On The splitting of the Einstein field equations with respect to a general $(1 + 3)$ threading of spacetime

Based on a general $(1+3)$ threading of the spacetime $(M,g)$, we obtain a new and simple splitting of a both the Einstein field equations (EFE) and the conservation laws in $(M,g)$. 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 Friedman equations in an almost FLRW universe.


Introduction
The present paper is a continuation of the paper [1], wherein an new approach on the (1+3) threading of spacetime with respect to an arbitrary timelike vector field has been developed. The study in [1] refers to Lorentz metrics given by (2.2) and subject to the condition that Φ 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 a 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 a threading of higher-dimensional universes. In this respect we mention the paper [2], wherein a (1 + 1 + 3)-threading of a 5D 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 (1 + 3) threading formalism of a spacetime (M, g) 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. (2.18)). 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. (3.13), (3.14), (3.15)). The structure equations on (M, g) induced by the (1 + 3) threading formalism are presented in Section 4 (cf. (4.5), (4.6)). They play an important role in the next sections, wherein we relate tensor fields on M with spatial tensor fields. In Section 5 we obtain simple expressions for the local components of the Ricci tensor of (M, g) with respect to the threading frame field , and for the scalar curvature (cf. (5.6), (5.7), (5.8a), (5.11)). The splittings of both the Einstein gravitational tensor field and the energy-momentum tensor field are stated in Section 6 (cf. (6.3), (6.11), (6.14), (6.16)). In Section 7 we obtain the spatial, mixed and temporal (EFE) (cf. (7.2), (7.3), (7.4)). Also, we state the equation (7.6) which in the particular case when the treading 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 a general (1 + 3) threading of spacetime is given in Section 8 (cf. (8.5), (8.6)). 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 (1 + 3) threading of an almost FLRW universe. We close the paper with conclusions and three appendixes.
2 The (1 + 3) threading formalism with respect to a non-normalized timelike vector field Recently, a new approach on the (1 + 3) threading of spacetime with respect to a non-normalized 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 4D spacetime. Let (M, g) be a 4D spacetime, and ξ be a timelike vector field on M that is not necessarily normalized. Then, the tangent bundle T M of M admits the decomposition where V M is the temporal distribution spanned by ξ, and SM is the spatial distribution that is complementary orthogonal to V M in T M . Throughout the paper we use the ranges of indices: i, j, k, ... ∈ {1, 2, 3} and a, b, c, ... ∈ {0, 1, 2, 3}. Also, for any vector bundle E over M denote by Γ(E) the F(M )-module of smooth sections of E, where F(M ) is the algebra of smooth functions on M .
The congruence of curves that is tangent to V M , determines a coordinate system (x a ) on M such that ξ = ∂/∂x 0 . Next, we put wherein Φ is a non-zero function on M . Remark 2.1 Note that in [1], Φ was supposed to be independent of x 0 . Here, we remove this condition on Φ, and thus the results stated in the present paper are valid for any Lorentz metric g on M .
In this approach we use the threading frame field {∂/∂x 0 , δ/δx i } and the threading coframe field {δx 0 , dx i } defined as follows: where we put The Lie brackets of the vector fields from the threading frame are given by Taking into account that the Levi-Civita connection ∇ on (M, g) is torsionfree, from (2.5a) we deduce that Thus, ω ij , i, j ∈ {1, 2, 3}, define the vorticity tensor field on (M, g). By using the Jacobi identity we deduce that the vorticity tensor field satisfies the identities is the cyclic sum with respect to (i, j, k). Now, we denote byḡ ij the local components of the Riemannian metric g induced by g on SM , with respect ti the basis {δ/δx i } in Γ(SM ), and obtainḡ (2.9) Then the lone element of g is expressed in terms of threading coframe {δx 0 , dx i } as follows: (2.10) Also, we define the expansion tensor field Θ ij , the expansion function Θ and the shear tensor field σ ij as follows: Raising and lowering indices i, j, k, ... are performed by usingḡ ij andḡ ij , as in the following examples: The expansion and vorticity tensor fields enable us to define the extrinsic curvature tensor field K of the spatial distribution by its local components or equivalently by (2.13) By using (2.5a) and (2.13), we see that K is a symmetric tensor field if and only if SM is integrable.

Remark 2.2
The extrinsic curvature tensor field was intensively used in the (3 + 1) decomposition of the spacetime (cf. [3], pp 509-516). As far as we know, the tensor field K given by (2.12) or (2.13) is considered here for the first time in a study of the (1 + 3) threading of spacetime.
Next, in order to justify the tensorial meaning of the above quantities, we define a spatial tensor field T of type (p, q) on M , as an F(M )-multilinear mapping where SM ⋆ is the dual vector bundle to SM . The local components of T with respect to a threading frame and coframe, are given by and satisfy As examples: {ω ij ,ḡ ij , θ ij , σ ij , K ij } and {a i , c i } define spatial tensor fields of type (0, 2) and (0.1) respectively. An important geometric object is the Riemannian spatial connection, which is a metric linear connection∇ on the spatial distribution, given by where S is the projection morphism of T M on SM with respect to the decomposition (2.1). Locally,∇ is given by where we putΓ Throughout the paper, the covariant derivatives defined by∇ will be denote by a vertical bar "|". As an example, for a spatial tensor field T = (T i j ) we have (2.17) A covariant derivative as in (2.17a) (resp. (2.17b)) is called a spatial covariant derivative (resp. temporal covariant derivative) of the spatial tensor field T . Finally, by direct calculations, using the Riemannian spatial connection and the above spatial tensor fields, we express the Levi-Civita connection ∇ on (M, g) as follows: 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 2.4 As the (1 + 3) threading of spacetime considered in this paper contains as a particular case the (1 + 3) threading with respect to a unit timelike vector field, we call it the general (1+3) 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 (1 + 3) threading of spacetime we find the so called three-dimensional derivative operator (cf.(4.19) of [4]). With respect to this operator we have the following remarks: (i) It is neither a linear connection on M , nor a linear connection on SM .
(ii) As a consequence of (i), for the general case when SM is not integrable, then a curvature tensor field for this operator could be not defined.
Contrary to this situation,∇ given by (2.14) is a metric linear connection on the vector bundle SM , and therefore it has a curvature tensor fieldR given byR and by using (3.1), (3.2), (2.15), and (2.5), we obtain Sine∇ is a metric linear connection, we havē As a consequence of (3.4) we deuce thatR i i 0k = 0, which implies via (3.3b) and (2.12). Thus, in any cosmological model of a 4D universe the expansion function must satisfy the system of P DE given by (3.6). Remark 3.1 Note thatR h i jk andR h i 0k define spatial tensor fields of type (1, 3) and (1, 2), respectively. Also, from (3.3b) we see that ∂Γ h i k /∂x 0 define a spatial tensor field of type (1, 2). However,Γ h i k do not define a spatial tensor field.
Remark 3.2 Comparing (3.3a) with (15.4) from [5], we see that the so called Zelmanov curvature tensor field is given by the first four terms from (3.3a). Moreover, from (15.5) of [5] we see that such a tensor field becomes a curvature tensor field, if and only if, SM is an integrable distribution.
Next, we extend the Riemannian spatial connection∇ on SM to a linear connection∇ on M given bỹ where T is the projection morphism of T M on V M with respect to (2.1). Clearly,∇ coincides with∇ on SM and therefore locally is given by (2.15) and We recall that the torsion and curvature tensor fields of∇ are given by Then, by direct calculations, using (3.9), (2.15), (3.8), (2.5), (2.18c) and (3.2), we deduce that (3.10) 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) is the cyclic sum with respect to (X, Y, Z). In order to use (3.12) ) and by using (3.12) and (3.10a), we infer that the spatial component in (3.11a) is expressed as follows: Taking temporal part in (3.11a) and any other triplet (X, Y, Z) from the threading frame we obtain the identities from (2.8) , and by using (3.7), (3.10a), (3.10c) and (3.10d), we obtain Finally, take X = ∂/∂x 0 , Y = δ/δx j , Z = δ/δx i and U = δ/δx h in (3.11b), and by using (3.7), (3.10) and (2.18c), we deduce the identitȳ The other identities obtained from (3.11b) are either trivial or they do not involve the curvature tensorR of∇. Thus, we are entitled to call (3.13), (3.14) and (3.15) the Bianchi identities for the Riemannian spatial connection.
We close the section with some comments on these identities. As for 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 M , then (3.13) and (3.14) become which are the well-known Bianchi identities on the 3-dimensional leaves of SM . Moreover, in this case, by using (2.12) and (3.3) we deduce that Finally, the identity (3.15) becomes 4 Structure equations induced by the (1+3) threading of spacetime Let (M, g) be a 4D spacetime and ∇ be the Levi-Civita connection defined by the Lorentz metric g. Denote by R the curvature tensor field of ∇ given by (3.9b) wherein we remove the tilde. Then, consider the following local components of R with respect to the threading frame {∂/∂x 0 , δ/δx i } : Now, comparing (A 4 ) and (A 8 ) from Appendix A with (4.1a) and (4.1b) respectively, we obtain By using (2.13), (2.6) and (2.6d), we deduce that Taking account of (4.3), (2.12) and (2.13) into (4.2), we infer that Next, by using the local components of the curvature tensor fields R andR of type (0, 4) (see (A 9 ) and (A 10 )), from (4.2) and (4.4) we obtain and With the theory of hypersurfaces of the spacetime in mind, we call (4.5a) and (4.6a) {resp. (4.5b), (4.6b) and (4.6c)} the Gauss equations (resp. Codazzi equations) for the spatial distribution SM in the ambient space (M, g). Also, all the equations from both (4.5) and (4.6) will be called structure equations induced by the (1 + 3) threading formalism. They have an important role in the next sections. Now, taking into account the symmetries of R we deduce some identities forR and for kinematic quantities. First, using well known identities for R and taking into account (4.5a) and (4.6a), we obtain the following identities forR: Also, taking into account that and by using (4.5b), (4.5c), (4.6b) and (4.6c), we deduce that Finally, using the identity and taking the symmetric and skew-symmetric parts in(4.5d) and (4.6d), we infer that and (a) (4.10) In particular, suppose that (M, g) is a vorticity-free spacetime, that is, the vorticity tensor field vanishes identically on M . Then, from (4.7c) and (4.8) we deduce that the curvature tensor field of the Riemannian spatial connection satisfies the identities (4.7a), (4.7b) and the following: Also, from (4.10b) we see that and (4.6) becomes (4.12) .
5 Ricci tensor field and scalar curvature of a spacetime expressed in terms of spatial tensor fields Let (M, g) be a 4D spacetime, and {E k } be an orthonormal basis in Γ(SM ). Then, {Φ −1 ∂ ∂x 0 , E k } is an orthonormal frame field on M . According to [7], p.87, the Ricci tensor of (M, g) is given by for all X, Y ∈ Γ(T M ). Now, we express E k as follows and obtainḡ Then we consider the following local components ofRic with respect to the threading frame {∂/∂x 0 , δ/δx i } : Now, by using (4.5), (4.6) and (4.9) in (5.5), we deduce that and where we put Next, we take the symmetric and skew-symmetric parts in (5.6a) and (5.7a), and obtain where we putR The spatial with local componentsR ij is called the spatial Ricci tensor of the spacetime (M, g). The above formulas for Ricci tensor enable us to obtain a new formula for the scalar curvature R of (M, g) in terms of spatial tensor fields. We start with R given by Then by direct calculations, using (5.8a) and (5.7c) into (5.10), we obtain where we putR =ḡ ijR ij . (5.12) We callR the spatial scalar curvature of the spacetime (M, g). Note that bothR ij andR are related to the geometry of the spatial distribution SM which is not necessarily supposed to be integrable.

The splitting of both the Einstein gravitational tensor field and the energy-momentum tensor field
We start with the Einstein gravitational tensor field G of (M, g) given by Then, with respect to the threading frame field {∂/∂x 0 , δ/δx i } we have (6.2) By using (5.8a), (5.7b), (5.7c) and (5.11) into (6.2), we obtain where we put The spatial tensor fieldḠ with local componentsḠ ij is called the 3D Einstein gravitational tensor field of the spacetime (M, g). Now, in order to give a coordinate-free formula for the energy-momentum tensor field, we consider a spatial 1-form ω and a spatial tensor field S of type (0, 2). Then, we define a 1-form and a tensor field of type (0, 2) on M , denoted by the same symbols and given by for all X, Y ∈ Γ(T M ). As an example, the Riemannian metricḡ on SM defines a symmetric tensor fieldḡ on M given bȳ Note thatḡ from (6.6) coincides with the tensor field h given by its local components in formula (4.10) of [4]. Also, we need the 1-form u induced by the unit vector field U = Φ −1 ∂/∂x 0 by the formula Based on these geometric objects, we claim that the energy-momentum tensor field T measured by an observer moving with the unit 4-velocity U has the following coordinate-free expression: Here, ρ and p are the relativistic energy density and the relativistic pressure respectively, while q is a 1-form on M defined by a spatial 1-form as in (6.5a) and π is a symmetric and trace-free tensor field on M defined by a spatial tensor field as in (6.5b). Now, take X = Y = U in (6.8) and obtain ρ = T (U, U ). (6.9) Then we put and (6.9) becomes Similarly, take X = U and Y = SY in (6.8) and deduce that (6.13) and taking Y = δ/δx i in (6.12), we infer that (6.15) and taking X = δ/δx j , Y = δ/δx i in (6.8), obtain Contracting (6.16) byḡ ij and taking into account that π is defined by a trace-free spatial tensor field, we infer that Now, taking into account(6.9) -(6.17) it is clear that (6.8) represents the coordinate-free version of (5.9) from [4], p.91. Thus q and π from (6.8) are the relativistic momentum density and the relativistic anisotropic (trace-free) stress tensor field, respectively.

A new splitting of Einstein field equations with respect to a general (1+3) threading of spacetime
We start this section with the coordinate-free form of Einstein field equations (EF E) expressed as follows: where Λ is the cosmological constant and G is the Newton constant. Now, take X = δ/δx j , Y = δ/δx i in (7.1) and by using (6.3a), (2.9) and (6.16), we obtain The equations from (7.2) will be called the spatial Einstein field equations (SEFE). Next, we take X = ∂/∂x 0 , Y = δ/δx i in (7.1) and by using (6.3a) and (6.14), we deduce that 3) The equations from either (7.3a) or (7.3b) are called mixed Einstein field equations (M EF E). Finally, take X = Y = ∂/∂x 0 in (7.1), and by using (6.3c), (2.2a) and (6.11), we infer that We call (7.4) the temporal Einstein field equation (T EF E). Thus the Einstein field equations (7.1) are splitting in three groups of equations given by (7.2), (7.3) and (7.4). 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 (7.2) byḡ ij and using (6.4) we deduce that the spatial scalar curvature is given by (7.5) Comparing (7.5) with (7.4) we obtain the Raychaudhuri-Ehlers equation induced by the general (1 + 3) threading of the spacetime: Note that (7.6) is the generalization of the equation (6.4) from [4], which was obtained for the particular case Φ = 1.

A new splitting of conservation laws with respect to a general (1 + 3) threading of spacetime
As is well known, the energy-momentum conservation equations are given by the vanishing of the divergence of T . In order to obtain their explicit form, we consider an orthonormal frame field {E k , U = Φ −1 ∂/∂x 0 }, and according to [7], p.86 we have for all X ∈ Γ(T M ). Then, take X = ∂/∂x 0 , in (8.1), and by using (B 1 ) and (B 2 ) obtain Similarly, take X = δ/δx i in (8.1) and by using (B 3 ) and (B 4 ) we deduce that Now, from (6.14) and (6.16), we infer that Then, by using (2.19), (6.11), (6.14), (6.16) and (8.4) in (8.2) and (8.3), and taking into account that both σ ij and π ij are trace-free spatial tensor fields, we obtain the energy conservation equation: ∂ρ ∂x 0 + (ρ + p) Θ + σ jk π jk + Φ q j |j + 2b j q j = 0, (8.5) and the momentum conservation equation: In order to compare with what is known in the literature with respect to the (1 + 3) threading of spacetime, we note that This is obtained by direct calculations using (2.18d). Also, we should remark that (8.7) states that though the velocity U is a timelike vector field, the acceleration ∇ U U is a spatial vector field. Remark 8.1 The above conservation laws are obtained in the most general setting. Indeed, if in particular Φ = 1, (8.5) and (8.6) become (5.11) and (5.12) from [4], p.92, respectively. If moreover, we have a perfect fluid on M , that is, q i = 0 and π ij = 0, then (8.5) and (8.6) become (1) and (2) from Proposition 5 in [7], p.339 .

Splitting of Einstein field equations in an almost F LRW universe
Let M be an F LRW universe, whose line element is given by where τ is the conformal time on M and the three-dimensional space given by τ = const., is an Euclidean space. The Lorentz metric given by (9.1) is called the background metric. For a more realistic model of the universe there have been studied perturbation of this metric (cf. [4], [8], [9]). The line element of the full metric g on M is expressed as follows: where δg ab determine the perturbation. In the present paper we consider the conformal-Newtonian gauge case, for which the full metric is given by where A and B are the well-known Bardeen invariants. The spacetime (M, g) with g given by (9.3) is called an almost F LRW universe. Now, by using (2.2b) for (9.3) we deduce that ξ i = 0, i ∈ {1, 2, 3}, which imply Also, according to the notation in (2.2a), we have Φ 2 = a 2 (1 + 2A). (9.5) From (9.4b) we see that the spatial distribution SM of (M, g) is integrable, but its leaves are not anymore Euclidean spaces. Moreover, by using (2.11a), (2.11b), (2.6d), (9.4d) and (9.5), we obtain where "|" denotes derivative with respect to τ , and H = a ′ /a is the Hubble parameter of the background metric. By calculations, using (9.6a), (9.6b) and (9.4d) into (2.11c), we infer that Taking into account that Φ −2 Θ ij are the local components of the second fundamental form of the leaves of SM (see (2.18a)), and using (9.7), we can state the following result on the kinematic quantities and geometry of (M, g).
Theorem 9.1 Let (M, g) be an almost F LRW universe. Then we have the following assertions: (i) (M, g) is both vorticity-free and shear-free spacetime.
(ii) The leaves of the spatial distribution are totally umbilical hypersurfaces of (M, g) with mean curvature vector Next, we assume that the energy-momentum tensor for the almost F LRW universe (M, g) takes the perfect fluid form, that is, Then, the Bardeen invariants coincide (cf. [4], p.259), that is, from now on in our calculations we put A instead of B. First, by using (2.19), (9.4c), (2.6b), (9.5) and (9.4e), we deduce that where we put A i = ∂A/∂x i . The local coefficients of the spatial Riemannian connection∇ are given by ( see (2.15), (C 1 ), (2.12), (9.7)) where we have A k = A k . Then, by using (9.10a) and (9.11a), we obtain the spatial covariant derivative and the divergence of the acceleration, given by where we put Now, by direct calculation using (9.6b), we infer that (9.13) Also, by using (9.7b), (9.11b) and (9.13), and taking into account that∇ is a metric connection, we obtain (9.14) Finally, by using (C 6 ) and (C 7 ) in (6.4), we deduce that the 3D Einstein gravitational tensor field is given bȳ (9.15) Now, we are in a position to present the splitting of (EF E) for the almost universe (M, g). First, we consider that the background energy-momentum tensorT take the perfect fluid form, that is, we have whereρ andp are the relativistic density and pressure, respectively. Then, the energy momentum tensor T of the perturbed universe (M, g) should have the perfect fluid from too. Hence, by (6.16), (6.14) and (6.11), we have where we put p =p + δp and ρ =ρ + δρ.

Conclusions
The idea to develop a (1+ 3) threading of a spacetime with respect to a nonnormalized vector field came up as a need for the study of the spacetimes whose metrics have the general form (2.2) with Φ 2 = 1. We only mention here all the metrics from both the theory of cosmological perturbations and the theory of black holes. The main difference between our approach and the methods developed in earlier papers consists in the fact that we deal with the spatial tensor fields as intrinsic objects from the geometry of the spatial distribution. In earlier literature, the spatial tensor fields have been considered as projections on the spatial distribution of tensor fields defined on M . This was the main obstacle in defining a correct spatial covariant differentiation for the general case when the spatial distribution is not integrable (cf. (4.19) of [4]).
The approach we develop in the paper is based on spatial tensor fields and on the Riemannian spatial connection which behave as 3D geometric objects with local components defined on a 4D spacetime. It is noteworthy that the three groups of (EF E) presented in the paper are expressed in terms of spatial tensor fields and their covariant derivatives induced by the Riemannian spatial connection. This enable us to write down a splitting of (EF E) for an almost F LRW universe, which might have an important role in the difficult task of finding models for such a universe. Moreover, the approach can be extended to the study of threading of higher-dimensional universes. This can be seen in a paper of the first author on the threading of a 5D universe (cf. [2]).
Then, by using (A 5 ), (A 6 ) and (A 1 ), and taking into account (3.3b) and both covariant derivatives of the extrinsic curvature tensor field, we deduce that Next, we consider the curvature tensor fields of type (0, 4) of the connections ∇ and∇, denoted by R andR, and given by for all X, Y, Z, U ∈ Γ(T M ). Then we have the following components with respect to the threading frame: These local components are used in both Sections 4 and 5 in order to deduce the final form for the structure equations and the local components of the Ricci tensor, respectively.