N ю 1 formalism in Einstein-Gauss-Bonnet gravity

N þ 1 formalism in Einstein-Gauss-Bonnet gravity

Takashi Torii^1,*and Hisa-aki Shinkai^2,+

1Department of General Education, Osaka Institute of Technology, Omiya, Asahi-ku, Osaka 535-8585, Japan

2Department of Information Systems, Osaka Institute of Technology, Kitayama, Hirakata, Osaka 573-0196, Japan (Received 16 September 2008; published 28 October 2008)

Towards the investigation of the full dynamics in a higher-dimensional and/or a stringy gravitational model, we present the basic equations of the Einstein-Gauss-Bonnet gravity theory. We show theðNþ 1Þ-dimensional version of the Arnowitt-Deser-Misner decomposition including Gauss-Bonnet terms, which shall be the standard approach to treat the space-time as a Cauchy problem. Because of the quasilinear property of the Gauss-Bonnet gravity, we find that the evolution equations can be in a treatable form in numerics. We also show the conformally transformed constraint equations for constructing the initial data. We discuss how the constraints can be simplified by tuning the powers of conformal factors.

Our equations can be used both for timelike and spacelike foliations.

DOI:10.1103/PhysRevD.78.084037 PACS numbers: 04.50.h, 04.20.Ex, 04.25.D, 11.25.Wx

I. INTRODUCTION

General relativity (GR) has been tested with many ex- periments and observations in both the strong and the weak gravitational field regimes (see e.g. [1]), and none of them are contradictory to GR. However, the theory also predicts the appearance of the space-time singularities under natu- ral conditions [2,3], which also indicates that GR is still incomplete as a physics theory that describes the whole of the gravity and the space-time structure.

We expect that the true fundamental theory will resolve these theoretical problems. Up to now, several quantum theories of gravity have been proposed. Among them, the superstring/M theory, formulated in higher dimensional space-time, is the most promising candidate. In these pros- pects, a considerable number of studies concerned with gravitational phenomena and cosmology have been made in the string theoretical framework beyond GR.

Since the present knowledge is still far from understand- ing the full aspects of the string theory, several kinds of approaches, which are fundamentally approximations, are usually taken. Among them, the perturbative approach plays important roles. There are two particular parameters which characterize the system in the superstring theory.

One is string coupling parameterg²_s ¼e, whereis the dilaton field. The other is the inverse string tension ⁰. When the tension is strong (i.e., small⁰) compared to the energy scale of the system, it is difficult to excite strings, and the size of the strings becomes small enough to be regarded as particles in the zeroth order approximation. In this limit GR (with other light fields) is recovered. This is called⁰expansion [4].

In the higher order terms of ⁰, curvature corrections appear. The Gauss-Bonnet (GB) term is the next leading order of the ⁰ expansion of type IIB superstring theory

[4,5] and has nice properties such that it is ghost-free combinations [6] and does not give higher derivative equa- tions but an ordinary set of equations with up to the second derivative in spite of the higher curvature combinations.

The models with the GB term and/or other higher cur- vature terms have been intensively studied in high energy physics. One of them is a series of studies in string cos- mology. The pre-big-bang scenario [7] is a fantastic sce- nario which tries to avoid the big-bang singularity by making use of T duality [8] (or scale factor duality).

Furthermore, the pre-big-bang scenario gives a natural inflation mechanism, since the solution in the pre-big- bang phase is inflating from the beginning at least in the string frame. Although these analyses show that the singu- larity problem has not been resolved yet completely, there are some cosmological solutions which do not start from an initial singularity [9–11].

In addition to cosmology, the string effects can be also seen in the study of black hole physics. As the size of a black hole becomes small, and the curvature around the black hole becomes large, it is expected that the curvature corrections cannot be negligible. The singularity inside of the event horizon would be also modified or even disappear by string effects. For these reasons, the static or stationary black hole solutions in effective string theories were inves- tigated in the systems both without higher curvature terms [12,13] and with such terms [14–16]. Besides the static or stationary solutions, there are some dynamical solutions which are motivated by gravitational collapse [17].

These analyses are performed on the assumption of highly symmetric space-time because the system is much more complicated than that in GR. To obtain a deeper understanding of the early stage of the Universe, singular- ity, and/or black holes, we should consider a less symmet- ric and/or dynamical space-time; the analyses require the direct numerical integration of the equations. None of the fully dynamical simulations in GB gravity has been performed.

*torii@ge.oit.ac.jp

+shinkai@is.oit.ac.jp

In this article, we present the basic equations of the Einstein-GB gravity theory. We show the ðNþ 1Þ-dimensional version of the Arnowitt-Deser-Misner (ADM) decomposition, which is the standard approach to treat the space-time as a Cauchy problem. The topic was first discussed by Choquet-Bruhat [18], but the full set of equations and the methodology have not yet been pre- sented. In four-dimensional GR, numerical simulations of binary compact objects are available in the past years, and many groups apply the modified ADM equations in order to obtain long-term stable and accurate simulations.

However, such modifications depend on the problem to consider, and the ‘‘robustest’’ formulation is not yet known (see e.g. [19]). Therefore, as the first step, we in this paper just present the fundamental space-time decomposition of the GB equations, focusing on the GB term.

The ADM decomposition is supposed to construct the space-time with foliations of the constant-time hypersur- faces. This method can be also applied to study the brane- world model [20], which states the visible space-time is embedded in higher dimensional ‘‘bulk’’ space-time. As was first investigated by Chamblinet al.[21], it is possible to study the bulk structure by switching the normal vector of the hypersurface from timelike to spacelike. We there- fore present all sets of equations for both cases for future convenience.

The outline of this paper is as follows. In Sec. II, we show that the set of equations is divided into two con- straints and evolution equations according to the standard procedure. In Sec. III, we present the conformal approach to solve the constraints which shall be used for preparing the initial data. In Sec. IV, we show the dynamical equa- tions in GR and in GB theory separately. Section V is devoted to the discussions and summary. We think these expressions are useful for future dynamical investigations.

II.ðNþ1ÞDECOMPOSITION IN EINSTEIN- GAUSS-BONNET GRAVITY

A. Model and basic equations

We start from the Einstein-Gauss-Bonnet action inðNþ 1Þ-dimensional space-timeðM; gÞwhich is described as [22]

S¼Z

Md^Nþ1Xpffiffiffiffiffiffiffig 1

2²ðR2þ_GBLGBÞ þLmatter

; (1)

with

LGB ¼R²4RRþRR; (2) where²is theðNþ1Þ-dimensional gravitational constant and R, R, R, and L_matter are the ðNþ 1Þ-dimensional scalar curvature, Ricci tensor, Riemann curvature, and matter Lagrangian, respectively. This action

reproduces the standard ðNþ1Þ-dimensional Einstein gravity, if we set the coupling constant _GBð0Þ equal to zero.

The action (1) gives the gravitational equation as Gþ_GBH¼²T; (3) where

G¼R¹₂gRþg; (4) H¼2½RR2RR2RR

þR R ¹₂gLGB; (5) T ¼ 2L_matter

g þgLmatter: (6)

B. Projections to hypersurface

In order to investigate the space-time structure as the foliations of the N-dimensional (spacelike or timelike) hypersurface , we introduce the projection operator to as

?¼g"nn; (7) where n is the unit-normal vector towithnn ¼", with which we definenas timelike (if"¼ 1) or space- like (if"¼1). Therefore,is spacelike (timelike) ifnis timelike (spacelike).

The projections of the gravitational equation (3) give the following three equations:

ðGþ_GBHÞnn¼²Tnn ¼²; (8) ðGþ_GBHÞn?¼²Tn?¼ ²J;

(9) ðGþ_GBHÞ??¼²T??

¼²S; (10) where we defined the components of the energy- momentum tensor as

T ¼nnþJnþJnþS; (11) and we also defineT ¼"þSfor later convenience.

Projection of the ðNþ1Þ-dimensional Riemann tensor onto theN-dimensional hypersurface can be written as

R ??? ?

¼R"ðKKKKÞ; (12) R ??? n¼ D½K; (13) R ?? nn¼LnKþKK; (14)

whereR is the Riemann tensor of the induced metric

ð¼ ?Þ,D is the covariant differentiation with re- spect to , Ln denotes the Lie derivative in the n direction, andKis the extrinsic curvature defined as

K¼ ¹₂Ln ¼ ??rn: (15)

Equation (12) is called the Gauss equation, and the con- traction of (13) ofandgives the Codacci equation.

Using these projections, the ðNþ1Þ-dimensional Riemann curvature and its contractions (the Ricci tensor and scalar curvature) are described by theN-dimensional variables on the hypersurfaceas

R¼R"ðKKKKnDKþnDKþnDKnDKnDK

þnDKþnDKnDKÞ þnnKKnnKKnnKKþnnKK þnnLnKnnLnKnnLnKþnnLnK; (16) R¼R"½KK2KKþnðDKDKÞ þnðDKDKÞ þnnKKþ"LnK

þnn LnK; (17)

R ¼R"ðK²3KK2 LnKÞ; (18) whereK¼K.

Substituting these relations into the field equation (3) or (8)–(10), we find the equations are decomposed as (a) the Hamiltonian constraint equation

Mþ_GBðM²4M_abM^abþM_abcdM^abcdÞ

¼ 2"²_Hþ2; (19) (b) the momentum constraint equation

N_iþ2_GBðMN_i2M_i^aN_aþ2M^abN_iabM_i^cabN_abcÞ

¼²J_i; (20)

and (c) the evolution equations for _ij

M_ij¹₂M _ij"ðK_iaK^a_jþ _ijK_abK^abLnK_ij þ _ij ^abLnK_abÞ þ2_GB½H_ijþ"ðMLnK_ij 2M_i^aLnK_aj2M_j^aLnK_aiW_ij^abLnK_abÞ

¼²S_{ij ij}; (21)

respectively, where

M_ijkl¼R_ijkl"ðK_ikK_jlK_ilK_jkÞ; (22)

M_ij ¼ ^abM_iajb¼R_ij"ðKK_ijK_iaK^a_jÞ; (23)

M¼ ^abM_ab ¼R"ðK²K_abK^abÞ; (24)

N_ijk¼D_iK_jkD_jK_ik; (25)

N_i¼ ^abN_aib¼D_aK_i^aD_iK; (26)

H_ij¼MM_ij2ðM_iaM^a_jþM^abM_iajbÞ þM_iabcM_j^abc2"½K_abK^abM_ij¹₂MK_iaK_j^aþK_iaK^a_bM^b_jþK_jaK^a_bM^b_i þK^acK_c^bM_iajbþN_iN_jN^aðN_aijþN_ajiÞ ¹₂N_abiN^ab_jN_iabN_j^{ab 1}_{4 ij}½M²4M_abM^abþM_abcdM^abcd " _ij½K_abK^abM2M_abK^acK_c^b2N_aN^aþN_abcN^abc; (27)

W_ij^kl¼M _ij ^kl2M_ij ^kl2 ijM^klþ2M_iajb ^{ak bl}: (28) We remark that the terms ofLnK_ijappear only in the linear

form in (21). This is due to the quasilinear property of the GB gravity.

C. Cauchy approach

Using the Bianchi identity, Choquet-Bruhat [18] showed that the set of equations forms the first-class system the

same as in GR; that is, a spacelike hypersurface which satisfies the constraints (19) and (20) will also satisfy the constraints after the evolution using (21).

The most standard procedures for following the dynam- ics of space-time consist of three steps: (i) solve the con- straint equations (19) and (20) for ð _ij; K_ij; _H; J_iÞ on ðt¼0Þ and prepare them as the initial data, (ii) evolve ð _ij; K_ij; _H; J_iÞusing (21) and the matter equations, and

(iii) monitor the accuracy of the evolutions by checking constraint equations on the evolvedðtÞ.

In the case of seeking the ‘‘dynamics’’ along a spacelike direction such as a study of the bulk structure in the brane-world model, the above strategy can be switched to the evolution incoordinates instead oft(using the set of equations of "¼ þ1). The initial data, in this case, is a timelike hypersurface which should satisfy the constraints.

Such initial data can be obtained either by solving the dynamics of the ‘‘brane’’ part or by taking double Wick rotation after the above step (i), depending on the models and motivations.

In the following sections, we describe a way of solving constraints (Sec. III) and a way of solving evolution equa- tions (Sec. IV).

III. CONFORMAL APPROACH TO SOLVE THE CONSTRAINTS

A. ‘‘Conformal approach’’

In order to prepare the initial data for dynamical evolu- tion, we have to solve two constraints: (19) and (20). The

standard approach is to apply a conformal transformation on the initial hypersurface [24]. The idea is to introduce a conformal factorc between the initial trial metric ^ijand the solution _ij, as

ij¼c^2m^ij; ^ij¼c^2m^^ij; (29) where m is a constant, and solve for c so the solution satisfies the constraints.

ForN-dimensional space-time, the Ricci scalar is trans- formed as

R¼c^{2mfR^}2ðN1Þmc^{1ðD^aD^}ac^Þ

þ ðN1Þ½2 ðN2Þmmc^{2ðD^}c^{Þ2g; (30)}

R_ij¼R^_ijm^ijc^{1D^}aD^^ac ^ðN2Þmc^{1D^}iD^_jc þ ðN2Þmðmþ1Þc^{2D^}ic^{D^}jc

m½ðN2Þm1c^{2ðD^}c^Þ2^ij; (31)

R_ijkl ¼c^{2mfR^}ijklþmc¹^il½D^_jD^_kc ^ðmþ1Þc^{1D^}jc^{D^}kc ^mc¹^ik½D^_jD^_lc ^ðmþ1Þc^{1D^}jc^{D^}lc þmc¹^jk½D^_iD^_lc ^ðmþ1Þc^{1D^}ic^{D^}lc ^mc¹^jl½D^_iD^_kc ^ðmþ1Þc^{1D^}ic^{D^}kc

þm²c^{2ðD^}c^Þ2ð^il^jk ^ik^jlÞg: (32)

Regarding the extrinsic curvature, we decompose K_ij into its trace partK¼ ^ijK_ijand the traceless partA_ij ¼ K_ijN¹ _ijK and assume the conformal transformation [25] as

A_ij¼c^{‘A^}ij; A^ij¼ c^{‘4mA^ij; (33)} K¼c^{K;^ (34)} where‘andare constants. For the matter terms, we also assume the relations¼c^p^ andJⁱ¼c^{qJ^i, wherep} and q are constants, while we regard the cosmological constant as common to both flames,¼ ^.

Up to here, the powers of conformal transformation‘, m, , p, and q are not yet specified. Note that, in the standard three-dimensional initial-data construction cases, the combination of m¼2, ‘¼ 2, ¼0, p¼5, and q¼10 is preferred since this simplifies the equations.

We also remark that if we choose ¼‘2m, then the extrinsic curvature can be transformed asK_ij¼c^{‘K^}ijand K^ij¼c^{‘4mK^ij.}

B. Hamiltonian constraint

Using these equations, the Hamiltonian constraint equa- tion (19) turns to be

2ðN1ÞmD^_aD^^ac ^ðN1Þ½2 ðN2ÞmmðD^c^Þ2c¹

¼R^c^N1

N "c^{2mþ2þ1K^2þ"}c^{2mþ2‘þ1A^}abA^^ab þ2"²^c^p2 ^þ_GB^c^{2mþ1: (35)} We will show the explicit form of the GB part ^¼M² 4M_abM^abþM_abcdM^abcdin Appendix A. At this moment, we observe that (35) can be simplified in the following two ways.

(A) If we specify ¼‘2m and m¼2=ðN2Þ, then (35) becomes

4ðN1Þ

N2 D^_aD^^ac ^{¼R^}c ^"c^{2‘þ14=ðN2Þ} ðK^²K^_abK^^abÞ þ2"²^c^p2 ^

þ_GB^c^{1þ4=ðN2Þ: (36)} In the case of the Einstein gravity (_GB ¼0) with ¼0, the combination ‘¼2=ðN2Þ and p¼ 1makes the right-hand side of (36) linear. If we choose‘¼ 2, which will make the momentum constraint simpler as we see later, (36) also remains as a simple equation.

(B) If we specify¼0andm¼2=ðN2Þ, then (35) becomes

4ðN1Þ

N2 D^_aD^^ac ^{¼R^}c ^"N1

N c^{1þ4=ðN2ÞK^2} þ"c^{2‘þ14=ðN2ÞA^}abA^^ab þ2"²^c^p2 ^

þ_GB^c^{1þ4=ðN2Þ: (37)}

C. Momentum constraint

In order to express the momentum constraint equation in a tractable form, additionally to the variables in Sec. III A, we introduce the transverse traceless part and the longitu- dinal part ofA^^ijas

D^_jA^^ij_TT ¼0; (38) A^^ij_L ¼A^^ijA^^ij_TT; (39) respectively. Since the conformal transformation of the divergenceD_jA_i^jbecomes

D_jA_i^j¼ c^{‘2mfD^}jA^_i^jþc^1½‘þmðN2ÞA^_i^jD^_jc^g;

(40) the momentum constraint (20) can be written as

c^{‘2mD^}aA^_iL^aþ ½‘þ ðN2Þmc^{‘2m1A^}iLaD^_ac N1

N D^_iðc^{KÞ þ^} 2_GB^i¼²c^{2mqJ^}i: (41) We will show the explicit form of the GB part ^i in Appendix B; meanwhile, we proceed with the standard procedure, that is, introducing a vector potential Wⁱ for A^^ij_L as

A^^ij_L ¼D^ⁱW^jþD^^jWⁱ 2

N ^^ijD^_kW^k: (42) Since the divergence ofA^^ij_L becomes

D^_jA^^ij_L ¼D^_aD^^aWⁱþN2

N D^ⁱD^_kW^kþR^ⁱ_kW^k; (43) the momentum constraint (41) becomes

D^_aD^^aW_iþN2

N D^_iD^_kW^kþR^_ikW^k þc^{1½‘þ ðN}2Þm

D^^aW^bþD^^bW^a 2

N ^^abD^_kW^k

^biD^_ac c^2m‘N1

N D^_iðc^{KÞ þ^} c^2m‘2_GB^i

¼²c^{4m‘qJ^}i: (44)

Similarly to the case of the Hamiltonian constraint equa- tion, we consider the following two cases.

(A) If we specify ¼‘2m and m¼2=ðN2Þ, then (44) becomes

D^_aD^^aW_iþN2

N D^_iD^_kW^kþR^_ikW^k þc^1ð‘þ2Þ

D^^aW^bþD^^bW^a 2

N ^^abD^_kW^k ^biD^_ac ^N1

N

‘ 4 N2

ðD^_ic^{ÞK^þD^}iK^

þc^{‘þ4=ðN2Þ}2_GB^i

¼²c^{8=ðN2Þ‘qJ^}i: (45) In the case of the Einstein gravity (_GB¼0), the choice of‘¼ 2cancels the mixing term between c ^{and Wi}. Further, when _GB ¼0, we have a chance to make two constraint equations, (36) and (45), decouple by assuming K^ ¼0 and q¼ 8=ðN2Þ þ2. However, when _GB0, this de- coupling feature is no longer available, since the term ^i includes c-related terms as we see in Eq. (B7) in Appendix B.

(B) If we specify¼0andm¼2=ðN2Þ, then (44) becomes

D^_aD^^aW_iþN2

N D^_iD^_kW^kþR^_ikW^kþc^1ð‘þ2Þ

D^^aW^bþD^^bW^a2

N ^^abD^_kW^k

^biD^_ac c^{4=ðN2Þ‘N1}

N D^_iK^ þc^4=ðN2Þ‘2_GB^i

¼²c^{8=ðN2Þ‘qJ^}i: (46) For the Einstein gravity (_GB¼0), the choice of

‘¼ 2again cancels the mixing term between c andWⁱ. The decoupling feature between (37) and (46) is available when _GB¼0, K^ ¼const, and q¼8=ðN2Þ þ2. However, when _GB0, this decoupling feature is no longer available, since the term ^iincludes c-related terms as we see in Eq. (B13) in Appendix B.

D. Procedures

For the readers’ convenience, we summarize the above procedure briefly. The initial data ð _ij; K_ij; ; JⁱÞ can be constructed by solving the Hamiltonian constraint (19) and the momentum constraint equations (20). This can be done by the following steps.

(1) Give the initial assumption (trial values) for ^ij,K, A^^TT_ij , and,^ J.^

(2) Solve (35) and (44) for c ^{and Wi} by fixing the exponents‘,m,,p, andq.

(3) By the following inverse conformal transforma- tions:

ij¼ c^2m^ij; (47) K_ij¼ c^{‘A^TT}ij þD^_iW_jþD^_jW_i 2

N ^ijD^_kW^k þ1

Nc^2mþ^ijK;^ (48) ¼ c^p;^ (49) Jⁱ¼c^{qJ^i; (50)} we obtain the solutions _ij, K_ij, , and Jⁱ, which satisfy the constraints.

E. Momentarily static case

The easiest construction of the initial data may be under the assumption of the momentarily static (or time- symmetric) situation K_ij¼J_i¼0. In such a case, the momentum constraint becomes trivial, and the Hamiltonian constraint (35) can be reduced as

2ðN1ÞmD^_aD^^ac ^ðN1Þ½2 ðN2ÞmmðD^c^Þ2c¹

¼R^c^þ2"²^c^p2 ^þ_GB^c^{2mþ1; (51)} where

^¼ ðN3Þmc^4mf4ðN2Þmc^{2½ðD^}aD^^ac^Þ2 ðD^_aD^_bc^{ÞðD^aD^b}c^Þ 4c^{1½R^ ðN}2Þ ½ðN3Þm2mc^{2ðD^}c^{Þ2D^}aD^^ac

þ8c^{1½R^abþ ðN}2Þmðmþ1Þc^{2D^a}c^{D^b}c D^_aD^_bc ^{þ ðN}1Þ₂m²½ðN4Þm4c^{4ðD^}c^Þ4 2c^2½ðN4Þm2ðD^c^{Þ2R^}

8ðmþ1Þc^{2R^abD^}ac^{D^}bc^{g þ}c^{4mR^}GB; (52) where

ðDnÞ_m ¼ ðDnÞ! ðDm1Þ!

¼ ðDnÞðDn1Þ. . .ðDmÞ; (53) andR^_GB ¼R^²4 ^R_abR^^abþR^_abcdR^^abcd.

WhenN ¼3, ^simply becomes ^¼c^{4mR^}GBso that (51) will be reduced as

8 ^D_aD^^ac ^{¼R^}c ^þ2"²^c^p2 ^þ_GBR^_GBc^3; (54) for the choice ofm¼2.

However, in generalN, it is hard to find an appropriatem which simplifies the equation, even if ^ijis taken to be the flat space-time that reduces ^as

^¼ ðN3Þm²c^4m2f4ðN2Þ½ðD^_aD^^ac^Þ2 ðD^_aD^_bc^{ÞðD^aD^b}c^{Þ þ}4ðN2Þ

½ðN3Þm2c^{1ðD^}c^{Þ2D^}aD^^ac ^þ8ðN2Þ ðmþ1Þc^{1ðD^a}c^{ÞðD^b}c^{ÞD^}aD^_bc ^{þ ðN}1Þ₂m ½ðN4Þm4c^{2ðD^}c^{Þ4g: (55)} Roughly speaking, Eq. (55) is a quadratic equation with respect to the second-order derivative of c, which means that there are two roots in general when a set of trial values

^ijand^ on the hypersurface is given. The meaning of the existence of two solutions is more clearly understood by assuming p¼0andq¼0in the conformal transforma- tion of the matter field. In this case, two different _ij’s are obtained through the different conformal transformations even for the same matter field distributions.

The nonuniqueness of the solution is due to the higher curvature combination of the GB terms. The existence of two solutions can also be seen for the black hole solutions in the Einstein-GB theory [14,16]. In the black hole case, the gravitational equations reduce to a quadratic equation under the suitable choice of a metric function. The two roots of the equation correspond to the different black hole solutions. In the _GB !0 limit, one of the black hole solutions approaches to the solutions in GR, while the metric component of the other solution diverges and there is no counterpart in GR. Hence the former is classified into the GR branch and the latter into the GB branch. In some mass parameter regions of the black hole, the metric com- ponents of these branches coincide at the certain radius. It is a multiple root of the field equation. Solving the field equations beyond this radius, we find that the metric be- comes imaginary, and this region is physically irrelevant.

The space-time becomes singular at this radius, and it is called a branch singularity, which is a new ingredient of GB gravity.

When we solve the Hamiltonian constraint equation (55), a similar situation may occur. The conformal factor c becomes imaginary, and the solution _ijis also imaginary in some regions on the hypersurface. The boundary of this region corresponds to the singularity. The situation is the same also in the non-time-symmetric case.

IV. DYNAMICAL EQUATIONS A. Dynamical equations in general relativity Equations in GR are obtained by putting_GB¼0. The evolution equations ofðNþ1Þ-dimensional ADM formu- lation are presented in Ref. [26], but here we generalize them to the set of equations for both spacelike and timelike foliations.

First, we rewrite the dynamical equations (21) by intro- ducing the metric components as

ds² ¼"²ðdx⁰Þ²þ _ijðdxⁱþⁱdx⁰Þðdx^jþ^jdx⁰Þ

¼ ð"²þ_a^aÞðdx⁰Þ²þ2_idxⁱdx⁰þ _ijdxⁱdx^j; (56) whereandⁱ are the lapse and shift functions, respec- tively. The components of the normal vector, then, are

n ¼ ð";0;. . .;0Þ; n¼ 1

ð1;ⁱÞ: (57) In the matrix from, the ðNþ1Þ and N metrics are ex- pressed as

g¼ "²þ_a^a _j

_{i ij}

!

; (58)

g¼ "

²

1 ⁱ

^j "^{2 ij}þ^ij

; (59) and

¼ _a^a _j

_{i ij}

; ¼ 0 0

0 ^ij

: (60) The trace of Eq. (21) minus half of Eq. (19) gives

abLnK_ab¼ "

2MK_abK^ab "²

N1T þ": (61) This is equivalent to the trace of Einstein equation (3) with the projection (14). Substituting Eq. (61) into Eq. (21), we find

LnK_ij¼ "M_ijK_iaK^a_jþ"²

S_ij 1

N1T ij

þ 2"

N1 ^ij: (62)

The extrinsic curvature and its Lie derivative are ex- pressed as

K_ij¼ 1

2ð@_{0 ij}þD_jiþD_ijÞ; (63)

LnK_ij¼ 1

ð@₀K_ijþD_iD_j^aD_aK_ijK_ajD_i^a K_aiD_j^aÞ; (64) respectively.

With the metric components (56), Eqs. (62) and (63) are

@_{0 ij}¼ 2K_ijþD_jiþD_ij; (65)

@₀K_ij¼ "M_ijK_iaK^a_jD_iD_jþ^aðD_aK_ijÞ þ ðD_j^aÞK_iaþ ðD_i^aÞK_aj

"²

S_ij 1 N1T ij

þ 2"

N1 ^ij: (66) If we have matter, we need to evolve them together with metric. The dynamical equations for matter terms can be derived from the conservation equationrT¼0.

B. Dynamical equations in Gauss-Bonnet gravity With the Gauss-Bonnet terms, the evolution equa- tion (21) cannot be expressed explicitly for each LnK_ij. That is, (21) is rewritten as

ð1þ2_GBMÞLnK_ij ð _ij ^abþ2_GBW_ij^abÞLnK_ab 8_GBM_ði^aLnKjajjÞ

¼ "ðM_ij¹₂M _ijÞ K_iaK^a_jþ _ijK_abK^ab

þ"²S_ij" _ij2"_GBH_ij; (67) and the second and third terms on the right-hand side include the linearly coupled terms between LnK_ij. Therefore, in an actual simulation, we have to extract each evolution equation of K_ij using a matrix form of Eq. (67) such as

k ¼Akþb; (68) wherek¼ ðLnK₁₁;LnK₁₂;. . .;LnK_NNÞ^TandAandbare the appropriate matrix and vector, respectively, derived from Eq. (67).

The procedure of inverting the matrixð1AÞis techni- cally available, but the invertibility of the matrix is not generally guaranteed at this moment. In the case of the standard ADM foliation in four-dimensional Einstein equations, the continuity of the time evolutions depends on the models and the choice of gauge conditions for the lapse function and shift vectors. If the combination is not appropriate, then the foliation hits the singularity which stops the evolution. The similar obstacle may exist also for the Gauss-Bonnet gravity. We expect that in the most cases Eq. (68) is invertible for K_ij, but we cannot deny the pathological cases which depend on the models and gauge conditions. Such a study must be done together with actual numerical integrations in the future.

We obtain, however, a similar form of the equation by introducing the ðNþ1Þ-dimensional Weyl curvature [27,28]. It is useful for the discussion of the brane-world cosmology although it is not written by only the

N-dimensional quantities, and we cannot adopt it as the evolution equation directly. The trace of Eq. (21) minus half of Eq. (19) gives

ðN1Þ"

2MþK_abK^abþh^abLnK_ab þ2ðn3Þ_GB

"

4ðM²4M_abM^abþM_abcdM^abcdÞ þMK_abK^ab2Kⁱ_jK^j_kM^k_i2N_aN^aþN_abcN^abc þMh^abLnK_ab2M^abLnK_ab

¼ "²T þ": (69) By the last term on the left-hand side of this equation, the term of the Lie derivative cannot be expressed by the other term explicitly.

Let us rewrite the dynamical equation (21) in a different form. From Eqs. (14), (17), and (18) with the decomposi- tion of the Riemann tensor as

R¼ 2

N1ðg½Rg½RÞ 2

NðN1Þg½gRþC; (70) where C is theðNþ1Þ-dimensional Weyl curvature, we find

LnK_ij¼"N1

N2E_ijþ "

N2

M_ij1 NM _ij

K_iaK^a_jþ1

N ^ijK_abK^abþ1 N ^ij

abLnK_ab; (71) where

E_ij:¼Cnn _{i j}: (72) However, because Eq. (71) is a trace-free equation,LnK cannot be fixed by Eq. (71). Inserting Eq. (71) into Eq. (69), we find

h^abLnK_ab¼ "

2MK_abK^ab"

Uð²T Þ þ2"ðN3Þ_GB

U I; (73)

where

U¼N1þ2ðN2ÞðN3Þ

N _GBM; (74) I¼ N6

4ðN2ÞM²þ N

N2M_abM^abþ1

4M_abcdM^abcd "

N_aN^aþ4N_abcN^abc2ðN1Þ

N2 M_abE^ab

: (75)

From Eq. (71) with Eq. (73), we then find LnK_ij ¼N1

N2E_ijþ "

N2

M_ij1 2M _ij

K_iaK^a_j

"

NUð²T Þ _ij2"ðN3Þ_GB NU I _ij:

(76) SinceE_ijis aðNþ1Þ-dimensional quantity,LnK_ijcannot be evaluated by the valuables onN-dimensional hypersur- face with this equation. This means that Eq. (76) cannot be used for the full dynamics as we mentioned. For example, however, for the Friedmann brane-world model, where the constant-time slice of the timelike hypersurface is homo- geneous and isotropic,E_ij can be written using quantities on the hypersurface. For such limited cases where the term E_ijcan be evaluated on the hypersurface, Eq. (76) is useful for simplifying the situation.

V. DISCUSSION

With the aim of numerical investigations of space-time dynamics in higher-dimensional and/or higher-curvature gravity models, we presented the basic equations of the Einstein-Gauss-Bonnet gravity theory.

We show theðNþ1Þ-dimensional decomposition of the basic equations, in order to treat the space-time as a Cauchy problem. With the aim of investigations of bulk space-time in recent brane-world models, we also prepared the equations for both timelike and spacelike foliations.

The equations can be separated into the constraints (the Hamiltonian constraint and the momentum constraint) and the evolution equations.

Two constraints should be solved for constructing the initial data. By showing the conformally transformed con- straint equations, we discussed how the constraints can be simplified by tuning the powers of conformal factors. If we have Gauss-Bonnet terms, however, the equations still remain in a complicated style.

For the evolution equations, we find thatLnK_ijcompo- nents are coupled. However, this mixture is only up to the linear order due to the quasilinear property of the Gauss- Bonnet terms, so that the equations can be in a treatable form in numerics.

We are now developing our numerical code and hope to present some results elsewhere in the near future.

ACKNOWLEDGMENTS

H. S. was partially supported by the Special Research Fund (Project No. 4244) of the Osaka Institute of Technology, Faculty of Information Science and Technology.

Note added in proof.—Regarding the invertibility of Eq. (68), Deruelle and Madore [29] gave an explicit ex- ample in a simple cosmological model where the equation corresponding to Eq. (68) is not invertible.

APPENDIX A: GB PART OF HAMILTONIAN CONSTRAINT EQUATION

In Eq. (35), the GB part of the Hamiltonian constraint equation ^¼M²4M_abM^abþM_abcdM^abcd is not written explicitly. It becomes

^¼ ðN3Þmc^4mf4ðN2Þmc^{2½ðD^}aD^^ac^{Þ2D^}aD^_bc^{D^aD^b}c 4c^{1½M^ ðN}2Þ½ðN3Þm2mc^{2D^}ac^{D^a}c^{D^}aD^^ac

þ8c^{1½M^abþ ðN}2Þmðmþ1Þc^{2D^a}c^{D^b}c^{D^}aD^_bc^{þ ðN}1Þ₂m²½ðN4Þm4c^{4ðD^}ac^{D^a}c^Þ2

2c^2½ðN4Þm2M^D^_cc^{D^c}c8ðmþ1Þc^{2M^abD^}ac^{D^}bc^{g þ}c^4mð^²4 ^ab^^abþ ^abcd^^abcdÞ; (A1)

where

^¼R^"

N1

N c^{2mþ2K^2}c^{2‘2mA^}abA^^ab

; (A2)

^ij¼R^_ij"

N1

N² c^2mþ2^ijK^²þN2

N c^{‘þK^A^}ij

c^{2‘2mA^}iaA^^a_j

; (A3)

^ijkl¼R^_ijkl"

1

N²c^2mþ2ð^ik^jl ^il^jkÞK^² þ 1

Nc^{‘þðA^}ik^jlA^_il^jkþA^_jl^ikA^_jk^ilÞ þc^{2‘2mðA^}ikA^_jlA^_ilA^_jkÞ

: (A4)

When ¼‘2m and m¼2=ðN2Þ, which corre- sponds to case A [Eq. (36)],

^¼2ðN3Þ

N2 c^8=ðN2Þ8c^{2½ðD^}aD^^ac^{Þ2D^}aD^_bc^{D^aD^b}c 4c^{1ðM^ þ}2c^{2D^}ac^{D^a}c^{ÞD^}aD^^ac þ8c^{1M^abþ 2N}

N2c^{2D^a}c^{D^b}c^{D^}aD^_bc⁸_ðN^NðN1Þ

2Þ² c^{4ðD^}ac^{D^a}c^{Þ2þ 8}

N2c^{2M^D^}cc^{D^c}c 8N

N2c^{2M^abD^}ac^{D^}bc^þc^8=ðN2Þð^²4 ^ab^^abþ ^abcd^^abcdÞ; (A5)

where

^¼R^"c^{2‘4=ðN2ÞðK^2K^}abK^^abÞ; (A6)

^ij¼R^_ij"c^{2‘4=ðN2ÞðK^K^}ijK^_iaK^^a_jÞ; (A7)

^ijkl¼R^_ijkl"c^{2‘4=ðN2ÞðK^}ikK^_jlK^_ilK^_jkÞ: (A8) When ¼0, m¼2=ðN2Þ, which corresponds to case B [Eq. (37)],is expressed as Eq. (A5) and

^¼R^"

N1

N c^{4=ðN2ÞK^2}c^{2‘4=ðN2ÞA^}abA^^ab

; (A9)

^ij¼R^_ij"

N1

N² c^4=ðN2Þ^ijK^²þN2

N c^{‘K^A^}ij

c^{2‘4=ðN2ÞA^}iaA^^a_j

; (A10)

^ijkl¼R^_ijkl"

1

N²c^4=ðN2Þð^ik^jl ^il^jkÞK^² þ 1

Nc^{‘ðA^}ik^jlA^_il^jkþA^_jl^ikA^_jk^ilÞ þc^{2‘4=ðN2ÞðA^}ikA^_jlA^_ilA^_jkÞ

: (A11)

APPENDIX B: GB PART OF MOMENTUM CONSTRAINT EQUATION

In Eq. (44), the GB part of the Hamiltonian constraint equation ^iis not written explicitly. It becomes