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.