# On the Splitting of the Einstein Field Equations with respect to General(1+3) Threading of Spacetime

Authors: Aurel Bejancu; Hani Reda Farran
Journal: Advances in Mathematical Physics
 (2016)

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## Abstract

Based on general(1+3) threading of the spacetime (M,g), we obtain a new and simple splitting of 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 Friedmann equations in an almost FLRW universe.


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## Body

## 1. Introduction

The present paper is a continuation of [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 (2a), (2b), and (2c) 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 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 (1+1+3) threading of a 5D universe has been developed.Now, we outline the content of the paper. In Section2 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. (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 (M,g) induced by the (1+3) 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 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. (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 (1+3) 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 (1+3) threading of an almost FLRW universe. We close the paper with Conclusions and Appendices A, B, and C.


## 2. The(1+3) Threading Formalism with respect to a Nonnormalized Timelike Vector Field

Recently, a new approach on the(1+3) 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 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 TM of M admits the decomposition (1)TM=VM⊕SM,where VM is thetemporal distribution spanned by ξ and SM is thespatial distribution that is complementarily orthogonal to VM in TM.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 toVM determines a coordinate system (xa) on M such that ξ=∂/∂x0. Next, we put(2a)ξ0=g∂∂x0,∂∂x0=-Φ2,(2b)ξi=g∂∂xi,∂∂x0,(2c)gij=g∂∂xi,∂∂xj,wherein Φ is a nonzero function on M.Remark 1.
Note that, in [1], Φ was supposed to be independent of x0. 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 thethreading frame field{∂/∂x0,δ/δxi} and thethreading coframe field{δx0,dxi} defined as follows: (3a)δδxi=∂∂xi-Ai∂∂x0,(3b)δx0=dx0+Aidxi,where we put (4)Ai=-Φ-2ξi.The Lie brackets of the vector fields from the threading frame are given by (5a)δδxj,δδxi=2ωij∂∂x0,(5b)∂∂x0,δδxi=ai∂∂x0,where we set (6a)ωij=12δAjδxi-δAiδxj=Φ-2ciξj-cjξi+12δξiδxj-δξjδxi,(6b)ci=Φ-1δΦδxi,(6c)ai=-∂Ai∂x0=Φ-2∂ξi∂x0-2Ψξi,(6d)Ψ=Φ-1∂Φ∂x0.Taking into account that the Levi-Civita connection ∇ on (M,g) is torsion-free, from (5a) we deduce that (7)ωij=12Φ-2g∇δ/δxiδδxi-∇δ/δxjδδxi,∂∂x0.Thus, ωij,i,j∈{1,2,3}, define thevorticity tensor field on (M,g). By using the Jacobi identity (8)X,Y,Z+Y,Z,X+Z,X,Y=0,∀X,Y,Z∈ΓTM,we deduce that the vorticity tensor field satisfies the identities (9a)∂ωij∂x0=12δaiδxj-δajδxi,(9b)∑i,j,kδωijδxk-ωijak=0,where ∑(i,j,k)‍ is the cyclic sum with respect to (i,j,k).Now, we denote byg¯ij the local components of the Riemannian metric g¯ induced by g on SM, with respect to the basis {δ/δxi} in Γ(SM), and obtain (10)g¯ij=g¯δδxi,δδxj=gij+Φ2AiAj=gij+Φ-2ξiξj.Then the lone element of g is expressed in terms of threading coframe {δx0,dxi} as follows: (11)ds2=-Φ2δx02+g¯ijdxidxj.Also, we define theexpansion tensor fieldΘij, theexpansion functionΘ, and theshear tensor fieldσij as follows: (12a)Θij=12∂g¯ij∂x0,(12b)Θ=Θijg¯ij,(12c)σij=Θij-13Θg¯ij.Raising and lowering indicesi,j,k,…, are performed by using g¯ij and g¯ij, as in the following examples: (13)ωjk=g¯kiωij,ωkh=g¯kig¯hjωij,ωij=g¯ikωjk,ωij=g¯ikg¯jhωkh.The expansion and vorticity tensor fields enable us to define theextrinsic curvature tensor fieldK of the spatial distribution by its local components (14)Kjh=Θjh+Φ2ωjh,or equivalently by (15)Kij=Θij+Φ2ωij.By using (5a) and (15), we see that K is a symmetric tensor field if and only if SM is integrable.Remark 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 (14) or (15) 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 aspatial tensor fieldT of type (p,q) on M, as an F(M)-multilinear mapping: (16)T:ΓSM⋆p×ΓSMq⟶FM,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 (17)Ti⋯k⋯=Tdxk,…,δδxi,…and satisfy(18)Ti⋯k⋯∂x~h∂xk=T~j⋯h⋯∂x~j∂xi,with respect to the coordinate transformations x~a=x~a(x0,xi) on M. For example, {ωij,g¯ij,θij,σij,Kij} and {ai,ci} define spatial tensor fields of types (0,2) and (0,1), respectively.An important geometric object is theRiemannian spatial connection, which is a metric linear connection ∇¯ on the spatial distribution, given by(19a)∇¯XSY=S∇XSY,∀X,Y∈ΓTM,where S is the projection morphism of TM on SM with respect to decomposition (1). Locally, ∇¯ is given by (20a)∇¯δ/δxjδδxi=Γ-ikjδδxk,(20b)∇¯∂/∂x0δδxi=Kikδδxk,where we put (21)Γ-ikj=12g¯khδg¯hjδxi+δg¯hiδxj-δg¯ijδxh.Throughout the paper, the covariant derivatives defined by ∇¯ will be denoted by a vertical bar “∣.” As an example, for a spatial tensor field T=(Tji), we have (22a)Ti∣kj=δTijδxk+TihΓ-hjk-ThjΓ-ihk,(22b)Ti∣0j=∂Tij∂x0+TihKhj-ThjKih.A covariant derivative as in (22a) (resp., (22b)) is called aspatial 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: (23a)∇δ/δxjδδxi=Γ-ikjδδxk+ωij+Φ-2Θij∂∂x0=Γ-ikjδδxk+Φ-2Kij∂∂x0,(23b)∇∂/∂x0δδxi=Θik+Φ2ωikδδxk+bi∂∂x0=Kikδδxk+bi∂∂x0,(23c)∇δ/δxi∂∂x0=Θik+Φ2ωikδδxk+ci∂∂x0=Kikδδxk+ci∂∂x0,(23d)∇∂/∂x0∂∂x0=Φ2bkδδxk+Ψ∂∂x0,where we put (24)bi=ai+ci,i∈1,2,3.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(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 thegeneral(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 onM nor a linear connection on SM.(ii)
As a consequence of (i), for the general case whenSM 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 SM, and therefore it has a curvature tensor field R¯ given by (25)R¯X,Y,Z=∇¯X∇¯YSZ-∇¯Y∇¯XSZ-∇¯X,YSZ,∀X,Y,Z∈ΓTM.Locally, we put (26a)R¯δδxk,δδxj,δδxi=R-ihjkδδxh,(26b)R¯δδxk,∂∂x0,δδxi=R-ih0kδδxh,and by using (25), (26a), (26b), (20a), (20b), (5a), and (5b), we obtain (27a)R-ihjk=δΓ-ihjδxk-δΓ-ihkδxj+Γ-iljΓ-lhk-Γ-ilkΓ-lhj-2Kihωjk,(27b)R-ih0k=Kih∣k-∂Γ-ihk∂x0+Kihak.Since ∇¯ is a metric linear connection, we have (28)R¯il0k+R¯li0k=0,where we put (29)R¯il0k=g¯lhR-ih0k.As a consequence of (28) we deduce that R-ii0k=0, which implies (30)Θ∣k=∂Θ∂xk-Ak∂Θ∂x0=Γ-iik∂x0-Θak,via (27b) and (14). Thus,in any cosmological model of a4Duniverse, the expansion function must satisfy the system ofPDEgiven by (30).Remark 5.
Note thatR-ihjk and R-ih0k define spatial tensor fields of types (1,3) and (1,2), respectively. Also, from (27b) we see that ∂Γ-ihk/∂x0 define a spatial tensor field of type (1,2). However, Γ-ihk 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 (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 by (31)∇~XY=∇¯XSY+T∇XTY,∀X,Y∈ΓTM,where T is the projection morphism of TM on VM with respect to (1). Clearly, ∇~ coincides with ∇¯ on SM and therefore locally is given by (20a), (20b), and (32a)∇~δ/δxi∂∂x0=ci∂∂x0,(32b)∇~δ/δx0∂∂x0=Ψ∂∂x0.We recall that the torsion and curvature tensor fields of ∇~ are given by (33a)T~X,Y=∇~XY-∇~YX-X,Y,(33b)R~X,Y,Z=∇~X∇~YZ-∇~Y∇~XZ-∇~X,YZ.Then, by direct calculations, using (33a), (33b), (20a), (20b), (32a), (32b), (5a), (5b), (23c), (26a), and (26b), we deduce that (34a)T~δδxj,δδxi=-2ωij∂∂x0,(34b)T~∂∂x0,δδxi=Kijδδxj-bi∂∂x0,(34c)R~δδxh,δδxk,δδxi=R-ijkhδδxj,(34d)R~δδxk,∂∂x0,δδxi=R-ij0kδδxj.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]) (35a)∑X,Y,Z∇~XT~Y,Z+T~T~X,Y,Z-R~X,Y,Z=0,(35b)∑X,Y,Z∇~XR~Y,Z,U+R~T~X,Y,Z,U=0,where ∑(X,Y,Z)‍ is the cyclic sum with respect to (X,Y,Z). In order to use (35a), we note that (36a)T~T~δδxk,δδxj,δδxi=-2Kihωjkδδxh+2biωjk∂∂x0,(36b)∇~δ/δxkT~δδxj,δδxi=-2ωij∣k+ωijck∂∂x0.Then, take X=δ/δxk, Y=δ/δxj, and Z=δ/δxi in (35a) and, by using (36a), (36b), and (34a), we infer that the spatial component in (35a) is expressed as follows: (37)∑i,j,kR-ihjk+2Kihωjk=0.Taking temporal part in (35a) and any other triplet (X,Y,Z) from the threading frame we obtain the identities from (9a) and (9b).Next, takeX=δ/δxk, Y=δ/δxj, Z=δ/δxi, and U=δ/δxh in (35b), and, by using (31), (34a), (34c) and (34d), we obtain (38)∑i,j,kR-hlij∣k+R-hl0i∣kωjk=0.Finally, take X=∂/∂x0, Y=δ/δxj, Z=δ/δxi, and U=δ/δxh in (35b), and, by using (31), (34a), (34b), (34c), (34d), and (23c), we deduce the identity (39)R-hlij∣0+R-hl0i∣j-R-hl0j∣i+R-hlikKjk-R-hljkKik+ajR-hl0i-aiR-hl0j=0.The other identities obtained from (35b) either are trivial or do not involve the curvature tensor R¯ of ∇¯. Thus, we are entitled to call (37), (38), and (39) theBianchi 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 onM, then (37) and (38) become (40)∑i,j,kR-iljk=0,∑i,j,kR-hlij∣k=0,which are the well-known Bianchi identities on the 3-dimensional leaves of SM. Moreover, in this case, by using (14) and (27a) and (27b), we deduce that (41a)Kij=Θij,(41b)R-ij0h=Θi∣hj-∂Γ-ijh∂x0+Θijah,(41c)R-ijkh=δΓ-ijkδxh-δΓ-ijhδxk+Γ-ilkΓ-ljh-Γ-ilhΓ-ljk.Finally, identity (39) becomes (42)R-hlij∣0+R-hl0i∣j-R-hl0j∣i+R-hlikΘjk-R-hljkΘik+ajR-hl0i-aiR-hl0j=0.


## 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 (33b) wherein we remove the tilde. Then, consider the following local components of R with respect to the threading frame {∂/∂x0,δ/δxi}:(43a)Rδδxk,δδxj,δδxi=Rihjkδδxh+Ri0jk∂∂x0,(43b)Rδδxk,∂∂x0,δδxi=Rih0kδδxh+Ri00k∂∂x0.Now, comparing (A.4) and (A.8) from Appendix A with (43a) and (43b), respectively, we obtain (44a)Rihjk=R-ihjk+Φ-2KijKkh-KikKjh,(44b)Ri0jk=Φ-2Kij∣k-Kik∣j+Kikcj-Kijck-2biωjk,(44c)Rih0k=R-ih0k+biKkh-bhKik,(44d)Ri00k=bi∣k+bibk-Φ-2Kik∣0+KijKkj-ΨKik.By using (15), (6a), (6b), (6c), and (6d), we deduce that (45a)Kij∣k=Θij∣k+Φ2ωij∣k+2ωijck,(45b)Kij∣0=Θij∣0+Φ2ωij∣0+2Ψωij.Taking account of (45a), (45b), (14), and (15) into (44a), (44b), (44c), and (44d), we infer that (46a)Rihjk=R-ihjk+ωij+Φ-2ΘijΘkh+Φ2ωkh-ωik+Φ-2ΘikΘjh+Φ2ωjh,(46b)Ri0jk=Φ-2Θij∣k-Θik∣j+Θikcj-Θijck+ωij∣k-ωik∣j+ωijck-ωikcj-2biωjk,(46c)Rih0k=R-ih0k+biΘkh-bhΘik+Φ2biωkh-bhωik,(46d)Ri00k=bi∣k+bibk-Φ-2Θik∣0-ΨΘik-ωik∣0-Ψωik-ωij+Φ-2ΘijΘkj+Φ2ωkj.Next, by using the local components of the curvature tensor fields R and R¯ of type (0,4) (see (A.9) and (A.10)), from (44a), (44b), (44c), (44d), (46a), (46b), (46c), and (46d) we obtain (47a)Riljk=R¯iljk+Φ-2KijKlk-KikKlj,(47b)Ri0jk=Kik∣j-Kij∣k+Kijck-Kikcj+2Φ2biωjk,(47c)Ril0k=R¯il0k+biKlk-blKik,(47d)Ri00k=Kik∣0+KijKkj-ΨKik-Φ-2bi∣k+bibk,(48a)Riljk=R¯ijk+Φ2ωij+Φ-2Θijωlk+Φ-2Θlk-ωik+Φ-2Θikωlj+Φ-2Θlj,(48b)Ri0jk=Θik∣j-Θij∣k+Θijck-Θikcj+Φ2ωik∣j-ωij∣k+ωikcj-ωijck+2biωjk,(48c)Ril0k=R¯il0k+biΘlk-blΘik+Φ2biωlk-blωik,(48d)Ri00k=Θik∣0-ΨΘik+Φ2ωik∣0+Ψωik+ωij+Φ-2ΘijΘkj+Φ2ωkj-bi∣k-bibk.With the theory of hypersurfaces of the spacetime in mind, we call (47a) and (48a) resp., (47b), (48b), and (48c) theGauss equations (resp.,Codazzi equations) for the spatial distribution SM in the ambient space (M,g). Also, all the equations from (47a), (47b), (47c), (47d), (48a), (48b), (48c), and (48d) will be calledstructure equations induced by the (1+3) threading formalism. They have an important role in the next sections.Now, taking into account the symmetries ofR, we deduce some identities for R¯ and for kinematic quantities. First, using well-known identities for R and taking into account (47a) and (48a), we obtain the following identities for R¯: (49a)R¯iljk+R¯ilkj=0,(49b)R¯iljk+R¯lijk=0,(49c)R¯iljk-R¯jkil=Φ-2KikKlj+KjiKkl-KijKlk-KjlKki=2Θikωlj+Θljωik+Θijωkl+Θklωji.Also, taking into account that (50)Ri0jk=-Rjk0iand by using (47b), (47c), (48b), and (48c), we deduce that (51)R¯jk0i=Kij∣k-Kik∣j+Kikcj-Kijck-2Φ2biωjk-bjKki+bkKji=Θij∣k-Θik∣j+Θijak-Θikaj+Φ2ωij∣k-ωik∣j+ωikaj-ωijak-2biωjk.Finally, using the identity (52)Ri00k=Rk00iand taking the symmetric and skew-symmetric parts in (47d) and (48d), we infer that (53a)Ri00k=12Kik∣0+Kki∣0+KijKkj+KkjKij-ΨKik+Kik-Φ-2bi∣k+bk∣i-Φ2bibk,(53b)Ri00k=Θik∣0-ΨΘik+Θijθkj+Φ4ωijωkj-Φ2bibk-12Φ2bi∣k+bk∣i,(54a)Kik∣0-Kki∣0+KijKkj-KkjKij-ΨKik-Kki-Φ2bi∣k-bk∣i=0,(54b)ωik∣0=-Ψωik+Φ2ωkjΘij-ωijΘkj+12bi∣k-bk∣i.In particular, suppose that (M,g) is avorticity-free spacetime; that is, the vorticity tensor field vanishes identically on M. 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: (55a)R¯iljk=R¯jkil,(55b)R¯jkoi=Θij∣k-Θik∣j+Θijak-Θikaj.Also, from (54b) we see that (56)bi∣k=bk∣i,and (48a), (48b), (48c), and (48d) become (57a)Riljk=R¯iljk+Φ-2ΘijΘlk-ΘikΘlj,(57b)Ri0jk=Θik∣j-Θij∣k+Θijck-Θikcj,(57c)Ril0k=R¯il0k+Θlkbi-Θikbl,(57d)Ri00k=Θik∣0-ΨΘik+ΘijΘkj-Φ2bi∣k+bibk.


## 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 {Ek} be an orthonormal basis in Γ(SM). Then, {Φ-1∂/∂x0,Ek} is an orthonormal frame field on M. According to [7, p. 87], the Ricci tensor of (M,g) is given by (58)RicX,Y=∑k=13REk,X,Ek,Y-Φ-2R∂∂x0,X,∂∂x0,Y,for all X,Y∈Γ(TM). Now, we express Ek as follows: (59)Ek=Ekiδδxi,and we obtain (60)g¯ij=∑k=13EkiEkj.Then we consider the following local components of Ric with respect to the threading frame {∂/∂x0,δ/δxi}: (61a)Rij=Ricδδxj,δδxi,(61b)Ri0=Ric∂∂x0,δδxi,(61c)R00=Ric∂∂x0,∂∂x0,and, by using (59), (60), (61a), (61b), (61c), and (A.9) into (58), we obtain (62a)Rij=g¯lkRiljk+Φ-2Ri00j,(62b)Ri0=g¯lkRil0k=-g¯lkRk0il,(62c)R00=-g¯ikRi00k.Now, by using (47a), (47b), (47c), (47d), (48a), (48b), (48c), (48d), (53a), and (53b) in (62a), (62b), and (62c), we deduce that (63a)Rij=R-ikjk+Φ-2Θ-ΨKij+Kij∣0-bi∣j-bibj,(63b)Ri0=R-ik0k+Θbi-Kikbk=Ki∣kk-Θ∣i+Θci-ckKIk+2Φ2ωikbk,(63c)R00=ΨΘ-Θ∣0-KhkKkh+Φ2b2+b∣kk,(64a)Rij=R-ikjk+Φ-2Θ-ΨΘij+Θij∣0+ωij∣0+Θ+Ψωij-bi∣j-bibj,(64b)Ri0=R-ik0k+Θbi-Θikbk-Φ2ωikbk,=Θi∣kk-Θ∣i+Θci-Θikck-Φ2ωi∣kk+ckωik-ωikbk,(64c)R00=ΨΘ-Θ∣0-σ2-13Θ2+Φ4ω2+Φ2b2+b∣kk,where we put(65)σ2=σhkσhk,ω2=ωhkωhk,b2=bkbk.Next, we take the symmetric and skew-symmetric parts in (63a) and (64a) and obtain (66a)Rij=R¯ij+Φ-2Θ-ΨΘij+Θij∣0-12bi∣j+bj∣i-bibj,(66b)12R-ikjk-R-jkik=12bi∣j-bj∣i-ωij∣0-Θ+Ψωij,where we put (67)R¯ij=12R-ikjk+R-jkik.The spatial with local components R¯ij are called thespatial Ricci tensor of the spacetime (M,g).The above formulas for Ricci tensor enable us to obtain a new formula for the scalar curvatureR of (M,g) in terms of spatial tensor fields. We start with R given by (68)R=∑k=13RicEk,Ek-Φ-2Ric∂∂x0,∂∂x0=g¯ijRij-Φ-2R00.Then by direct calculations, using (66a) and (64c) into (68), we obtain (69)R=R¯+Φ-243Θ2-2ΨΘ+2Θ∣0-Φ2ω2-2b2-2b∣kk,where we put (70)R¯=g¯ijR¯ij.We call R¯ thespatial scalar curvature of the spacetime (M,g). Note that both R¯ij and R¯ are related to the geometry of the spatial distribution SM 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 fieldG of (M,g) given by (71)GX,Y=RicX,Y-R2gX,Y,∀X,Y∈ΓTM.Then, with respect to the threading frame field {∂/∂x0,δ/δxi}, we have (72a)Gij=Gδδxj,δδxi=Rij-R2gij,(72b)Gi0=G∂∂x0,δδxi=Ri0,(72c)G00=G∂∂x0,∂∂x0=R00+Φ2R2.By using (66a), (64b), (64c), and (69) into (72a), (72b), and (72c), we obtain (73a)Gij=G¯ij+b2+b∣kk+12Φ2ω2g¯ij-12bi∣j+bj∣i-bibj+Φ-2Θ-ΨΘij+Θij∣0-Θ∣0-ΨΘ+23Θ2+12σ2g¯ij,(73b)Gi0=R-ik0k-R-jkik+Θbi-Θikbk-Φ2ωik=Θi∣kk-Θ∣i+Θci-Θikck-Φ2ωi∣kk+ckωik-2ωikbk,(73c)G00=12Φ2R¯+23Θ2-σ2+Φ4ω2,where we put (74)G¯ij=R¯ij-R¯2g¯ij.The spatial tensor field G¯ with local components G¯ij is called the 3DEinstein 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-formω 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 (75a)ωX=ωSX,(75b)SX,Y=SSX,SY,for all X,Y∈Γ(TM). As an example, the Riemannian metric g¯ on SM defines a symmetric tensor field g¯ on M given by (76)g¯X,Y=g¯SX,SY,∀X,Y∈ΓTM.Note that g¯ from (76) 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∂/∂x0 by the formula (77)uX=Φ-1gX,∂∂x0,∀X∈ΓTM.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: (78)TX,Y=ρuXuY+qXuY+qYuX+pg¯X,Y+πX,Y,∀X,Y∈ΓTM.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 (75a) and π is a symmetric and trace-free tensor field on M defined by a spatial tensor field as in (75b). Now, take X=Y=U in (78) and obtain (79)ρ=TU,U.Then we put (80)T00=T∂∂x0,∂∂x0,and (79) becomes (81)T00=Φ2ρ.Similarly, take X=U and Y=SY in (78) and deduce that (82)qSY=-TU,SY.Now, we put (83a)Ti0=T∂∂x0,δδxi,(83b)qi=qδδxi,and, taking Y=δ/δxi in (82), we infer that (84)Ti0=-Φqi.Finally, denote (85a)Tij=Tδδxj,δδxi,(85b)πij=πδδxj,δδxi,and, taking X=δ/δxj and Y=δ/δxi in (78), we obtain (86)Tij=pg¯ij+πij. Contracting (86) by g¯ij and taking into account that π is defined by a trace-free spatial tensor field, we infer that (87a)p=13g¯ijTij,(87b)πij=Tij-13Thkg¯hkg¯ij.Now, taking into account (79)–(87b), it is clear that (78) represents the coordinate-free version of (5.9) from [4, p. 91]. Thus q and π from (78) are therelativistic momentum density and therelativistic anisotropic (trace-free) stress tensor field, respectively.


## 7. A New Splitting of Einstein Field Equations with respect to General(1+3) Threading of Spacetime

We start this section with the coordinate-free form of Einstein field equations(EFE) expressed as follows: (88)GX,Y+ΛgX,Y=8πGTX,Y,∀X,Y∈ΓTM,where Λ is the cosmological constant and G is the Newton constant. Now, take X=δ/δxj and Y=δ/δxi in (88) and, by using (73a), (10), and (86), we obtain (89)G¯ij+b2+b∣kk+12Φ2ω2+Λg¯ij-12bi∣j+bj∣i-bibj+Φ-2Θ-ΨΘij+Θij∣0-Θ∣0-ΨΘ+23Θ2+12σ2g¯ij=8πGpg¯ij+πij.The equations from (89) will be called thespatial Einstein field equations (SEFE). Next, we take X=∂/∂x0 and Y=δ/δxi in (88) and, by using (73a) and (84), we deduce that (90a)R-ik0k+Θbi-Θikbk-Φ2ωik=-8πGΦqi,(90b)Θi∣kk-Θ∣i+Θci-Θikck-Φ2ωi∣kk+ckωik-2ωikbk=-8πGΦqi. The equations from either (90a) or (90b) are calledmixed Einstein field equations(MEFE). Finally, take X=Y=∂/∂x0 in (88), and, by using (73c), (2a), and (81), we infer that (91)Φ2R¯+23Θ2-σ2+Φ4ω2=2Φ2Λ+8πGρ.We call (91) thetemporal Einstein field equation(TEFE). 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 g¯ij and using (74) we deduce that the spatial scalar curvature is given by (92)12R¯=2b2+b∣kk+32Φ2ω2+3Λ-Φ-2Θ2+Θ∣0+32σ2-2ΨΘ-24πGp.Comparing (92) with (91), we obtain theRaychaudhuri-Ehlers equation induced by the general (1+3) threading of the spacetime: (93)Θ∣0+13Θ2+σ2-ΨΘ-Φ2b2+b∣kk+Φ2ω2+Λ-4πGρ+3p=0.Note that (93) is the generalization of (6.4) from [4], which was obtained for the particular case Φ=1.


## 8. 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 ofT. In order to obtain their explicit form, we consider an orthonormal frame field {Ek,U=Φ-1∂/∂x0}, and according to [7, p. 86] we have (94)divTX=∑k=13∇EkTEk,X-∇UTU,X=0,for all X∈Γ(TM). Then, take X=∂/∂x0, in (94), and by using (B.1) and (B.2) obtain (95)Φ2T0∣jj+2bj-cjTj0-ΘjkTjk-∂T00∂x0+2Ψ-ΘT00=0.Similarly, take X=δ/δxi in (94) and by using (B.3) and (B.4) we deduce that (96)Φ2Ti∣jj-ωijT0j+Tijbj+Ψ-ΘTi0-Ti0∣0+T00bi-ΘijT0j=0.Now, from (84) and (86), we infer that (97a)T0j=-Φqj,(97b)Tij=pδij+πij.Then, by using (24), (81), (84), (86), (97a), and (97b) in (95) and (96) and taking into account that both σij and πij are trace-free spatial tensor fields, we obtain theenergy conservation equation: (98)∂ρ∂x0+ρ+pΘ+σjkπjk+Φq∣jj+2bjqj=0and themomentum conservation equation:(99)qi∣0+43Θqi+σij+Φ2ωijqj+Φδpδxi+πi∣jj+p+ρbi+πijbj=0.In order to compare with what is known in the literature with respect to the (1+3) threading of spacetime, we note that (100)∇UU=bkδδxk.This is obtained by direct calculations using (23d). Also, we should remark that (100) states that though the velocity U is a timelike vector field, the acceleration ∇UU is a spatial vector field.Remark 7.
The above conservation laws are obtained in the most general setting. Indeed, if in particularΦ=1, (98) and (99) become (5.11) and (5.12) from [4, p. 92], respectively. If, moreover, we have a perfect fluid on M, that is, qi=0 and πij=0, then (98) and (99) become (1) and (2) from Proposition  5 in [7, p. 339].


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

LetM be an FLRW universe, whose line element is given by (101)ds~2=g~abdxadxb=a2τ-dτ2+δijdxidxj,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 (101) is called thebackground 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 thefull metricg on M is expressed as follows: (102)ds2=gabdxadxb=g~abdxadxb+δgabdxadxb,where δgab determine the perturbation. In the present paper we consider the conformal-Newtonian gauge case, for which the full metric is given by (103)ds2=a2τ-1+2Adτ2+1-2Bδijdxidxj,where A and B are the well-knownBardeen invariants. The spacetime (M,g) with g given by (103) is called analmostFLRWuniverse.Now, by using (2b) for (103), we deduce that ξi=0,i∈{1,2,3}, which imply(104a)δδxi=∂∂xi,(104b)ωij=0,(104c)ai=0,(104d)g¯ij=a21-2Bδij,(104e)g¯ij=1a21-2Bδij.Also, according to the notation in (2a), we have (105)Φ2=a21+2A.From (104b) we see that the spatial distribution SM of (M,g) is integrable, but its leaves are not anymore Euclidean spaces. Moreover, by using (12a), (12b), (6d), (104d), and (105), we obtain (106a)Θij=a21-2BH-B′δij,(106b)Θ=3H-B′1-2B,(106c)Ψ=H+A′1+2A, where “∣” denotes derivative with respect to τ and H=a′/a is theHubble parameter of the background metric. By calculations, using (106a), (106b), and (104d) into (12c), we infer that (107a)σij=0,∀i,j∈1,2,3.(107b)Θij=13Θg¯ij,∀i,j∈1,2,3.Taking into account that Φ-2Θij are the local components of the second fundamental form of the leaves of SM (see (23a)) and using (107a) and (107b), we can state the following result on the kinematic quantities and geometry of (M,g).Theorem 8.
Let(M,g) be an almost FLRW universe. Then one has 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 (108)H=13Φ-2Θ∂∂τ.Next, we assume that the energy-momentum tensor for the almostFLRW universe (M,g) takes the perfect fluid form, that is, (109)qi=0,πij=0,∀i,j∈1,2,3.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 (24), (104c), (6b), (105), and (104e), we deduce that(110a)bi=ci=Ai1+2A,(110b)b2=1a21-4A21+2A∑k=13Ak2,where we put Ai=∂A/∂xi. The local coefficients of the spatial Riemannian connection ∇¯ are given by (see (20a), (20b), (C.1), (14), (107a), and (107b)) (111a)Γ-ikj=11-2AδijAk-δikAj-δjkAi,(111b)Kij=Θij=13Θδij,where we have Ak=Ak. Then, by using (110a) and (111a), we obtain the spatial covariant derivative and the divergence of the acceleration, given by (112a)bi∣j=11+2AAij+8A1-4A2AiAj-11-2Aδij∑k=13Ak2,(112b)b∣kk=1a21-4A2∑k=13Akk+2A-31-4A2Ak2,where we put (113)Aij=∂2A∂xi∂xj.Now, by direct calculation using (106b), we infer that(114a)Θ∣0=∂Θ∂τ=3H′-2A′2+1-2AA′′1-2A2,(114b)Θ∣i=∂Θ∂xi=-31-2A22A′Ai+1-2AAi′.Also, by using (107b), (111b), (114a), and (114b) and taking into account that ∇¯ is a metric connection, we obtain (115a)Θij∣0=13Θ∣0g¯ij=a21-2A1-2A2H′-1-2AA′′-2A′2δij,(115b)Θi∣kk=13Θ∣i=-11-2A22A′Ai+1-2AAi′.Finally, by using (C.7) and (C.8) in (74), we deduce that the 3D Einstein gravitational tensor field is given by (116)G¯ij=11-2A23AiAj+1-2AAij-δij∑k=132Ak2+1-2AAkk.Now, we are in a position to present the splitting of EFE for the almost universe (M,g). First, we consider that the background energy-momentum tensor T~ takes the perfect fluid form; that is, we have (117a)T~ij=p~g~ij=p~a2δij,(117b)T~io=0,(117c)T~00=a2ρ~,where ρ~ and p~ are the relativistic density and pressure, respectively. Then, the energy-momentum tensor T of the perturbed universe (M,g) should have the perfect fluid form too. Hence, by (86), (84), and (81), we have (118a)Tij=pg¯ij=pa21-2Aδij,(118b)Tio=0,(118c)T00=Φ2ρ=a21+2Aρ,where we put (119)p=p~+δp,ρ=ρ~+δρ.After some long calculations using (116), (110a), (110b), (112a), (112b), (114a), (106a), (106b), (106c), (107a), (107b), (104d), (105), and (118a) into (89), we deduce that the SEFE for the almost FLRW universe (M,g) are given by (120)21-4A22AAij+1+4A+12A21-4A2AiAj+δij11+2A2A′′+6+4A1+2AHA′+6A-11-4A2A′2-1-2AH2+2H′-11-4A24A∑k=13Akk+3+4A+12A21-4A2∑k=13Ak2=8πGp-Λa21-2Aδij.Next, since (M,g) is vorticity-free and qi=0, (90b) becomes (121)Θ∣i-Θci=0,via (107b) and (115b). Then, by using (106b), (110a), and (114b) into (121), we obtain the following MEFE for (M,g): (122)1+6AA′Ai+1-4A2Ai′+1-2A2HAi=0.Also, by using (C.8), (106b), and (105) into (91) and taking into account that σ2=ω2=0, we deduce that the TEFE for (M,g) is given by (123)1+2A1-2A3∑k=133Ak2+21-2AAkk+3H-A′1-2A2=a21+2AΛ+8πGρ.Finally, Raychaudhuri-Ehlers equation (93) becomes (124)3H′-11-2AA′′-11-4A2HA′-4A1-2A1-4A2A′2-11-2A∑k=13Akk-21-4A2∑k=13Ak2=a21+2AΛ-4πGρ+3p.Summing up the above results, we state the following theorem.Theorem 9.
Let(M,g) be an almost FLRW universe with the energy-momentum tensor of a perfect fluid. Then one has the following assertions:(i)
TheSEFE, MEFE, and TEFE of (M,g) are given by (120), (122), and (123), respectively.(ii)
The Raychaudhuri-Ehlers equation in(M,g) is given by (124).In particular, suppose thatA=0 and Λ=0. Then, (123) and (124) become (125)H2=8πG3a2ρ,H′=-4πG3a2ρ+3p,respectively. Note that, in this case, (122) is trivial, and (120) is a consequence of (125) and (9.24). As (125) and (9.24) are the well-known Friedmann equations for an FLRW universe (cf. (8.4) and (8.5) in [8]), we are entitled to call (123) and (124) theperturbed Friedmann equations in an almost FLRW universe.


## 10. Conclusions

The idea to develop(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 of (2a), (2b), and (2c) with Φ2≠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 onM. 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]). In particular, if the spatial distribution is integrable, the ADM formalism (cf. [3]) can be applied. However, even in this case, the threading formalism is totally different from the ADM formalism. This is because in the ADM formalism the frame field is used,(126)∂∂xi,δδx0=∂∂x0-Ni∂∂xi,which is adapted to the 3D spatial foliation. Thus, all the formulas like (89), (90a), (90b), (91), (98), and (99) from our paper are expressed with respect to the frame field (126). The advantages of the threading formalism of spacetime with respect to an arbitrary timelike vector field are the following: (i)
It can be applied to any Lorentz metric given by (2a), (2b), and (2c) with Φ2≠1. In particular, the perturbation theory and the black holes theory are suitable for this formalism.(ii)
The spatial distribution is not supposed to be necessarily integrable, and therefore this formalism can be used in a study of any cosmological model.(iii)
It provides, for the first time in literature, a spatial covariant differentiation for spacetimes with nonzero vorticity.The approach we develop in the paper is based on spatial tensor fields and on the Riemannian spatial connection which behave as3D geometric objects with local components defined on a 4D spacetime. It is noteworthy that the three groups of EFE presented in the paper are expressed in terms of spatial tensor fields and their covariant derivatives induced by the Riemannian spatial connection. This enables us to write down a splitting of EFE for an almost FLRW 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]).



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