A Master Equation for Gravitational Perturbations of Maximally Symmetric Black Holes in Higher Dimensions

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. * )

Gravitational perturbations of higher dimensional rotating black holes: Tensor perturbations

“…If we set a = 0 then we have a Schwarzschild(-AdS) black hole and our tensor perturbations should form a subset of the tensor perturbations considered in [4,5,6]. Demanding agreement between our results and those of [6] yields…”

“…5 Asymptotically anti-de Sitter black holes 5.1 Form of the potential and behaviour of solutions Set ω = mΩ H > 0. The potential vanishes at the horizon and is monotonically increasing just outside the horizon.…”

Hidden Symmetries of Higher-Dimensional Rotating Black Holes

“…A certain progress in this direction was achieved recently (see, e.g., [139], [157], [183]). These results are very important for the study of the stability of such black holes and different aspect of the Hawking radiation produced by them.…”

“…Such a separability would provide us the important tools for studying the stability, quasinormal modes, and different aspects of the Hawking radiation of higher-dimensional black holes. A certain progress in this direction was already achieved (see, e.g., [139], [157], [183]). However, most of the results obtained in these directions so far (see also references in Chapter 6) assumed some additional restrictions on the parameters characterizing black hole solutions.…”

Maxwell’s equations in the Myers-Perry geometry

“…For the Schwarzschild-Tangherlini geometry [35], separation of the Klein-Gordon and Maxwell equations is rather straightforward, 2 but the study of gravitational waves in such backgrounds is an active area of research [36][37][38][39][40][41]. In contrast, the vast majority of efforts in studying rotating geometries has been dedicated to scalar and spinor fields [42][43][44][45][46][47][48][49][50][51][52][53][54][55][56] with a few notable exceptions [57][58][59][60].…”