Hyperbolicity and Causality of Einstein-Gauss-Bonnet Gravity in Warped Product Spacetimes
IICTS-USTC/PCFT-21-02
Hyperbolicity and Causality of Einstein-Gauss-Bonnet Gravity in Warped Product Spacetimes
Li-Ming Cao a ,b ∗ and Liang-Bi Wu b † a Peng Huanwu Center for Fundamental Theory, Hefei, Anhui 230026, China and b Interdisciplinary Center for Theoretical Study and Department of Modern Physics,University of Science and Technology of China, Hefei, Anhui 230026, China (Dated: January 14, 2021)In Einstein-Gauss-Bonnet gravity, for a group of warped product spacetimes, we get a generalized masterequation for the perturbation of tensor type. We show that the “effective metric” or “acoustic metric” for thetensor perturbation equation can be defined even without a static condition. Since this master equation doesnot depend on the mode expansion, the hyperbolicity and causality of the tensor perturbation equation can beinvestigated for every mode of the perturbation. Based on the master equation, we study the hyperbolicityand causality for all relavent vacuum solutions of this theory. For each solution, we give the exact hyperboliccondition of the tensor perturbation equations. Our approach can also applied to dynamical spacetimes, andVaidya spacetime have been investigated as an example.
I. INTRODUCTION
Lovelock theories are the most general diffeomorphism covariant theories only involving a metric tensor with second orderequations of motion [1]. In four dimension and generic values of the coupling constants, the theory reduces to the GeneralRelativity with a cosmological constant [2]. The equations of motion of such a theory in four dimensions are the Einsteinequations. However, when the higher order terms of spacetime curvature exist, there will be a lot of properties which aredifferent from the Einstein equations. For example, the gravitational propagation velocity may exceed the speed of light unlikethe Einstein theories where graviton always travels slower than light [3].Einstein-Gauss-Bonnet gravity, whose Lagrangian contains the quadratic term of spacetime curvature, as the lowest Lovelocktheory, is a simplest model to display the difference between the general Lovelock gravity theory and Einstein gravity theory inhigher dimensions. The theory is fascinating since it has been realized in the low-energy limit of heterotic string theory [4–7].Moverover, this theory is studied in many aspects, such as black holes [8–11] and AdS/CFT correspondence [12, 13].It is well known that Einstein equations are second order quasi-linear equations, i.e., the coefficients of the second orderpartial derivatives of the metric in the equations of motion only depend on the metric and the first order partial derivatives ofthe metric. However, the equations of motion of the Einstein-Gauss-Bonnet gravity do not have such a property. In general, theequations of motion of Einstein-Gauss-Bonnet gravity are second order full nonlinear equations but not quasi-linear equations.This reason, the Einstein-Gauss-Bonnet gravity and more general Lovelock gravity may have very different behaviors from theEinstein gravity theory in the viewpoint of partial differential equations.Many physical systems are described by partial differential equations. The Cauchy problem is required to be well-posed bythe determinism. The basic causal properties of a system of partial differential equations are determined by its characteristichypersurfaces [14]. As for the Einstein equations, a hypersurface which is characteristic is equivalent to it is null. This means thatgravity travels at the speed of light in general relativity. In Einstein-Gauss-Bonnet theory, however, the superluminal propagationof gravitons will arise because of the noncanonical kinetic terms [15]. Along the characteristic hypersurfaces, one can define theeffective metric. Izumi has proved some important conclusions by using the method of the characteristics in the Gauss-Bonnettheory [16]. In that paper, He claims that on an evaporating black hole where the geometrical null energy condition is expectednot to hold, classical gravitons can escape from the black hole defined with null curves. This kind of study has close relations tothe hyperbolicity and causality of the gravitational equations.In principle, to get the hyperbolicity of the equations of motion for a given gravity theory, we have to study the leading partialderivative terms of the equations, and get the principle symbol which is a matrix in usual. The classification of the partialdifferential equations depends on the property of this matrix. However, in gravity theory, usually, people are interested in thepropagation of a gravitational fluctuation on a fixed background, i.e, the gravitational perturbation theory or linearized gravitytheory. Although the linearized gravity theory is very different from the full theory, it really provides some useful information onthe hyperbolicity and causality of the full theory [17]. So the study of the gravitational perturbation equation provides a practicalway to investigate the hyperbolicity and causality of the theory.It is well known that any gravitational perturbation theory is gauge dependence. The perturbation variables between one gaugeand another are related by a gauge transformation. Problems of choosing gauge will be faced when we study the perturbation ofa spacetime. One way is to find physically preferred gauges. Another way is to use the gauge-invariant variable, for example,the Kodama-Ishibashi gauge-invariant variables [20]. By using these gauge-invariant variables, one can get the master equationswhich have tensor, vector, and scalar parts [20, 22–24]. Based on these, the stability of the higher dimensional black holes ∗ e-mail address: [email protected] † e-mail address: [email protected] a r X i v : . [ g r- q c ] J a n such as Gauss-Bonnet black holes are studied by Takahashi and Soda [26–28]. A typical way to calculate the effective metricis using the master equations. This means that one can determine characteristics from the equations of motion of linearizedperturbations of a background [17, 29, 30]. In the paper [29], Papallo and Reall work out a speed limit for small black holes andinvestigate a Shapiro time delay in the Einstein-Gauss-Bonnet theory. In the paper [17], Reall and Takahashi generalized theresult of Izumi [16] to prove that any Killing horizon is a characteristic hypersurface for all gravitational degrees of freedom ofa Lovelock theory. They also investigated the hyperbolicity of Ricci flat type N spacetimes and the static, maximally symmetricblack hole solutions in the Lovelock theory. Beside the method by gauge-invariant variables, recently, it has been shown that(weakly coupled) Lovelock and Horndeski theories possess a well-posed Cauchy problem base on the conventional harmonicgauge or modified harmonic gauge [18, 19].However, unlike the Kodama-Ishibashi formalism in Einstein gravity theory, the gauge-invariant gravitational perturbation inEinstein-Gauss-Bonnet theory and more general Lovelock theory is far from completed — even the simplest tensor perturbationis still staying at the static spacetimes [20]. So, for black hole spacetimes, the study of the hyperbolicity and causality of theEinstein-Gauss-Bonnet theory is mainly limited to static cases up to date [17]. In this paper, we conquer the general masterequation of tensor perturbation for general warped product spacetimes in Einstein-Gauss-Bonnet theory. Based on this newmaster equation, we show that effective metric of the tensor perturbation equation defined by Reall in [17] can be generalized tothe cases without a static condition.Since the new master equation does not depend on the mode expansion (For example, the expansion by harmonic tensors on aclosed manifold with integers k = (cid:96) ( (cid:96) +1) · · · which is proportional to the eigenvalues of Laplacian operator), the hyperbolicityand causality can be investigated for any mode of the perturbation. This generalized the method in [17] in which only large (cid:96) mode has been studied. Based on the master equation, we study the hyperbolicity and causality of all relevant vacuum solutionsof the theory. For each solution, we give the exact hyperbolic condition of the tensor perturbation equations. For example, wecan give an analytic hyperbolic condition of the tensor perturbation equation on D = 6 dimensional black holes background.Our approach can also be applied to dynamical spacetimes, and Vaidya spacetime have been investigated as an example.The paper is organized as follows. In section II, we give a brief review on the Einstein-Gauss-Bonnet theory, and the effectivemetric defined by Reall, and explain how to get the effective metric from the master equation of tensor part. In section III, wecalculate the tensor perturbation equations for a general warped product spacetime instead of the static solutions. Moreover, wegive the condition of keeping the hyperbolicity of the tensor perturbation equations and give the condition of the tensor modewhich can travel faster than light. In section IV-VI, according to the classification of the vacuum solutions in [31], we check thehyperbolicity of the tensor perturbation equation on three types of spacetimes — the Boulware-Deser-Wheeler-Cai solution, theNariai-type spacetime, and the dimensionally extended constant curvature black hole. We also check whether there are casesthat the graviton travels faster than the light. In the section VII, we apply our methods to investigate hyperbolicity of the tensorperturbation equation on Vaidya spacetime which is not a vacuum solution any more. Section VIII is devoted to conclusion anddiscussion.We use the following notion for indices. The capital letters { M, N, L, P · · · } are the indices for the D = n + 2 -dimensionalspacetime. The lowercase letters a , b are the indices for the manifold ( M , g ab ) , while the lowercase letters { i , j , k , l , · · · } are the indices for the manifold ( N n , γ ij ) . The convention of the curvature is given by ( ∇ M ∇ N − ∇ N ∇ M ) v L = R MNLP v P ,which is the same as in the reference [24]. II. BASIC THEORYA. Einstein-Guass-Bonnet theory
Here, we start by a brief review of Einstein-Gauss-Bonnet gravity with a cosmological constant [1, 2]. The action in the D -dimansional spacetime with a metric g MN is given by S = (cid:90) d D x √− g (cid:20) κ D ( R −
2Λ + αL GB ) (cid:21) + S matter , (2.1)where κ D is the coupling constant of gravity, and R and Λ are the D -dimensional Ricci scalar and the cosmological constant,respectively. S matter stands for the matter fields. The Gauss-Bonnet term is given by L GB = R − R MN R MN + R MNP Q R MNP Q . (2.2)The symbol α is the coupling constant of the Gauss-Bonnet term. This type of action can be derived from the low-energy limitof heterotic string theory [4–7]. This reason, α is considered to be positive. The equation of motion of this theory is given by G MN + αH MN + Λ g MN = κ D T MN , (2.3)where G MN = R MN − g MN R , (2.4)and H MN = 2 (cid:2) RR MN − R ML R LN − R KL R MKNL + R KLPM R NKLP (cid:3) − g MN L GB . (2.5)The second order tensor T MN , which can be obtained from S matter , is the energy-momentum tensor for matter fields. In the four-dimensional spacetime, the Gauss-Bonnet term does not make a contribution to the equations of motion since it is identically atotal derivative. It is worth pointing out that the field equations (2 . contain up to the second derivative of the metric just aswhat the Lovelock theorem said [1].Now, we consider a D = 2 + n -dimensional spacetime ( M D , g MN ) , which has a local direct product manifold M D ∼ = M × N n and a metric, g MN dx M dx N = g ab ( y ) dy a dy b + r ( y ) γ ij ( z ) dz i dz j , (2.6)where coordinates x M = { y , y ; z , · · · , z n } . The two element tuple ( M , g ab ) forms a two dimensional Lorentzian mani-fold, and ( N n , γ ij ) is an n − dimensional Riemann manifold. The metric compatible covariant derivatives associated with g MN , g ab , and γ ij are denoted by ∇ M , D a , and ˆ D i , respectively. In the following discussion, the Riemann manifold ( N n , γ ij ) isassumed to be an Einstein manifold, i.e, ˆ R ij = ( n − Kγ ij , (2.7)where ˆ R ij is the Ricci tensor of ( N n , γ ij ) , and K = 0 , ± . Actually, we will consider more restrictive case in which thisEinstein manifold is a maximally symmetric space, and K is the sectional curvature of the space.An ( n + 2) -dimensional spacetime in GB gravity with the metric (2 . , in which ( N n , γ ij ) is an Einstein manifold, has threegroups of field equations (cid:40) (cid:20) α ( n − n − K − ( Dr ) r (cid:21) R − n − (cid:3) rr + ( n − n − K − ( Dr ) r − α ( n − n − (cid:34)(cid:16) (cid:3) rr (cid:17) − ( D a D b r )( D a D b r ) r + ( n − n − K − ( Dr ) ] r − ( n − K − ( Dr ) ] r (cid:3) rr (cid:35) + α ˆ C klmn ˆ C klmn r (cid:41) γ ij − αr ˆ C iklm ˆ C jklm = − κ D pγ ij , (2.8) (cid:20) α ( n − n − K − ( Dr ) r (cid:21) (cid:18) D a D b r − (cid:3) rg ab (cid:19) = − rn κ D (cid:18) T ab − T cd g cd g ab (cid:19) , (2.9)and − (cid:3) rr + ( n − K − ( Dr ) r + 2 α ( n − n − K − ( Dr ) ] r (cid:40) ( n − K − ( Dr ) ]2 r − (cid:3) rr (cid:41) − n + α ˆ C ijkl ˆ C ijkl nr = − κ D n g ab T ab , (2.10)where the energy-momentum tensor T MN has been decomposed into T MN = diag (cid:8) T ab ( y ) , r p ( y ) γ ij (cid:9) . The Weyl tensor of ( N , γ ij ) is denoted by ˆ C ijkl . In the case of vacuum and maximally symmetric ( N , γ ij ) , i.e., T MN = 0 and ˆ C ijkl = 0 , similarequations can be found in [31]. It is worth mentioning that the first equation (2.8) comes from the ( ij ) − components of Eq.(2.3),the second equation (2.9) comes from the traceless part of the ( ab ) − components of Eq.(2.3), and the third equation (2.10) comesfrom the trace part of the ( ab ) − components of Eq.(2.3). B. Effective metrics
In gravity theory, the so called “effective metric” or “acoustic metric” is important to determine the type of the linearizedgravitational equations which are usually second order partial differential equations. In the following sections, this kind ofdiscussion will be applied to the Einstein-Gauss-Bonnet theory. By requiring that the effective metric to be Lorentzian, theequation of the tensor perturbation is hyperbolic in the usual sense.Principally, to study the causality and hyperbolicity of a gravity theory, we have to consider the full nonlinear equations ofmotion of the theory. However, for some background spacetime, the linear perturbation equations can provide a lot of informationon the hyperbolicity or causality of the theory, see [17] for details. To understand the effective metric, let us give a brief reviewon the hyperbolicity of a second order differential equations.Consider a second order linear differential equation for an N -dimensional vector φ I in a patch of a spacetime with a coordinatesystem { x M , M = 1 , · · · , D } : P MNI J ∂ M ∂ N φ J + P MI J ∂ M φ J + V I J φ I = 0 , (2.11)where I , J = 1 , · · · , N , and P MN = ( P MNI J ) = ( P NMI J ) = P NM ,P M = ( P MI J ) , and V = ( V I J ) are N × N real matrices. For a covector ξ at a point p , the principal symbol of the equations is given by P ( p, ξ ) = P MN ( p ) ξ M ξ N . (2.12)This is an N × N matrix and it plays an important role in the classification of the partial differential equations in usual theoryof differential equation. The characteristic polynomial Q ( p, ξ ) is defined as Q ( p, ξ ) = det P ( p, ξ ) . (2.13)The hypersurface φ = const is a characteristic hypersurface if Q ( p , dφ ) = 0 , and the normal vector at each point of thecharacteristic hypersurface is called a characteristic direction. The normal cone at p is defined by the equation Q ( p , ξ ) = 0 atthe point p , and it is important in the discussion of the causality of the theory [14, 16, 17, 32].For the gravitational theories with the second order derivatives of metrics, the linearized gravitational equations can be putinto the form (2.11). Of course, now φ I corresponds to h MN = δg MN , and the indices I and J now correspond to ( M N ) and ( LP ) respectively. Actually, this can be realized by rearranging h MN into a column vector.In general, the characteristic polynomial Q ( p, ξ ) is quite complicated. However, for the metric (2.6) with an maximallysymmetric space ( N n , γ ij ) (especially for a static case), it is assumed that Q ( p , ξ ) can be factorized into a form Q ( p , ξ ) = (cid:16) G M N S ( p ) ξ M ξ N (cid:17) p s · (cid:16) G M N V ( p ) ξ M ξ N (cid:17) p V · (cid:16) G M N T ( p ) ξ M ξ N (cid:17) p T , (2.14)where p S , p V , and p T are the number of degrees of freedom of scalar, vector, and tensor perturbation respectively, and thesecond order tensor G MNS , G MNV , and G MNT are the so-called “effective metric” associated with the scalar, vector, and tensorperturbation [17].In the function space formed by h MN , the scalar, vector, and tensor perturbation of the metric (2.6) belong to three differentsubspaces which are orthogonal to each other [20, 21], so we can consider these three kinds of perturbation separately. In thecase only the tensor perturbation is involved, we have Q ( p , ξ ) = (cid:16) G MNT ( p ) ξ M ξ N (cid:17) p T . (2.15)Naively speaking, one can say: since the scalar and vector perturbation are shut down, i.e., p S = 0 = p V , so Eq.(2.14) reduceto the above equation. According to the paper [17], a hypersurface is characteristic if, and only if, it is null with respect tothe effective metric G MNT . The tensor perturbation equation is hyperbolic if that the effective metric G MNT has a Lorentziansignature.For several static examples, it has been shown that the factorization (2.14) or (2.15) is really a reasonable assumption [17]. Inthis paper, for the tensor perturbation of the general metric (2.6), we will show that Q ( p , ξ ) can be put into the form (2.15) evenwithout the static condition if ( N n , γ ij ) is maximally symmetric. However, without the condition of the maximal symmetry of ( N n , γ ij ) , we have no relation (2.15) in general. III. HYPERBOLICITY AND CAUSALITY OF GENERAL TENSOR PERTURBATION EQUATIONSA. General tensor perturbation equations
Considering a metric perturbation g MN → g MN + h MN , the linear perturbation equations of Gauss-Bonnet gravity are givenby δG MN + Λ h MN + αδH MN = κ D δT MN , (3.1)where δG MN and δH MN are the perturbations of the Einstein tensor and the Gauss-Bonnet tensor of the spacetime ( M D , g MN ) respectively, and δT MN is the perturbation of the energy-momentum tensor.Here, we will get the effective metric for tensor perturbations around a fixed spacetime with the metric (2.6). The tensorperturbations around backgrounds (with the Einstein manifold ( N , γ ij ) ) are transverse and traceless part of h ij [20], i.e. h ij = h TT ij , which means that we are considering perturbation h MN which satisfies h ab = 0 , h ai = 0 , h = h ii = 0 , ˆ D i h ij = 0 . (3.2)It is well known that the above tensor perturbation of the spacetime with the metric (2.6) is gauge invariant. Of course thescalar and vector perturbations are quite different, to get the gauge invariant perturbation variables, one has to consider thecombinations of different parts of h MN .Substituting h MN (satisfying (3.2)) into Eq.(3.1), after lengthy and tedious calculation, the tensor perturbation equation hasthe following form (cid:16) P abijkl D a D b + P mnijkl ˆ D m ˆ D n + P aijkl D a + V ijkl (cid:17)(cid:16) h kl r (cid:17) = − κ D r δT ij , (3.3)where the detailed expressions for the coefficients P abijkl , P mnijkl , P aijkl , V ijkl can be found in Appendix.A. When ( N , γ ij ) is maximally symmetric, the Weyl tensor ˆ C ijkl is vanishing, and we have P abijkl = P ab δ ik δ jl ,P mnijkl = P mn δ ik δ jl ,P aijkl = P a δ ik δ jl ,V ijkl = V δ ik δ jl , (3.4)where P ab , P mn , P a , V have following forms P ab = g ab + 4 α ( n − (cid:26) D a D b rr + (cid:20)
12 ( n − K − ( Dr ) r − (cid:3) rr (cid:21) g ab (cid:27) , (3.5) P mn = (cid:40) α (cid:20) R − n − (cid:3) rr + ( n − n − K − ( Dr ) r (cid:21) (cid:41) γ mn r , (3.6) P a = n D a rr + 2( n − α (cid:40) D a D b rr + (cid:104) R − n − (cid:3) rr +( n − n − K − ( Dr ) r (cid:105) g ab (cid:41) D b rr , (3.7)and V = R − n − (cid:3) rr + n ( n − Kr − ( n − n − Dr ) r − Λ+ α (cid:40) − n − n −
2) ( D a D b r )( D a D b r ) r + 4( n − n − (cid:18) (cid:3) rr (cid:19) + 2 n ( n − K · Rr − n − n −
2) ( Dr ) · Rr − n ( n − K · (cid:3) rr + 4( n − n − n −
3) ( Dr ) · (cid:3) rr − n ( n − ( n − K · ( Dr ) r + ( n − n − n − n − K r + ( n − n − n − n − (cid:20) ( Dr ) r (cid:21) (cid:41) . (3.8)So, when ( N , γ ij ) is maximally symmetric, the tensor perturbation equation can written as (cid:16) P ab D a D b + P kl ˆ D k ˆ D l + P a D a + V (cid:17)(cid:16) h ij r (cid:17) = − κ D r δT ij . (3.9)Obviously, this equation reduces to the tensor perturbation equation in Einstein gravity theory when α is vanishing [20, 24]. Inthe static case, one can check the above equations exactly reduce to the one in reference [25]. B. Hyperbolicity of the tensor perturbation equations
Comparing Eq.(3.3) and Eq.(2.11), it is easy to find that indices I and J correspond to ( ij ) and ( kl ) . We can rearrange h ij = h TT ij into a column vector φ I such that δ ik δ jl in Eq.(3.3) can be expressed as δ I J . So the characteristic polynomial Q ( p, ξ ) now has a form (2.15), i.e., Q ( p, ξ ) = (cid:16) P MN ( p ) ξ M ξ N (cid:17) p T , where P MN can be put into a matrix form ( P MN ) = (cid:34) P ab Qr γ ij (cid:35) . (3.10)Here P ab has been defined in Eq.(3.5), and Q is the coefficient of γ ij /r in Eq.(3.6), i.e., Q = 1 + 2 α (cid:20) R − n − (cid:3) rr + ( n − n − K − ( Dr ) r (cid:21) . (3.11)Since h ij = h TT ij , the dimension of the vector φ I is the number of degrees of freedom of gravitational perturbation, i.e., we have p T = 12 D ( D − − D .
Thus the effective metric is nothing but P MN in Eq.(3.10), i.e., we have G MNT = P MN . It should be noted here: the effective metric is not so simple if that the Einstein manifold ( N , γ ij ) is not maximally symmetric,see Appendix A. From now on, we only consider the cases with maximal symmetry.The effective metric P MN should be Lorentzian to maintain the hyperbolicity of the theory. Therefore, we have to set P = det( P ab ) < , and Q > . (3.12)By using the equations of motion and the null frame { (cid:96) a , n a } , we have (see Appendix.B) P = det( P ab ) = − (cid:40) − α ( n − (cid:20) (cid:3) rr − ( n − K − ( Dr ) r (cid:21) (cid:41) +16 α ( n − κ D ( T nn T (cid:96)(cid:96) ) (cid:40) n (cid:20) α ( n − n − K − ( Dr ) r (cid:21) (cid:41) − , (3.13)where T (cid:96)(cid:96) = T ab (cid:96) a (cid:96) b and T nn = T ab n a n b are the components for the energy-momentum tensor along the null directions. Ofcourse, the above equation is valid only in the case where α ( n − n − K − ( Dr ) r (cid:54) = 0 . It will be discussed separately when the above condition is not satisfied (see section.VI).Eq.(3.13) suggests that P is always negative or vanishing for vacuum solutions. When matter fields are present and satisfy nullenergy condition, P might be positive. However, for the radiation matter with a null direction (cid:96) a or n a , P is also nonpositive.This happens in Vaidya spacetimes. C. Causality of the tensor perturbation equations
Once the system is hyperbolic, we can discuss the speed of a gravitational fluctuation. In other words, we can check whethergravity travels faster than light or not. For simplicity, we suppose D a D b r − (cid:3) rg ab = 0 . (3.14)This condition implies that K a = (cid:15) ab D b r corresponds to a Killing vector field of the spacetime (where (cid:15) ab is the volume elementof ( M , g ab ) ). If gravity travels faster than light, we have g MN ξ M ξ N < , (3.15)where ξ M satisfies the following condition P MN ξ M ξ N = 0 . (3.16)This means that ξ M is the a characteristic direction, and normal cone by g MN is smaller than the normal cone by P MN [17].Substituting the P MN in Eq.(3.10) into Eq.(3.16) , we have (cid:26) − α ( n − (cid:20) (cid:3) rr − ( n − K − ( Dr ) r (cid:21)(cid:27) g ab ξ a ξ b + Qg ij ξ i ξ j = 0 . (3.17)Since g ij = γ ij /r is positive defined, Eq.(3.15) implies that g ab ξ a ξ b < . (3.18)Only in the case Q > , it makes sense of discussing the velocity of the graviton. So we have − α ( n − (cid:20) (cid:3) rr − ( n − K − ( Dr ) r (cid:21) > . (3.19)For convenience, we define I = 1 − α ( n − (cid:20) (cid:3) rr − ( n − K − ( Dr ) r (cid:21) . (3.20)Therefore, from Eqs.( . ) and (3.17), we arrive at Q − I > . (3.21)From the expressions of Q and I , it is not hard to find that Q − I is proportional to α , so we can define Q − I = 2 αJ , where J is given by J = R − ( n − (cid:3) rr − n − K − ( Dr ) r . (3.22)It is easy to find that I approaches Q if we take the limit α → . So, under this limit, the gravitational wave travels at a speed oflight. In conclusion, if gravity travels faster than light, we have the following three conditions Q = 1 + 2 α (cid:20) R − n − (cid:3) rr + ( n − n − K − ( Dr ) r (cid:21) > , (3.23) I = 1 − α ( n − (cid:20) (cid:3) rr − ( n − K − ( Dr ) r (cid:21) > , (3.24)and J = R − ( n − (cid:3) rr − n − K − ( Dr ) r > . (3.25)In the following sections, we will discuss the hyperbolicity and causality of the tensor perturbation equations for exact solutionsin Einstein-Gauss-Bonnet gravity theory. We will investigate the above three inequalities on these exact spacetimes. IV. BOULWARE-DESER-WHEELER-CAI SOLUTION
The static vacuum solution in Einstein-Gauss-Bonnet theory has been found long time ago by Boulware and Deser [8] andWheeler [9]. The solutions have been extended to the case with a cosmological constant by Cai [10] about twenty years ago.The metric of the solution is given by ds = − F ( r ) dt + F − ( r ) dr + r γ ij dz i dz j , (4.1)where F ( r ) = K + r α ∓ (cid:118)(cid:117)(cid:117)(cid:116) α (cid:32) κ D MnV Kn r n +1 + ˜Λ (cid:33) . (4.2)In the abvoe equation, “ − ” corresponds to the so-called “General Relativity branch” which reduces to the solution in generalrelativity when α approaches to zero, while “ + ” corresponds to “Gauss-Bonnet branch” which has no general relativity limit as α approaches to zero. The parameter M is mass parameter, and ˜ α = ( n − n − α and ˜Λ = 2Λ / ( n ( n + 1)) . In the followingdiscussion, It is convenient to define ˜ M as ˜ M = 2 κ D MnV Kn , where V Kn is the volume of the maximally symmetric space ( N , γ ij ) with a unit radius. Since it is a vacuum solution, fromEq.(3.13), we always have P < (excluding some special values of P = 0 which form a zero measure set). Therefore, thehyperbolicity is determined by the sign of Q . Some calculation shows Q = 1 − α (cid:20) F (cid:48)(cid:48) ( r ) + 2( n − F (cid:48) ( r ) r − ( n − n − K − F ( r ) r (cid:21) . (4.3)This result is the same as the one in reference [30] when n = 3 and n = 4 .Another important thing is to study the existence of the superluminal modes. Before the detailed discussion, we give threeuseful formulas which can be expressed as Q GR + Q GB = 0 , (4.4) I GR + I GB = 0 , (4.5) J GR + J GB = 0 , (4.6)where GR stands for the General Relativity branch while GB stands for the Gauss-Bonnet branch. It is easy to get the aboveequations, and we will not give the proof here. These results imply that the discussion on the GR branch is enough. So, in thispaper, we only consider the solution which can reduce to the one in general relativity under the limit of α → .The details of the solution (4.1) is complicated. For different values of the parameters, the solution might be a globally regularsolution, a black hole, a naked singularity, or a branch singularity. The classification of this solution has been done in [34] andreferences therein. A. Black hole solutions
In this subsections, we show that the hyperbolicity is broken outside the event horizon in some dimensions, for example D = 6 , and this is consistent with the results in [17]. Beside this, we give the precise conditions when the hyperbolicity isbroken outside the event horizon.For simplicity, when the event horizon is present, we introduce three dimensionless quantities, x , a , and λx = r + r , a = ˜ αr , λ = r ˜Λ , (4.7)where r + is the radius of the outermost event horizon. Here, we have assumed α > , so a is always positive. By thesedefinitions, Q , I , and J can be expressed as Q = 1 − ax ( n − n − (cid:104) x F xx − n − xF x − ( n − n − K − F ( x )) (cid:105) , (4.8) I = 1 + 2 ax ( n −
1) [ xF x + ( n − K − F ( x ))] , (4.9) J = 1 r (cid:2) − x F xx + ( n − xF x − n − K − F ( x )) (cid:3) , (4.10)where F ( x ) = K + 12 ax (cid:110) − (cid:112) aλ + [(1 + 2 Ka ) − − aλ ] x n +1 (cid:111) . (4.11)To simplify the discussion, it is convenient to define ˆ J = r J = − x F xx + ( n − xF x − n − K − F ( x )) . (4.12)Note that J and ˆ J have the same sign. Here the F x denotes the derivative of F ( x ) with respect to x . It should be noted here that F x is negative at x = 1 because r + is the radius of the outermost event horizon of the black hole. Λ = 0 and K = 1 case For n = 3 , when ˜ M > ˜ α , there is a black hole horizon. For n ≥ , when ˜ M > , there is a black hole horizon [34]. Here,we will investigate the sign of Q outside the (outmost) event horizons of the black holes. In these regions of the spacetimes, x ∈ [0 , , and Q can be written as Q = ( n − n − y + 2( n + 1)(2 n − y − ( n + 1) n − n − y , (4.13)where y = [1 + 4 a (1 + a ) x n +1 ] . y ∈ [1 , a ] . Since the denominator of Q in Eq.(4.13) is always positive, we will focus on the numeratorof Q , denoted by ˆ Q , given by ˆ Q ( y ) = ( n − n − y + 2( n + 1)(2 n − y − ( n + 1) . (4.14)We investigate the sign of ˆ Q in the following discussion. The n = 3 , , and n ≥ will be discussed separately.For n = 3 , it is easy to find that ˆ Q ( y ) = 24 y − ≥ −
16 = 8 > . So in D = 5 , the tensor perturbation is hyperbolic outside the event horizon of the spacetime.For n = 4 , we have ˆ Q ( y ) = − y + 50 y − . This equation implies ˆ Q > , y ∈ (cid:104) , (cid:112)
25 + 10 √ (cid:17) , ˆ Q < , y ∈ (cid:16)(cid:112)
25 + 10 √ , + ∞ (cid:17) . Therefore, ˆ Q is positive for all y ∈ [1 , a ] when a < (cid:113)
25 + 10 √ . However, ˆ Q is negative if y ∈ ( (cid:112)
25 + 10 √ , a ] when a > (cid:113)
25 + 10 √ , and ˆ Q is positive if y ∈ [1 , (cid:112)
25 + 10 √ .For n = 5 , we find ˆ Q = 84 y − ≥ −
36 = 48 > . So, in the case of D = 7 , the equation is always hyperbolic outside the horizon.For n ≥ , from the expression (4.14), it is not hard to find that ˆ Q ( y ) has only one zero point t in (0 , + ∞ ) , where t = (cid:115) ( n + 1) (cid:2) √ n − n + 24 − (2 n − (cid:3) ( n − n − . (4.15)However, it is not hard to prove that t < . This means that Q is positive for all y ∈ [1 , a ] .The above discussion shows that tensor perturbation equation is hyperbolic outside the event horizon for all possible parameter a if D (cid:54) = 6 . The case of D = 6 or n = 4 is very special, the hyperbolicity is maintained outside the event horizon only when a = ˜ αr < (cid:16)(cid:113)
25 + 10 √ − (cid:17) ≈ . . This suggests that the radius of the event horizon of the black hole has to satisfy r + > (cid:112) ˜ α/ . . So, to ensure the hyperbolicity of the equation, the black hole can not be too small.1 Λ < and K = 1 case For n = 3 , when ˜ M > ˜ α , there is a black hole horizon. For n ≥ , when ˜ M > , there is a black hole horizon [34]. Outsidethe event horizon, we have x ∈ [0 , , and Q has a form Q ( y ) = ( n − n − y + 2( n + 1)(2 n − aλ ) y − ( n + 1) (1 + 4 aλ ) n − n − y , (4.16)where y = [1 + 4 aλ + 4 a ( − λ + 1 + a ) x n +1 ] . (4.17)Since x ∈ [0 , , we have y ∈ [ √ aλ, a ] .(i). Firstly, let us consider the case with aλ > . Similar to the case with Λ = 0 , the numerator of Q in Eq.(4.16) can bedefined as ˆ Q ( y ) = ( n − n − y + 2( n + 1)(2 n − aλ ) y − ( n + 1) (1 + 4 aλ ) . (4.18)For n = 3 , we have ˆ Q ( y ) = 24(1 + 4 aλ ) y − aλ ) ≥ aλ ) > . For n = 4 , we have ˆ Q ( y ) = − y + 50(1 + 4 aλ ) y − aλ ) . From this equation, we get ˆ Q > , y ∈ (cid:104) √ aλ , √ aλ · (cid:112)
25 + 10 √ (cid:17) , ˆ Q < , y ∈ (cid:16) √ aλ · (cid:112)
25 + 10 √ , + ∞ (cid:17) . Thus, we have ˆ Q > for all y ∈ [ √ aλ , a ] when a < √ aλ · (cid:113)
25 + 10 √ . (4.19)However, we have ˆ Q < if y ∈ ( √ aλ · (cid:112)
25 + 10 √ , a ] when a > √ aλ · (cid:113)
25 + 10 √ , and ˆ Q > if y ∈ [ √ aλ , √ aλ · (cid:112)
25 + 10 √ .For n = 5 , obviously we have ˆ Q ( y ) = 84(1 + 4 aλ ) y − aλ ) ≥ aλ ) > . For n ≥ , from the expression of ˆ Q ( y ) in (4.18), we find that ˆ Q ( y ) has only one zero ponit ( √ aλ ) t in (0 , + ∞ ) ,where t is given by Eq.(4.15). Noted that t < , we have ( √ aλ ) t < √ aλ . So, we always have Q > when n ≥ .So, as the case Λ < and K = 1 , the D = 6 or n = 4 is special. If we hope the hyperbolicity is present outside theevent horizon of the black hole, we have to impose a condition on a and λ , i.e., the inequality (4.19), see Fig.1. Thehyperbolicity is broken above the solid black line.(ii). Secondly, when aλ = 0 , from x ∈ (0 , we have y ∈ (0 , a ] , and Q has a simple form Q = ( n − n − y n − n − . (4.20)It is obvious that Q = 0 when n = 3 , , Q < when n = 4 , and Q > when n ≥ .2 - - - - - - λ a FIG. 1: n = 4 , the case for Λ < and K = 1 . In the shadow region, inequalities (4.19) and aλ > are satisfied. For a given λ , we havean upper bound for a . The dotted line is for a = 3 . , and this corresponds to the upper bound of a in the case with Λ = 0 and K = 1 . (iii). Finally, in the case with aλ < , it is not hard to find x ∈ (cid:32)(cid:20) − (1 + 4 aλ )4 a ( − λ + 1 + a ) (cid:21) n +1 , (cid:35) . then y ∈ (0 , a ] . Actually, physically, this case is not so interesting because the solution has some (branch) singularitywhen y approaches to . However, it is also curious to us to that the possibility of the existence of some wave equationon this strange background. From the expression of ˆ Q ( y ) in Eq.(4.18) and the condition aλ < , it is easy to findwe have ˆ Q ( y ) < for n = 3 , , . So Q ( y ) < when n ≤ . The linearized gravitational wave does not exist in when D ≤ .For n ≥ , ˆ Q ( y ) has only one zero point s in (0 , + ∞ ) , where s = (cid:115) − (1 + 4 aλ ) ( n + 1) (cid:2) √ n − n + 24 + (2 n − (cid:3) ( n − n − . (4.21)This implies we always have Q < in (0 , a ] when a < s . However, we can get Q > in ( s, a ] when a > s , and Q is still negative in (0 , s ) . So, for this background with a branch singularity, we can get some waveequations when D ≥ . Λ < and K = 0 case For any n ≥ , when ˜ M > , there is a black hole horizon [34]. Consider the region outside the event horizon, then we have x ∈ [0 , . By defining y = (cid:112) aλ − aλx n +1 , the Q and ˆ Q have the same forms as in Eq.(4.16) and (4.18).(i). In the case with aλ > , we have y ∈ (cid:2) √ aλ, (cid:3) . Similar to the case with K = 1 , n = 3 , , , and n ≥ willbe discussed separately.For n = 3 , we have ˆ Q ( y ) = 24(1 + 4 aλ ) y − aλ ) ≥ aλ ) > . n = 4 , we have ˆ Q ( y ) = − y + 50(1 + 4 aλ ) y − aλ ) . This equation implies ˆ Q > , y ∈ (cid:104) √ aλ , √ aλ · (cid:112)
25 + 10 √ (cid:17) , ˆ Q < , y ∈ (cid:16) √ aλ · (cid:112)
25 + 10 √ , + ∞ (cid:17) . So we have ˆ Q > for all y in [ √ aλ , if < √ aλ · (cid:113)
25 + 10 √ . (4.22)However, ˆ Q < for y in ( √ aλ · (cid:112)
25 + 10 √ , if > √ aλ · (cid:113)
25 + 10 √ . Of course, now we also have ˆ Q > for y in [ √ aλ , √ aλ · (cid:112)
25 + 10 √ .For n = 5 , we have ˆ Q ( y ) = 84(1 + 4 aλ ) y − aλ ) ≥ aλ ) > . For n ≥ , from the expression (4.18), we find that ˆ Q ( y ) has only one zero ponit ( √ aλ ) t in (0 , + ∞ ) , where t is thesame as the one in Eq.(4.15). Noted that t < , we have ( √ aλ ) t < √ aλ . So we have Q > .Above discussions show that D = 6 is special, one might get negative Q if the condition (4.22) is broken. The details canbe found in Fig.2. Obviously, for a given λ , we have an upper bound of a . However, this upper bound approaches infinitywhen λ approaches zero. This is very different from the case with K = 1 . Physically, this suggests: by choosing the valueof λ , the black hole can be arbitrary small without breaking the hyperbolicity of the tensor perturbation equation. - - - - - - λ a FIG. 2: n = 4 , the case for Λ < and K = 0 . In the shadow region, inequalities (4.22) and aλ > are satisfied. (ii). In the case with aλ = 0 , we have x ∈ (0 , , and then y ∈ (0 , . Q has a same form as in Eq.(4.20). It is obviousthat Q = 0 when n = 3 , , and Q < when n = 4 , and Q > when n ≥ .(iii). In the case aλ < , we have x ∈ (cid:32)(cid:18) aλ (cid:19) n +1 , (cid:35) , y ∈ (0 , . From the expression (4.18) and the condition aλ < , we have we have ˆ Q ( y ) < for n = 3 , , .Of course, this also means Q < for n ≤ .For n ≥ , ˆ Q ( y ) has only one zero point s in (0 , + ∞ ) , where s is given by Eq.(4.21). Therefore, when < s , we have Q < in (0 , ; when > s , we have Q < in (0 , s ) and Q > in ( s, . Λ < and K = − case For n ≥ , when ˜ M − n + 1) (cid:40) − n (cid:34) − ( n −
1) + (cid:113) ( n − + 4˜ α ˜Λ( n − n + 1)( n + 1)˜Λ (cid:35) n − × (cid:20) − α ˜Λ + n + 4 n ˜ α ˜Λ + (cid:113) ( n − + 4˜ α ˜Λ( n − n + 1) (cid:21) (cid:41) > , (4.23)there is a black hole horizon [34]. With this condition, the outer event horizon is not degenerate. Furthermore, to ensure theexistence of the event horizon, we also have r + > √ α , i.e., a < / . Since the mass parameter ˜ M is not necessary to bepositive when K = − , we will discuss the hyperbolicity as follows.(1): Firstly, let us consider the case with positive mass parameter ˜ M , i.e., the case with ˜ M > . ˜ M > implies a > λ .So we get aλ < (1 − a ) . The quantity Q and its numerator ˆ Q still have forms (4.16) and (4.18), but now y is defined as y = (cid:112) aλ + 4 a ( − λ − a ) x n +1 . (i). If aλ > , then y ∈ (cid:2) √ aλ, − a (cid:3) . For n = 3 and n = 5 , with the same logic, we have ˆ Q > , and then Q > . For n = 4 , we have ˆ Q ( y ) = − y + 50(1 + 4 aλ ) y − aλ ) , (4.24)and then ˆ Q > , y ∈ (cid:104) √ aλ , √ aλ · (cid:112)
25 + 10 √ (cid:17) , ˆ Q < , y ∈ (cid:16) √ aλ · (cid:112)
25 + 10 √ , + ∞ (cid:17) . Therefore, ˆ Q > for all y in [ √ aλ, − a ] when − a < √ aλ · (cid:113)
25 + 10 √ . (4.25)We also have ˆ Q < for y in ( √ aλ · (cid:112)
25 + 10 √ , − a ] when − a > √ aλ · (cid:113)
25 + 10 √ , and ˆ Q > for y in [ √ aλ , √ aλ · (cid:112)
25 + 10 √ . For n ≥ , we find that ˆ Q ( y ) has only one zero point ( √ aλ ) t in (0 , + ∞ ) . Noted that t < , we have ( √ aλ ) t < √ aλ . So, we have Q > .So the D = 6 is special. To ensure Q > , the allowed parameters have been given in the shadow region of Fig.3. Thedashed line in Fig.3 is for a = 1 + λ . So the region above this line corresponds to ˜ M > .(ii). When aλ = 0 , x ∈ (0 , , and then y ∈ (0 , − a ] . Now Q has a form in Eq.(4.20). It is obvious that Q = 0 when n = 3 , , and Q < when n = 4 , and Q > when n ≥ .(iii). When aλ < , we have x ∈ (cid:32)(cid:20) − (1 + 4 aλ )4 a ( − λ − a ) (cid:21) n +1 , (cid:35) , - - - - - - λ a FIG. 3: n = 4 , the case for Λ < and K = − . In the shadow region, inequalities (4.25) , aλ > , a > λ , and a < / aresatisfied. Above the dashed line, the mass parameter ˜ M is positive. Bellow the solid black line, the hyperbolicity is satisfied outside the eventhorizon. and then y ∈ (0 , . Obviously, we have ˆ Q ( y ) < when n = 3 , , . For n ≥ , ˆ Q ( y ) has only one zero point s in (0 , + ∞ ) , where s is given by Eq.(4.21). Therefore, we have Q < for all y in (0 , when − a < s . When − a > s ,we have Q < for y in (0 , s ) and Q > for y in ( s, − a ] .(2): In the case with ˜ M = 0 . Noted that ≤ aλ < , we have Q = √ aλ ≥ . (4.26)(3): In the case with ˜ M < . Obviously, we have < aλ < . ˜ M < implies a < λ , and we have √ aλ > − a . Consider the region outside the event horizon, i.e., the region with x ∈ [0 , , then we can define y = (cid:112) aλ + 4 a ( − λ − a ) x n +1 . It is easy to find y ∈ (cid:2) − a, √ aλ (cid:3) , and ˆ Q has the same form as the one in (4.18). When K = − , the black hole mighthave inner horizon [34]. Inner and outer event horizons both exist when (2˜ α ) n − ˜ α (1 + 4˜ α ˜Λ) + ˜ M < , (4.27)and only one event horizon exists when (2˜ α ) n − ˜ α (1 + 4˜ α ˜Λ) + ˜ M > . (4.28)For n = 3 , we have ˆ Q ( y ) = 24(1 + 4 aλ ) y − aλ ) . This implies ˆ Q > , y ∈ (cid:16)(cid:112) aλ ) / , √ aλ (cid:105) , ˆ Q < , y ∈ (cid:16) , (cid:112) aλ ) / (cid:17) . ˆ Q > for all y in [1 − a, √ aλ ] when − a > (cid:112) aλ ) / . We have ˆ Q < for y in [1 − a, (cid:112) aλ ) / when − a < (cid:112) aλ ) / , and ˆ Q > for y in ( (cid:112) aλ ) / , √ aλ ] .For n = 4 , we have ˆ Q ( y ) = − y + 50(1 + 4 aλ ) y − aλ ) . This equation gives ˆ Q > , y ∈ (cid:16) √ aλ · (cid:112) − √ , √ aλ (cid:105) , ˆ Q < , y ∈ (cid:16) , √ aλ · (cid:112) − √ (cid:17) . So we have ˆ Q > for all y in [1 − a, √ aλ ] when − a > √ aλ · (cid:113) − √ . (4.29)However, we have ˆ Q < for y in [1 − a, √ aλ · (cid:112) − √ when − a < √ aλ · (cid:113) − √ , and ˆ Q > for y in ( √ aλ · (cid:112) − √ , √ aλ ] .For n = 5 , we have ˆ Q ( y ) = 84(1 + 4 aλ ) y − aλ ) . This equation tells us we have ˆ Q > , y ∈ (cid:16)(cid:112) aλ ) / , √ aλ (cid:105) , ˆ Q < , y ∈ (cid:16) , (cid:112) aλ ) / (cid:17) . So we have ˆ Q > for all y in (cid:2) − a, √ aλ (cid:3) when − a > (cid:112) aλ ) / . We have ˆ Q < for y in [1 − a, (cid:112) aλ ) / when − a < (cid:112) aλ ) / , and ˆ Q > for y in ( (cid:112) aλ ) / , √ aλ ] .For n ≥ , we have ˆ Q ( y ) = ( n − n − y + 2( n + 1)(2 n − aλ ) y − ( n + 1) (1 + 4 aλ ) . It is obvious that ˆ Q ( y ) has only one zero point ( √ aλ ) t in (0 , + ∞ ) , where t is given by Eq.(4.15). Noted that t < , wehave ( √ aλ ) t < √ aλ . Therefore, we have ˆ Q > for y in (cid:2) − a, √ aλ (cid:3) when − a > ( √ aλ ) t . We also have ˆ Q < for y in (cid:2) − a, √ aλt (cid:1) when − a < ( √ aλ ) t , ˆ Q > for y in (cid:0) ( √ aλ ) t, √ aλ (cid:3) .So the situation is complicated when ˜ M < . In this case, hyperbolicity can be broken outside the event horizon for alldimension D ≥ . Here, as an example, Fig.4 for D = 6 or n = 4 has been given to show the details of the range of theparameters. The dotted line in the shadow region of Fig.4 is for the conditions (4.27) and (4.28). When condition (4.27) issatisfied, the solution has two event horizon. To ensure r + is the radius of outer event horizon, we have to impose a condition F x ( x ) < at x = 1 (noted that x = r + /r ), and this gives a > λ + 3 . This corresponds to the dotdashed line in Fig.4. - - - - - - - λ a FIG. 4: n = 4 , the case for Λ < and K = − . In the shadow region, inequalities (4.23), (4.29), a > λ + 3 , and a < λ are satisfied.Above the dotted line in the shadow region, the black hole has only one horizon. Bellow this dotted line, the black hole has two horizons. Thedotdashed line is for a > λ + 3 . The solid black line is for the hyperbolicity. Λ > and K = 1 case Although cosmological horizon exists when Λ > , the discussion on the hyperbolicity is similar to the case with K = 1 and Λ < . Maybe the most significant difference is the existence conditions for the black hole solutions.For n = 3 , when ˜ M > ˜ α and α ˜Λ − α ˜ M > , there is a black horizon . For n ≥ M − n + 1) (cid:40) − n (cid:34) ( n −
1) + (cid:113) ( n − + 4˜ α ˜Λ( n − n + 1)( n + 1)˜Λ (cid:35) n − × (cid:20) − α ˜Λ + n + 4 n ˜ α ˜Λ + (cid:113) ( n − + 4˜ α ˜Λ( n − n + 1) (cid:21) (cid:41) < , (4.30)with ˜ M > , black hole solution exists [34]. The above inequality suggests a bound on the black hole, i.e., the radius of blackhole horizon must be less than the radius of cosmological horizon. Outside the event horizon of this solution, we have x ∈ [0 , .let y = [1 + 4 aλ + 4 a ( − λ + 1 + a ) x n +1 ] , then y ∈ (cid:2) √ aλ, a (cid:3) . By this definition, the numerator of Q is the same as the one in Eq.(4.16). Except Λ > , all ofthe discussion and results are the same as the case with K = 1 and Λ < .8For n = 3 , n = 5 , and n ≥ , Q is always positive outside the black hole horizon. Actually, Q is positive for x ∈ [0 , + ∞ ) ,where + ∞ corresponds to r = 0 , i.e., the singularities of the spacetimes. D = 6 or n = 4 is also special, the numerator of Q is still given by (4.19), so we have ˆ Q > , y ∈ (cid:104) √ aλ, √ aλ · (cid:112)
25 + 10 √ (cid:17) , ˆ Q < , y ∈ (cid:16) √ aλ · (cid:112)
25 + 10 √ , + ∞ (cid:17) . Thus, we have
Q > for all y ∈ (cid:2) √ aλ, a (cid:3) when a < √ aλ · (cid:113)
25 + 10 √ . (4.31)However, we have Q < if y ∈ (cid:0) √ aλ · (cid:112)
25 + 10 √ , a (cid:3) when a > √ aλ · (cid:113)
25 + 10 √ , and Q > if y ∈ [ √ aλ , √ aλ · (cid:112)
25 + 10 √ .It should be point out here: There are two horizons in this case, i.e., black hole horizon and cosmological horizon. Theparameter x is defined by the radius of black hole horizon, i.e., x = r + /r , so we have F x ( x ) < at x = 1 . For n = 4 , F x ( x ) < at x = 1 gives a constraint a > λ − . (4.32)Above discussion show that D = 6 is special, we have to impose a condition (4.31) on the parameter a and λ to ensure thepositiveness of Q outside the event horizon. The allowed range of the parameters have been depicted in Fig.5. The black solidline in Fig.5 is for the condition (4.31). The region below this line has a positive Q . The dotdashed line in Fig.5 is for (4.32). λ a FIG. 5: n = 4 , the case for Λ > and K = 1 . In the shadow region, inequalities (4.30), (4.31), and a > λ − are satisfied. B. regular solution
A globally regular solution of the theory only happens when ˜ M = 0 (The case with M = 0 , K = − , and Λ < is anexception, which has a horizon and has been discussed in the previous part of the paper). For ˜ M = 0 , we have Q = (cid:115) n − n − α Λ n ( n + 1) . (4.33)9So, for the regular solution, we always have Q ≥ . Of course, the above result also implies n − n − α Λ n ( n + 1) ≥ . (4.34)For Λ ≥ , this inequality is obviously right. However, for Λ < , the Gauss-Bonnet coupling α has an upper bound which canbe read out from the above immediately. C. The existence of superluminal mode
When the tensor perturbation equation is hyperbolic, there are travel modes in the theory, and we can discuss the speed of agravitational fluctuation. In this subsection, we study the causality of this kind of tensor perturbation.We investigate the possible superluminal modes by solving the inequalities (3.23-3.25). Actually from these inequalities, wecan get the precise conditions for the existence of a superluminal mode. For the black hole solutions of the theory, we find thatthe superluminal mode is allowed only in the case with ˜ M > . Further, the superluminal mode could exist near the infinity ofthe spacetime. These can be found as follows.Now, we find that I and ˆ J can be expressed as I = 2 a ( n − K + aK − λ ) x n +1 + ( n − aλ )( n − aλ + 4 a ( K + aK − λ ) x n +1 ] , (4.35)and ˆ J = 2( n + 1)( K + aK − λ ) x n − (cid:2) − a ( n − K + aK − λ ) x n +1 + 1 + 4 aλ (cid:3) [1 + 4 aλ + 4 a ( K + aK − λ ) x n +1 ] . (4.36)Firstly, we consider the cases with aλ > . With this condition, we discuss n = 3 , n ≥ separately.(i). For n = 3 , we have I > . When ˜ M > , i.e., K + aK − λ > , we have ˆ J > . Therefore, there is always superluminalmode outside the black hole horizon if ˜ M > . When ˜ M ≤ , i.e., K + aK − λ ≤ , we always have ˆ J ≤ . So thesuperluminal mode is absent when ˜ M < .(ii). For n ≥ , when ˜ M > , i.e., K + aK − λ > , we have I > . By solving ˆ J > , we obtain ≤ x < (cid:20) aλa ( n − K + aK − λ ) (cid:21) n +1 . (4.37)This is the condition for the existence of superluminal mode. When ˜ M = 0 , i.e., K + aK − λ = 0 , the superluminalmode is absent. Actually, in this case, we have I > and ˆ J = 0 . When ˜ M < , i.e., K + aK − λ < , we always have ˆ J < no matter whether I is positive or not. Therefore, there is no superluminal mode when ˜ M < .In the case with aλ ≤ , we always have K + aK − λ ≥ . This suggests that ˆ J ≤ . So there is no superluminalmode in this case.At the end of this section, based on the exact range in (4.37), we list the detailed conditions for the existence of superluminalmodes when K = 1 , λ = 0 , a = 24 . When r > r c , where r c is determined by the rightmost term in the inequality (4.37), therewill be superluminal mode. This result is consistence with the one in [17].Dimensions critical value r c ( r + = 1 ) D = 6 , n = 4 D = 7 , n = 5 D = 8 , n = 6 D = 9 , n = 7 V. NARIAI-TYPE SPACETIME
When r = r = constant, equations of motion imply that R is a constant. It means that ( M , g ab ) is a two-dimensional con-stant curvature spacetime. This kind solution is called the Nariai-type spacetime [31], and the metric in the standard coordinates0is given by ds = − (1 − σρ ) dt + dρ − σρ + r γ ij dz i dz j , (5.1)where σ = (cid:20) ( n −
1) + 2( n − n − n − αKr − r + 2( n − n − αK (cid:21) K . (5.2)From Eq. (2 . , we know that r is the real and positive root of the following algebraic equation
2Λ = n ( n − Kr + n ( n − n − n − αK r . (5.3)Of course, this equation does not always exist a real and positive root r . The simplest case is the solution with K = 0 .Obviously, Λ and σ have to be vanishing when K = 0 , and r is an arbitrary positive constant. So the solution (5.1) reduces toa very simple form. In the case with K (cid:54) = 0 , the condition for real and positive r depends on the dimension n . For n = 3 , thiscondition is K Λ > . For n ≥ , the condition becomes [31] Λ > , K = ± , Λ = 0 , K = − , − n ( n − / [8( n − n − α ] < Λ < , K = − . From the metric (5.1), it is easy to find R = 2 σ , (cid:3) r = 0 , ( Dr ) = 0 . Therefore, P , Q , I , and J have following forms P = − (cid:20) α ( n − n − Kr (cid:21) , (5.4) Q = r + 4( n − n + 6) Kr α + 4( n − n − ( n − K α r [ r + 2 α ( n − n − K ] . (5.5) I = 1 + 2 α ( n − n − Kr , (5.6)and J = 4 Kr + 2( n − n − α . (5.7)Obviously, P can be vanishing only in the case with negative K . Roughly speaking, we find that the superluminal modes appearin the cases K = ± with some special α and Λ . Details can be found in the following subsections. A. K = 1 case From the expression (5.4), one has
P < . It is also easy to find that (5.5) implies Q = r + 4( n − n + 6) r α + 4( n − n − ( n − α r [ r + 2 α ( n − n − > , (5.8)It is easy to find that I and J are both positive when K = 1 . Therefore, there is always superluminal modes in this case.1 B. K = 0 case When K = 0 , we have σ = 0 and Λ = 0 . The metric can be represented as ds = − dt + dρ + r δ ij dz i dz j . (5.9)In this case, P , Q , I , J have very simple forms i.e., P = − , Q = 1 , I = 1 , J = 0 . Therefore, there is no superluminal modesin this case. C. K = − case This case is a little bit complicated. However, Eq.(5.4) implies that P is always nonpositive. Now Q has a form Q = r − n − n + 6) r α + 4( n − n − ( n − α r [ r − α ( n − n − . (5.10)So Q might be negative. From the expressions (5.6) and (5.7), we find that the signs of I and J depend on the parameters anddimensions. We will discuss the cases with Λ = 0 , Λ > , and Λ < separately. Λ = 0 case
In the case with n = 3 , there is a contradiction in Eq.(5.3). For n ≥ , we have r = ( n − n − α . In this case, Q = − n − n − n + 1)( n − < . (5.11)Since Q < in this case, the discussion of the causality is meaningless. Λ > case In this case, Eq.(5.3) gives r = n (cid:40) − ( n −
1) + (cid:114) ( n − + 8Λ n ( n − n − n − α (cid:41) . (5.12)It is not hard to find that there is no α and Λ satisfying the condition P < and Q > . So Q is always negative, the tensorperturbation equation is not hyperbolic. Λ < case In this case, Eq.(5.3) tells us r = n (cid:40) − ( n − ± (cid:114) ( n − + 8Λ n ( n − n − n − α (cid:41) . (5.13)Solving the condition P < , Q > , I > and J > , we have the following results:dimenion hyperbolicity condition causality condition n = 3 α Λ < − or − < α Λ < − < α Λ < n ≥ − n ( n − n − n − < α Λ < M ( n ) − n ( n − n − n − < α Λ < M ( n ) n ≥ − ) − n ( n − n − n − < α Λ < − n ( n + 1)8( n − n − or N ( n ) < α Λ < N ( n ) < α Λ < (+) and ( − ) correspond to the sign ± in (5.13), and M ( n ) = − n ( n − n + 7 n + 16 n − − n ( n − √ n − n + 248( n − n − ( n − , (5.14) N ( n ) = − n ( n − n + 7 n + 16 n −
36) + 2 n ( n − √ n − n + 248( n − n − ( n − . (5.15)In the “(+)” case of n ≥ , it is easy to find that the range of α Λ become narrower and narrower when n increases. It is alsonot hard to find that M ( n ) and N ( n ) are both monotonically increase when n ≥ . Therefore, if − < α Λ < , the hyperbolicity and causality will be satisfied in all dimensions. VI. DIMENSIONALLY EXTENDED CONSTANT CURVATURE BLACK HOLE
Consider the equations of motion, especially the Eq.(2.9), when α K − ( Dr ) r = 0 , one gets a solution with α ˜Λ = 0 . This corresponds to the dimensionally extended constant curvature black hole given byBanados, Teitelboim and Zanelli [31, 35]. This kind of solution is given by ds = − h ( r ) e δ ( t,r ) dt + h − ( r ) dr + r γ ij dz i dz j , (6.1)where h ( r ) = K + r α , (6.2)and δ ( t, r ) is an arbitrary function. For this spacetime, we find that P is always vanishing, and Q = − α (cid:20) h (cid:48) δ (cid:48) + 2 hδ (cid:48)(cid:48) + 2 h ( δ (cid:48) ) + 2( n − hδ (cid:48) r (cid:21) , (6.3)where (cid:48) stands for ∂/∂r . Obviously, Q is also vanishing if δ does not depend on r , i.e., δ = δ ( t ) . So the principle symbol for thetensor perturbation equation in this background is totally degenerated.Here we show why that P is always vanishing. From the solution, it is not hard to find D a D b rD a D b r = 12 (cid:104) ( h (cid:48) ) + 2 hh (cid:48) δ (cid:48) + 2 h ( δ (cid:48) ) (cid:105) . (6.4)By using (cid:3) r = h (cid:48) + hδ (cid:48) , (6.5)we get D a D b rD a D b r = 12 (cid:20)(cid:0) (cid:3) r (cid:1) + 12 h ( δ (cid:48) ) (cid:21) . (6.6)Therefore, substituting this result into the expression (43) in Appendix B, we obtain P = 4( n − r α h ( δ (cid:48) ) − (cid:26) − α ( n − (cid:20) (cid:3) rr − ( n − K − ( Dr ) r (cid:21)(cid:27) . (6.7)Consider that function h is given by (6.2), we obtain P = 0 . With the above relations, we also get the expression of Q , i.e.,Eq.(6.3).3The above calculation shows the tensor perturbation equation is not hyperbolic in this case, and there is no gravitational waveon this dimensionally extended constant curvature black hole spacetime. VII. VAIDYA SPACETIMES
A Vaidya spacetime is a solution of the theory with radiation matter, the metric of the spacetime can be expressed as ds = − F ( v , r ) dv + 2 dvdr + r γ ij dz i dz j . (7.1)In general, the energy-momentum tensor of the radiation matter satisfy T (cid:96)(cid:96) (cid:54) = 0 , and T nn = 0 , or T (cid:96)(cid:96) = 0 , and T nn (cid:54) = 0 . Therefore, from Eq.(3.13), we have P ≤ . From the above metric, we have R = − F (cid:48)(cid:48) , (cid:3) r = F (cid:48) , ( Dr ) = F , (7.2)where (cid:48) stands for ∂/∂r as before. Substituting the quantities in (7.2) into Eq.(3.11), we can get Q = 1 + 2 α (cid:20) − F (cid:48)(cid:48) − n − F (cid:48) r + ( n − n − K − Fr (cid:21) , (7.3)where F ( v , r ) = K + r α (cid:34) ∓ (cid:115) κ D ˜ αM ( v ) nV Kn r n +1 + 4˜ α ˜Λ (cid:35) . (7.4)The trapping horizon or apparent horizon of Vaidya spacetime is given by F ( v, r ) = 0 , and we can get the radius of the apparenthorizon r A ( v ) . Instead of the radius of event horizon r + in Eq.(4.7), in the untrapped region of the spacetime, the radius of theapparent horizon r A ( v ) can be used to define x , a , and λ , i.e., we have x = r A r , a = ˜ αr A , λ = r A ˜Λ . Although these quantities depend on v , in the discussion of the hyperbolicity and causality, the algebraic structure is the same asthe case of the static, and we can get nearly the same conclusions as the static cases. We will not repeat this kind of discussionhere. VIII. SUMMARY AND DISCUSSION
In this paper, we have obtained the general master equations of tensor type for the warped spacetimes with the metric (2.6),i.e., Eq.(3.3) or Eq.(3.9). These new master equations do not depend on the mode expansion. Of course, one can introduce theharmonic tensor T ij on the maximally symmetric space ( N , γ ij ) [33], i.e., the functions satisfying ˆ∆ T ij = ( − k + 2 nK ) T ij , and expand h ij = h TT ij as h ij /r = h T T ij (the summation on k implied), then we can get the equation for each mode h T .Actually, in Appendix A, we have get the more general master equations (30) for the D = m + n dimension warped spacetimesin which ( N , γ ij ) is an Einstein manifold.Based on the master equation, especially the effective potential in [25], Reall has provided a smart way to study the hyperbol-icity and causality of the perturbation equations in Einstein-Gauss-Bonnet theory [17]. This method is focused on the large k limit in the mode expansion. Now, from our formula (3.9), it is obvious this limit can help people extract the information about P ij ˆ D i ˆ D j from the effective potential. However, this also implies the method is only valid for the large k modes. This is notnecessary in the discussion based on our new master equations.Furthermore, by using this new master equation, we show that effective metric or acoustic metric of the tensor perturbation4equation defined by Reall in [17] can be generalized to the cases without a static condition. In fact, we have get the effective met-ric P MN in Eq.(3.10). With this effective metric in hand, we can study the hyperbolicity and causality of the tensor perturbationon all vacuum solutions of the theory. We have given the explicit conditions that P MN is Lorentzian, i.e., Q > in Eq.(3.23).Under the assumption (3.14), when condition (3.24) and (3.25) are satisfied, the causality is broken, and superluminary modeexists.For each vacuum solution which can be written in the form of Eq.(2.6), the exact hyperbolic condition of the tensor perturba-tion equations has been given. Among the black hole solutions, when ˜ M > and α ˜Λ > , D = 6 or n = 4 is very specialbecause the hyperbolicity might be an issue outside the event horizon only in this case. We have found the analytic hyperboliccondition of the tensor perturbation equation on this background. In the case Λ = 0 , only K = 1 solution exists, and we findthat hyperbolicity can be broken outside the event horizon, when the black hole is small enough, i.e., r + < (cid:112) ˜ α/ . . √ α . This point has been noticed by Reall in [17]. Here, we have found the precise value of r + in the hyperbolic condition. In thecase of Λ < , K = 1 , we find the constraint by the hyperbolicity becomes tighter when | Λ | increases. However, in the caseof Λ < , K = 0 , the situation is quite different: the radius of the black hole can be arbitrary small without breaking thehyperbolicity outside the event horizon if we turn down the abstract value of the cosmological constant. For the positive Λ , thehyperbolic condition also provide constraint of the parameter α , and detailed constraints have been depicted in the figures of thepaper. It should be noted here, for Λ < and K = − , in the case where mass parameter ˜ M < , the situation is complicated,and the hyperbolicity of the tensor perturbation might be broken for all D ≥ . Although the case of α ˜Λ < is not sophysically attractive, for the completeness of the paper, we have also studied this situation with a very short discussion. For thisbackground spacetime, some wave equations exist when the spacetime dimension is greater than seven, i.e., D > .Nariai type solutions and dimensionally extended constant curvature black hole solutions are also considered in this paper,and the constraints by the hyperbolicity and causality on the tensor perturbation have been given explicitly. Our approach canalso applied to dynamical spacetimes, and Vaidya spacetime have been investigated as an example.We have discussed the case m = 2 , but from the Eq.(30) in Appendix A, we can also discuss the cases with m = 1 , and m > . For instance, the case with m = 4 , which is interested for many people. For example, our formulas can be used to thespecial Kaluza-Klein compactification in [36, 37], in which the topology of the spacetime is locally M D ∼ = M × N n with amaximally symmetric space N n with negative sectional curvature.On the other hand, the approach of using the effective metric to study the hyperbolicity and the causality can be applied toother gravity theories, for example, scalar-tensor gravity and F ( R ) gravity theories. Acknowledgement
This work was supported in part by the National Natural Science Foundation of China with grants No.11622543,No.12075232, No.11947301, and No.12047502. This work is also supported by the Fundamental Research Funds for theCentral Universities under Grant No: WK2030000036.
Appendix.A1. Background spacetimes
In this appendix, we derive the perturbation equation for a general warped spacetime M D ∼ = M m × N n with metric g MN dx M dx N = g ab ( y ) dy a dy b + r ( y ) γ ij ( z ) dz i dz j , (1)where coordinates x M = { y , · · · y m ; z , · · · , z n } . The tuple ( M m , g ab ) forms a m − dimension Lorentzian manifold, and ( N n , γ ij ) is an n − dimensional Riemann manifold. This Riemann manifold ( N n , γ ij ) is also assumed to be an Einstein man-ifold, i.e., the Ricci tensor is given by Eq.(2.7). As in section II, the metric compatible covariant derivatives associated with g MN , g ab , and γ ij are denoted by ∇ M , D a , and ˆ D i , respectively.According to the metric (1), we get the nontrivial components of Riemann tensor R KMNL as follows: R dabc = m R dabc ,R jaib = − D a D b rr δ ji ,R lijk = ˆ R lijk − ( Dr ) ( δ lj γ ki − δ li γ kj ) . (2)Here, m R dabc and ˆ R lijk are the Riemann tensors of ( M m , r ab ) and ( N n , γ ij ) , respectively, and ( Dr ) = g ab D a rD b r . Utilizing5Eqs. (2) , we obtain the nonvanished components of the Ricci tensor of the spacetime ( M D , g MN ) , which are given by R ab = m R ab − n D a D b rr ,R ij = (cid:20) − m (cid:3) rr + ( n − K − ( Dr ) r (cid:21) r γ ij . (3)Then the scalar curvature of the spacetime has a form R = m R − n m (cid:3) rr + n ( n − K − ( Dr ) r . (4)In the above Eqs.(2)-(4), m R and m R ab are the scalar curvature and the Ricci tensor of the manifold ( M m , g ab ) , respectively.The symbol (cid:3) = g ab D a D b is the d’Alembertian in ( M m , g ab ) . With the metric (1), any energy-momentum tensor T MN canbe decomposed into T MN = diag (cid:8) T ab ( y ) , r p ( y ) γ ij (cid:9) , in which both T ab and p only depend on the coordinates { y a } .
2. General tensor perturbation equations
For the background spacetime with the metric (1), we discuss the tensor perturbation by setting h ab = 0 , h ai = 0 ,δT ab = 0 , δT ai = 0 , (5)and the tensor h ij is transverse trace free i.e. ˆ D i h ij = 0 , and h = g MN h MN = g ij h ij = γ ij h ij r = r γ ij h ij = 0 . (6)As a result, in Einstein-Gauss-Bonnet gravity theory, the nontrivial components of the perturbation equation (3.1) are δG ij + Λ h ij + αδH ij = κ D δT ij . (7)To get the detailed forms of these equations, we have to calculate the perturbation of the geometric quantities associated to thespacetime. Consider the perturbation of the metric g MN → g MN + h MN , one can get δR PMNL = − (cid:104) ( ∇ M ∇ N − ∇ N ∇ M ) h PL + ( ∇ M ∇ L h PN − ∇ N ∇ L h PM ) − ( ∇ M ∇ P h NL − ∇ N ∇ P h ML ) (cid:105) , (8)and δR MNLP = g SP δR SMNL + R SMNL h SP . By using Eq. (5) and Eq. (8), we find that the perturbation of the Riemann tensor satisfies δR abcd = 0 , (9) δR aibj = 2 D a rD b rr h ij − (cid:20) D a rD b h ij r + D b rD a h ij r + r D a D b (cid:18) h ij r (cid:19)(cid:21) − D a D b rr h ij , (10)and δR ijkl = r γ ml R mijn h nk + ˆ R nijk h mn ) − r γ ml (cid:18) ˆ D i ˆ D k h mj − ˆ D j ˆ D k h mi − r ˆ D i ˆ D m h jk + 1 r ˆ D j ˆ D m h ik (cid:19) − r γ ml D a r (cid:20) rγ ik D a h mj − rγ jk D a h mi − δ mi r ( D a h jk ) + δ mj r ( D a h ik ) (cid:21) + ( Dr ) ( γ kj h il − γ ki h jl ) . (11)6The perturbation of the Ricci tensor can be put into forms δR ab = 2 D a rD b r ( γ ij h ij ) r − γ ij r ( D a rD b h ij + D b rD a h ij ) − D a D b h , (12)and δR ij = − r m (cid:3) (cid:18) h ij r (cid:19) − n D a rr D a h ij + 12 r (cid:16) ˆ D i ˆ D k h jk + ˆ D j ˆ D k h ik (cid:17) − r ˆ∆ h ij + (cid:20) ( n − Kr + ( Dr ) r − m (cid:3) rr (cid:21) h ij − r ˆ R k li j h kl −
12 ˆ D i ˆ D j h − r D a rD a hγ ij . (13)By these, one can get the perturbation for the scalar curvature δR = − m (cid:3) h − r ˆ∆ h + 1 r ˆ D i ˆ D j h ij −
12 ( n + 2) D a rr γ ij D a h ij − n r D a rD a h + ( n + 2)( Dr ) r h − K ( n − hr , (14)where ˆ∆ = γ ij ˆ D i ˆ D j denotes the Laplace-Beltrami operator of ( N n , γ ij ) . Furthermore, by using Eqs.(9)-(14) and consideringthat the tensor h ij is transverse trace free, we obtain the necessary terms to calculate the perturbation of the Gauss-Bonnet tensor H ij : RδR ij = (cid:34) m R − n m (cid:3) rr + n ( n − K − ( Dr ) r (cid:35)(cid:40) − r m (cid:3) (cid:18) h ij r (cid:19) − n D a rr D a h ij − r ˆ∆ h ij + (cid:20) ( n − Kr + ( Dr ) r − m (cid:3) rr (cid:21) h ij − r ˆ R k li j h kl (cid:41) , (15) R Mj δR iM = (cid:20) − m (cid:3) rr + ( n − K − ( Dr ) r (cid:21) (cid:26) − r m (cid:3) (cid:18) h ij r (cid:19) − n D a rr D a h ij − r ˆ∆ h ij + (cid:20) ( n − Kr + ( Dr ) r − m (cid:3) rr (cid:21) h ij − r ˆ R k li j h kl (cid:27) , (16) R iM δR Mj = − (cid:20) − m (cid:3) rr + ( n − K − ( Dr ) r (cid:21) h ij + (cid:20) − m (cid:3) rr + ( n − K − ( Dr ) r (cid:21) (cid:26) − r m (cid:3) (cid:18) h ij r (cid:19) − n D a rr D a h ij − r ˆ∆ h ij + (cid:20) ( n − Kr + ( Dr ) r − m (cid:3) rr (cid:21) h ij − r ˆ R k li j h kl (cid:27) , (17) R MN δR iMjN = (cid:18) m R ab − n D a D b rr (cid:19) (cid:40) D a rD b rr h ij − (cid:20) D a rD b h ij r + D b rD a h ij r + r D a D b (cid:18) h ij r (cid:19)(cid:21) − D a D b rr h ij (cid:41) + (cid:20) − m (cid:3) rr + ( n − K − ( Dr ) r (cid:21) (cid:40) K ( n − h ij r − r ˆ∆ h ij + r γ kj D a rD a h ki − n − r D a rD a h ij + ( Dr ) r h ij (cid:41) , (18)7 R iMjN δR MN = − r (cid:20) − m (cid:3) rr + ( n − K − ( Dr ) r (cid:21) (cid:104) ˆ R m ni j h mn + ( Dr ) h ij (cid:105) + 1 r ˆ R k li j (cid:26) − r m (cid:3) (cid:18) h kl r (cid:19) − n D a rr D a h kl − r ˆ∆ h kl + (cid:20) ( n − Kr + ( Dr ) r − m (cid:3) rr (cid:21) h kl − r ˆ R m nk l h mn (cid:27) + ( Dr ) r (cid:40) − r m (cid:3) (cid:18) h ij r (cid:19) − n D a rr D a h ij − r ˆ∆ h ij + (cid:104) ( n − Kr + ( Dr ) r − m (cid:3) rr (cid:105) h ij − r ˆ R k li j h kl (cid:41) , (19) R MNPi δR jMNP = − D a D b rr (cid:26) D a rD b rr h ij − (cid:20) D a rD b h ij r + D b rD a h ij r + r D a D b (cid:18) h ij r (cid:19)(cid:21) − D a D b rr h ij (cid:27) + 1 r ˆ R mkli (cid:40) r γ pl (cid:16) ˆ R pjmn h nk + ˆ R njmk h pn (cid:17) − r γ pl (cid:104) ˆ D j ˆ D k h pm − ˆ D m ˆ D k h pj − r ˆ D j ˆ D p h mk + 1 r ˆ D m ˆ D p h jk (cid:105)(cid:41) − r γ pl D a r (cid:104) rγ jk ( D a h pm ) − rγ mk ( D a h pj ) − δ pj r ( D a h mk ) + δ pm r ( D a h jk )+( Dr ) ( γ km h jl − γ kj h ml ) (cid:105) − Dr ) r K ( n − h ij + ( Dr ) r ˆ∆ h ij + ( n − Dr ) ] r h ij + ( Dr ) D a r r ( n − (cid:18) rγ mi D a h mj + D a h ij r (cid:19) , (20) R jMNP δR MNPi = − D a D b rr (cid:40) D a rD b rr h ij − (cid:20) D a rD b h ij r + D b rD a h ij r + r D a D b (cid:18) h ij r (cid:19)(cid:21) (cid:41) + 1 r ˆ R mklj (cid:40) r γ pl ( ˆ R pimn h nk + ˆ R nimk h pn ) − r γ pl (cid:104) ˆ D i ˆ D k h pm − ˆ D m ˆ D k h pi − r ˆ D i ˆ D p h mk + 1 r ˆ D m ˆ D p h ik (cid:105)(cid:41) − r γ pl D a r (cid:104) rγ ik ( D a h pm ) − rγ mk ( D a h pi ) − δ pi r ( D a h mk )+ δ pm r ( D a h ik ) + ( Dr ) ( γ km h il − γ ki h ml ) (cid:105) − Dr ) r K ( n − h ij + ( Dr ) r ˆ∆ h ij + ( Dr ) D a r r ( n − (cid:18) rγ mj D a h mi + D a h ij r (cid:19) + (4 − n )[( Dr ) ] h ij r − ˆ R jmnp ˆ R istu (cid:0) h ms γ nt γ pu + h nt γ ms γ pu + h pu γ ms γ nt (cid:1) + 4( Dr ) r (cid:104) R s ti j h st + K ( n − h ij (cid:105) , (21)and δL GB = − R ijkl ˆ R nijk h nl r − ˆ R ijkl r (cid:16) ˆ D i ˆ D k h jl − ˆ D j ˆ D k h il − ˆ D i ˆ D l h jk + ˆ D j ˆ D l h ik (cid:17) . (22)From Eq.(2.5), we have δH ij = 2( RδR ij + R ij δR ) − R Mj δR iM + R iM δR Mj ) − R MN δR iMjN + R iMjN δR MN )+2( R MNPi δR jMNP + R jMNP δR MNPi ) − h ij L GB − g ij δL GB . (23)For the Einstein manifold, the relation between the Weyl tensor and the Riemann tensor is given by ˆ C ijkl = ˆ R ijkl − K ( γ ik γ lj − γ il γ kj ) . (24)8Substituting Eqs.(15)-(22), and Eq. (24) into Eq.(23), after lengthy calculation, we finally obtain the exact formula of δH ij : δH ij r = 4 (cid:40) m G ab − ( n − D a D b rr − (cid:20)
12 ( n − n − K − ( Dr ) r − ( n − m (cid:3) rr (cid:21) g ab (cid:41) D a D b (cid:18) h ij r (cid:19) + 8 (cid:40) m G ab − ( n − D a D b rr −
14 ( n − (cid:34) m R − n − m (cid:3) rr + ( n − n − K − ( Dr ) r (cid:35) g ab (cid:41) × D b rr D a (cid:18) h ij r (cid:19) + 2 (cid:20) − m Rr + 2( n − m (cid:3) rr − ( n − n − K − ( Dr ) r (cid:21) ˆ∆ (cid:18) h ij r (cid:19) + 4 (cid:40) − m R · m (cid:3) rr + 2( n − (cid:16) m (cid:3) rr (cid:17) + n m R · Kr − ( n − m R · ( Dr ) r − n (3 n − Kr · m (cid:3) rr +3( n − n − n −
3) ( Dr ) r · m (cid:3) rr + ( n − n − n − (cid:104) ( Dr ) r (cid:105) +( n − n − n − (cid:16) Kr (cid:17) − n ( n − n − Kr · ( Dr ) r (cid:20) m R ab − ( n − D a D b rr (cid:21) D a D b rr − L GB (cid:41) (cid:18) h ij r (cid:19) + 2 W ij r . (25)Here L GB = m L GB + 8 n m G ab D a D b rr − n ( n − n − m (cid:3) rr · K − ( Dr ) r + n ( n − n − n − (cid:34) K − ( Dr ) r (cid:35) − n ( n −
1) ( D a D b r )( D a D b r ) r + 2 n ( n − m R · K − ( Dr ) r + 4 n ( n − (cid:16) m (cid:3) rr (cid:17) + ˆ C ijkl ˆ C ijkl r , (26)where m L GB is the Gauss-Bonnet term in the Lorentizan manifold ( M m , g ab ) . The W ij in Eq.(25) contains the terms about theWeyl tensor ˆ C ijkl , and it can be expressed as W ij = 2 ˆ C k li j m (cid:3) (cid:18) h kl r (cid:19) + 2 n − r ˆ C k li j D a rD a (cid:18) h kl r (cid:19) + 2 r ˆ C k li j ˆ∆ (cid:18) h kl r (cid:19) − ˆ C mkli r (cid:104) ˆ D j ˆ D k h ml − ˆ D m ˆ D k h jl − ˆ D j ˆ D l h mk + ˆ D m ˆ D l h jk (cid:105) − ˆ C mklj r (cid:104) ˆ D i ˆ D k h ml − ˆ D m ˆ D k h il − ˆ D i ˆ D l h mk + ˆ D m ˆ D l h ik (cid:105) + ˆ C pqkl r (cid:104) ˆ D p ˆ D k h ql − ˆ D q ˆ D k h pl − ˆ D p ˆ D l h qk + ˆ D q ˆ D l h pk (cid:105) γ ij + 4 r ˆ C k li j ˆ C m nk l h mn + 2 r ˆ C k mj n ˆ C n lm i h kl + γ ij r ˆ C pqkl ˆ C pqkn h nl + 2 (cid:104) − m R + 2( n − m (cid:3) rr − ( n − n + 16) Kr +( n − n −
4) ( Dr ) r (cid:105) ˆ C k li j (cid:16) h kl r (cid:17) . (27)At the same time, we can also get the perturbation of the Einstein tensor δG ij δG ij r = − m (cid:3) (cid:18) h ij r (cid:19) − n D a rr D a (cid:18) h ij r (cid:19) + ˆ∆ L r (cid:18) h ij r (cid:19) − (cid:34) m R − n − m (cid:3) rr + n ( n − Kr − ( n − n −
2) ( Dr ) r (cid:35) (cid:18) h ij r (cid:19) , (28)9where ˆ∆ L is the Lichnerowicz operator acting on the symmetric rank-2 tensor on ( N n , γ ij ) . The relation between this operatorand usual Laplace operator is given by the following formula, ˆ∆ L s ij = − ˆ∆ s ij + ˆ R ki s kj + ˆ R kj s ik − R k li j s kl , (29)where s ij is an arbitrary symmetric tensor field tensor on ( N n , γ ij ) . When ( N n , γ ij ) is maximal symmetry manifold, it shouldbe noted here that ˆ∆ L s ij = ( − ˆ∆ + 2 nK ) s ij . From Eqs.(25), (27), (28), and (29), we find that Eq.(7) becomes (cid:16) P abijkl D a D b + P mnijkl ˆ D m ˆ D n + P aijkl D a + V ijkl (cid:17)(cid:16) h kl r (cid:17) = − κ D r δT ij , (30)where P abijkl = P ab δ ik δ jl − αr g ab ˆ C ikjl , (31) P aijkl = P a δ ik δ jl − α ( n − D a rr ˆ C ikjl r , (32) P mnijkl = P mn δ ik δ jl + 4 αr (cid:0) ˆ C jknl δ im + ˆ C iknl δ jm + ˆ C jmln δ ik + ˆ C imln δ jk − ˆ C mknl γ ij − ˆ C ikjl γ mn (cid:1) , (33) V ijkl = V δ ik δ jl + 2 ˆ C ikjl r + α (cid:40) (cid:34) m R − n − m (cid:3) rr + ( n − n + 16) Kr − ( n − n −
4) ( Dr ) r (cid:35) ˆ C ikjl r − r ˆ C imjn ˆ C mknl + 4 r ˆ C mnjk ˆ C mnil − r ˆ C mnpl ˆ C mnpk γ ij + ˆ C mnpq ˆ C mnpq r δ ik δ jl (cid:41) , (34)In the above equations P ab = g ab + 2( n − α (cid:26) D a D b rr + (cid:20) ( n − K − ( Dr ) r − m (cid:3) rr (cid:21) g ab (cid:27) − α · m G ab , (35) P mn = (cid:40) α (cid:20) m R − n − m (cid:3) rr + ( n − n − K − ( Dr ) r (cid:21) (cid:41) γ mn r , (36) P a = n D a rr + 2( n − α (cid:40) D a D b rr + (cid:104) m R − n − m (cid:3) rr +( n − n − K − ( Dr ) r (cid:105) g ab (cid:41) D b rr − α · m G ab D b rr , (37)0and V = m R − n − m (cid:3) rr + n ( n − Kr − ( n − n − Dr ) r − Λ+ α (cid:40) m L GB + 8( n − · m G ab D a D b rr − n − n −
2) ( D a D b r )( D a D b r ) r + 4( n − n − (cid:18) m (cid:3) rr (cid:19) + 2 n ( n − K · m Rr − n − n −
2) ( Dr ) · m Rr − n ( n − K · m (cid:3) rr + 4( n − n − n −
3) ( Dr ) · m (cid:3) rr − n ( n − ( n − K · ( Dr ) r + ( n − n − n − n − K r + ( n − n − n − n − (cid:20) ( Dr ) r (cid:21) (cid:41) . (38)Eq.(30) is the most general master equation of tensor type for the warped spacetime with the metric (1). In general, the compo-nents of h ij = h TT ij are coupled to each other if ( N , γ ij ) is not maximally symmetric. If we restrict to the case with m = 2 , wehave G ab = 0 , L GB = 0 , and the above equations reduce to Eqs.(3.5), (3.6), (3.7), (3.8) in Sec.III. If we further restrict to thecase that ( N , γ ij ) is maximally symmetric, Eq.(30) reduces to Eq.(3.9) in Sec.III. Appendix.B
In two-dimensional Lorentz manifold, the volume element (cid:15) ab and the metric g ab can be related by (cid:15) ac (cid:15) bd = g ad g cb − g ab g cd . (39)Hence, the determinant of P ab can be expressed as P = 12 (cid:15) ac (cid:15) bd P ab P cd = 12 P ab P ab −
12 [Tr( P )] , (40)where Tr( P ) = g ab P ab is the trace of the tensor P ab . Simple calculations show Tr( P ) = 2 + α (cid:20) n − n − K − ( Dr ) r − n − (cid:3) rr (cid:21) , (41)and P ab P ab = 2 + 2 α (cid:20) n − n − K − ( Dr ) r − n − (cid:3) rr (cid:21) + α (cid:40) n − (cid:3) rr (cid:20) n − n − K − ( Dr ) r − n − (cid:3) rr (cid:21) +2 (cid:20) n − n − K − ( Dr ) r − n − (cid:3) rr (cid:21) +16( n − ( D a D b r )( D a D b r ) r (cid:41) . (42)1By using expressions of Tr ( P ) and P ab P ab in the above equations, we arrive at P = − − α (cid:20) n − n − K − ( Dr ) r − n − (cid:3) rr (cid:21) + α (cid:40) n − ( D a D b r )( D a D b r ) r − n − (cid:18) (cid:3) rr (cid:19) +8( n − ( n − (cid:3) r [ K − ( Dr ) ] r − n − ( n − [ K − ( Dr ) ] r (cid:41) . (43)Consider Eq.(2.9), we obtain D a D b r − (cid:3) rg ab = rn κ D (cid:18) g cd T cd g ab − T ab (cid:19) · (cid:34) α ( n − n − K − ( Dr ) r (cid:35) − . (44)and ( D a D b r )( D a D b r ) −
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