Remark on the effective potential of the gravitational perturbation in the black hole background projected on the brane

The polar perturbation is examined when the spacetime is expressed by a 4d metric induced from higher-dimensional Schwarzschild geometry. Since the spacetime background is not a vacuum solution of 4d Einstein equation, the various general principles are used to understand the behavior of the energymomentum tensor under the perturbation. It is found that although the general principles fix many components, they cannot fix two components of the energy-momentum tensor. Choosing two components suitably, we derive the effective potential which has a correct 4d limit. Email:[email protected] 1 The spacetime stability/instability problem is an important issue to determine the final state of the gravitational collapse. It is also important in the astrophysical side for the detection of the gravitational wave. Long ago this issue was firstly examined by Regge and Wheeler [1] in the background of the 4d Schwarzschild geometry. In four spacetime dimension there are two types of the gravitational perturbations called axial (or odd-parity) and polar (or even-parity) perturbations [2]. The Schwarzschild geometry in general admits the perturbed equations to be separable and the resultant radial equations can be transformed into the Schrödinger-like form. Thus the effective potential for each perturbation was explicitly derived in Ref. [3,4]. The transformation into the Schrödinger-like expression is important to predict the various physical phenomena because we have much background in quantum mechanics. Recent quantum gravity such as string theories [5] and the brane-world scenarios [6,7] in general introduce the extra dimensions to reconcile general relativity with quantum physics. In this context, recently, much attention is paid to the higher-dimensional spacetime. The various spacetime metrics of the higher-dimensional black holes were derived in Ref. [8,9]. The gravitational perturbations in the higher-dimensional Schwarzschild background were also discussed in Ref. [10]. When the spacetime dimension is more than four, there are three types of the gravitational perturbations called scalar, vector, and tensor perturbations. The scalar and vector perturbations correspond to the polar and the axial perturbations in four dimension. There is no correspondence of the tensor perturbation in four dimension. Recently, the axial perturbation is discussed when the spacetime is a 4d metric induced from the higher-dimensional Schwarzschild black hole [11]. In the following we will briefly review Ref. [11]. Subsequently the polar perturbation will be addressed in this background. We start with an 4d metric which is induced from the (4+n)-dimensional Schwarzschild black hole ds = −h(r)dt + h(r)dr + r(dθ + sin θdφ) (1) where h(r) = 1− (rH/r) . As mentioned, there are two types of the gravitational metric 2 perturbations in four spacetime dimension called axial and polar perturbations [1,2]. In the former case the metric is changed into ds + δsA where δsA = [H0(r)dtdφ+H1(r)drdφ] e iωt sin θ dPl dθ (cos θ) (2) while the metric for the latter case is ds + δsP where δsP = [ H(r)hdt +H(r)hdr + rK(r)(dθ + sin θdφ) + 2H1(r)dtdr ] ePl(cos θ) (3) where Pl(cos θ) is an usual Legendre polynomial. The various r-dependent functions are assumed to be small for the linearization. When n = 0, the metric in Eq.(1) is an usual 4d Schwarzschild metric which is a vacuum solution of the Einstein equation. In this 4d case, therefore, the gravitational linearized fluctuation can be expressed as δRμν = 0 where Rμν is a Ricci tensor. Using this representation the effective potentials for the perturbations were derived long ago [3,4] VA(r) = h [ 2λ+ 2 r2 − 3 r2 ( rH r ) ] (4) VP (r) = h (2λr + 3rH) [ 8λ(λ+ 1) + 12 + 12λ ( rH r ) + 18λ ( rH r )2 + 9 ( rH r )3 ] where λ = (l− 1)(l+ 2)/2. The most interesting feature of the potentials is that they are related to each other as following [2] VP,A(r) = ±β df dr∗ + βf 2 + κf (5) where the upper(lower) sign corresponds to the polar(axial) perturbation and r∗ is a “tortoise” coordinate defined dr/dr∗ = h. In Eq.(5) β = 3rH , κ = 4λ(λ+ 1) and f = h r(2λr + 3rH) . (6) In fact this relation was found when the Newman-Penrose formalism [12] is applied to the gravitational perturbations. This explicit relation between VP and VA enables us to realize 3 that both potentials have the same transmission coefficient. In this context the following several questions arise naturally: (i) Do the effective potentials exist when the spacetime is a metric induced from the higher-dimensional Schwarzschild black hole? (ii) If the potentials exist, is there any relation between them like Eq.(5)? (iii) Do they have same transmission coefficient like 4d case? We would like to address the first issue as much as possible in this letter. The most difficult problem one copes with when the spacetime is induced from the higher-dimensional one is the fact that Eq.(1) is not a vacuum solution of the 4d Einstein equation if n 6= 0. Thus what we can do is to assume that Eq.(1) is a non-vacuum solution, i.e. Eμν = Tμν where Eμν and Tμν are Einstein and energy-momentum tensors, respectively. Since, however, Tμν is not originated from the real matter living on the brane, but appears effectively in the course of the projection of the bulk metric to the brane, we do not know how Tμν is transformed when the perturbations (2) or (3) is turned on. The only way, in our opinion, to get rid of this difficulty is to get an information on the energy-momentum tensor as much as possible from the general principles. The fluctuation equation in the background of the non-vacuum solution is generally expressed as δEμν = δTμν . Thus the difficulty mentioned above is how to obtain δTμν from the general principles without knowing the exact nature of the matter. The axial perturbation in the background of Eq.(1) was discussed in Ref. [11]. In this case the non-vanishing components of δTμν are δTtφ, δTrφ and δTθφ. The general principles used in Ref. [11]: (i) covariant conservation of T μν (ii) linear dependence of the Einstein equation (iii) the correct 4d limit of the effective potential. It turned out that the first principle generates the unique non-trivial constraint to the energy-momentum tensor. But 1In Ref. [2] the general criterion for the same transmission coefficient of the different onedimensional potentials was derived in terms of the KdV equation, which is well-known in the solitonic theories.