Covariant gravitational dynamics in 3+1+1 dimensions

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Covariant gravitational dynamics in 3+1+1 dimensions

Zoltán Keresztes, László Gergely

To cite this version:

Zoltán Keresztes, László Gergely. Covariant gravitational dynamics in 3+1+1 dimensions. Classical and Quantum Gravity, IOP Publishing, 2010, 27 (10), pp.105009. �10.1088/0264-9381/27/10/105009�.

�hal-00594825�

Covariant gravitational dynamics in 3+1+1 dimensions

Zolt´an Keresztes^1,2, L´aszl´o ´A. Gergely^1,2

1 Department of Theoretical Physics, University of Szeged, Tisza Lajos krt 84-86, Szeged 6720, Hungary

2 Department of Experimental Physics, University of Szeged, D´om T´er 9, Szeged 6720, Hungary (Dated: February 25, 2010)

We develop a 3+1+1 covariant formalism with cosmological and astrophysical applications. First we give the evolution and constraint equations both on the brane and off-brane in terms of 3- space covariant kinematical, gravito-electro-magnetic (Weyl) and matter variables. We discuss the junction conditions across the brane in terms of the new variables. Then we establish a closure condition for the equations on the brane. We also establish the connection of this formalism with isotropic and anisotropic cosmological brane-worlds. Finally we derive a new brane solution in the framework of our formalism: the tidal charged Taub-NUT-(A)dS brane, which obeys the closure condition.

PACS numbers:

I. INTRODUCTION

In recent years the idea of exploring the possibility of the gravitational interaction acting in more than 4-dimensions (4d) attracted a lot of attention. In particular, one of the simplest such models arises from the curved generalization of the one-brane Randall-Sundrum model [1] (for a review see [2]), where gravity acts in 5-dimensions (5d), however standard model fields are confined to the 4d brane. Nonstandard model fields can occur in the 5d space-time. A Z₂-symmetric embedding looks natural only if the brane is envisaged as a boundary; otherwise generic asymmetric embeddings should be allowed. The generic dynamics was given in Refs. [3] and [4] in a 5d-covariant approach.

Although promising at the level of allowing new degrees of freedom, which seem to be adjustable to explain for example dark matter [5]-[8], the complexity of the brane-world dynamics also represents a major impediment in obtaining simple exact solutions or in monitoring the evolution of perturbations. Therefore new approaches to gravitational dynamics in brane-worlds are worth to study.

In Ref. [9] the 5d space-time was foliated first by a family of 4d space-times and a second family of 4d space-like hypersurfaces (see Fig 1. in Ref. [9]). Such an 5d space-time is double foliable. This structure of the 5d space-time allowed to describe the gravitational degrees of freedom in terms of tensorial, vectorial and scalar quantities with respect to the 3-space emerging as the intersection of the two hypersurfaces. They represent the gravitons, a gravi- photon, and a gravi-scalar and are given by quantities with well-defined geometrical meaning, namely the tensorial, vectorial and scalar projections of the spatial 4d metric and their canonically conjugated momenta: the extrinsic curvature (second fundamental form of the 3-spaces with respect to the temporal normal), normal fundamental form and normal fundamental scalar. The evolution equations for these variables were given `a la ADM both on the brane and outside it. An extension of this formalism towards a full Hamiltonian description was advanced in Ref. [10]. It has been shown, that among all gravitational variables on the brane, only the momentum of the gravi-vector has a jump across the brane, related to the energy transport (heat flow) on the brane.

The formalism presented in Refs. [9]-[10] however is not straightforward to be applied in brane cosmology. A formal difference would be the definition of the time derivative. In Refs. [9]-[10] this was defined in the tradition of canonical gravity as a Lie-derivative taken along the temporal direction (which is not necessarily orthogonal to the 3-space) projected to the 3-space, whereas in cosmology by tradition the time derivative is defined as a covariant derivative along the normal to the 3-surfaces (this derivative happens to enter only in expressions projected onto the 3-space).

Obviously this mismatch in the definitions of the time derivatives is not crucial, as the two definitions differ only in terms taken on the 3-space. However the formalism developed in Refs. [9] and [10] relying on the double foliability of the 5d space-time is unable to deal with the possiblevorticity of the word-lines of observers.

In this paper we develop a formalism overcoming this inconvenience and derive the full set of evolution and constraint equations governing gravitational dynamics in a 3+1+1 covariant form. Provided the space-like normalnhas vanishing vorticity on a hypersurface (the time evolution vector can still have vorticity), the formalism becomes suitable for describing gravitational dynamics on the brane (then nbecomes the brane normal). In this sense the formalism is a generalization of the brane 3+1 covariant cosmology [11] (which in turn is a generalization of the general relativistic 3+1 covariant cosmology [12]-[15]).

This newly developed formalism also generalizes the s+1+1 covariant brane-world dynamics developed in Ref. [9].

Both the vector fieldnand the time evolution vectoruare allowed to have vorticity in the present formalism. Though observational evidence suggest that the directly detectable 3+1 part of the universe is best described by a Friedmann brane with perfect fluid, when discussing cosmological perturbations, the vorticity ofushould be allowed, similarly

as in existing formalisms of covariant cosmology in both general relativity and brane-worlds. We can also argue for the need of keeping the vorticity of n. One reason would be, that if it is vanishing at some initial instant, it should stay zero; and the formalism should be able to handle its ”evolution”. Secondly, and more important, the vorticity of nshould not necessarily vanish at other locations, than the brane, a gauge freedom worth to explore.

We note that a lower-dimensional, 2+1+1 formalism was developed in Refs. [16], [17] and applied in the general relativistic covariant perturbations of Schwarzschild black holes and rotationally symmetric space-time; then for investigating spherically symmetric static solutions inf(R) gravity theories in Ref. [18].

In general relativity the important topic of cosmological perturbations, related to both the Cosmic Microwave Background and structure formation, has a rich literature, from which (without claiming completeness) we mention Refs. [19]-[27], all based on the 3+1 covariant approach.

Brane-world cosmological perturbations are equally important, however additional difficulties emerge due to the complexity of brane-world theory and the impediment to predict and perform observations on the brane. At a technical level, the latter is obstructed by the lack of closure of the perturbation equations on the brane. Despite impressive developments [28]-[40], many questions remain unanswered. Although we cannot overcome well-known difficulties, we expect our new formalism will provide the most convenient and complete tool-chest for approaching the problems.

We also foresee the possibility of important applications for brane-world exact solutions.

We establish the generic 3+1+1 covariant formalism in Sec. II, by defining the kinematical quantities and the decomposition of the Weyl and energy-momentum tensors. We also relate the curvature and Ricci tensors and the 3-dimensional (3d) scalar curvature to kinematic, non-local (Weyl) and matter-defined variables. In Appendix A the commutation relations among all types of derivatives emerging in the formalism are given.

Sec. III contain the full 3+1+1 decomposed covariant gravitational dynamics and constraints, together with the available matter field evolutions. In Appendix B we discuss the gauge freedom in the frame choice and give the transformations of all relevant quantities under infinitesimal frame transformations.

Taking into account that the brane is a 4d time-like hypersurface, in Sec. IV we discuss the decomposition of the Lanczos equation and of the sources of the effective Einstein equation. Then we specify the generic evolution and constraint equations on the brane, expressed in terms of quantities defined on the brane. These equations arise from combinations of the equations given in Section III, evaluated on the brane. Appendix C contains the brane equations for an asymmetric embedding. We continue Section IV by specializing to a symmetrically embedded brane, by taking into account the Lanczos equation. Then we conclude the section by deriving a closure condition for the equations on the brane.

Sec. V contains the derivation of the most important cosmological equations to a hypersurface (a generic brane), then the discussion of the particular case of cosmological symmetries and perfect fluid source on the brane. As a test of our formalism we recover the Friedmann, Raychaudhuri and energy balance equations and compare them with the relevant results of Ref. [4]. In subsection V B and Appendix D we also relate our formalism to anisotropic brane-world cosmologies.

Sec. VI contains an astrophysical application of our formalism devoted to locally rotationally symmetric (LRS) space-times, at the end of which we recover a new brane solution with NUT charge. This solution obeys the previously derived closure condition.

Sec. VII contains the Concluding Remarks.

Notations. Quantities defined on the 3-space orthogonal to both u^a and n^a carry no distinguishing mark and the 3d metric is denotedh_ab. Quantities defined on the brane carry the pre-index ⁽⁴⁾, the only exception being the 4d metric g_ab. Quantities defined on the full 5-space carry a distinguishingemark. Exceptionally, other quantities also carry the distinguishingemark. These are (a) the 3+1+1 decomposed components of the 5d energy-momentum tensor, which are defined on the 3-space orthogonal to bothu^a andn^a, and (b) the kinematic and extrinsic curvature type quantities related to another singled out spatial vector e^a in Sec. VI. Calligraphic symbols denote 3+1+1 decomposed components of the 5d Weyl tensor. Whenever possible, identical symbols are used for quantities related to the temporal vector fieldu^a and the brane normaln^a, the latter distinguished by abmark. We denote the average of a quantityf taken over the two sides of the brane byhfi, and its jump by ∆f. Angular bracketsh ion abstract indices denote tensors which are projected in all indices with the metric hab, symmetrized and trace-free. Round brackets ( ) and square brackets [ ] on indices denote the symmetric and antisymmetric parts, respectively.

II. THE 3+1+1 COVARIANT FORMALISM A. Decomposition of the metric

Let u^a = dx^a/dτ and n^a = dx^a/dy be a time-like and a space-like vector field in the 5d space-time, with τ and y the affine parameters of the respective non-null integral curves. They obey the normalization conditions

FIG. 1: Elements of the 3+1+1 decomposition of the 5d space-time geometry with metric egab and compatible connection∇.e The 4d brane with normaln^a has induced metricgab=egab−nanband extrinsic curvature⁽⁴⁾Kab=g_a^cg^d_b∇e_cnd. The 4-velocity field u^a of observers in the brane defines local 3d orthogonal spatial patches, hypersurface-forming only when the vorticity ωab= 0. The other kinematical characteristics ofu^aare the expansion Θ and shearσab. The expansion, shear and vorticity of n^a are Θ,b bσab andωbab, the latter vanishing on the brane. The temporal, off-brane and 3d covariant derivative are shown for the vorticity 1-formωa.

−u^aua= 1 =n^ana and the perpendicularity conditionu^ana = 0. The 5d metric is decomposed as e

gab=nanb+gab , (1)

with

gab=−uaub+hab , (2)

the metric on the 4d temporal leaves y =const (with the brane at y = 0) and the spatial part h_ab of this metric obeyingu^ah_ab=n^ah_ab= 0. We denote by ε_abc the volume element associated withh_ab. The 4-vectoru^a represents the time-like velocity field of brane observers (see Fig 1).

We employ three type of derivatives, all associated with projections of the 5d connection∇e_i. A dot and a prime denote covariant derivatives along the integral curves ofu^a andn^a, respectively, whileDis the 3d covariant derivative compatible with the metrich:

T˙b..c = u^a∇e_aTb..c, (3)

T_b..c⁰ = n^a∇e_aTb..c, (4)

D_aT_b..c = h_a^dh_bⁱ..h_c^j∇e_dT_i..j . (5) Note that the D-derivative is the same as introduced in general relativity employing the corresponding projection of the ∇-derivative (∇ being the connection compatible with gab). This is, because both generate the covariant derivatives formed with the connection compatible withhab. Concerning the time-derivative defined above, except for scalars, it differs from the corresponding general relativistic time derivative employed in the 3+1 covariant formalism (which is defined with ∇) in n^a and u^a terms. Nevertheless, when projected to the 3-manifold with h_b^a, the two definitions agree.

B. Kinematic quantities

We introduce the kinematic quantities through the decomposition of the 5d covariant derivative ofu^a,n^a as e

∇_au_b = −u_aA_b+Kub _an_b+Kn_an_b+n_aK_b+L_an_b+K_ab, (6a) e

∇_anb = naAbb+Knaub+ Kub aub−uaKbb+Laub+Kbab, (6b) where

Aa = ˙u_hai, Aba=n⁰_hai , Ka = u⁰_hai, Kba = ˙n_hai, K = n^bu⁰_b , Kb =u^bn˙b , La = h_a^cn^d∇e_cud ,

Kab = Daub , Kbab=Danb . (7) As a rule, an overhat is used for kinematical quantities related to the vector field n^a in order to distinguish them from the similar kinematical quantities related to the vector fieldu^a. HereAa is the acceleration. All scalars, vectors and tensors in the above decomposition are defined on the 3d manifold orthogonal both to n^a andu^a. The tensorial expressionsK_abandKb_abcan be further decomposed into (trace-, trace-free symmetric and antisymmetric) irreducible parts as follows:

Kab = Θ

3hab+σab+ωab, (8)

b

Kab = Θb

3hab+σbab+bωab, (9)

where we have defined the expansion, vorticity and shear of the vector fieldsu^a andn^a as Θ = D^aua , ωab=D_[au_b] , σab=D_hau_bi ,

b

Θ = D^ana , ωbab=D_[an_b] , bσab=D_han_bi . (10) The antisymmetric 3d tensors ωab and ωbab can be also encoded in the vorticity vectors ω^a = ε^abcωbc/2 andωb^a = ε^abcωbbc/2. When the vorticities of both vector fieldn^a andu^avanishωab=ωbab= 0, the tensorial expressionsKaband Kbabare symmetric and they represent the two extrinsic curvatures of a 3d hypersurface. The conditionbωab= 0 is also necessary in order to have a brane aty = 0, but not sufficient. Indeed the brane is a 3+1 dimensional hypersurface which can exist only if the higher-dimensional vorticity of its normalωbab=∇_[an_b] vanishes. This condition translates into 0 =g_[a^cg_b]^d∇e_cnd =−u_[a

b

K_b]+L_b]

+ωbab, therefore (due to Frobenius’ theorem) the necessary and sufficient conditions for the existence of the 3+1 brane are

b

ωab|_y=0 = 0, La|_y=0 = −Kba

_y=0 . (11)

In summary, the independent components of ∇e_aub are expressed by three scalars (K, K,b Θ), four 3-vectors (Aa, Ka, La, ωa) and a symmetric trace-free 3-tensor (σab). The corresponding decomposition of ∇e_anb consists of the three scalars (K, K,b Θ), four 3-vectors (b Ab_a, Kb_a, L_a, ωb_a) and a symmetric trace-free 3-tensor (bσ_ab). The irreducible decompositions of the covariant derivatives ∇e_aub and ∇e_anb have therefore 20 independent components each (due to the constraintsu^b∇e_aub= 0 andn^b∇e_anb= 0).

C. Gravito-electro-magnetic quantities

The non-local gravitational properties of the 5d space-time are carried by the 5d Weyl tensor, the principal directions of which lead to a classification scheme generalizing the general relativistic Petrov classification [41]. Here we are

interested in the 3+1+1 decomposition¹ of Ceabcd, which can be given in terms of the quantities (with a total of 35 independent components):

E = Ce_abcdn^au^bn^cu^d , H_k = 1

2ε_k^abCeabcdu^cn^d , F_kl=Ceabcdh_(k^au^bh_l)^cn^d , b

E_k = Ceabcdh_k^an^bu^cn^d , E_k=Ceabcdh_k^au^bn^cu^d , b

E_kl = Ceabcdh_hk^an^bh_li^cn^d , E_kl=Ceabcdh_hk^au^bh_li^cu^d , b

H_kl = 1

2ε_(k ^abh_l)^cCe_abcdn^d , H_kl= 1

2ε_(k ^abh_l)^cCe_abcdu^d . (12) We note that all tensorial quantities defined above are trace-free (from the properties of the Weyl tensor), and further in the tensorsF_kl,H_kl andHb_klthe bracketsh iare equivalent with the round brackets ( ). The Weyl tensor in terms of the quantities defined in (12) is

e

Cabcd = −2 E_d[ah_b]c− E_c[ah_b]d + 2

b

E_d[ah_b]c−Eb_c[ah_b]d

−2

3Eh_c[ah_b]d +2

ε_cd ⁱHb_i[an_b]+ε_ab ⁱHb_i[cn_d]

−2 ε_cd ⁱH_i[au_b]+ε_ab ⁱH_i[cu_d]

+2

ncn_[aEb_b]d−ndn_[aEb_b]c

+ 2 ucu_[aE_b]d−udu_[aE_b]c

−2 ucn[aF_b]d−udn[aF_b]c+ncu[aF_b]d−ndu[aF_b]c

− n_[cu_a]ε_bdk+u_[cn_b]ε_adk

H^k− u_[dn_a]ε_bck+n_[du_b]ε_ack H^k

−2 u_[cn_d]εabk+u_[an_b]εcdk

H^k−2 E_[ah_b][cn_d]+E_[ch_d][an_b]

−2 b

E_[ahb][cud]+Eb_[chd][aub]

+ 4Eu[anb]u[cnd]

−2

n_cu_dn_[aEb_b]−u_cn_dn_[aEb_b]+n_au_bn_[cEb_d]−u_an_bn_[cEb_d]

+2 ucndu_[aE_b]−ncudu_[aE_b]+uanbu_[cE_d]−naubu_[cE_d]

−2

3E ndn_[ah_b]c−ncn_[ah_b]d +2

3E udu_[ah_b]c−ucu_[ah_b]d

. (13)

This relation generalizes the general relativistic 3+1 covariant decomposition of the 4d Weyl tensorCabcd (which has only 10 independent components), where only two tensorsE_kl=C_abcdh_hk^au^bh_li^cu^d andH_kl= ¹₂ε_(k^abh_l)^cC_abcdu^d appear, the electric and magnetic parts of the Weyl curvature. On the brane the relation between the set of variables E_kl andH_kl and the variablesEkl andHkl is given by

E_ab = Eab+1 2Eb_ab−1

2 Kb +Θb 3

! b σab+1

2bσchabσ_bi^c+1

2KbhaKbbi, (14)

H_ab = Hab−ε_ha ^cdbσ_bicKbd . (15)

D. Decomposition of the energy-momentum tensor The 5d gravitational dynamics is governed by the Einstein equation

e

G_ab=−eΛeg_ab+eκ²h e

T_ab+τ_abδ(y)i

, (16)

where eκ² denotes the 5d coupling constant. The sources of gravity are the 5d cosmological constant Λ, the regulare part of the 5d energy-momentum tensorTeaband a distributional energy-momentum tensor with support on the brane:

τab=−λgab+Tab . (17)

1 The case of a genericn+ 1 decomposition of the Weyl tensor was discussed in Ref. [42].

Hereλis the constant brane tension andTab describes standard model matter fields on the brane², decomposed with respect to a brane observer with 4-velocityu^a as

T_ab=ρu_au_b+q_(au_b)+ph_ab+π_ab. (18) The quantities ρ, q_a, p and π_ab are the energy density, the energy current vector, the isotropic pressure and the symmetric trace-free anisotropic pressure tensor of the matter on the brane.

We decompose the regular part of the 5d energy-momentum tensor relative tou^a andn^a as:

e

Tab=ρue aub+ 2qe(aub)+ 2que (anb)+phe ab+πne anb+ 2πe(anb)+eπab , (19) whereρeis the relativistic energy density relative tou^a,pethe isotropic pressure,eqa andqethe relativistic momentum densities on the 3-space and in the directionn^a. The quantitiesπ,e πea and the symmetric and trace-free tensoreπabare related to the scalar, vectorial and tensorial components (with respect to the 3d space) of the 5d anisotropic pressure tensor, which is

3 (eπ−p)e

4 nanb+ 2eπ_(an_b)+ep−πe

4 hab+πeab . (20)

By employing the definitions given in this Section we give the commutation relations among the temporal, off-brane and brane covariant derivatives in Appendix A.

E. The Gauss equation and its contractions

We define the local 3d curvature tensorR_abcd of the space orthogonal to bothu^a andn^a as D[aDb]Vc−ωabV˙hci+ωbabV_hci⁰ =1

2R_abcdV^d , (21)

resulting in

R_abcd = h_aⁱh_b^jh_c^kh_d^lRe_ijkl−(D_au_c) (D_bu_d) + (D_au_d) (D_bu_c)

+ (Danc) (Dbnd)−(Dand) (Dbnc) . (22) This is a natural generalization of the definition used in general relativity [22], [44], [45]. By the definitions of the kinematical, gravito-electro-magnetic and matter variables, we have

R_abcd =

−2 3

Θ²−Θb²

− E+Λ +e eκ²(ρe+pe−π)e

h_c[ah_b]d 3

−2 E_d[ah_b]c− E_c[ah_b]d + 2

b

E_d[ah_b]c−Eb_c[ah_b]d

−2κe²

3 h_d[aπe_b]c−h_c[aπe_b]d

−2Θ 3

σ_a[c+ω_a[c

h_d]b+ σ_b[d+ω_b[d h_c]a

−2 σ_a[c+ω_a[c

σ_d]b−ω_d]b +2bΘ

3

bσ_a[c+ωb_a[c

h_d]b+ σb_b[d+bω_b[d h_c]a

+ 2 bσ_a[c+bω_a[c b

σ_d]b−ωb_d]b

, (23)

2 For DGP / induced gravity type models [43],T_abshould be replaced byT_ab−` γ/κ²´

G_ab, withG_abthe Einstein tensor constructed from the metricg_abandγ the dimensionless induced gravity parameter. Randall-Sundrum type brane-worlds are recovered forγ→0, on theε=−1 branch.

and the local 3d Ricci tensor and Ricci scalar become R_ac = h^bdR_abcd=

−2 3

Θ²−Θb²

−2E+Λ +e eκ²(ρe+pe−eπ) hac

3 +Eac−Eb_ac+eκ²

3eπac−2Θ

3 (σac+ωac) +2bΘ

3 (bσac+ωbac) +σabσ_c^b−ωbω^bhac+ωaωc−2σ_[a^bω_c]b

−bσ_abbσ_c^b+ωb_bbω^bh_ac−ωb_aωb_c+ 2bσ_[a^bωb_c]b , (24) R = h^acR_ac=−2E+Λ +e κe²(ρe+pe−π)e −2

3

Θ²−Θb²

−2ω_aω^a+σ_abσ^ab+ 2bω_aωb^a−σb_abσb^ab . (25) Eq. (25) can be referred as a generalized Friedmann equation in the 5d space-time, as will become evident in Section V. The general relativistic counterpart is presented in Refs. [44], [45].

III. 3+1+1 COVARIANT DYNAMICS

Thefull set of the 3+1+1 covariant evolution and constraint equations for the kinematic, gravito-electro-magnetic and matter variables are given by the projections of the Bianchi and Ricci identities foru^a and n^a. These equations hold as the main result of the paper, and they are presented in the following three subsections. In particular, the first subsection contains all Ricci identities; the second subsection contains twice contracted Bianchi identities, which by virtue of the Einstein equations describe evolutions for the energy density and currents; while the third subsection contains the rest of independent Bianchi identities. A related Appendix B gives the transformation rules under a frame change for the totality of the kinematic, gravito-electro-magnetic and matter variables, to linear order accuracy.

A. Ricci identities The Ricci identity foru^a gives the following independent equations:

0 = K˙ +Kb⁰+AbaA^a+K²−Kb²+LaK^a−KbaK^a−LaKb^a +E − Λe

6 +eκ²

6 (ρe−πe+ 3p)e , (26)

0 = ˙Θ−D^aAa+Θ²

3 +ΘbKb−A^aAa−2ωaω^a+σabσ^ab− E +

b

K^aKa+K^aLa+L^aKba

−Λe 2 +eκ²

2 (ρe+πe+p)e , (27)

0 = K˙_hai−A⁰_hai+

K+Θ 3

Ka+

K−Θ 3

Kba+E_a

+Kb b

A_a+A_a

+ (ω_ba+σ_ba)

K^b−Kb^b

−eκ²

3eπ_a , (28)

0 = ˙L_hai+DaKb +

K+Θ 3

La+ Kb +Θb 3

! Aa+

K−Θ

3

b Ka

+ (ωbab+bσab)A^b−(ωab+σab) b

K^b−L^b

+E_a−eκ²

3πea , (29)

0 = ˙ω_hai−1

2ε_a^cdD_cA_d+Kbωb_a+2Θ

3 ω_a−σ_abω^b+1 2ε_a^cd

K_cKb_d+K_cL_d+Kb_cL_d

, (30)

0 = ˙σhabi−DhaAbi+2Θ

3 σab+Kbσbab−AhaAbi+KhaLbi

+Kb_haK_bi+L_haKb_bi+ω_haω_bi+σ_chaσ_bi^c+E_ab−eκ²

3eπab , (31)

0 = Θ⁰−D^aKa−

K−Θ 3

Θ + (Kb ^a+L^a)Aba−A^a(Ka−La)−2ωbaω^a+bσabσ^ab−eκ²q ,e (32)

0 = L⁰_hai−DaK+

K−Θ 3

Aba− Kb −Θb 3

!

La+ Kb+Θb 3

! Ka

+ (bσab+ωbab) K^b+L^b

−(ωab+σab)Ab^b−Eb_a+eκ²

3qea , (33)

0 = ω_hki⁰ −1

2ε_k^abDaKb−1 2ε_k^ab

Ab+Abb

(Ka−La)−

K−Θ 3

b

ωk (34)

+Θb 3ωk+1

2ε_k^abωaωbb−1

2bσkaω^a−1

2σkaωb^a+1

2ε_k^abσcbbσ_a^c−1

2H_k , (35)

0 = σ_habi⁰ −D_haK_bi−A_hb K_ai−L_ai

+Ab_hb K_ai+L_ai +Θb

3σab

−

K−Θ 3

b

σab+ω_haωb_bi+ω_chaσb_bi^c+σ_ha^dωb_bid+σ_chabσ_bi^c+F_ab , (36) 0 = ε_k^abDaLb+ 2 Kb+Θb

3

! ωk+ 2

K−Θ

3

b

ωk+ε_k^abωaωbb−bσkaω^a+σkabω^a−ε_k^abσcbσb_a^c+H_k , (37)

0 = D^aωa−Aaω^a+ωb^a(Ka−La) , (38)

0 = Dhcωki+εabhkD^bσ_ci^a+ 2Ahcωki−2Khcωbki−Lhcωbki+ε_hk^abbσcibLa+H_kc, (39) 0 = D^bσab+ε_a^ckDcωk−2

3DaΘ +2bΘ

3 La−ε_a^ckLcbωk

+2ε_a^ck(A_cω_k−K_cωb_k)−σb_abL^b+Eb_a+2eκ²

3 eq_b . (40)

The Ricci identity forn^a gives the following independent equations:

0 = Θb˙ −D^aKba+ Kb +Θb 3

! Θ−

b

K^a−L^a

Aa+Ab^a b Ka+La

−2ωaωb^a+bσabσ^ab−κe²eq , (41)

0 = ˙bω_hki−1

2ε_k^abDaKbb+1 2ε_k^ab

b

Ab+Ab Kba+La

+ Kb+Θb 3

! ωk

+Θ 3ωbk+1

2ε_k^abωbaωb−1

2bσkaω^a−1

2σkaωb^a+1

2ε_k^abbσcbσ_a^c+1

2H_k , (42)

0 = ˙bσ_habi−D_haKb_bi+Ab_hb b

K_ai+L_ai

−A_hb b

K_ai−L_ai +Θ

3σb_ab + Kb +Θb

3

!

σab+ωhabωbi−ωchabσ_bi^c−σ_ha^dbωbid+σchaσb_bi^c+F_ab, (43)

0 = Kb_hai⁰ −A˙bhai− Kb −Θb 3

! b

Ka− Kb+Θb 3

!

Ka−K

Aa+Aba

+ (bω_ba+bσ_ba) b

K^b−K^b

+Eb_a−eκ²

3qe_a , (44)

0 = Θb⁰−D^aAb_a+Θb²

3 −ΘK+Ab^aAb_a−2ωb_aωb^a+bσ_abbσ^ab+E

− b

K^aK_a−Kb^aL_a−L^aK_a +Λe

2 +eκ²

2 (πe+ρe−p)e , (45)

0 = ωb⁰_hai−1

2ε_a^cdDcAbd−Kωa+2Θb

3 ωba−bσabωb^b−1 2ε_a^cd

KcKbd+KbcLd+KcLd

, (46)

0 = σb⁰_habi−DhaAbbi+2bΘ

3 bσab−Kσab+AbhaAbbi+KbhaLbi

−Kb_haK_bi+L_haK_bi+bω_haωb_bi+σb_chaσb_bi^c+Eb_ab+κe²

3πeab, (47)

0 = D^aωb_a+Ab_abω^a−ω^a b

K_a+L_a

, (48)

0 = D_hcωb_ki+ε_abhkD^bbσ_ci^a−2Ab_hcωb_ki+ 2Kb_hcω_ki−L_hcω_ki+ε_hk^abσ_cibLa+Hb_kc , 0 = D^bbσab+ε_a^ckDcωbk−2

3DaΘ +b 2Θ

3 La−ε_a^ckLcωk (49)

−2ε_a^ck b

A_cωb_k−Kb_cω_k

−σ_abL^b− E_a−2eκ²

3 eπ_a . (50)

B. Conservations laws

The twice-contracted 5d Bianchi identities imply∇e^aTeab= 0, which can be decomposed into the projections taken withu, n andh, respectively:

0 = ˙ρe+qe⁰+D^aqea+ρe(K+ Θ) +Keπ+ Θpe+πe^abσab

−qe

2Kb−Θb +qe^a

2Aa−Aba

+eπ^a(La+Ka) , (51)

0 = ˙qe+eπ⁰+D^aeπa+eπ b Θ−Kb

−Kbρe−Θbpe−eπ^abσbab

+qe(2K+ Θ)−eπ^a

2Ab_a−A_a

−qe^a b

K_a−L_a

, (52)

0 = ˙qe_hki+eπ_hki⁰ +Dkpe+D^aπeak+4bΘ

3 eπk+4Θ

3 qek−Kbπek+Kqek+ρAe k+πeAbk

−pe b

A_k−A_k

+eq^aω_ak+qe^aσ_ak+eπ^aωb_ak+πe^aσb_ak+qe b

K_k+K_k

−πe_ak b

A^a−A^a

. (53)

The first of these equations is the continuity equation, as can be easily verified in the homogeneous, isotropic case (eq=eq^a =πe^a =eπ^ab= 0 and Θ = 3H) and for K= 0. The ensemble of the equations represent an incomplete set of evolution equations for the 5d matter (there are no evolution equations forp,e eπ,πea,eπab).

C. Bianchi identities

The equations independent from Eqs. (51)-(53) arising from the 5d Bianchi identities are:

0 = ˙E −D^aEb_a+4

3ΘE+Eb_abσ^ab− F_abσb^ab+ 3Haωb^a− E_a

2Kb^a+L^a

−2bE_aA^a

−eκ²

2 (eρ−πe+p)e^·−2eκ²

3 D^aqea−2eκ²

3 Θ (eρ+p)e

−2eκ²

3 Θbqe−4eκ²

3 qeaA^a−2eκ² 3 πea

2Kb^a+L^a

−2eκ²

3 eπabσ^ab, (54)

0 = E˙_hki+D^aF_ka+1

2ε_k^abDaH_b− E_ka b

K^a+L^a

−Eb_kaL^a− Kb +Θb 3

! b E_k+4

3ΘEk

+4E

3Kbk+F_kaA^a−1

2(σka+ωka)E^a−(bσka−2ωbka)Eb^a−2Hkaωb^a−Hb_kaω^a +ε_k^abHb_caσ_b ^c−3

2ε_k^abH_aAb+2eκ² 3

˙e

Phki+2eκ²

3 Dkeq+2κe²

3 (eπ−p)e Kbk

+2κe²

3 (ρe+eπ)Lk−2eκ²

3 Kb +Θb 3

! e qk−2eκ²

3 (ωbka+σbka)qe^a

−2κe²

3 eπkaKb^a+2κe²

3 (ωka+σka)eπ^a+2eκ²Θ

9 πek+2eκ²qe

3 Ak , (55)

0 = H˙_hki−ε_k^abDaE_b−Hb_kaA^a+H_kaKb^a− E_kaωb^a−8

3Eωbk+F_kaω^a+ε_k^abF_acσ_b^c+ ΘHk

+3

2(ω_ka−σ_ka)H^a+ε_k^ab 3

2

Eb_aKb_b−A_aE_b

−L_aEb_b

−ε_k^abE_acbσ_b^c+κe²

3 ε_k^abD_aπe_b

−eκ²

3ε_k^abeqaLb+2eκ²

3 qωe k+eκ²

3ε_k^abeπacbσ_b^c+2eκ²

3 (eπ−p)e ωbk+eκ²

3eπkaωb^a , (56)

0 = E˙b_hki+E_hki⁰ −DkE − E_kaAb^a−Eb_kaA^a+KEb_k−KEb k+2 3

ΘEb_k+ΘEb k

+2 (ωka+σka)Eb^a+ 2 (bωka+bσka)E^a+F_ka

K^a+Kb^a +3

2εkabH^a b

K^b−K^b

−4 3E

Ak−Abk

−eκ²

6Dk(ρe−πe+p) +e eκ² 3D^aπeak

+2eκ² 9

Θbeπk+ Θqek

−κe²

3 [(3bωka+bσka)πe^a+ (3ωka+σka)qe^a] , (57)

0 = E˙_hkji− F_hkji⁰ −ε_abhkD^aH_ji^b−1

2D_hkEb_ji+

K−Θ 3

b

E_kj + (K+ Θ)E_kj +2E

3σ_kj+ 2 Kb−Θb 3

!

F_kj −Eb_hk

A_ji−3 2Ab_ji

+1

2E_hk

4Kb_ji−L_ji−3K_ki +3

2H_hkωb_ji+ 2Eb_ahkσ_ji^a+3

2ε_hj ^abσb_kiaH_b+E_hk^a ω_jia−3σ_jia

− F_hj^a ωb_kia−σb_kia +ε_hk ^abH_jia

2Ab−Abb

−ε_hk ^abHb_jia(Kb+Lb) +2eκ² 3

e˙

π_hkji+2eκ²

3 D_hkqe_ji+2κe² 9 Θeπjk

+2eκ² 3 eπhk

3Kbji+Lji

+ 2eκ²eqhkAji+2κe²

3 qebσjk+2eκ²

3 (ρe+p)e σjk+2eκ²

3 eπ_hj^a ωkia+σkia

, (58)

0 = F˙_hkji− E_hkji⁰ +D_hkE_ji+ Kb −Θb 3

!

E_kj+KbEb_kj+4E 3 σb_kj+

2K+Θ 3

F_kj

−3

2H_hkω_ji−Eb_hj 3

2Kb_ki−L_ki−K_ki

+F_hj^a ω_kia+σ_kia

−2E_hk

b A_ji−3

4A_ji

−E_hj^a ωb_kia+σb_kia

+ε_hk^abH_jia b

Kb−2Kb

+ε_hj^abHb_kiaAb+3

2ε_hj^abσ_kiaH_b+eκ² 3eπ_hkji⁰

−eκ²

3D_hkπe_ji+eκ²

9Θbeπkj −κe²

3 (eπ−p)e bσkj−κe² 3 eqσkj

+eκ²

3eπ_hj^a ωb_kia+bσ_kia +2eκ²

3 eπ_hjAb_ki+eκ²

3qej 2K_ki−L_ki

, (59)

0 = H˙_hkji+εabhkD^aE_ji^b+KbHb_kj+ ΘHkj −3

2H_hkKbji−3

2Eb_hjωki− ωahk+ 3σahk

H_ji^a

+ε_hk^abF_jia b Kb+ 2Lb

+ε_hk^ab b

E_jia−2Ejia

Ab−1

2ε_hk^abσjiaEb_b+ε_hk^abbσjiaE_b +3E_hjbω_ki−eκ²

3ε_abhkD^aπe_ji^b−eκ² qe_hjω_ki+eπ_hjbω_ki

−eκ²

3ε_hk ^ab σ_jiaqeb+bσ_jiaeπb

, (60)

0 = Eb˙_hkji− F_hkji⁰ −D_hkEb_ji+

K+Θ 3

Eb_kj +KEkj +4E

3 σkj+ 2Kb −Θb 3

!

F_kj−3 2H_hkωb_ji

−F_hj^a ωbkia+σbkia

− E_hj 3

2Kki+Lki−Kbki

−2Eb_hk

Aji−3 4Abji

+Eb_hj^a ωkia+σkia

−ε_hk^abHb_jia

Kb−2Kbb

−ε_hj^abH_kiaAbb+3

2ε_hj^abbσ_kiaH_b+eκ² 3

e˙

π_hkji+eκ²

3D_hkqe_ji+eκ²

3 (ρe+ep)σkj

+eκ²qe

3 bσ_kj +eκ²Θ

9 πe_kj+2κe²

3 qe_hjA_ki+eκ² 3 eπ_hj

2Kb_ki+L_ki +eκ²

3eπ_hj^a ω_kia+σ_kia

, (61)

0 = Hb˙_hkji+ε_abhkD^aF_ji^b −1

2D_hjH_ki+ Kb +Θb 3

!

H_kj+2Θ

3 Hb_kj−2H_ahkbσ_ji^a +Hb_hj^a ω_kia−σ_kia

+3

2E_hjω_ki−3ωb_hkEb_ji−ε_hk^abE_jia b Kb−Lb

−3

2H_hkA_ji+ε_hk^abEb_jia

2Kbb+Lb

+3

2ε_hk^abσ_jiaE_b−ε_hk^abF_jiaAb , (62) 0 = E⁰+D^aE_a+4

3ΘE −b 2EaAb^a−Eb_a(2K^a−L^a) + 3ωaH^a+F_abσ^ab

−E_abσb^ab−eκ²

2 (ρe−πe+ep)⁰+2eκ²

3 D^aeπa+2eκ²Θ 3 eq

−2eκ²Θb

3 (pe−eπ)−4κe²

3 eπaAb^a−2eκ²

3 qea(2K^a−L^a)−2eκ²

3 eπabσb^ab , (63)

0 = Eb_hki⁰ −D^aF_ka+1

2ε_k ^abDaH_b−Eb_ka(K^a−L^a) +E_kaL^a−1

2(σbka+ωbka)Eb^a

−(σka−2ωka)E^a+ 2Hb_kaω^a+H_kabω^a+F_kaAb^a+

K−Θ 3

E_k−4E 3 Kk

+4bΘ

3 Eb_k−ε_k ^abH_cabσ_b^c+3

2ε_k ^abH_aAbb+2eκ²

3 eq_hki⁰ −2eκ² 3 Dkeq +2eκ²

3 (ρe+p)e Kk−2κe²

3 (eπ+ρ)e Lk+2eκ² 3

K−Θ

3

e

πk+2eκ²Θb 9 eqk

−2eκ²

3 (ωka+σka)eπ^a+2eκ²

3 (ωbka+bσka)qe^a+2κe²qe

3 Abk+2eκ²

3 eπkaK^a , (64)

0 = H⁰_hki+ε_k^abD_aEb_b+H_kaAb^a−Hb_kaK^a+Eb_kaω^a− F_kaωb^a+ε_k^abEb_acσ_b^c−ε_k^abF_acbσ_b^c+3

2(bω_ka−bσ_ka)H^a +bΘHk−8

3Eωk−ε_k^ab 3

2AbaEb_b−3

2E_aKb−LaE_b

−eκ²

3ε_k^abDaqeb+eκ²

3ε_k^abeπaLb

−2eκ²eq

3 ωb_k+κe²

3 ε_k^abeπ_acσ_b^c−2eκ²

3 (ρe+p)e ω_k+eκ²

3eπ_kaω^a , (65)

0 = E_hki⁰ +D^aEb_ka−2

3DkE+ΘEb k+4E

3 Abk− E_kaAb^a−3Hb_kaωb^a+

K−2 3Θ

b

E_k+F_ka(K^a−2L^a) +1

2(3σbka+ωbka)E^a+ (σka+ 3ωka)Eb^a−3

2ε_k^abH_aKb+ε_k^abHb_cabσ_b^c+2eκ²

3 eπ_hki⁰ +eκ² 3 D^aeπka

−κe²

3Dk(ρe+πe−p) +e 2eκ²qe

3 (Kk−2Lk) +2eκ² 3

K+Θ

3

e qk−2eκ²

3 eπkaAb^a +2eκ²

3 (eπ−p)e Abk+2eκ²

3 Θbπek+eκ²

3 [(bωka+ 3bσka)πe^a−(3ωka+σka)qe^a] , (66)