respect to a nonnormalized timelike vector field has been developed (cf. [1]). In the most general setting that we explain in this section, we recall the main geometric objects introduced in [1]. Also, we introduce the extrinsic curvature tensor field for the spatial distribution and use it in the expressions of the Levi-Civita connection on a spacetime.

Let be a spacetime and be a timelike vector field on that is not necessarily normalized. Then, the tangent bundle of admits the decomposition where is the temporal distribution spanned by and is the spatial distribution that is complementarily orthogonal to in .

Throughout the paper we use the ranges of indices: and . Also, for any vector bundle over , denote by the -module of smooth sections of , where is the algebra of smooth functions on .

The congruence of curves that is tangent to determines a coordinate system on such that . Next, we putwherein is a nonzero function on .

Remark 1. Note that, in [1], was supposed to be independent of . Here, we remove this condition on , and thus the results stated in the present paper are valid for any Lorentz metric on .

In this approach we use the threading frame field and the threading coframe field defined as follows: where we put The Lie brackets of the vector fields from the threading frame are given by where we set Taking into account that the Levi-Civita connection on is torsion-free, from (5a) we deduce that Thus, , define the vorticity tensor field on . By using the Jacobi identity we deduce that the vorticity tensor field satisfies the identities where is the cyclic sum with respect to .

Now, we denote by the local components of the Riemannian metric induced by on , with respect to the basis in , and obtain Then the lone element of is expressed in terms of threading coframe as follows: Also, we define the expansion tensor field , the expansion function , and the shear tensor field as follows:

Raising and lowering indices , are performed by using and , as in the following examples: The expansion and vorticity tensor fields enable us to define the extrinsic curvature tensor field of the spatial distribution by its local components or equivalently by By using (5a) and (15), we see that is a symmetric tensor field if and only if is integrable.

Remark 2. The extrinsic curvature tensor field was intensively used in the decomposition of the spacetime (cf. [3, pp 509–516]). As far as we know, the tensor field given by (14) or (15) is considered here for the first time in a study of the threading of spacetime.

Next, in order to justify the tensorial meaning of the above quantities, we define a spatial tensor field of type on , as an -multilinear mapping: where is the dual vector bundle to . The local components of , with respect to a threading frame and coframe, are given by and satisfywith respect to the coordinate transformations on . For example, and define spatial tensor fields of types and , respectively.

An important geometric object is the Riemannian spatial connection, which is a metric linear connection on the spatial distribution, given bywhere is the projection morphism of on with respect to decomposition (1). Locally, is given by where we put Throughout the paper, the covariant derivatives defined by will be denoted by a vertical bar “.” As an example, for a spatial tensor field , we have A covariant derivative as in (22a) (resp., (22b)) is called a spatial covariant derivative (resp., temporal covariant derivative) of the spatial tensor field .

Finally, by direct calculations, using the Riemannian spatial connection and the above spatial tensor fields, we express the Levi-Civita connection on as follows: where we put

Remark 3. It is worth mentioning that all the equations we state in the paper are expressed in terms of spatial tensor fields and their covariant derivatives defined by the Riemannian connection.

Remark 4. As the threading of spacetime considered in this paper contains as a particular case the threading with respect to a unit timelike vector field, we call it the general threading of spacetime. The advantage of this general setting on the splitting of spacetime is that it can be applied to any Lorentz metric of a spacetime.

3. Bianchi Identities for the Riemannian Spatial Connection

In earlier literature on the threading of spacetime, we find the so-called three-dimensional derivative operator (cf. of [4]). With respect to this operator, we have the following remarks:(i)It is neither a linear connection on nor a linear connection on .(ii)As a consequence of (i), for the general case when is not integrable, then a curvature tensor field for this operator could be not defined.

Contrary to this situation, given by (19a) is a metric linear connection on the vector bundle , and therefore it has a curvature tensor field given by Locally, we put and by using (25), (26a), (26b), (20a), (20b), (5a), and (5b), we obtain Since is a metric linear connection, we have where we put As a consequence of (28) we deduce that , which implies via (27b) and (14). Thus, in any cosmological model of a universe, the expansion function must satisfy the system of given by (30).

Remark 5. Note that and define spatial tensor fields of types and , respectively. Also, from (27b) we see that define a spatial tensor field of type . However, do not define a spatial tensor field.

Remark 6. Comparing (27a) with (15.4) from [5], we see that the so-called Zelmanov curvature tensor field is given by the first four terms from (27a). Moreover, from of [5] we see that such a tensor field becomes a curvature tensor field, if and only if is an integrable distribution.

Next, we extend the Riemannian spatial connection on to a linear connection on given by where is the projection morphism of on with respect to (1). Clearly, coincides with on and therefore locally is given by (20a), (20b), and We recall that the torsion and curvature tensor fields of are given by Then, by direct calculations, using (33a), (33b), (20a), (20b), (32a), (32b), (5a), (5b), (23c), (26a), and (26b), we deduce that Now, in order to find some Bianchi identities for the Riemannian spatial connection, we recall that the Bianchi identities for the linear connection are given by (cf. [6, p. 135]) where is the cyclic sum with respect to . In order to use (35a), we note that Then, take , , and in (35a) and, by using (36a), (36b), and (34a), we infer that the spatial component in (35a) is expressed as follows: Taking temporal part in (35a) and any other triplet from the threading frame we obtain the identities from (9a) and (9b).

Next, take , , , and in (35b), and, by using (31), (34a), (34c) and (34d), we obtain Finally, take , , , and in (35b), and, by using (31), (34a), (34b), (34c), (34d), and (23c), we deduce the identity The other identities obtained from (35b) either are trivial or do not involve the curvature tensor of . Thus, we are entitled to call (37), (38), and (39) the Bianchi identities for the Riemannian spatial connection.

We close the section with some comments on these identities. As far as we know, the above Bianchi identities are stated here for the first time in the literature. They represent a generalization of usual Bianchi identities on a 3-dimensional Riemannian manifold. Indeed, if the spatial distribution is integrable, that is, the vorticity tensor field vanishes identically on , then (37) and (38) become which are the well-known Bianchi identities on the 3-dimensional leaves of . Moreover, in this case, by using (14) and (27a) and (27b), we deduce that Finally, identity (39) becomes

4. Structure Equations Induced by the Threading of Spacetime

Let be a spacetime and be the Levi-Civita connection defined by the Lorentz metric . Denote by the curvature tensor field of given by (33b) wherein we remove the tilde. Then, consider the following local components of with respect to the threading frame :Now, comparing (A.4) and (A.8) from Appendix A with (43a) and (43b), respectively, we obtain By using (15), (6a), (6b), (6c), and (6d), we deduce that Taking account of (45a), (45b), (14), and (15) into (44a), (44b), (44c), and (44d), we infer that Next, by using the local components of the curvature tensor fields and of type (see (A.9) and (A.10)), from (44a), (44b), (44c), (44d), (46a), (46b), (46c), and (46d) we obtain With the theory of hypersurfaces of the spacetime in mind, we call (47a) and (48a) , (47b), (48b), and (48c) the Gauss equations (resp., Codazzi equations) for the spatial distribution in the ambient space . Also, all the equations from (47a), (47b), (47c), (47d), (48a), (48b), (48c), and (48d) will be called structure equations induced by the threading formalism. They have an important role in the next sections.

Now, taking into account the symmetries of , we deduce some identities for and for kinematic quantities. First, using well-known identities for and taking into account (47a) and (48a), we obtain the following identities for : Also, taking into account that and by using (47b), (47c), (48b), and (48c), we deduce that Finally, using the identity and taking the symmetric and skew-symmetric parts in (47d) and (48d), we infer that In particular, suppose that is a vorticity-free spacetime; that is, the vorticity tensor field vanishes identically on . Then, from (49c) and (51) we deduce that the curvature tensor field of the Riemannian spatial connection satisfies identities (49a) and (49b) and the following: Also, from (54b) we see that and (48a), (48b), (48c), and (48d) become

5. Ricci Tensor Field and Scalar Curvature of a Spacetime Expressed in terms of Spatial Tensor Fields

Let be a spacetime and be an orthonormal basis in Then, is an orthonormal frame field on . According to [7, p. 87], the Ricci tensor of is given by for all Now, we express as follows: and we obtain Then we consider the following local components of Ric with respect to the threading frame : and, by using (59), (60), (61a), (61b), (61c), and (A.9) into (58), we obtain Now, by using (47a), (47b), (47c), (47d), (48a), (48b), (48c), (48d), (53a), and (53b) in (62a), (62b), and (62c), we deduce that where we putNext, we take the symmetric and skew-symmetric parts in (63a) and (64a) and obtain where we put The spatial with local components are called the spatial Ricci tensor of the spacetime .

The above formulas for Ricci tensor enable us to obtain a new formula for the scalar curvature of in terms of spatial tensor fields. We start with given by Then by direct calculations, using (66a) and (64c) into (68), we obtain where we put We call the spatial scalar curvature of the spacetime . Note that both and are related to the geometry of the spatial distribution which is not necessarily supposed to be integrable.

6. The Splitting of Both the Einstein Gravitational Tensor Field and the Energy-Momentum Tensor Field

We start with the Einstein gravitational tensor field of given by Then, with respect to the threading frame field , we have By using (66a), (64b), (64c), and (69) into (72a), (72b), and (72c), we obtain where we put The spatial tensor field with local components is called the Einstein gravitational tensor field of the spacetime .

Now, in order to give a coordinate-free formula for the energy-momentum tensor field, we consider a spatial 1-form and a spatial tensor field of type . Then, we define a 1-form and a tensor field of type on , denoted by the same symbols and given by for all . As an example, the Riemannian metric on defines a symmetric tensor field on given by Note that from (76) coincides with the tensor field given by its local components in formula (4.10) of [4]. Also, we need the 1-form induced by the unit vector field by the formula Based on these geometric objects, we claim that the energy-momentum tensor field measured by an observer moving with the unit 4-velocity has the following coordinate-free expression: Here, and are the relativistic energy density and the relativistic pressure, respectively, while is a 1-form on defined by a spatial 1-form as in (75a) and is a symmetric and trace-free tensor field on defined by a spatial tensor field as in (75b). Now, take in (78) and obtain Then we put and (79) becomes Similarly, take and in (78) and deduce that Now, we put and, taking in (82), we infer that Finally, denote and, taking and in (78), we obtain Contracting (86) by and taking into account that is defined by a trace-free spatial tensor field, we infer that Now, taking into account (79)–(87b), it is clear that (78) represents the coordinate-free version of from [4, p. 91]. Thus and from (78) are the relativistic momentum density and the relativistic anisotropic (trace-free) stress tensor field, respectively.

7. A New Splitting of Einstein Field Equations with respect to General Threading of Spacetime

We start this section with the coordinate-free form of Einstein field equations expressed as follows: where is the cosmological constant and is the Newton constant. Now, take and in (88) and, by using (73a), (10), and (86), we obtain The equations from (89) will be called the spatial Einstein field equations (SEFE). Next, we take and in (88) and, by using (73a) and (84), we deduce that The equations from either (90a) or (90b) are called mixed Einstein field equations . Finally, take in (88), and, by using (73c), (2a), and (81), we infer that We call (91) the temporal Einstein field equation . Thus Einstein field equations (88) are splitting into three groups of equations given by (89), (90a), (90b), and (91). It is worth mentioning that these equations are expressed in terms of spatial tensor fields and their covariant derivatives induced by the Riemannian spatial connection.

Next, by contracting (89) by and using (74) we deduce that the spatial scalar curvature is given by Comparing (92) with (91), we obtain the Raychaudhuri-Ehlers equation induced by the general threading of the spacetime: Note that (93) is the generalization of () from [4], which was obtained for the particular case .

8. A New Splitting of Conservation Laws with respect to a General Threading of Spacetime

As is well known, the energy-momentum conservation equations are given by the vanishing of the divergence of . In order to obtain their explicit form, we consider an orthonormal frame field , and according to [7, p. 86] we have for all . Then, take , in (94), and by using (B.1) and (B.2) obtain Similarly, take in (94) and by using (B.3) and (B.4) we deduce that Now, from (84) and (86), we infer that Then, by using (24), (81), (84), (86), (97a), and (97b) in (95) and (96) and taking into account that both and are trace-free spatial tensor fields, we obtain the energy conservation equation: and the momentum conservation equation:In order to compare with what is known in the literature with respect to the threading of spacetime, we note that This is obtained by direct calculations using (23d). Also, we should remark that (100) states that though the velocity is a timelike vector field, the acceleration is a spatial vector field.

Remark 7. The above conservation laws are obtained in the most general setting. Indeed, if in particular , (98) and (99) become and from [4, p. 92], respectively. If, moreover, we have a perfect fluid on , that is, and , then (98) and (99) become and from Proposition  5 in [7, p. 339].

9. Splitting of Einstein Field Equations in an Almost FLRW Universe

Let be an universe, whose line element is given by where is the conformal time on and the three-dimensional space given by = const. is an Euclidean space. The Lorentz metric given by (101) is called the background metric. For a more realistic model of the universe, there has been studied perturbation of this metric (cf. [4, 8, 9]). The line element of the full metric on is expressed as follows: where determine the perturbation. In the present paper we consider the conformal-Newtonian gauge case, for which the full metric is given by where and are the well-known Bardeen invariants. The spacetime with given by (103) is called an almost universe.

Now, by using (2b) for (103), we deduce that , which implyAlso, according to the notation in (2a), we have

From (104b) we see that the spatial distribution of is integrable, but its leaves are not anymore Euclidean spaces. Moreover, by using (12a), (12b), (6d), (104d), and (105), we obtain where “” denotes derivative with respect to and is the Hubble parameter of the background metric. By calculations, using (106a), (106b), and (104d) into (12c), we infer that Taking into account that are the local components of the second fundamental form of the leaves of (see (23a)) and using (107a) and (107b), we can state the following result on the kinematic quantities and geometry of .

Theorem 8. Let be an almost universe. Then one has the following assertions:(i) is both vorticity-free and shear-free spacetime.(ii)The leaves of the spatial distribution are totally umbilical hypersurfaces of with mean curvature vector

Next, we assume that the energy-momentum tensor for the almost universe takes the perfect fluid form, that is, <