No abstract

We show that in four or more spacetime dimensions, the Einstein equations for gravitational perturbations of maximally symmetric vacuum black holes can be reduced to a single second-order wave equation in a two-dimensional static spacetime, irrespective of the mode of perturbations. Our starting point is the gauge-invariant formalism for perturbations in an arbitrary number of dimensions developed by the present authors, and the variable for the final second-order master equation is given by a simple combination of gauge-invariant variables in this formalism. Our formulation applies to the case of non-vanishing as well as vanishing cosmological constant Λ. The sign of the sectional curvature K of each spatial section of equipotential surfaces is also kept general. In the four-dimensional Schwarzschild background with Λ = 0 and K = 1, the master equation for a scalar perturbation is identical to the Zerilli equation for the polar mode and the master equation for a vector perturbation is identical to the Regge-Wheeler equation for the axial mode. Furthermore, in the four-dimensional Schwarzschild-anti-de Sitter background with Λ < 0 and K = 0, 1, our equation coincides with those recently derived by Cardoso and Lemos. As a simple application, we prove the perturbative stability and uniqueness of four-dimensional non-extremal spherically symmetric black holes for any Λ. We also point out that there exists no simple relation between scalar-type and vector-type perturbations in higher dimensions, unlike in four dimension. Although in the present paper we treat only the case in which the horizon geometry is maximally symmetric, the final master equations are valid even when the horizon geometry is described by a generic Einstein manifold, if we employ an appropriate reinterpretation of the curvature K and the eigenvalues for harmonic tensors. * )

In the present paper the gauge-invariant formalism is developed for perturbations of the braneworld model in which our universe is realized as a boundary of a higher-dimensional spacetime. For the background model in which the bulk spacetime is (n + m)-dimensional and has the spatial symmetry corresponding to the isometry group of a n-dimensional maximally symmetric space, gauge-invariant equations are derived for perturbations of the bulk spacetime. Further, for the case corresponding to the brane-world model in which m = 2 and the brane is a boundary invariant under the spatial symmetry in the unperturbed background, relations between the gauge-invariant variables describing the bulk perturbations and those for brane perturbations are derived from Israel's junction condition under the assumption of Z2 symmetry. In particular, for the case in which the bulk spacetime is a constant-curvature spacetime, it is shown that the bulk perturbation equations reduce to a single hyperbolic master equation for a master variable, and that the physical condition on the gauge-invariant variable describing the intrinsic stress perturbation of the brane yields a boundary condition for the master equation through the junction condition. On the basis of this formalism, it is pointed out that it seems to be difficult to suppress brane perturbations corresponding to massive excitations for a brane motion giving a realistic expanding universe model.

We investigate the classical stability of the higher-dimensional Schwarzschild black holes against linear perturbations, in the framework of a gauge-invariant formalism for gravitational perturbations of maximally symmetric black holes, recently developed by the authors. The perturbations are classified into the tensor, vector, and scalar-type modes according to their tensorial behaviour on the spherical section of the background metric, where the last two modes correspond respectively to the axial- and the polar-mode in the four-dimensional situation. We show that, for each mode of the perturbations, the spatial derivative part of the master equation is a positive, self-adjoint operator in the $L^2$-Hilbert space, hence that the master equation for each tensorial type of perturbations does not admit normalisable negative-modes which would describe unstable solutions. On the same Schwarzschild background, we also analyse the static perturbation of the scalar mode, and show that there exists no static perturbation which is regular everywhere outside the event horizon and well-behaved at spatial infinity. This checks the uniqueness of the higher-dimensional spherically symmetric, static, vacuum black hole, within the perturbation framework. Our strategy for the stability problem is also applicable to the other higher-dimensional maximally symmetric black holes with non-vanishing cosmological constant. We show that all possible types of maximally symmetric black holes (thus, including the higher-dimensional Schwarzschild-de Sitter and Schwarzschild-anti-de Sitter black holes) are stable against the tensor and the vector perturbations.Comment: 19 pages, 9 figures, references and comments on the generalised black hole case are added, minor changes in text, version to appear in PT

We extend the formulation for perturbations of maximally symmetric black holes in higher dimensions developed by the present authors in a previous paper to a charged black hole background whose horizon is described by an Einstein manifold. For charged black holes, perturbations of electromagnetic fields are coupled to the vector and scalar modes of metric perturbations non-trivially. We show that by taking appropriate combinations of gauge-invariant variables for these perturbations, the perturbation equations for the Einstein-Maxwell system are reduced to two decoupled second-order wave equations describing the behaviour of the electromagnetic mode and the gravitational mode, for any value of the cosmological constant. These wave equations are transformed into Schrödinger-type ODEs through a Fourier transformation with respect to time. Using these equations, we investigate the stability of generalised black holes with charge. We also give explicit expressions for the source terms of these master equations with application to the emission problem of gravitational waves in mind. * )