Covariant gravitational dynamics in 3+ 1+ 1 dimensions

Covariant gravitational dynamics in 3+1+1 dimensions

arXiv:0909.0490v2 [gr-qc] 11 May 2010

1

´ Gergely1,2 Zoltán Keresztes1,2 , L´ aszló A. 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 Tér 9, Szeged 6720, Hungary (Dated: May 12, 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 3space 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 graviphoton, 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 à 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 possible vorticity 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 normal n has 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 n becomes 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 field n and the time evolution vector u are 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 of u should be allowed, similarly

2 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 n should 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 in f (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 ua and na carry no distinguishing mark and the 3d metric is denoted hab . Quantities defined on the brane carry the pre-index (4) , the only exception being the 4d metric gab . Quantities defined on the full 5-space carry a distinguishing e mark. 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 both ua and na , and (b) the kinematic and extrinsic curvature type quantities related to another singled out spatial vector ea 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 field ua and the brane normal na , the latter distinguished by abmark. We denote the average of a quantity f taken over the two sides of the brane by hf i, and its jump by ∆f . Angular brackets h i on 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 ua = dxa /dτ and na = dxa /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

3

e FIG. 1: Elements of the 3+1+1 decomposition of the 5d space-time geometry with metric e gab and compatible connection ∇. e c nd . The 4-velocity The 4d brane with normal na has induced metric gab = e gab − na nb and extrinsic curvature (4)Kab = gac gbd ∇ field ua of observers in the brane defines local 3d orthogonal spatial patches, hypersurface-forming only when the vorticity ω ab = 0. The other kinematical characteristics of ua are the expansion Θ and shear σab . The expansion, shear and vorticity of b σ na are Θ, b ab and ω b ab , the latter vanishing on the brane. The temporal, off-brane and 3d covariant derivative are shown for the vorticity 1-form ω a .

−ua ua = 1 = na na and the perpendicularity condition ua na = 0. The 5d metric is decomposed as

with

geab = na nb + gab ,

(1)

gab = −ua ub + hab ,

(2)

the metric on the 4d temporal leaves y =const (with the brane at y = 0) and the spatial part hab of this metric obeying ua hab = na hab = 0. We denote by εabc the volume element associated with hab . The 4-vector ua represents the time-like velocity field of brane observers (see Fig 1). e i . A dot and a prime We employ three type of derivatives, all associated with projections of the 5d connection ∇ a a denote covariant derivatives along the integral curves of u and n , respectively, while D is the 3d covariant derivative compatible with the metric h: e a Tb..c , T˙b..c = ua ∇ ′ e a Tb..c , Tb..c = na ∇

Da Tb..c =

e d Ti..j had hb i ..hc j ∇

(3) (4) .

(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 with hab . 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 na and ua terms. Nevertheless, when projected to the 3-manifold with hb a , the two definitions agree.

4 B.

Kinematic quantities

We introduce the kinematic quantities through the decomposition of the 5d covariant derivative of ua , na as e a ub = −ua Ab + Ku b a nb + Kna nb + na Kb + La nb + Kab , ∇ b b + La ub + K b ab , bb + Kna ub + Ku b a ub − ua K e a nb = na A ∇

where

Aa = u˙ hai , Ka = u′hai , K = nb u′b ,

(6a) (6b)

ba = n′ , A hai

b a = n˙ hai , K

b = ub n˙ b , K

e c ud , La = hac nd ∇ b ab = Da nb . Kab = Da ub , K

(7)

As a rule, an overhat is used for kinematical quantities related to the vector field na in order to distinguish them from the similar kinematical quantities related to the vector field ua . Here Aa is the acceleration. All scalars, vectors and tensors in the above decomposition are defined on the 3d manifold orthogonal both to na and ua . The tensorial b ab can be further decomposed into (trace-, trace-free symmetric and antisymmetric) irreducible expressions Kab and K parts as follows: Θ hab + σ ab + ω ab , 3 b Θ = hab + σ bab + ω b ab , 3

Kab =

(8)

b ab K

(9)

where we have defined the expansion, vorticity and shear of the vector fields ua and na as Θ = D a ua , b = D a na , Θ

ω ab = D[a ub] ,

σ ab = Dha ubi ,

ω b ab = D[a nb] ,

σ bab = Dha nbi .

(10)

The antisymmetric 3d tensors ωab and ω b ab can be also encoded in the vorticity vectors ω a = εabc ωbc /2 and ω ba = abc a a ε ω b bc /2. When the vorticities of both vector field n and u vanish ωab = ω b ab = 0, the tensorial expressions Kab and b ab are symmetric and they represent the two extrinsic curvatures of a 3d hypersurface. The condition ω K b ab = 0 is also necessary in order to have a brane at y = 0, but not sufficient. Indeed the brane is a 3+1 dimensional hypersurface which can exist only if the higher-dimensional vorticity of its normal ω b ab = ∇[a nb] vanishes. This condition translates c de b into 0 = g g ∇c nd = −u[a Kb] + Lb] + ω b ab , therefore (due to Frobenius’ theorem) the necessary and sufficient [a

b]

conditions for the existence of the 3+1 brane are

ω b ab |y=0 = 0 ,

b a La |y=0 = − K

y=0

.

(11)

e a ub are expressed by three scalars (K, K, b Θ), four 3-vectors In summary, the independent components of ∇ e a nb consists (Aa , Ka , La , ω a ) and a symmetric trace-free 3-tensor (σ ab ). The corresponding decomposition of ∇ b b b b of the three scalars (K, K, Θ), four 3-vectors (Aa , Ka , La , ω b a ) and a symmetric trace-free 3-tensor (b σ ab ). The e a ub and ∇ e a nb have therefore 20 independent components irreducible decompositions of the covariant derivatives ∇ e a ub = 0 and nb ∇ e a nb = 0). each (due to the constraints ub ∇ 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

5 eabcd , which can be given in terms of the quantities (with a total of 35 interested in the 3+1+1 decomposition1 of C independent components): eabcd na ub nc ud , E = C 1 eabcd uc nd , Fkl = C eabcd h a ub h c nd , Hk = εk ab C l) (k 2 a b c d a b c eabcd h n u n , Ek = C eabcd h u n ud , Ebk = C k k eabcd h a nb h c nd , Ekl = C eabcd h a ub h c ud , Ebkl = C hk li hk li 1 1 eabcd nd , Hkl = ε ab h c C eabcd ud . b kl = ε ab h c C H l) l) 2 (k 2 (k

(12)

We note that all tensorial quantities defined above are trace-free (from the properties of the Weyl tensor), and further b kl the brackets h i are equivalent with the round brackets ( ). The Weyl tensor in terms in the tensors Fkl , Hkl and H of the quantities defined in (12) is eabcd = −2 Ed[a hb]c − Ec[a hb]d + 2 Ebd[a hb]c − Ebc[a hb]d − 2 Ehc[a hb]d C 3 bi[a nb] + ε i H bi[c nd] − 2 ε i Hi[a ub] + ε i Hi[c ud] +2 εcd i H ab cd ab +2 nc n[a Ebb]d − nd n[a Ebb]c + 2 uc u[a Eb]d − ud u[a Eb]c −2 uc n[a Fb]d − ud n[a Fb]c + nc u[a Fb]d − nd u[a Fb]c − n[c ua] εbdk + u[c nb] εadk Hk − u[d na] εbck + n[d ub] εack Hk −2 u[c nd] εabk + u[a nb] εcdk Hk − 2 E[a hb][c nd] + E[c hd][a nb] −2 Eb[a hb][c ud] + Eb[c hd][a ub] + 4Eu[a nb] u[c nd] −2 nc ud n[a Ebb] − uc nd n[a Ebb] + na ub n[c Ebd] − ua nb n[c Ebd] +2 uc nd u[a Eb] − nc ud u[a Eb] + ua nb u[c Ed] − na ub u[c Ed] 2 2 (13) − E nd n[a hb]c − nc n[a hb]d + E ud u[a hb]c − uc u[a hb]d . 3 3

This relation generalizes the general relativistic 3+1 covariant decomposition of the 4d Weyl tensor Cabcd (which has only 10 independent components), where only two tensors Ekl = Cabcd hhk a ub hlic ud and Hkl = 21 ε(k ab hl)c Cabcd ud appear, the electric and magnetic parts of the Weyl curvature. On the brane the relation between the set of variables Ekl and Hkl and the variables Ekl and Hkl is given by ! b 1 1b 1 b Θ 1b b σ bab + σ K+ Eab = Eab + Eab − (14) bcha σ bbic + K ha Kbi , 2 2 3 2 2 Hab = Hab − εha D.

cd

bd . σ bbic K

(15)

Decomposition of the energy-momentum tensor

The 5d gravitational dynamics is governed by the Einstein equation i h e ab = −Λe e gab + κ G e2 Teab + τ ab δ (y) ,

(16)

e the regular where e κ2 denotes the 5d coupling constant. The sources of gravity are the 5d cosmological constant Λ, e part of the 5d energy-momentum tensor Tab and a distributional energy-momentum tensor with support on the brane: τ ab = −λgab + Tab .

1

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

(17)

6 Here λ is the constant brane tension and Tab describes standard model matter fields on the brane2 , decomposed with respect to a brane observer with 4-velocity ua as Tab = ρua ub + q(a ub) + phab + π ab .

(18)

The quantities ρ, qa , 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 to ua and na as: Teab = e ρua ub + 2e q(a ub) + 2e q u(a nb) + pehab + π ena nb + 2e π (a nb) + π eab ,

(19)

where e ρ is the relativistic energy density relative to ua , pe the isotropic pressure, qea and qe the relativistic momentum densities on the 3-space and in the direction na . The quantities π e, π ea and the symmetric and trace-free tensor π eab are related to the scalar, vectorial and tensorial components (with respect to the 3d space) of the 5d anisotropic pressure tensor, which is 3 (e π − pe) pe − π e na nb + 2e π(a nb) + hab + π e ab . 4 4

(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 tensor Rabcd of the space orthogonal to both ua and na as

resulting in

1 ′ b ab Vhci = Rabcd V d , D[a Db] Vc − ω ab V˙ hci + ω 2 eijkl − (Da uc ) (Db ud ) + (Da ud ) (Db uc ) Rabcd = ha i hb j hc k hd l R + (Da nc ) (Db nd ) − (Da nd ) (Db nc ) .

(21)

(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 hc[a hb]d 2 e+κ b2 − E + Λ e2 (e ρ + pe − π e) Rabcd = − Θ2 − Θ 3 3 2e κ2 −2 Ed[a hb]c − Ec[a hb]d + 2 Ebd[a hb]c − Ebc[a hb]d − hd[a π eb]c − hc[a π eb]d 3 2Θ σ a[c + ω a[c hd]b + σ b[d + ω b[d hc]a − 2 σ a[c + ωa[c σ d]b − ωd]b − 3 b 2Θ + σ ba[c + ω b a[c hd]b + σ bb[d + ω b b[d hc]a + 2 σ b a[c + ω b a[c σ bd]b − ω b d]b , (23) 3 2

For DGP / induced gravity type models [43], Tab should be replaced by Tab − γ/κ2 Gab , with Gab the Einstein tensor constructed from the metric gab and γ the dimensionless induced gravity parameter. Randall-Sundrum type brane-worlds are recovered for γ → 0, on the ε = −1 branch.

7 and the local 3d Ricci tensor and Ricci scalar become 2 hac e+κ b 2 − 2E + Λ e2 (e ρ + pe − π e) Rac = hbd Rabcd = − Θ2 − Θ 3 3 2 b 2Θ 2Θ κ e eac − (σ ac + ωac ) + (b σ ac + ω b ac ) +Eac − Ebac + π 3 3 3 +σ ab σ c b − ω b ω b hac + ω a ω c − 2σ [ab ω c]b −b σ ab σ bc b + ω bbω b b hac − ω baω b c + 2b σ [a b ω b c]b , 2 2 b 2 e +κ Θ −Θ R = hac Rac = −2E + Λ e2 (e ρ + pe − π e) − 3 −2ωa ω a + σ ab σ ab + 2b ωa ω ba − σ b ab σ bab .

(24)

(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

The full 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 for ua and na . 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 for ua gives the following independent equations: ba b′ + A ba Aa + K 2 − K b 2 + La K a − K b a K a − La K 0 = K˙ + K e Λ κ e2 +E − + (e ρ−π e + 3e p) , 6 6 2 bK b − Aa Aa − 2ωa ω a + σ ab σ ab − E ˙ − Da Aa + Θ + Θ 0 = Θ 3 e e2 b a Ka + K a La + La K ba − Λ + κ + K (e ρ+π e + pe) , 2 2 Θ Θ b 0 = K˙ hai − A′hai + K + Ka + K − K a + Ea 3 3 e2 bb − κ b A ba + Aa + (ω ba + σ ba ) K b − K +K π ea , ! 3 b Θ b Θ Θ b b ˙ La + K + Aa + K − Ka 0 = Lhai + Da K + K + 3 3 3 e2 b b − L b + Ea − κ + (b ω ab + σ bab ) Ab − (ω ab + σ ab ) K π ea , 3 2Θ 1 1 b d + Kc Ld + K b c Ld , bω ba + ω a − σ ab ωb + εa cd Kc K 0 = ω˙ hai − εa cd Dc Ad + K 2 3 2

(26)

(27)

(28)

(29) (30)

8 2Θ bσ σ ab + K bab − Aha Abi + Kha Lbi 3 e2 b ha Kbi + Lha K b bi + ωha ω bi + σ cha σ c + Eab − κ +K π e ab , bi 3 Θ b ba − Aa (Ka − La ) − 2b Θ + (K a + La ) A ωa ω a + σ bab σ ab − e κ2 qe , 0 = Θ′ − D a Ka − K − 3 0 = σ˙ habi − Dha Abi +

0 =

L′hai

Θ b − Da K + K − Aa − 3

b b−Θ K 3

!

La +

b b+Θ K 3

!

(32)

Ka

e2 bb − Eba + κ + (b σ ab + ω b ab ) K b + Lb − (ω ab + σ ab ) A qea , 3 1 1 bb (Ka − La ) − K − Θ ω 0 = ω ′hki − εk ab Da Kb − εk ab Ab + A bk 2 2 3 b 1 1 1 1 1 Θ bb − σ bka ω a − σ ka ω b a + εk ab σ cb σ ba c − Hk , + ω k + εk ab ω a ω 3 2 2 2 2 2 b bhb Kai + Lai + Θ σ ab 0 = σ ′habi − Dha Kbi − Ahb Kai − Lai + A 3 Θ b bid + σ cha σ bbic + Fab , − K− σ b ab + ω ha ω b bi + ω cha σ bbi c + σ ha d ω 3 ! b Θ Θ ab b ωk + 2 K − ω b k + εk ab ω a ω bb − σ bka ωa + σ ka ω b a − εk ab σ cb σ bac + Hk , 0 = εk Da L b + 2 K + 3 3 0 = Da ω a − Aa ω a + ω b a (Ka − La ) , bcib La + Hkc , b ki − Lhc ω b ki + εhk ab σ 0 = Dhc ωki + εabhk Db σ cia + 2Ahc ω ki − 2Khc ω

b 2 2Θ 0 = Db σ ab + εa ck Dc ω k − Da Θ + La − εa ck Lc ω bk 3 3 2e κ2 +2εa ck (Ac ω k − Kc ω bk ) − σ bab Lb + Eba + qeb . 3

The Ricci identity for na gives the following independent equations: ! b Θ ˙ a b ba K b a + La − 2ω a ω b a − La Aa + A b b ba + σ b ab σ ab − κ e2 qe , Θ− K 0 = Θ − D Ka + K + 3 ! b 1 Θ 1 ab ab b+ b a + La + K bb + Ab K bb + ε 0 = ω b˙ hki − εk Da K ωk A 2 2 k 3 Θ 1 1 1 1 1 ω b k + εk ab ω b a ωb − σ bka ω a − σ ka ω b a + εk ab σ bcb σ a c + Hk , 3 2 2 2 2 2 Θ ˙ b b b b bab 0 = σ bhabi − Dha Kbi + Ahb Kai + Lai − Ahb Kai − Lai + σ 3 ! b b + Θ σ ab + ω ha ω b bid + σ cha σ bbic + Fab , b bi − ω cha σ bbic − σ ha d ω + K 3 ! ! b b Θ Θ ˙ ′ b −A bhai − K b− ba ba − K b+ 0 = K Ka − K Aa + A K hai 3 3 e2 b b − K b + Eba − κ + (b ωba + σ b ba ) K qea , 3 b2 ba A ba − 2b b ′ − Da A ba + Θ − ΘK + A ωaω ba + σ b ab σ bab + E 0 = Θ 3 e e2 b a Ka − K b a La − La Ka + Λ + κ (e π +e ρ − pe) , − K 2 2 +

(31)

(33) (34) (35)

(36) (37) (38) (39)

(40)

(41)

(42)

(43)

(44)

(45)

9 b 1 1 bd + K b c Ld + Kc Ld , bd − Kωa + 2Θ ω ba − σ bab ω b b − εa cd Kc K 0 = ω b ′hai − εa cd Dc A 2 3 2 b bbi + 2Θ σ bha A bbi + K b ha Lbi 0 = σ b′habi − Dha A bab − Kσ ab + A 3 κ e2 b ha Kbi + Lha Kbi + ω eab , −K b ha ω b bi + σ bcha σ bbic + Ebab + π 3 b a + La , ba ω 0 = Da ω ba + A ba − ωa K b kc , bhc ω b hcω ki − Lhc ω ki + ε ab σ cib La + H 0 = Dhc ω b ki + εabhk Db σ bcia − 2A b ki + 2K hk

2 b + 2Θ La − εa ck Lc ω k 0 = Db σ b ab + εa ck Dc ω b k − Da Θ 3 3 κ2 bc ω b c ω k − σ ab Lb − Ea − 2e π ea . −2εa ck A bk − K 3 B.

(46)

(47) (48)

(49)

(50)

Conservations laws

e a Teab = 0, which can be decomposed into the projections taken The twice-contracted 5d Bianchi identities imply ∇ with u, n and h, respectively: 0 = e ρ˙ + qe′ + Da qea + e ρ (K + Θ) + K π e + Θe p+π eab σ ab ba + π b −Θ b + qea 2Aa − A ea (La + Ka ) , −e q 2K b −K b − Ke b ρ − Θe bp − π 0 = qe˙ + π e′ + Da π ea + π e Θ eab σ bab b a − La , ba − Aa − qea K +e q (2K + Θ) − π e a 2A

b 4Θ 4Θ b π k + K qek + e bk 0 = qe˙ hki + π e′hki + Dk pe + Da π eak + π ek + qek − Ke ρAk + π eA 3 3 ba − Aa . b k + Kk − π bk − Ak + qea ω ak + qea σ ak + π e ak A ea ω b ak + π ea σ bak + qe K −e p A

(51)

(52)

(53)

The first of these equations is the continuity equation, as can be easily verified in the homogeneous, isotropic case (e q = qea = π ea = π e 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 for pe, π e, π ea , π eab ). C.

Bianchi identities

The equations independent from Eqs. (51)-(53) arising from the 5d Bianchi identities are: 4 b a + La − 2Eba Aa 0 = E˙ − Da Eba + ΘE + Ebab σ ab − Fab σ bab + 3Ha ω b a − Ea 2 K 3 2e κ2 a 2e κ2 κ e2 ρ−π e + pe)· − D qea − Θ (e ρ + pe) − (e 2 3 3 2e 2e κ2 b 4e κ2 2e κ2 b a κ2 − Θe q− qea Aa − π e a 2K + La − π e ab σ ab , 3 3 3 3

(54)

10 1 b a + La − Ebka La − 0 = E˙hki + Da Fka + εk ab Da Hb − Eka K 2

b b+Θ K 3

!

4 Ebk + ΘEk 3

1 4E b b ka ωa Kk + Fka Aa − (σ ka + ω ka ) E a − (b σ ka − 2b ωka ) Eba − 2Hka ω ba − H 3 2 κ2 e 2e κ2 2e κ2 ˙ bk bca σ c − 3 ε ab Ha Ab + 2e D q e + (e π − pe) K P + +εk ab H k hki b 2 k 3 3 3 ! b 2e κ2 2e κ2 2e κ2 b Θ qek − + K+ (e ρ+π e ) Lk − (b ω ka + σ b ka ) qea 3 3 3 3 +

−

κ2 2e κ2 Θ 2e κ2 qe 2e κ2 b a + 2e π eka K (ω ka + σ ka ) π ea + π ek + Ak , 3 3 9 3

8 b a − Eka ω b ka Aa + Hka K ba − E ω 0 = H˙ hki − εk ab Da Eb − H b k + Fka ωa + εk ab Fac σ b c + ΘHk 3 κ e2 3 b b 3 bb c + εk ab Da π Ea Kb − Aa Eb − La Ebb − εk ab Eac σ eb + (ωka − σ ka ) Ha + εk ab 2 2 3

2e κ2 κ2 e 2e κ2 κ e2 κ e2 ab εk qea Lb + qeωk + εk ab π eac σ bb c + (e π − pe) ω bk + π e ka ω ba , 3 3 3 3 3 ˙ ′ b k ba − Ebka Aa + K Ebk − KE b k + 2 ΘEbk + ΘE − Dk E − Eka A 0 = Ebhki + Ehki 3 a a a b a + 3 εkab Ha K b b − Kb +2 (ωka + σ ka ) Eb + 2 (b ωka + σ bka ) E + Fka K + K 2 2 2 4 e κ e bk − κ − E Ak − A Dk (e ρ−π e + pe) + Da π eak 3 6 3 κ e2 2e κ2 b Θe π k + Θe qk − [(3b ωka + σ bka ) π ea + (3ωka + σ ka ) qea ] , + 9 3 −

1 Θ b ′ 0 = E˙hkji − Fhkji − εabhk Da Hjib − Dhk Ebji + K − Ekj + (K + Θ) Ekj 2 3 ! b 3b 1 Θ 2E b b− Fkj − Ebhk Aji − A + E 4 K − L − 3K + σ kj + 2 K ji hk ji ji ki 3 3 2 2 3 3 + Hhk ω b kia − σ b kia b kia Hb + Ehk a ωjia − 3σ jia − Fhja ω b ji + 2Ebahk σ jia + εhj ab σ 2 2 2e κ2 2e κ2 κ2 ˙ bjia (Kb + Lb ) + 2e bb − ε ab H π e + D q e + Θe π jk +εhk ab Hjia 2Ab − A hkji hk ji hk 3 3 9 2e κ2 2e κ2 2e κ2 2e κ2 a b ji + Lji + 2e + π ehk 3K κ2 qehk Aji + qeσ b jk + (e ρ + pe) σ jk + π ehj ωkia + σ kia , 3 3 3 3 0 = F˙ hkji −

′ Ehkji

+ Dhk Eji +

!

(57)

(58)

4E Θ Fkj σ bkj + 2K + 3 3 3 b ω kia + σ kia − 2Ehk Aji − Aji 4 ab

e2 ′ bkia Ab + 3 ε ab σ kia Hb + κ H π e 2 hj 3 hkji

κ e2 κ e2 b κ e2 κ e2 π kj − Dhk π eji + Θe (e π − pe) σ bkj − qeσ kj 3 9 3 3 2 2e κ e2 a κ e κ2 bki + qej 2Kki − Lki , + π e π ehj A ω b kia + σ bkia + 3 hj 3 3 −

(56)

b Ebkj + Ekj + K

3b Kki − Lki − Kki + Fhj a 2 b b − 2Kb + ε ω b kia + σ b kia + εhk ab Hjia K hj

3 − Hhk ω ji − Ebhj 2

−E hj a

b b−Θ K 3

(55)

(59)

11 b ji − 3 Ebhj ωki − ω ahk + 3σ ahk H a bH bkj + ΘHkj − 3 Hhk K 0 = H˙ hkji + εabhk Da Ejib + K ji 2 2 b b + 2Lb + ε ab Ebjia − 2Ejia Ab − 1 ε ab σ jia Ebb + ε ab σ b jia Eb +εhk ab Fjia K hk hk 2 hk κ e2 κ e2 e jib − κ e2 qehj ω ki + π ehj ω b ki − εhk ab σ jia qeb + σ bjia π eb , +3Ehj ω b ki − εabhk Da π 3 3

! b Θ b 3 4E Θ ˙ ′ b b b 0 = E hkji − Fhkji − Dhk Eji + K + Fkj − Hhk ω Ekj + KEkj + σ kj + 2K − b ji 3 3 3 2 3b 3 a b b Kki + Lki − Kki − 2Ehk Aji − Aji + Ebhj a ω kia + σ kia −Fhj ω b kia + σ bkia − Ehj 2 4 2 κ e ˙ κ e2 κ e2 bb + 3 ε ab σ bjia Kb − 2K b b − ε ab Hkia A −εhk ab H b H + π e + D q e + (e ρ + pe) σ kj b kia hkji hk ji hj 2 hj 3 3 3 κ2 κ2 Θ e 2e κ2 κ e2 κ e2 qe b ki + Lki + e σ bkj + π ekj + qehj Aki + π ehj 2K π ehj a ωkia + σ kia , + 3 9 3 3 3 ! b 2Θ b Θ 1 ˙b b b+ Hkj + Hkj − 2Hahk σ bjia − Dhj Hki + K 0 = Hhkji + εabhk Da Fji 2 3 3 b a ωkia − σ kia + 3 Ehj ω ki − 3b bji − ε ab Ejia K b b − Lb +H ω E hk hj hk 2 3 3 b b + Lb + ε ab σ jia Eb − ε ab Fjia Ab , − Hhk Aji + εhk ab Ebjia 2K hk 2 2 hk 4 ba − Eba (2K a − La ) + 3ωa Ha + Fab σ ab b − 2Ea A 0 = E ′ + Da Ea + ΘE 3 2e κ2 a 2e κ2 Θ κ e2 ′ (e ρ−π e + pe) + D π ea + qe −Eab σ b ab − 2 3 3 b κ2 4e κ2 ba 2e 2e κ2 2e κ2 Θ (e p−π e) − π ea A − qea (2K a − La ) − π eab σ bab , − 3 3 3 3 1 ′ − Da Fka + εk 0 = Ebhki 2

1 Da Hb − Ebka (K a − La ) + Eka La − (b σ ka + ω b ka ) Eba 2 ba + K − Θ Ek − 4E Kk b ka ω a + Hka ω − (σ ka − 2ωka ) E a + 2H b a + Fka A 3 3 2 2 b 3 κ ′ 2e κ 4Θ b bb + 2e Ek − εk ab Hca σ b b c + εk ab Ha A qe − Dk qe + 3 2 3 hki 3 b 2e κ2 Θ 2e κ2 Θ 2e κ2 2e κ2 + K− π ek + (e ρ + pe) Kk − (e π+e ρ) L k + qek 3 3 3 3 9

−

(61)

(62)

(63)

ab

2e κ2 2e κ2 2e κ2 qe b 2e κ2 Ak + (ωka + σ ka ) π ea + (b ωka + σ bka ) qea + π e ka K a , 3 3 3 3

3 ′ ba − H bka K a + Ebka ω a − Fka ω b a + εk ab Ebac σ b c − εk ab Fac σ bb c + (b + εk ab Da Ebb + Hka A 0 = Hhki ω ka − σ b ka ) Ha 2 κ e2 3 κ e2 8 ab 3 b b b Aa Eb − Ea Kb − La Eb − εk ab Da qeb + εk ab π e a Lb +ΘHk − Eω k − εk 3 2 2 3 3 −

(60)

2e κ2 qe κ e2 2e κ2 e2 κ ω b k + εk ab π eac σ b c − (e ρ + pe) ωk + π eka ωa , 3 3 3 3

(64)

(65)

12 2 2 ′ ba − 3H b ka ω bk − Eka A b k + 4E A b a + K − Θ Ebk + Fka (K a − 2La ) + Da Ebka − Dk E + ΘE 0 = Ehki 3 3 3 3 2e κ2 ′ e2 κ 1 bca σ σ ka + ω b ka ) E a + (σ ka + 3ωka ) Eba − εk ab Ha Kb + εk ab H bb c + π e hki + Da π eka + (3b 2 2 3 3 Θ 2e κ2 2e κ2 qe 2e κ2 κ e2 ba K+ qek − ρ+π e − pe) + (Kk − 2Lk ) + π eka A − Dk (e 3 3 3 3 3 +

2e κ2 κ e2 κ2 b bk + 2e Θe πk + (e π − pe) A [(b ωka + 3b σka ) π ea − (3ωka + σ ka ) qea ] , 3 3 3

b 2Θ ′ ′ b Ebkj − Ehk A bji b b + 1 Dhk Eji + 2E σ bjk − Ekj + Θ + εabhk Da H − Ehkji 0 = Ebhkji ji 2 3 3 1 −3Hhk ω ji − Ebhk 2Kji − Lji + 2Fahk σ jia − Ehk a ω b jia − 3b σ jia b jia − σ b jia + Ebhk a ω 2 e2 ′ κ e2 Θ bjia A bb − ε ab Hjia (2Kb − Lb ) − κ π e − qehk 4Kji − Lji − Fkj + 2εhk ab H hkji hk 3 3 3 2 2 2b κ e2 4e κ2 κ e κ e κ e Θ κ e2 bji + qeσ jk + + Dhk π b jia + σ b jia , eji − π ehk A (e π − pe) σ b jk − π e jk − π ehk a ω 3 3 3 3 9 3 b 2 Θ Θ 3 1 ′ b kj + bji H Hkj − Hhk A 0 = Hhkji + εabhk Da Fjib + Dhj Hki − K − 2 3 3 2 3 b ahk σ a + H a ω − Ebhj ω b ki + 3ω hk Eji − 2H b kia − σ bkia + εhk ab Ebjia (Kb + Lb ) ji hj 2 3 bb , bjia Ebb + εhk ab Fjia A −εhk ab Ejia (2Kb − Lb ) − εhk ab σ 2 a 3 b ′ + εabhk Da Eb b − 3 Hhk Kji − KHkj + Θ bH b kj − ω b + Ehj ω 0 = H b ahk + 3b σ ahk H b ki ji hkji ji 2 2 bb + 1 ε ab σ bjia Eb − εhk ab σ jia Ebb − εhk ab Fjia (Kb − 2Lb ) −εhk ab Ejia − 2Ebjia A 2 hk κ e2 κ2 e e hj ω b ki + εhk ab σ jia qeb + σ b jia π eb , e jib + e κ2 qehj ω ki + π −3Ebhj ωki + εabhk Da π 3 3 bab σ ab + Hab σ 0 = Dk Hk − H bab − 3Ea ω a − 3Eba ω ba , b Θ Θ 1 0 = Da Ebak − Da Eak − Dk E − Ek − Ebk + 3Hka ω a − εk ab Hac σ b c 3 3 3 i 1h κ e2 b ka ω b ac σ + (3b b a + εk ab H bb c − Da π ωka + σ b ka ) E a + (3ωka + σ ka ) Eba − 3H eak 2 3 b 2e κ2 Θ 2e κ2 Θ κ e2 κ e2 a κ e2 ρ−π e + pe) − qek − π ek + κ e2 qea ωka + qea σ ka + κ e2 π ea ω b ka + π e σ bka , + Dk (e 6 9 9 3 3 b 1 Θ 4E bka La − 2Fka ω 0 = Da Hak + εk ab Da Ebb − Hk − ωk − H b a − (b σ ka − 2b ω ka ) Ha 2 3 3 κ2 e 1 − Ebka − 3Eka ω a − εk ab Eac − Ebac σ b c − εk ab Ea Lb + εk ab Da qeb 2 3 2 κ e2 ab 2e κ2 2e κ2 κ e2 κ e − εk π e a Lb + qeω bk + (e ρ + pe) ωk − π eka ωa − εk ab π e a c σ bc , 3 3 3 3 3 4E b ak − 1 εk ab Da Eb − Θ Hk − Hka La + 2Fka ω a + Eka − 3Ebka ω ba − ω bk 0 = Da H 2 3 3 κ e2 1 bb c Ebca − Eca − εk ab Da π eb − (σ ka − 2ωka ) Ha − εk ab La Ebb + εk ab σ 2 3 κ e2 2e κ2 qe 2e κ2 κ2 e κ e2 − εk ab La qeb − ba c . ωk − (e π − pe) ω bk − π eka ω b a + εk ab π ecb σ 3 3 3 3 3

(66)

(67)

(68)

(69) (70)

(71)

(72)

(73)

13 IV.

3+1 GRAVITATIONAL DYNAMICS ON THE BRANE

In this section we consider distributional energy-momentum tensor sources on the brane, in addition to the regular energy-momentum tensor Teab . Such a distributional source comes together with a discontinuity in the extrinsic curvature, as related by the Lanczos equation. A.

The Lanczos equation

a b e The extrinsic curvature of the brane is (4)Kab = ∇(c nd) = g(c gd) ∇a nb , equal to the symmetrized version of the last b four terms of Eq. (6b). Replacing Kab by the expression (9), and specializing to the brane cf. Eqs. (11), the extrinsic curvature is expressed as

b b a ub − 2u(a K b b) + Θ hab + σ bab . Kab = Ku 3

(4)

(74)

As we approach the brane from left or right, the limiting values of the extrinsic curvature could be different, according to the embedding and 5d metric in the two regions. Therefore we introduce averages and differences of the extrinsic curvature. The Lanczos equation [46], [47] relates the jump of the extrinsic curvature across the brane to the distributional matter layer: τ ∆(4)Kab = −e κ2 τ ab − gab . (75) 3

The ua ub , ua hbc , trace and trace-free parts of the hac hbd projections give:

κ e2 (λ − 2ρ − 3p) , 3 = κ e2 qa ,

b = ∆K

ba ∆K b = −e ∆Θ κ2 (λ + ρ) , 2

∆b σ ab = −e κ π ab .

(76) (77) (78) (79)

The Lanczos equation is necessary in order to derive the gravitational dynamics on the brane, given by a scalar (the twice-contracted Gauss), a 4d vectorial (the Codazzi) and a 4d tensorial (the effective Einstein) equations [3]. The latter has been first derived in [48], later generalized to include bulk matter and asymmetric embedding contributions (Eq. (1) in [3]). B.

The 3+1 decomposition of the source terms of the effective Einstein equation

We give the 3+1 covariant decomposition of the source terms of the effective Einstein equation. This equation is [3]: ! D E κ e2 he πi gab + κ2 Tab + κ e4 Sab − (4)Eab + hLab i + hPab i . (80) Gab = − Λ − 2

The sources are: the stress-energy tensor Tab representing standard model matter [decomposed in Eq. (18)]; the source term Sab quadratic in Tab (dominant at high energies), hPab i the pull-back to the brane of the bulk matter; hLab i a source term originating in the asymmetry of the embedding and (4)Eab the contribution of the electric part (relative to the vector na ) of the 5d Weyl tensor. We have defined the 4d coupling constant κ2 and the brane cosmological constant Λ as 6κ2 = κ e4 λ ,

D E e . 2Λ = κ λ + Λ 2

(81) (82)

14 The quadratic source term is decomposed as Sab =

1 1 2ρ2 − 3π cd π cd ua ub + 2ρ2 + 4pρ − 4qc q c + π cd π cd hab 24 24 1 ρ 1 c ρ + 3p 1 + qha qbi + q(a ub) − q π c(a ub) − π ab − π cha π bic . 4 3 2 12 4

(83)

Some of the numerical coefficients here are corrected with respect to the corresponding expression (7) given in Ref. [11]. The electric part of the 5d Weyl tensor expressed in terms of gravito-electro-magnetic quantities defined in Section II C is D E E D E D 1 (4) (84) Eab = hEi ua ub + hab − 2 Eb(a ub) + Ebab . 3 The asymmetry source term is decomposed as D E 1 D b E2 3 hLab i = Θ − hb σ cd i σ bcd ua ub 3 2 D E

D cE 4 b b K Θ hb)c − 2 σ bb)c −u(a 3 D E D E D E2 9 D E 1 DbE D b E b + hb bc − Θ bc K + 6 Θ K −9 K σ cd i σ b cd hab 9 2 + * D ED E b Θ b b b − K hb σ ab i − hb σ a c i hb σ bc i . + Ka Kb + 3

(85)

As the induced metric is continuous, the average of the trace L = g ab Lab is the trace of the average: D E D E 2 D E2 D E D ED E bb − bb K b +2 Θ b b . hLi = hb σ cd i σ bcd − 2 K Θ K (86) 3 D E D E D E

b = K b =0= K b c = hb For a symmetric embedding (4)Kab = 0, therefore cf. Eq. (74) Θ σ cd i, so that hLab i = 0. Finally,

C.

6Pab = 3 (e ρ + pe) ua ub + 8e q(a ub) + (e ρ + pe) hab + 4e π ab . κ e2

(87)

Gravitational dynamics on the brane: generic embedding

In order to obtain the evolution and constraint equations on the brane, we select a subset of the Ricci and Bianchi equations given in subsections III A and III C, by combining them in such a way, that the off-brane derivatives of the kinematical and gravito-electro-magnetic quantities drop out. Additionally, the equations of this subset contain only quantities appearing in the effective Einstein equation and in the 4d theory. bab from (37), (50), (43) and (49) respectively and we employ the definitions (14), First we express Ha , Ea , Fab and H (15) in order to introduce Eab and Hab in place of Eab and Hab . Inserting these into the system of equations given by the following equations: (41), (29), (54), (57)-(66), (27), (30), (31), (38), (39), (40), (58)-(61), (60), (71) and (72), evaluated at the brane, we obtain a system of equations to be referred as the brane Eqs. These equations are either evolution or constraint equations on the brane and for a generic asymmetric embedding are presented in Appendix b K b a , Θ, ω a , σ ab , E, Eba , Eab , Hab . C. The evolutions refer to the quantities Θ, D.

Gravitational evolution and constraint equations on a symmetrically embedded brane

In this subsection we restrict ourselves to symmetrically embedded branes. The Z2 -symmetric embedding arises when there is a perfect symmetry between the 5d space-time regions on the two sides of the brane. In this case the extrinsic curvatures on the two sides of the brane are opposite. This is due to the fact, that the normal vectors to the brane on its two sides are na and −na , respectively. Therefore ∆(4)Kab = 2(4)Kab and K ab = 0.

15 We present a system of evolution and constrain equations, which hold on the brane and contain no off-brane derivatives.We obtain these equations by specifying Eqs. (C1)-(C14) for a symmetrically embedded brane; then b K ba, K b and σ replacing Θ, bab with the corresponding matter variables as given by the projections (76)-(79) of the Lanczos equation, again specified for a symmetrically embedded brane. Finally, we employ the definitions (81) and (82), whenever possible. For improved clarity we group all general relativistic contributions on the left hand side of the equations, keeping the brane-world contributions on the right hand side: ρ˙ + Da qa + (ρ + p) Θ + 2q a Aa + π ab σ ab = −2e q,

4Θ q˙hai + Da p + Db π ab + qa + (ρ + p) Aa + π ab Ab − ω ab q b + σ ab q b = −2e πa , 3

(88) (89)

" 4 2 e4 ab · κ ab ab a ˙ b b 0 = E − D Ea + ΘE + Eab σ − 2Ea A + π π habi + π ab Da qb + q a Db π ab − q a Da ρ 3 4 3 # 2Θ Θ qa q a − σ ab q a q b +2Ab qa π ab + π ab π ab + (ρ + p) σ ab π ab + σ ca π b c π ab + 3 3 κ e2 2e κ2 a 2e κ2 κ4 e 4e κ2 2e κ2 (e ρ−π e + pe)· − D qea − Θ (e ρ + pe) + 2κ2 qe + ρe q− qea Aa − π eab σ ab , 2 3 3 3 3 3 4 1 4E ˙ 0 = Ebhki + ΘEbk − Dk E − Ak − Da Ebka − Ebka Aa − (ω ka − σ ka ) Eba 3 3 3 " κ e4 2 2Θ + −π˙ hkai q a − (ρ + p) Db π kb − q a σ ck π a c + (ρ + p) Dk ρ − (ρ + p) qk 4 3 3 −

2 1 −2q a Dk qa + q a Da qk + qk Da qa − 2q a Ahk qai + σ ba π k b q a − qk σ ab π ab 3 3 # 1 ab a b a a b c ab c +π Dk π ab − π b D π ka − π k Da ρ − εcab q ω π k − εk qc ω a π b 3 " κ e2 1 κ e2 + −2e π′hki + Dk (e ρ + 3e π − 3e p) + (2λ − ρ − 3p) π e k − 2K qek 3 2 3 # 5e κ2 2 b a a b b π − pe) Ak + π ka A − 2 (e π ka π e , − qe 2Kk + Kk + 2e 3 2

2 2 ˙ − Da Aa + Θ − Aa Aa − 2ωa ω a + σ ab σ ab + κ (ρ + 3p) − Λ Θ 3 2 4 2 4 κ e κ e κ e (e ρ+π e + pe) , = E + q a qa − ρ (2ρ + 3p) − 4 12 2 2Θ κ2 σ˙ habi − Dha Abi + σ ab − Aha Abi + ω ha ω bi + σ cha σ bic + Eab − π ab 3 2 2 4 4 4 κ e κ e κ e 1 κ e qha qbi − π cha π bic − (ρ + 3p) π ab − Ebab + π eab , = 8 8 24 2 3

2Θ 1 ω a − σ ab ω b = 0 , ω˙ hai − εa cd Dc Ad + 2 3

2 κ e4 κ e4 2e κ2 Db σ ab − Da Θ + εa ck Dc ωk + 2εa ck Ac ω k + κ2 qa = − ρqa + π ab q b − Eba − qea , 3 6 4 3 Dhc ωki + εabhk Db σ cia + 2Ahc ω ki + Hab = 0 , a

D ω a − Aa ω

a

= 0,

(90)

(91)

(92)

(93)

(94)

(95)

(96) (97)

16 E˙ hkji − εabhk Da Hjib + ΘEkj + E ahk ωjia − 3σjia + 2εhk ab Hjia Ab " # Θ κ2 a π˙ hkji + Dhk qji + (ρ + p) σ kj + 2qhk Aji + π kj + π hk ω jia + σ jia + 2 3

=

Θ 2E 1 1 1b ˙ σ kj E hkji − Dhk Ebji + Ebhj a ωkia + σ kia + Ebkj − Ebhk Aji + 2 " 2 2 6 3 κ e4 · + (ρ + 3p) π˙ hkji + 6π hja π˙ kia + (ρ + 3p) π kj − 2qhk Dji (ρ − 3p) − 2ρDhk qji + 3π hj a Dki qa 24

+6qhk Db π jib + 3qa Dhk π jia − 2ρ (ρ + p) σ kj + 7Θqhk qji + 2 (ρ + 3p) qhk Aji + Θπ hj a π kia Θ + (ρ + 3p) π kj + (ρ + 3p) π hk a ωjia + σ jia + 3qhk σ jib q b + 3π hj a ωkic π a c + 3σ jk q a qa 3 # −9qhk ωjia q a + 6qhk π jia Aa + 6π ahj Aki q a + 3π c a π hk c σ jia

# " κ e2 ˙ κ e2 Θ a + π ehkji + Dhk qeji − π e q + 4e qhk Aji + (e ρ + pe) σ jk + π ejk + π ehj ω kia + σ kia , 3 2 hk ji 3

(98)

κ2 εabhk Da π jib + 3ωhk qji + εhk ab σ jia qb H˙ hkji + εabhk Da Ejib + ΘHkj − 3σ ahk Hjia − ω ahk Hjia − 2εhk ab Ejia Ab − 2 " 4 1 κ e 1 3 = − εabhk Da Ebjib + εhk ab σ jia Ebb + Ebhj ω ki + ε cd π jic Dd (ρ + 3p) − (ρ + 3p) εabhk Da π jib 2 2 2 24 hk −3εabhk Da π jic π cb − 3εabhk π jic Da π cb + 6ρω hk qji + 2ρεhk ab σ jia qb + 3εabhk qji Da q b + 3εabhk q b Da qji # κ e2 c a ab +3εhk σ jib π a qc − 9πhj ωki qa + εabhk Da π ejib + 3e qhj ω ki + εhk ab σ jia qeb , 3

(99)

" # 2 2 κ2 a b cd − Dk ρ + D π ak + Θqk − σ kb q − 3εk qc ω d D Eak − 3Hka ω + εk Hac σ b + 2 3 3 " 1 1 Θ κ e4 4 1 ρDk ρ + π ak Da (ρ + 3p) = Da Ebak − Dk E − Ebk + (3ωka + σ ka ) Eba + 2 3 3 2 24 3 a

a

ab

c

+ (ρ + 3p) Da π ak + 3πa c Da π ck − 4π ab Dk π ab + 3π kb Da π a b + 2q a Dk qa − 3qa Da qk

4 −3qk D qa + 9εk qc ω a π d − 3π a q σ kb − ρΘqk + 2ρσ kb q b + 6ρεk cd qc ωd + 2Θπ k a qa 3 " # 2 κ e 1 2 a a − D π eak − Dk (e ρ−π e + pe) + Θe qk − qe (3ωka + σ ka ) , 3 2 3 a

ad

c

b a

# (100)

κ2 ab εk Da qb + 2 (ρ + p) ω k − π k c ωc − εk ab π ac σ b c 2 " 4 1 ab 4E 1 1 κ e a ab c 3q c ω c qk − 2εk ab qb Da ρ = − εk Da Ebb + ω k − Ebak ω − εk Ebac σ b + 2 3 2 2 24 Da Hak + 3Eka ωa − εk ab Eac σ b c +

−2ρεk ab Da qb − 3εabk q c Db π c a − 3εk

ab

π a c Db qc − 4ρ (ρ + p) ω k + 3qa q a ωk − (ρ + 3p) π k c ω c #

−3εk ac σ ab q b qc + 3π ab π ab ωk − 3π ca π k c ω a − (ρ + 3p) εk ab π ac σ b c − 3εk ab π da π c d σ b c # " κ e2 c a ab ab π eka ω − εk Da qeb − 2 (e ρ + pe) ω k + εk π ea σ bc . + 3

(101)

17 For vanishing 5d matter these equations reduce to the corrected form of the Eqs. (26)-(29) and Appendix A of Ref. [11]. Eqs. (88) and (89) express the interchange of energy density and energy current between the brane and the 5d space-time (due to the nonvanishing right hand sides). In the absence of the 5d sources, these equations become evolution equations for the brane matter alone. Similar relations for the effective nonlocal energy density and effective nonlocal energy current are given by Eqs. (90) and (91). Even for the chosen Z2 -symmetric embedding and with the simplifying assumption Teab = 0 (thus 0 = e ρ = qe = qea = π e=π ea = π eab = pe) the above equations do not close on the brane, due to the absence of evolution equations for Ebab , π ab , and p. Assumptions fixing π ab (typically by kinetic considerations employing the Boltzmann equation) and p (by choice of a continuity equation) are required already in general relativity for closing the system, however on the brane (even with empty 5d space-time) an additional assumption for Ebab is equally required [11]. From these considerations it is immediate to establish the closure in the special case Ebab = 0 = π ab , provided the equation of state is known. As the closure is difficult to achieve even in the simple 5d vacuum case, in order to tackle realistic problems we need to consider the ensemble of all dynamical and constraint equations given in the preceding section. E.

Closure conditions

The general relativistic 3+1 covariant formalism contains 10 gravito-electro-magnetic variables with 10 evolution equations (besides there are also 12 kinematic variables with 9 evolution equations and 15 constraints altogether). The 3+1+1 brane-world formalism developed in this paper contains 35 gravito-electro-magnetic variables with 35 evolution equations (beside 35 kinematical variables with 28 evolution equations and 77 constraints altogether). The subset of equations on the brane, given in subsection IV D has 10+9 gravito-electro-magnetic variables with only 10+4 evolution equations (there are also 12 kinematical variables with 9 evolution equations and 15 constraints altogether.) The 9 new gravito-electro-magnetic variables are the quantities appearing on the right hand side of Eq. (84); among them the last term Ebab , representing 5 independent variables, has no evolution equation. It has been known that the system of equation is closed by the condition Ebab = 0 [11]. In this subsection we want to explore the extra information we have in the complete system of evolution equations derived. In particular, Eq. (61) contains the desired temporal evolution, however it also contains terms not appearing in subsection IV D. Remembering, that on the brane ω b ab = 0 we could impose ! b Θ 3 ′ a b b b Fhkji = 2K − Fkj − Fhj σ bkia + K Ekj + KEkj − Ehj K + Lki − Kki 3 2 ki 3 bb + 3 ε ab σ bjia Kb − 2K b b − ε ab Hkia A bji − ε ab H bkia Hb , (102) + Ebhk A hj hk 2 2 hj

such that Eq. (61) becomes

Θ 4E ˙ 0 = Ebhkji − Dhk Ebji + Ebkj + σ kj − 2Ebhk Aji + Ebhj a ω kia + σ kia + Mkj , 3 3

(103)

with the 5d matter contributions Mkj =

κ e2 ˙ κ e2 κ e2 κ e2 qe κ e2 Θ π e hkji + Dhk qeji + (e ρ + pe) σ kj + σ b kj + π ekj 3 3 3 3 9 e2 κ e2 2e κ2 b ki + Lki + κ qehj Aki + π ehj 2K π ehj a ωkia + σ kia . + 3 3 3

(104)

A particular solution of Eq. (102) would be

b kj = Ej = Hj = K = A bj = 0 . Fkj = H

(105)

None of the quantities above appear in the brane equations presented in subsection IV D, thus those equations are not altered by the choice (105), and the system becomes closed by Eq. (103).3

3

The no-go theorem for closure given in Ref. [32] does not apply here, as it was derived for perturbations of 5d Anti de Sitter spacetimes.

18 The 5d Schwarzschild-Anti de Sitter space-time containing a Friedmann brane emerges as special case of the spacetimes obeying Eqs. (103) and (105), as they have Mab = 0 = K and b a = ωa = ω ba , 0 = Ea = Eba = Ha = La = Ka = K b a = Aa = A b ab = σ ab = σ 0 = Eab = Fab = Hab = Ebab = H bab .

(106)

At the end of this section we emphasize, that Eq. (103) closing the system of brane equations should allow for far more solutions than the trivial one for Ebab . We will construct a specific example in Section VI. V.

COSMOLOGY

In this section we consider generic embeddings. By employing the definitions (81) and (82), also the conditions (11), arising from the existence of the brane, we can derive average and difference equations on the brane. We give here but the most relevant such dynamical equations. The rest of the equations is straightforward to derive, although it may be lengthy. The average taken on the two sides of the brane of Eq. (25) reduces to the generalized brane Friedmann equation. D E D E2 2 b2 = Θ b + ∆Θ b /4 and use a similar relation for σ b from the Lanczos equations We also rewrite Θ b ab . We take ∆Θ (76)-(79). Eq. (25) then becomes

D E2 b 4 D E κ Θ R Θ ρ κ e 1 1 e2 2 ab a ab + −Λ−κ ρ 1 + − hb σ ab i σ bab + he ρ + pe − π ei . (107) − σ ab σ + ωa ω = − hEi − π ab π + 2 3 2λ 2 8 3 2 2 2

The generalized brane Raychaudhuri equation is obtained from the average of Eq. (27), by employing the same sequence of simplifications, as for the Friedmann equation. We obtain: 2 2 ˙ + Θ + σ ab σ ab − 2ωa ω a − Da Aa − Aa Aa + κ (ρ + 3p) − Λ Θ 3 2 2 D ED E D ED E κ κ e4 κ e4 ρ b a − e he ba K b + K b K (2ρ + 3p) + qa q a − Θ ρ+π e + pei . = hEi − 12 4 2

(108)

Finally we give the generalized brane energy-balance equation from the jump across the brane of the Eq. (41) by the same procedure: ρ˙ + Θ (ρ + p) + Da q a + 2Aa q a + π ab σ ab = −∆e q.

(109)

These equations hold for arbitrary branes. Again, general relativistic contributions are on the left hand sides, braneworld contributions on the right hand sides. A.

Friedmann brane with perfect fluid

In this case the conditions ω a = 0 = σ ab = ∆b σ ab hold, arising from the particular geometry and matter source chosen, also R = 6k/a2 , where a is the scale factor. Moreover Θ/3 = H ≡ a/a, ˙ where H is the Hubble parameter. The Friedmann, Raychaudhuri and energy-balance equations become: k ρ 2 3 H + 2 = Λ + κ2 ρ 1 + − hEi a 2λ D E2 b D E κ Θ e2 1 + − hb σ ab i σ bab + he ρ + pe − π ei . (110) 3 2 2 κ2 κ e4 ρ κ e4 (ρ + 3p) − Λ = hEi − (2ρ + 3p) + qa q a 3 H˙ + H 2 + 2 12 4 2 D ED E D ED E κ b K b + K b a − e he ba K ρ+π e + pei , (111) − Θ 2 ρ˙ + 3H (ρ + p) = −∆e q. (112)

19 D E b = 0 = hb For symmetric embedding the equations simplify by Θ σ ab i. For generic asymmetric embedding Eq. (110) can be rewritten as hLi κ ρ e2 k + κ2 U − − he πi , 3 H 2 + 2 = Λ + κ2 ρ 1 + a 2λ 4 2

where hLi is given by Eq. (86) and U is defined by D E2 b D E Θ e2 κ 1 D b E D b E 1 D b E D b bE 1 Θ K − Kb K − hb + σ ab i σ bab − hEi + he ρ + pei . κ2 U = 6 2 2 4 2

(113)

(114)

The quantity U is nothing but the effective energy density introduced in Ref. [4], encompassing the trace-free parts of the Weyl, embedding and 5d matter sources in the effective Einstein equation, given here in terms of the 3+1+1 kinematic, gravito-electro-magnetic and 5d matter variables. B.

Anisotropic brane-worlds

Full brane-world solutions with homogeneous, but anisotropic 5d space-times are also known. In Ref. [49] a vacuum 5d static and anisotropic space-time with cosmological constant admitting a moving Bianchi I brane was analyzed. The Z2 -symmetric junction conditions could be obeyed only by anisotropic stresses on the brane, hence the brane cannot support a perfect fluid. Isotropy of the brane fluid could be achieved only for isotropic 5d space-time and brane. This setup was generalized in Ref. [50] by allowing for a non-static 5d space-time. Assuming separability of the metric components, new 5d solutions combining the 4d Kasner solution with the static 5d solutions of Ref. [49] were obtained. For the reader’s convenience we give in Appendix D the list of kinematical, gravito-electro-magnetic and matter variables for the 5-dimensional models presented in Ref. [49]. Working out the respective quantities for other metrics would be a similar straightforward exercise. VI.

STATIONARY VACUUM SPACE-TIMES WITH LOCAL ROTATIONAL SYMMETRY

In this section we discuss an application of our formalism, by assuming vacuum in both 4d and 5d, but keeping the respective cosmological constants. The embedding of the brane is symmetric. Then the effective Einstein equation reduces to 1 Gab = Λgab − E ua ub + hab + 2Eb(a ub) − Ebab . (115) 3

We are interested in stationary space-times, therefore f˙ = 0 for any scalar field f . The stationarity implies a singled-out temporal Killing vector, therefore the 3+1+1 decomposition of the gravitational dynamics developed in this paper turns particularly useful for the study of gravitational dynamics on the brane (which defines the other singled-out direction). We also specialize our treatment to space-times with local rotational symmetry (LRS) on the brane. Such a symmetry singles out an additional space-like unit vector field ea , in the sense that there is a unique preferred spatial direction at each point that assigns the local axis of symmetry. Once such a special vector field is chosen, a further decomposition of the spatial quantities would lead to a generic 2+1+1+1 formalism. For this, the metric hab should be further decomposed as hab = ea eb + qab ,

(116)

where qab is the induced 2-metric on the surface perpendicular to both ea and ua , and lying in the brane. In what follows we will see that these symmetries assure that the structure of the space-time can be described solely in terms of scalars, thus no vectors and tensorial quantities are needed. This is a powerful feature of the formalism. Furthermore, the symmetries assure, that all scalars f depend only on the coordinate parametrizing the integral curves of the rotation axis field ea . We denote this coordinate derivative as f ⋆ ≡ ea Da f (a spatial covariant derivative along these integral curves).

20 A.

Independent kinematic quantities related to the vector field ea 1.

Decomposition

For the purposes of the present application it is enough to give here the decomposition of the covariant brane derivative of the vector field ea in terms of kinematical quantities and extrinsic curvature type quantities analogous to the ones appearing in the decomposition of the vector fields ua and na . We keep the familiar notations, with the remark that the quantities belonging to the decomposition of ∇a eb will carry a distinguishingemark. We thus have

with

eb + e b + ea A e a ub − L e a ub − ua K ∇a eb = Lu

e Θ qab + σ eab + ω e ab , 2

e = uc ud ∇ e c ed , Θ e = q ab Da eb , L d c e a = u ha ∇ e c ed , K e b = u c hb d ∇ e c ed , L

e c ed , σ eab ω e ab = q[a c qb] d ∇

eb = ec qb d ∇ e c ed , A e e c ed − Θ qab . = qa c qb d ∇ 2

(117)

(118)

e and L e a are not independent from the previously introduced sets of variables, they can be expressed We remark that L in term of projections of the kinematic quantities related to ua : e = −ea Aa , L e a = −ed Θ had + σ ad + ωad L . (119) 3 e K eb, A eb , σ By contrast, the quantities Θ, eab and ω e ab are independent of the rest of the kinematical and extrinsic curvature-type variables. Similarly to ωa and ω b a , we can also define a rotation vector ω ea =

2.

1 εabc ω e bc . 2

(120)

LRS symmetry

The preferred spacelike unit vector field ea satisfies ua ea = 0, ea ea = 1. We employ here various results following from the LRS symmetry, following Ref. [51]. The symmetry and normalization implies: ea Da eb = 0 ,

e˙ hbi = 0 ,

(121)

i.e. ea is geodesic with respect to the connection compatible with hab and is Fermi propagated along the world line of a brane observer. The above equations and normalization further imply that Eq. (120) can be rewritten into the standard form e e ed = εabc Db ec . ω e a = εa bc q[b e qc] d ∇

(122)

Due to LRS, all spacelike vector fields must be proportional to ea , thus

Aa = Aea , ω a = ωea , ω ea = ω e ea . a V a a ⋆ Eb = Eb e , D Θ = Θ ea , Da E = E ⋆ ea ,

(123)

The vector field ea and the induced metric qab define a unique spacelike tracefree symmetric tensor field eab as eab = ea eb −

qab , 2

(124)

satisfying: ua eab = 0 eab eab

, ea eab = eb , ea a = 0 , 2eac ec b = eab + hab , e 3Θ 3 , Db eab = ea , e˙ habi = 0 . = 2 2

(125)

21 Again, due to LRS all 3d tracefree symmetric tensor fields are proportional to eab : 2E 2H 2Eb 2σ σ ab = √ eab , Eab = √ eab , Hab = √ eab , Ebab = √ eab . 3 3 3 3

(126)

b E and H, replacing the In Eqs. (123) and (126) we have introduced suitably normalized scalars A, ω, ω e , σ, EbV , E, vectorial and tensorial variables. From LRS, by use of Eqs. (121), (123), (126) and (125) also follows that Θ 2σ e e ea , (127) L = −A , La = − +√ 3 3 ea = A ea = 0 , σ K eab = 0 . (128) Thus there are only two kinematic quantities related to ea left, which are both non-trivial and independent from those e and ω introduced the Sec. II. These are Θ e. B.

LRS class I type conditions

The general relativistic classification of the LRS models presented in [51] is recovered for Eba = 0 = Ebab and E < 0. For brane-worlds, when the above conditions do not hold even under the simplifying assumptions of this chapter, this classification should be refined, however for the application we are interested in, we shall still assume the conditions ω e = Θ = σ = 0 characterizing the LRS class I of general relativity. From these conditions, Eqs. (95), (122), (123) and εaij ei ej = 0 we also find EbV = 0. Therefore we have verified, that Eq. (103), which closes the system of brane equations for the considered stationary vacuum space-times containing LRS class I type branes, trivially holds. 1.

Dynamics

e characterizing ea can be inferred from the Ricci identity for ea : The evolution of the single kinematic quantity Θ e2 2E 2Λ 2E Eb e⋆ + Θ = −√ + Θ +√ − . 2 3 3 3 3

(129)

Other nontrivial brane Eqs. are (91), (92), (93), (96), (97), (100), and (101). Under the assumptions of this section they simplify to: " ! # e √ 3 Θ ⋆ ⋆ E + 4AE + 2 3 Eb + + A Eb = 0 , (130) 2 e + A A + 2ω2 + Λ = −E , A⋆ + Θ (131) ! √ e √ 3Eb Θ A − ω 2 − 3E = , (132) A⋆ + A − 2 2 √3ω e = 0, (133) H + 2A − Θ 2 e −A ω = 0 , ω⋆ + Θ (134) e Eb e Eb⋆ 3Θ E⋆ 3Θ E − 3ωH = + − √ , 2 2 4 2 3 e + 6Eω = 4ωE √ − ωEb . 2H ⋆ + 3ΘH 3 E⋆ +

Eliminating A⋆ from Eqs. (131) and (132) we obtain the algebraic equation √ e √ 3ΘA 3Eb 2 0= + 3ω + Λ + 3E + E + . 2 2

(135)

(136)

(137)

22 Also for any solution of the system (129)-(134), Eqs. (135) and (136) are identically obeyed. This can be seen as follows. By taking the ⋆-derivative of Eq. (137) and employing Eqs. (129)-(134) we obtain Eq. (135). Similarly, the ⋆-derivative of Eq. (133), combined with Eqs. (129)-(134) gives (136). b Θ, e A, As we have 6 independent (2 algebraic and 4 first order differential) equations left for the 7 variables (E, E, ω, E, H), we need to impose an additional ansatz, chosen as 2E Eb = − √ . 3

(138)

This condition will considerably simplify the algebraic equation Eq. (137). 2.

Discussion

The algebraic Eqs. (133) and (137) give H and E in terms of the rest of variables. By Eq. (138) the system (129)-(131) and (134) reduces to e2 e ⋆ + Θ − ΘA e − 2ω2 Θ 2 e E ⋆ + 2E Θ e + A A + 2ω2 + Λ A⋆ + Θ e −A ω ω⋆ + Θ

= 0,

(139)

= 0,

(140)

= −E ,

(141)

= 0.

(142)

From the newly arised two algebraic Eqs. (140) and (142) we express ⋆ e = ln E −1/2 Θ , ⋆ . A = ln ωE −1/2

(143) (144)

In terms of the auxiliary variables

x = ln E −1/2 , y = ln ωE

−1/2

(145) ,

(146)

the remaining Eqs. (139) and (141) become (x⋆ )2 − x⋆ y ⋆ = 2ey−x , 2 2 y ⋆⋆ + (y ⋆ ) + x⋆ y ⋆ = − 2ey−x + e−2x + Λ .

x⋆⋆ +

(147) (148)

They form a coupled second order system, which would eventually give ω and E in full generality. The solution of this system is however not immediate, therefore in what follows we will employ a metric ansatz in order to find a particular solution. C.

Taub-NUT-(A)dS solution with tidal charge

We take the following metric ansatz, compatible with the chosen symmetries: ds2 = − where

g (r) 2 f (r) 2 (dt + ωk dϕ) + dr + g (r) dθ2 + Ω2k dϕ2 , g (r) f (r)

Ωk =

 

sin θ , 1 ,  sinh θ ,

k=1 k=0 , k = −1

(149)

(150)

23 and ω k is another function of θ. The axis of LRS is given by a f 1/2 ∂ ea = 1/2 . ∂r g

(151)

Choosing the 1-form ua as u=−

f 1/2 (dt + 2ωk dϕ) , g 1/2

(152)

and employing EbV = 0, Eqs. (126), (138), (151), the electric part of the 5d Weyl tensor is found as (4) t

E

t

=

(4) r

E

r

= − (4)E θ θ = − (4)E ϕϕ = −E ,

(4) t

E

ϕ

= −2Eωk .

Both from Eba ∝ EbV = 0 and from Hab being proportional to eab we find Ωk

(153)

d2 ω k dΩk dω k =0. − dθ dθ dθ2

(154)

dωk = −2l dθ

(155)

Equivalently, by an integration: Ω−1 k

where l is constant. A second integration gives  2l cos θ + L ,  −2lθ + L , ωk (θ) =  − 2l cosh θ + L ,

k=1 k=0 , k = −1

(156)

where L is another integration constant. Locally, this constant can be absorbed in a new time variable t + Lϕ → t, which translates to the choice L = 0. Direct computation, employing Eq. (156) shows that the metric ansatz is compatible with the chosen LRS class I conditions Θ = σ = a = EbV = 0 .

(157)

The nontrivial kinematic and Weyl quantities, by employing Eqs. (149) and (156) are 1/2 1/2 lf 1/2 d f dg e = f , ω = 3/2 , , A = Θ 3/2 dr dr g g g 1/2 1/2 2 f k 3l f d f dg E = 5/4 − 3 − +Λ , dr g 3/4 dr g g g 2 2 1 d f 4l2 f 2 df dg dg 2k f 2Eb √ = + − + + , 2 2 3 3 3g dr 3g dr dr 3g dr 3g 3g 3 2E 1 d2 f 4l2 f 1 d f 3/2 dg k √ = − − − , 2 2 3 1/2 6g dr dr g dr 3g 3g 3f 3 2H d √ = −l f g −2 . dr 3

(158)

These quantities are constrained by Eqs. (133), (137), (138), and (139)-(142). From here by straightforward algebra we find two independent equations for the metric functions f (r) and g (r): g 3/2

d2 g 1/2 = l2 , dr2 d2 f = 2k − 4Λg . dr2

(159) (160)

24 By multiplying the first equation with dg/dr and integrating, we get

dg dr

2

+ 4l2 = C1 g ,

(161)

with C1 > 0 an integration constant. A further integration then gives g (r) =

C1 4l2 2 , (r + C2 ) + 4 C1

(162)

with C2 another integration constant. The constant C2 can be absorbed into r by a redefinition of its origin, therefore, without restricting generality we choose C2 = 0. Also the constant C1 disappears in the following rescaling of the 1/2 1/2 2 coordinates and parameters: 2t/C1 , C1 r/2, 4l /C1 → t, r, l2 . Formally this is the same as choosing C1 = 4. Eq. (160) then gives the other metric function as

Λ f (r) = C4 + C3 r + k − 2l2 Λ r2 − r4 , 3

(163)

with C3 and C4 emerging as integration constants. With the reparametrizations C3 = −2m and C4 = q − kl2 + Λl4 we find the brane solution given by Eqs. (149), (150) and 4 r f (r) = k r2 − l2 − 2mr + q − Λ + 2l2 r2 − l4 , 3 g (r) = r2 + l2 ,  2l cos θ ,  −2lθ , ω k (θ) =  − 2l cosh θ ,

k=1 k=0 . k = −1

(164)

This is quite similar to the charged-Taub-NUT-(A)dS solution of a general relativistic Einstein-Maxwell system with mass m, electric charge Q, NUT charge l, and cosmological constant Λ, however the constant q replacing Q2 is not restricted to positive values. A glance at the effective Einstein equation (115) shows that this constant could possibly arise only from the electric part of the 5d Weyl tensor, Eq. (153). Indeed from the fourth Eq. (158) we get E=−

q 2

(l2 + r2 )

.

(165)

As q originates in the Weyl sector of the higher dimensional space-time, the derived solution has the interpretation of a Taub-NUT-(A)dS brane with tidal charge. VII.

CONCLUDING REMARKS

In this paper we have developed a generic 3+1+1 covariant formalism for characterizing 5d gravitational dynamics on and outside a brane. The singled-out directions are the off-brane and temporal directions, thus the 3-spaces are constant time sections of the brane. Generalizing previous approaches, like 3+1+1 with the extra requirement of double foliability [9], [10], 3+1 in general relativity [12]-[15] and in brane-worlds [11], finally 2+1+1 in general relativity [16], [17], we presented gravitational evolution and constraint equations in terms of kinematic, gravito-electro-magnetic and matter variables, defined as scalars, 3-vectors or 3-tensors. The number of variables being higher than for other lower dimensional formalisms, the 5d matter and especially the 5d Weyl tensor leads to a multitude of projections without counterpart in the mentioned approaches. Only the kinematical set of variables is similar. We have compared and checked our results with those presented in a recent work by Nzioki, Carloni, Goswami and Dunsby [18] on the 2+1+1 decomposition of f (R) gravity and we give the correspondence between our and the notations of Ref. [18] in Table 1. Our generic formalism contains the complete set of dynamical equations in the 5d space-time. All equations, with the exception of those in IV to VI are independent of the particular form of the dynamics on the brane. As such, they can be employed both to discuss DGP / induced gravity type [43] branes (a project deferred for future work) and one-brane Randall-Sundrum type branes. For the latter, we have given both the full set of equations on the brane, and those containing off-brane evolutions.

25 TABLE I: Comparison of the notations for the kinematic quantities with Ref. [18]. na → ea Aa → Aa ba → aa A La → Σa − εab Ωb hab → Nab

K → 31 Θ + Σ Ka → Σa + εab Ωb Θ → Θ − 23 Σ ω ab → Ωεab σ ab → Σab

b → −A K b a → αa K b Θ→φ ω b ab → ξεab σ b ab → ζ ab

The brane equations are more general than previously published results by the inclusion of arbitrary 5d sources. Although we have recovered the known fact that in the generic case this system does not close on the brane (excepting the trivial case Ebab = 0), by deriving the complementary system of equations of the 5d dynamics we could establish a more generic condition for closure. This is given by either of Eqs. (102), (105) which carry much richer possibilities than the previously known Ebab = 0 case. The initial value problem in general relativity is usually discussed in ADM-like variables (including modifications enhancing stability, like the use of variables with factored-out conformal factors [54]), therefore a similar treatment would be possible to develop in the framework of the Hamiltonian approach presented in [9, 10]. A 3+1 covariant approach for discussing the evolution of cosmic microwave background anisotropies in the cold dark matter model was employed in Ref. [24]. The complete set of infinitesimal frame transformations given in the present paper may turn useful in the study of perturbations in a 3+1+1 setup. We have decomposed the Lanczos equations and all source terms of the effective Einstein equation in terms of the 3+1+1 covariant variables. The ensemble of these results opens the possibility for applications, with both cosmological and other symmetries. We have given generalized Friedmann, Raychaudhuri and energy balance equations for a generic brane, and by specifying for cosmological symmetries we have established correspondence with previous related work [4]. We have also established the correspondence with the anisotropic brane-world presented in Ref. [49]. We have also employed the 3+1+1 covariant formalism to discuss stationary space-times with local rotation symmetry of class I, imposed on the brane. We have shown that such space-times obey the closure condition presented here. The symmetries and metric ansatz (149) implemented in the 3+1+1 covariant formalism led to a simple decoupled system of second order differential equations for the metric functions, the solution of which gave a new exact solution of the effective Einstein equation, the tidal charged Taub-NUT-(A)dS brane, given by Eqs. (150) and (164). In the spherically symmetric and rotating cases tidal charged brane solutions were already found [52]-[53], which correspond to electrically charged general relativistic Einstein-Maxwell solutions, when a formal identification of the electric charge squared with the tidal charge is carried on. Here we also found that replacing the electric charge squared in the electrically charged Taub-NUT-(A)dS solution of an Einstein-Maxwell system with a tidal charge originating in the Weyl curvature of the 5d space-time leads to a brane solution with the same symmetries. Unless the electric charge squared, the tidal charge however is allowed to have both positive and negative values, thus allowing to either weaken gravity, or contribute towards its confinement on the brane. VIII.

ACKNOWLEDGEMENTS

We thank Roy Maartens and Chris Clarkson for discussions on the paper and for suggesting ways to simplify the ´ was notations. This work was supported by the Hungarian Scientific Research Fund (OTKA) grant no. 69036. LAG further supported by the Pol´ anyi and Sun Programs of the Hungarian National Office for Research and Technology (NKTH) and Collegium Budapest. Appendix A: Commutation relations

In this appendix we enlists some useful differential identities, obtained by computing the commutators among the e a ≡ D/dy (prime), and Da (induced metric compatible covariant derivative) on e a ≡ D/dτ (dot), na ∇ derivatives ua ∇ scalars, brane-vectors and symmetric brane-tensors of second rank.

26 For a scalar field φ the following commutation relations hold: ˙ − uc ∇ e a (φ) e c (φ′ ) = Kφ′ + K b φ˙ + K a − K b a Da φ , na ∇

(A1)

b e b (Di φ) = − (Ka − La ) φ˙ + A ba φ′ + Θ Da φ + (b Da φ′ − ha i nb ∇ ωab + σ b ab ) Db φ , 3 e b (Di φ) = −Aa φ˙ + K b a + La φ′ + Θ Da φ + (ωab + σ ab ) Db φ , Da φ˙ − ha i ub ∇ 3 b ab φ′ . D[a Db] φ = ω ab φ˙ − ω

(A2) (A3) (A4)

For a 3-vector field V a the following commutation relations hold:

b V˙ hbi + K a − K b a Da Vb e a (V˙hji ) − h j uc ∇ e c (V ′ ) = −εkab Ha V b + KV ′ + K hb j na ∇ hbi hji b b aV a A bb + Ka V a Ab − A ba V a K bb , −Aa V a Kb + K

(A5)

b b c + Lc V ′ + Ac V˙ hdi − Θ Dc Vd + Θ K e a (Di Vj ) − Dc V˙ hdi = − K b d Vc − Θ Ad Vc hc i hd j u a ∇ hdi 3 3 3 1 κ e2 1 − (σ ac − ωac ) Da Vd + Eba V a hcd − Ebd Vc − εdab Hc b V a + qea V a hcd 2 3 ! 2 b Θ κ e2 baV a + K b d (b ωca + σ bca ) V a hcd + ω b cd + σ bcd K − Vc qed − 3 3 Θ (A6) hcd + ω cd + σ cd Aa V a − Ad (ω ca + σ ca ) V a , + 3 b b bc V ′ − Θ Dc Vd − Θ Kd Vc + Θ A bd Vc e a (Di Vj ) − Dc V ′ = − (Lc − Kc ) V˙ hdi − A hc i hd j na ∇ hdi hdi 3 3 3 1 e2 1 b bV a − κ − (b σ ac − ω b ac ) Da Vd − Ea V a hcd + Ed Vc − εdab H π ea V a hcd c 2 2 3 Θ κ e2 ed + hcd + ω cd + σ cd Ka V a − Kd (ωca + σ ca ) V a + Vc π 3 3 ! b Θ ba V a + A bd (b ωca + σ bca ) V a , (A7) hcd + ω b cd + σ b cd A − 3 ′ b ab Vhci + hc[a Eb]d V d + Ec[a Vb] − hc[a Ebb]d V d − Ebc[a Vb] D[a Db] Vc = ω ab V˙ hci − ω b 1 b 2 hc[a Vb] − Θ σ c[a − ω c[a Vb] + Θ σ − Θ2 − Θ bc[a − ω b c[a Vb] 9 3 3 b Θ Θ 1 − hc[a σ b]d + ω b]d V d + hc[a σ b b]d + ω b b]d V d − Ehc[a Vb] 3 3 3 − σ c[a − ω c[a ω b]d + σ b]d V d + σ b c[a − ω b c[a ω b b]d + σ bb]d V d e κ2 e κ e2 κ e2 Λ (e ρ−π e + pe) hc[a Vb] + hc[a π eb]d V d + π ec[a Vb] . + hc[a Vb] + 6 6 3 3

(A8)

For a symmetric trace-free 3-tensor field Tab the following commutation relations hold:

b T˙hcdi + K a Da Tcd − K b a Da Tcd e b (T ′ ) = KT ′ + K e a (T˙hiji ) − h i h j ub ∇ hhc i hdi j na ∇ hcdi hiji hc di

bhc Tdia K b a + 2Ahc Tdia K a −2Khc Tdia Aa + 2A b hc Tdia A ba − 2εabhd T a Hb , −2K ci

(A9)

27 b ba T ′ − Θ Dd Tbc + Ehb Tcia e d (Dk Tij ) − Da T ′ = − (La − Ka ) T˙hbci − A ha k hhb i hci j nd ∇ hbci hbci 3 b i + 2Θ hahb Tcid K d − (b ωad + σ bad ) Dd Tbc − 2Thb d εcidi H a 3 b b 2Θ 2Θ b bd Ahb Tcia − hahb Tcid A −hahb Tci d Ed + 3 3 bhb T d ω ad + σ b ad ) A −2 (ω ad + σ ad ) Khb Tci d + 2 (b ci d bd +2 ω ahb + σ ahb TcidK − 2 ω b ahb + σ bahb Tcid A −

2Θ 2e κ2 2e κ2 Khb Tcia − hahb Tci d π ed + π e T , 3 3 3 hb cia

(A10)

e d (Dk Tij ) − Da T˙hbci = − K b a + La T ′ + Aa T˙hbci − Θ Dd Tbc − Ebhb Tcia ha k hhb i hci j ud ∇ hbci 3 − (ω ad + σ ad ) Dd Tbc − 2Thb d εcidi Ha i + hahb Tci d Ebd b b 2Θ b hb Tcia − 2Θ Ahb Tcia b d + 2Θ K hahb Tcid K 3 3 3 b hb T d − 2 (ω ad + σ ad ) Ahb T d +2 (b ωad + σ bad ) K ci ci d b −2 ω b ahb + σ bahb Tcid K + 2 ω ahb + σ ahb Tcid Ad −

+

2Θ 2e κ2 2e κ2 hahb Tcid Ad + hahb Tci d qed − qehb Tcia . 3 3 3

(A11)

These results apply for arbitrary (not necessarily small) scalar, vector and tensor fields. Appendix B: Infinitesimal frame transformations

An infinitesimal frame transformation from the diad (ua , na ) to the diad (ua , na ) can be defined as a generalization of the corresponding general relativistic procedure [25] as: ua = ua + υ a + νna , with ua υ a = na υ a = 0 , na = na + la + mua , with ua la = na la = 0 ,

(B1) (B2)

where υ a , la , ν, m are all O (1).4 The new diad also obeys ua ua = −1 , na na = 1 , ua na = 0 ,

(B3)

ν=m.

(B4)

which implies

For υ a = la = 0 the above parameters define an infinitesimal orthogonal transformation (this is a 2-dimensional infinitesimal Lorentz boost). The parameters υa and la represent infinitesimal translations. These transformations represent gauge degrees of freedom, worth to explore in order to achieve particular tasks, for example to fulfill physical conditions, to conveniently close subsets of differential equations etc. The fundamental algebraic tensors hab and εabc change accordingly: hab = hab + 2u(a υ b) − 2n(a lb) ,

e

(B5) d

εabc = εabc − nc εabe + 2n[a εb]ce l + uc εabd + 2u[a εb]cd υ . The new 3-metric obeys hab na = hab ua = 0.

4

(B6)

The quantities denoted O (1) all vanish for identical transformation. We also assume a first order accuracy, thus all quantities O (2) = O (1)2 will be dropped.

28 The covariant derivatives of the new diad vectors can be invariantly decomposed as b a nb + Kna nb + na K b + La nb + Θ hab + ω ab + σ ab , e a ub = −ua Ab + Ku ∇ 3 b e a nb = na A bb + Kna ub + Ku b a ub − ua K b b + La ub + Θ hab + ω b ab + σ bab , ∇ 3

(B7) (B8)

which imply the following transformations for the kinematic quantities: ba υ a + (Ka + La ) la − mK b + m′ , K = K −A b = K b − Aa la + K b a − La υ a − mK − m ˙ , K ba , b = Θ b + mΘ + Da la − la A ba − υ a La − K Θ

b + Da υ a + υa Aa − la (Ka + La ) , Θ = Θ + mΘ b c − lb Ab nc + Θ υc + (ω ac + σ ac ) υ a + m Kc + K b c + υ˙ hci , Ac = Ac + υ b Ab uc + Kl 3 b bc = A bc − lb A bb nc − Kυc + υb A bb uc + Θ lc + (b b c + l′ , A ω ac + σ bac ) la + m Kc + K hci 3 Θ bc + υ ′ , K c = Kc − Klc + uc υb Kb − nc lb Kb + lc + (σ ac + ω ac ) la + m Ac + A hci 3 b bc = K b c + Kυ b c − nc l b K b b + uc υ b K b b + Θ υ c + (b bc + l˙hci , K σ ac + ω b ac ) υ a + m Ac + A 3 b b c − Klc + uc υ a La − nc la La − Θ υ c Lc = Lc − Kυ 3 Θ σ cb + ω b cb ) υ b + (σ ca + ωca ) la , + lc + Dc m − (b 3 b di − Ldi − lhc A bdi + 2υa σ σ bcd = σ bcd + mσ cd + Dhc ldi + υhc K ba(d uc) − 2la σ ba(d nc) , σ cd = σ cd + mb σ cd + Dhc υ di − lhc Kdi + Ldi + υ hc Adi − 2la σ a(d nc) + 2υ a σ a(d uc) , b d] − Ld] − l[c A bd] + 2υa ω ω b cd = ω b cd + mω cd + D[c ld] + υ [c K b a[d uc] − 2la ω b a[d nc] , a a ωcd = ω cd + mb ω cd + D[c υ d] − l[c Kd] + Ld] + υ [c Ad] − 2l ω a[d nc] + 2υ ωa[d uc] .

(B9) (B10) (B11) (B12) (B13) (B14) (B15) (B16)

(B17) (B18) (B19) (B20) (B21)

29 Similarly, the transformation laws for the gravito-electro-magnetic quantities are E = E + 2Ea la + 2Eba la , 4 E k = Ek − mEbk − Elk + Ea υa uk − Ea la nk + εkab Ha υ b + Eka la − Fka υ a , 3 4 Ebk = Ebk − mEk + Eυ k − Eba la nk + Eba υ a uk − εkab Ha lb + Ebka υ a − Fka la , 3 1 1 bka υ a − Hka la , Hk = Hk − εk ab Eba lb − εk ab Ea υ b − nk Ha la + uk Ha υ a − εk ab Ea υb + H 2 2 3 F kl = Fkl + 2u(k Fl)a υ a − 2n(k Fl)a la − Ehk υli 2 3 b a υ b − εab(k H a lb , + Ebhk lli + mEbkl + mEkl − εab(k H l) l) 2 b a lb , Ebkl = Ebkl + 2mFkl + 2υhk Ebli − Ehk lli + 2u(k Ebl)a υ a − 2n(k Ebl)a la − 2εabhk H li

E kl = Ekl + 2mFkl − 2lhk Eli + Ebhk υ li + 2uhk Elia υa − 2nhk Elia la − 2εabhk Hlia υ b , bkl + 3 Hhk lli − ε ab Fl)a lb + 2ε ab El)a υ b Hkl = Hkl + mH (k (k 2 a a ab b −ε(k El)a υ b − 2n(k Hl)a l + 2u(k Hl)a υ , b kl = H b kl + mHkl + 3 Hhk υ li + ε ab Fl)a υ b − 2ε ab Ebl)a lb H (k (k 2 b l)a υ a − 2n(k H b l)a la . +ε(k ab El)a lb + 2u(k H

(B22) (B23) (B24) (B25)

(B26) (B27) (B28)

(B29)

(B30)

The matter variables transform as

e ρ = e ρ − 2υa qea − 2me q,

(B31)

π e = π e + 2la π ea − 2me q, 2 2 ea , pe = pe − υ a qea − la π 3 3 qe = qe + la qea − υ a π e a − m (e ρ+π e) ,

(B32) (B33) (B34) b

c

c

(B35)

c

(B36) (B37)

qea = qea − e ρυ a − qela − 2e pυ a − 2e π ab υ + ua υ qec − na l qec − me πa , b

c

π ea = π ea − qeυa + 2e pla − π ela + 2e π ab l − na l π e c + ua υ π ec − me qa , d π eab = π ab + 2e pυ ha ubi − 2e plha nbi − 2e qha υ bi + 2π ha ubi υ d − 2π ha c nbi lc − 2e π ha lbi .

We have checked that in the particular case la = 0 = m, by applying the Lanczos equations (76)-(79) for eliminating b Θ, b La = −K b a and σ the quantities K, bab , by suppressing the quantities related to the brane normal, in particular imposing ω b ab = 0, we recover the linearized form of the transformation laws for the kinematical and dynamical quantities (Θ, σ ab , ωab , Aa ) and (ρ, p, qa , π ab ) of Ref. [25]. Similarly, by employing Eqs. (14) and (15), we obtain the required transformations for the Weyl projections (Eab , Hab ). If we would like to apply the generic transformations derived in this Appendix in a brane-world scenario, we have to impose la = 0 = m on the brane (in order to preserve the vector field na at y = 0, which defines the brane), however the derivatives along the off-brane direction (the derivatives denoted by prime) of these quantities can be different from zero even at y = 0. Appendix C: Gravitational evolution and constraint equations on an asymmetrically embedded brane

The brane equations describing the gravitational dynamics are

30

! b Θ b a Aa + σ b Θ − 2K bab σ ab − κ e2 qe , K+ 3 ! b 4Θ Θ 2 ˙b bb + κ b b + σ ab K b − Db σ ba − K b+ b− Θ e2 π ea , Aa − σ bab Ab − ω ab K bab + K 0 = K hai − Da K 3 3 3 b˙ − Da K ba + 0 = Θ

4 2 ba b bb − K b a Db σ 0 = E˙ − Da Eba + ΘE + Ebab σ ab − 2Eba Aa + σ b ab + K bab σ b˙ habi − σ b ab Da K Da Θ 3 3 ! b Θ 2Θ b b a baσ b aK bb b + Θ σ ab σ bab + σ −2Ab K bb c σ b ab + σ ca σ b ab + Ka K − σ ab K bab σ bab + K 3 3 3 −

κ e2 4e κ2 2e κ2 a 2e κ2 2e κ2 b 2e κ2 · Θe q− (e ρ−π e + pe) − D qea − Θ (e ρ + pe) − qea Aa − π eab σ ab , 2 3 3 3 3 3

(C1) (C2)

(C3)

1 4E 4 ˙ ba Ak − Da Ebka − Ebka Aa − (ω ka − σ ka ) Eba + σ b˙ hkai K 0 = Ebhki + ΘEbk − Dk E − 3 ! 3 3 ! ! b b b 2Θ 2 Θ Θ Θ c b a b+ bk b σ ck σ ba + K b a Da K b+ b+ b+ b k − 2K b a Dk K − K D σ bkb + K ba + Dk Θ K K K 3 3 3 3 3

1b ab 1 a b ba b ba + 2K b k σ ab σ + K bk b K bab + σ b ab Dk σ bab − σ bb a Db σ b ka − σ b k Da Θ k D Ka − 2K Ahk Kai − σ ba σ 3 3 !3 b 2e κ2 ′ κ e2 2e κ2 b Θ b aωbσ b cωaσ bk c + εk ab K bb c − +εcab K π ek K− π ehki + Dk (e ρ + 3e π − 3e p) + 3 6 3 3 2e κ2 2e κ2 κ2 2e κ2 b κ2 ba − 2e bk − 5e − K qek − qe 2Kk + Kk + π eka A (e π − pe) A σ b ka π ea , 3 3 3 3 3 2 e2 e+κ bK b − Aa Aa − 2ωa ω a + σ ab σ ab − K b aK ba − E − 1 Λ ˙ − Da Aa + Θ + Θ (e ρ+π e + pe) , 0 = Θ 3 2 2 2Θ 1 ω a − σ ab ωb , 0 = ω˙ hai − εa cd Dc Ad + 2 3 2Θ 1 0 = σ˙ habi − Dha Abi + σ ab + 3 2

b b−Θ K 3

!

1b b σ bab − Aha Abi − K ha Kbi 2

1 e2 κ 1 +ωha ω bi + σ cha σ bi c + σ bcha σ bbi c + Eab + Ebab − π eab , 2 2 3 0 = Da ω a − Aa ωa , 0 = Dhc ω ki + εabhk Db σ cia + 2Ahc ω ki + Hab ,

b κ2 2 2Θ b b + Eba + 2e b a + 2εa ck Ac ω k + σ 0 = Db σ ab − Da Θ + εa ck Dc ω k − K b ab K qea , 3 3 3

(C4) (C5) (C6)

(C7) (C8) (C9) (C10)

31 2E Θ 1 1˙ σ kj 0 = E˙ hkji − Ebhkji − εabhk Da Hjib + Dhk Ebji + ΘEkj − Ebkj + Ebhk Aji − 2 2 6 3 ! b 1 b Θ 1 K− σ b˙ hkji − Ebhja ω kia + σ kia + Ehka ωjia − 3σ jia + 2εhk ab Hjia Ab − 2 2 3 !· b b 1 b Θ a˙ b hk Dji K b ji + 1 σ ba + K b hk Db σ b −Θ b − Θ Dhk K K− σ bkj + K b a Dki K bjib −b σ hj σ bkia − 2 3 3 2 hj ! ! b b b 1b 7Θ b b Θ Θ Θ a b b hk Aji − Θ σ b + Ka Dhk σ σ kj − K+ K bji + Khk Kji + K − b aσ bkia 2 3 3 6 3 6 hj ! ! b b Θ 1b Θ 1 b Θ a b− b b 1 bhja ω kic σ σ b hk ωjia + σ jia − σ bkj − K K− K bac − hk σ jib K − σ 2 3 6 3 2 2

1 κ e2 ˙ b aK ba + 3 K b hk ω jia K ba + K b hk σ ba − 1σ − σ jk K bjia Aa + σ b ahj Aki K bc a σ bhk c σ jia − π ehkji 2 2 2 3 2 2 2 2 2 2 κ e κ κ e κ e κ e κ e b ji − 4e ehk K qehk Aji − (e ρ + pe) σ jk − Θe π jk − π ehja ωkia + σ kia , − Dhk qeji + π 3 3 3 3 9 3 1 a b a a a b 0 = H˙ hkji + εabhk D Eji + εabhk D Ebji + ΘHkj − 3σ ahk Hji − ωahk Hji 2 ! b Θ 3 1 1 ab cd ab b− bjic Dd K −2εhk Ejia Ab − εhk σ jia Ebb − Ebhj ωki − εhk σ 2 2 2 3 ! b 1 b Θ 1 1 + εabhk Da σ bjib + εabhk Da σ K− bjic σ bcb + εabhk σ bjic Da σ bcb 2 3 2 2 b Θ b b − 1 εabhk K b ji Da K b b − 1 εabhk K b b Da K b ji εhk ab σ jia K 3 2 2 e2 κ e2 1 bc − 3 σ ba − κ ba c K bhja ω ki K εabhk Da π e bji − κ e2 qehj ω ki − εhk + εhk ab σ jib σ 2 2 3 3 b hk K b ji + +Θω

ab

σ jia qeb ,

(C11)

(C12)

b 1 1 Θ 2Θ 1 b Dk Θ 0 = Da Eak − Da Ebak + Dk E − 3Hka ω a + εk ab Hac σ cb + Ebk − (3ωka + σ ka ) Eba − 2 3 3 2 9 ! ! b b 1 1 b Θ 1 c a Θ 2 ab 1 a b − σ − Da σ bak − σ K− bak D K − ba D σ bck + σ b Dk σ bab − σ bkb Da σ ba b 2 3 2 3 2 3 2

b Θ 1 b ba 2b b 1 ba ba + 1 K b aDa K bk + 1 K b k Da K b a + 3 εk ad K bb b cωaσ Dk K Kk + σ kb K bd c − σ b a K σ kb − ΘΘ − K 3 2 2 2 2 9 3 e2 a κ e2 2e κ2 κ e2 b k cd K b c ωd + Θ σ ba + κ +Θε bk a K D π e ak − Dk (e ρ−π e + pe) + Θe qk − qea (3ωka + σ ka ) , (C13) 3 3 6 9 3

4E 1 1 1 bc b 1 ω c Kk ωk + 3Eka ω a − εk ab Eac σ b c + Ebak ω a + εk ab Ebac σ b c − K 0 = Da Hak + εk ab Da Ebb − 2 3 2 2 2! b b b 1 1 b + Θ ωk − 1 K b aωk b − Θ εk ab Da K b b − 1 εabk K b c Db σ b c + 2Θ K baK b b Da Θ − εk ab K bc a − εk ab σ b a c Db K 3 3 2 2 3 3 2 ! ! b b 1 Θ 1 1 1 1 b Θ b− b bK bc − σ σ bk c ω c + εk ac σ ab K εk ab σ bac σ b c K− K bab σ b ab ω k + σ bca σ bk c ω a + + 2 3 2 2 2 2 3 1 κ e2 κ e2 2e κ2 κ e2 + εk ab σ bda σ bc d σ b c − π e ka ω a + εk ab Da qeb + (e ρ + pe) ω k − εk ab π e a c σ bc . 2 3 3 3 3

We note that this system of equations is valid on both sides of the brane.

(C14)

32 Appendix D: Kinematical, gravito-electro-magnetic and matter variables for Bianchi I brane-worlds

In this Appendix we rewrite the Bianchi I brane-world, presented in Ref. [49] in terms of our variables. The brane-world solution contains an unspecified function V (y) of a Gauss normal coordinate y (which is however not related to the brane normal). The kinematical quantities appearing on and outside the brane are presented in Tables II and III, respectively, while the gravito-electro-magnetic quantities on and outside the brane are given in Tables IV and V. Tables III and V contain quantities, which were not computed in Ref. [49]. TABLE II: Brane kinematical quantities (first column) for the brane-world [49] (second column; notations and Eq. numbers are from this reference).

kinematic quantity

for the metric in Ref. [49]

Aa Θ σab ωa

0 Θ, as given by Eq. (63) σ AB , as given by Eqs. (64)-(65) 0

TABLE III: Off-brane kinematical quantities (first column) for the brane-world [49] (second column; notations are from this reference).

kinematic quantity

for the metric in Ref. [49] ′ C0 u ε VV′ + + 2 4u u 1−V 1−V 2 V′ u′ − √ 1 2 V4u + V uC0 + 1−V 2 1−V ′ C0 3u √ ε − 4u u 2

b K

√

K b Θ

1−V

ba A ba K

0 0 0 0 0

Ka La ω ba

3 P

σ b ab

i=1

σ bi eia eib , σ bi =

3u

√ε

1−V 2

(C0 + 3Ci )

TABLE IV: Brane gravito-electro-magnetic quantities (first column) for the brane-world [49] (second column; notations and Eq. numbers are from this reference).

Weyl quantity


E Eba Ebab

−κ2 U given by Eq. (53) 0 −κ2 PAB given by Eqs. (54)-(55)

The notations for the brane matter variables (after the straightforward change in the indices from lower case to upper case letters) are identical in this paper and in Ref. [49], with qa = 0. There are no off-brane matter variables.

33 TABLE V: Off-brane gravito-electro-magnetic quantities (first column) for the brane-world [49] (second column; notations are from this reference).

Weyl quantity


Ea Ha

0 0

Eab

3 P

i=1

Ei eia eib , Ei = − 6u1 2 (C0 + 3Ci ) u′ − 32 C − 2 C02 + 3Ci2 + 4u2

Fab

3 P

i=1

Fi eia eib , Fi =

1

(1−V 2 )

[(C0 + 3Ci ) u′ − C + 4Ci (C0 − Ci )]

εV

(1−V 2 )u2

Hab b ab H

C0 +3Ci 4

u′ + Ci (C0 − Ci ) −

C 4

0 0

References

[1] L. Randall and R. Sundrum, An alternative to compactification, Phys. Rev. Lett. 83, 4690 (1999), e-print: arXiv:hep-th/9906064 [2] R. Maartens, Brane-world gravity, Living Rev. Rel. 7, 7 (2004), e-print: arXiv:gr-qc/0312059 ´ Gergely: Generalized Friedmann branes, Phys. Rev. D 68 124011 (2000), e-print: arXiv:gr-qc/0308072 [3] L. A. ´ Gergely: Friedmann branes with variable tension, Phys. Rev. D 78, 084006 (2008), e-print: arXiv:0806.3857 [gr-qc]. [4] L. A. The correspondence to the notations of this paper is: Λ0 , a2 hab , L → (Λ, hab , hLi) [5] M. K. Mak and T. Harko, Can the galactic rotation curves be explained in brane world models?, Phys.Rev. D 70, 024010 (2004), e-print: arXiv:gr-qc/0404104 [6] T. Harko and K. S. Cheng, Galactic metric, dark radiation, dark pressure and gravitational lensing in brane world models, Astrophys.J. 636, 8 (2006), e-print: arXiv:gr-qc/0509576 [7] C. G. Boehmer and T. Harko, Galactic dark matter as a bulk effect on the brane, Class. Quantum Grav. 24, 3191 (2007), e-print: arXiv:0705.2496 [8] T. Harko and K. S. Cheng, The virial theorem and the dynamics of clusters of galaxies in the brane world models, Phys. Rev. D 76, 044013 (2007), e-print: arXiv:0707.1128 ´ Gergely and Z. Kov´ [9] L. A. acs: Gravitational dynamics in s+1+1 dimensions, Phys. Rev. D 72 064015 (2005), e-print: b a = −La = −K a and arXiv:gr-qc/0507020. In the setup developed there the 5d space-time is double-foliable, therefore K ωa =ω b a = 0 hold in the notations of the present paper. The conversion of notations to the notations of the present paper a a (s+1) a a b b b is: n , l , gab , gab , αa , β a , L, K, La = Ka , Lab → u , n , gab , hab , Aa , Aa , K, K, − La , Kab . ´ [10] L. A. Gergely and Z. Kov´ acs: Gravitational dynamics in s+1+1 dimensions II. Hamiltonian theory, Phys. Rev. D 77 024003 (2008), e-print: arXiv:0709.2131 [11] R. Maartens: Cosmological dynamics on the brane, Phys. Rev. D 62 084023 (2000), e-print: arXiv:hep-th/0004166. The correspondence to the notations of this paper is: −e κ4 /κ4 U, e κ4 /κ4 Qa , − κ e4 /κ4 Pab , − ω a → E , Eba , Ebab , ω a [12] J. Ehlers: Beitr¨ age zur relativistischen Mechanik kontinuierlicher Medien, Abh. Mainz Akad. Wiss. u. Litt. (Math. Nat. kl) 11 792 (1961) [13] G. F. R. Ellis, in General Relativity and Cosmology, Ed. Sachs R K, New York, Academic (1971) [14] G. F. R. Ellis and H. van Elst: Cosmological models in Theoretical and Observational Cosmology, in NATO Adv. Study Inst. Ser. C. Math. Phys. Sci. 541 1-116, Ed. Lachiéze-Rey M, Kluwer, Dordrecht (1999), e-print: arXiv:gr-qc/9812046 [15] G. F. R. Ellis: Cosmological models, in Modern Cosmology edited by Bonometto S, Gorini V and Moschella U (IOP publishing Ltd, London) (2002) [16] C. A. Clarkson and R. K. Barrett: Covariant Perturbations of Schwarzschild Black Holes, Class. Quant. Grav. 20 3855-3884 (2003), e-print: arXiv:gr-qc/0209051v3 [17] C. Clarkson: A covariant approach for perturbations of rotationally symmetric spacetimes, Phys. Rev. D 76, 104034 (2007), e-print: arXiv:0708.1398v1 [gr-qc] [18] A. M. Nzioki, S. Carloni, R. Goswami, and P. K. S. Dunsby: A new framework for studying spherically symmetric static solutions in f(R) gravity, e-print: arXiv:0908.3333 [gr-qc]

34 [19] S. W. Hawking: Perturbations of an Expanding Universe, Astrophys. J. 145 544 (1966) [20] G. F. R. Ellis and M. Bruni: Covariant and gauge-invariant approach to cosmological density fluctuations, Phys. Rev. D 40 1804 (1989) [21] G. F. R. Ellis, J. Hwang, and M. Bruni: Covariant and gauge-independent perfect-fluid Robertson-Walker perturbations, Phys. Rev. D 40 1819 (1989) [22] G. F. R. Ellis, M. Bruni, and J. Hwang: Density-gradient-vorticity relation in perfect-fluid Robertson-Walker perturbations, Phys. Rev. D 40 1819 (1989) [23] P. Dunsby, M. Bruni and G. F. R. Ellis, Cosmological perturbations and the physical meaning of gauge-invariant variables, Astrophys. J. 395 34 (1992) [24] A. Challinor and A. Lasenby, Cosmic microwave background anisotropies in the Cold Dark Matter model: A covariant and gauge-invariant approach, Astrophys. J. 513 1 (1999) [25] R. Maartens, T. Gebbie, and G. F. R. Ellis: Cosmic microwave background anisotropies: Nonlinear dynamics, Phys. Rev. D 59 083506 (1999), e-print: arXiv:astro-ph/9808163 [26] T. Gebbie and G. F. R. Ellis: 1+3 Covariant Cosmic Microwave Background anisotropies I : Algebraic relations for mode and multipole representations, Annals Phys. 282 285 (2000), e-print: arXiv:astro-ph/9804316 [27] T. Gebbie, P. Dunsby, and G. F. R. Ellis, 1+3 Covariant Cosmic Microwave Background anisotropies II: The almost Friedmann Lemaitre model, Annals Phys. 282 321 (2000) [28] S. Mukohyama: Gauge-invariant gravitational perturbations of maximally symmetric spacetimes, Phys.Rev. D 62 084015 (2000), arXiv:hep-th/0004067 [29] S. Mukohyama: Perturbation of junction condition and doubly gauge-invariant variables, Class. Quantum Grav. 17 4777 (2000), e-print: arXiv:hep-th/0006146 [30] K. Koyama and J. Soda, Evolution of Cosmological Perturbations in the Brane World, Phys. Rev. D 62 123502 (2000) [31] C. Gordon and R. Maartens, Density perturbations in the brane-world, Phys. Rev. D 63 044022 (2001) [32] S. Mukohyama: Integro-differential equation for brane-world cosmological perturbations, Phys. Rev. D 64 064006 (2001); Erratum-ibid. D 66 049902 (2002), arXiv:hep-th/0104185 [33] B. Leong, P. Dunsby, A. Challinor, and A. Lasenby: (1+3) Covariant Dynamics of Scalar Perturbations in Braneworlds, Phys. Rev. D 65 104012 (2002), e-print: arXiv:gr-qc/0111033 [34] B. Gumjudpai, R. Maartens, and C. Gordon: Density perturbations in a brane-world universe with dark radiation, Class. Quantum Grav. 20 3295 (2003), e-print: arXiv:gr-qc/0304067 [35] P. K. S. Dunsby, N. Goheer, M. Bruni, and A. Coley: Are braneworlds born isotropic?, Phys. Rev. D 69 101303 (2004), e-print: arXiv:hep-th/0312174 [36] N. Goheer, P. K. S. Dunsby, A. Coley, and M. Bruni: Large-scale perturbations on the brane and the isotropy of the cosmological singularity, Phys. Rev. D 70 123517 (2004), e-print: arXiv:hep-th/0408092 [37] K. Koyama, D. Langlois, R. Maartens, and D. Wands: Scalar perturbations from brane-world inflation, JCAP 0411 002 (2004), e-print: arXiv:hep-th/0408222 [38] C. Deffayet: A note on the wellposedness of scalar brane world cosmological perturbations, Phys. Rev. D 71 023520 (2005), e-print: arXiv:hep-th/0409302 [39] S. Pal, Structure formation on the brane: a mimicry, Phys. Rev. D 74, 024005 (2006), e-print: arXiv:gr-qc/0606085 [40] S. Pal, Brane cosmology, Weyl fluid, and density perturbations, Phys. Rev. D 78, 043517 (2008), e-print: arXiv:0806.2505 [gr-qc] [41] A. Coley: Classification of the Weyl Tensor in Higher Dimensions and Applications, Class.Quant.Grav. 25 033001 (2008), e-print: arXiv:0710.1598 [42] J. M. M. Senovilla: General Electric-Magnetic decomposition of fields, positivity and Rainich-like conditions, in the Proceedings of the Spanish Relativity Meeting (2000), e-print: arXiv:gr-qc/0010095 [43] G. Dvali, G. Gabadadze, M. Porrati: 4D Gravity on a Brane in 5D Minkowski Space, Phys. Lett. B 485, 208 (2000) e-print: arXiv:hep-th/0005016 C. Deffayet: Cosmology on a Brane in Minkowski Bulk, Phys. Lett. B 502, 199 (2001) e-print: arXiv:hep-th/0010186 C. Deffayet: On Brane World Cosmological Perturbations, Phys. Rev. D 66, 103504 (2002) e-print: arXiv:hep-th/0205084 K. Maeda, S. Mizuno, T. Torii: Effective Gravitational Equations on Brane World with Induced Gravity, Phys. Rev. D 68, 024033 (2003) e-print: arXiv:gr-qc/0303039 V. Sahni, Y. Shtanov: Braneworld models of dark energy, JCAP 0311, 014 (2003) e-print: arXiv:astro-ph/0202346 ´ Gergely, R. Maartens: Asymmetric brane-worlds with induced gravity, Phys. Rev. D 71, 024032 (2005) e-print: L. A. arXiv:gr-qc/0411097 K. Koyama: Are there ghosts in the self-accelerating brane universe?, Phys. Rev. D 72, 123511 (2005) e-print: arXiv:hep-th/0503191 D. Gorbunov, K. Koyama, S. Sibiryakov: More on ghosts in DGP model, Phys. Rev. D 73, 044016 (2006) e-print: arXiv:hep-th/0512097 C. Charmousis, R. Gregory, N. Kaloper, A. Padilla: DGP Specteroscopy, JHEP 0610, 066 (2006) e-print: arXiv:hep-th/0604086 K. Izumi, K. Koyama, T. Tanaka: Unexorcized ghost in DGP brane world, JHEP 0704, 053 (2007) e-print: arXiv:hep-th/0610282 [44] J. D. Barrow, R. Maartens, and C. G. Tsagas: Cosmology with inhomogeneous magnetic fields, Phys. Rept. 449 131 (2007) [45] C. G. Tsagas, A. Challinor, and R. Maartens: Relativistic cosmology and large-scale structure, Phys. Rept. 465 61 (2008),

35 e-print: arXiv:0705.4397 [46] W. Israel: Singular Hypersurfaces and Thin Shells in General Relativity, Nuovo Cimento B XLIV B, 4349 (1966). errata :ibid. XLVIII B, 2583 (1966) [47] C. Lanczos: Ein vereinfachendes Koordinatensystem f¨ ur Einsteinschen Gravitationsgleichungen, Phys. Zeils., 23, 539 (1922); Flachenhafte Verteilung der Materie in der EinsteinEinsteinschen Gravitationstheorie, Ann. Phys. (Lepzig) 74, 518 (1924) [48] T. Shiromizu, K. Maeda, and M. Sasaki: The Einstein Equations on the 3-Brane World, Phys. Rev. D 62 024012 (2000), e-print: arXiv:gr-qc/9910076 [49] A. Campos, R. Maartens, D. Matravers, C. F. Sopuerta: Braneworld cosmological models with anisotropy, Phys. Rev. D 68, 103520 (2003), e-print: arXiv:hep-th/0308158 [50] A. Fabbri, D. Langlois, D. A. Steer, R. Zegers: Brane cosmology with an anisotropic bulk, JHEP 0409, 025 (2004), e-Print: hep-th/0407262 [51] H. van Elst and G. F. R. Ellis: The Covariant Approach to LRS Perfect Fluid Space-time Geometries, Class. Quantum Grav. 13, 1099 (1996), e-print: arXiv:gr-qc/9510044 [52] N. Dadhich, R. Maartens, P. Papadopoulos, V. Rezania: Black holes on the brane, Phys. Lett. B 487 1 (2000), e-print: arXiv:hep-th/0003061 [53] A. N. Aliev and A. E. G¨ umr¨ uk¸cu ¨˜ glu: Charged Rotating Black Holes on a 3-Brane, Phys. Rev. D 71 104027 (2005), e-print: arXiv:hep-th/0502223 [54] M. Shibata and T. Nakamura: Evolution of three-dimensional gravitational waves: Harmonic slicing case, Phys. Rev. D, 52, 5428 (1995) T. W. Baumgarte and S. L. Shapiro: Numerical integration of Einstein’s field equations, Phys. Rev. D 59, 024007 (1998)