In recent years, there has been considerable interest in theories formulated in anti-de Sitter (AdS) spacetime. However, AdS spacetime fails to be globally hyperbolic, so a classical field satisfying a hyperbolic wave equation on AdS spacetime need not have a well defined dynamics. Nevertheless, AdS spacetime is static, so the possible rules of dynamics for a field satisfying a linear wave equation are constrained by our previous general analysis-given in paper II-where it was shown that the possible choices of dynamics correspond to choices of positive, self-adjoint extensions of a certain differential operator, A. In the present paper, we reduce the analysis of electromagnetic, and gravitational perturbations in AdS spacetime to scalar wave equations. We then apply our general results to analyse the possible dynamics of scalar, electromagnetic, and gravitational perturbations in AdS spacetime. In AdS spacetime, the freedom (if any) in choosing self-adjoint extensions of A corresponds to the freedom (if any) in choosing suitable boundary conditions at infinity, so our analysis determines all of the possible boundary conditions that can be imposed at infinity. In particular, we show that other boundary conditions besides the Dirichlet and Neumann conditions may be possible, depending on the value of the effective mass for scalar field perturbations, and depending on the number of spacetime dimensions and type of mode for electromagnetic and gravitational perturbations.

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

Generic extensions of the standard model predict the existence of ultralight bosonic degrees of freedom. Several ongoing experiments are aimed at detecting these particles or constraining their mass range. Here we show that massive vector fields around rotating black holes can give rise to a strong superradiant instability which extracts angular momentum from the hole. The observation of supermassive spinning black holes imposes limits on this mechanism. We show that current supermassive black hole spin estimates provide the tightest upper limits on the mass of the photon (mv 4 × 10 −20 eV according to our most conservative estimate), and that spin measurements for the largest known supermassive black holes could further lower this bound to mv 10 −22 eV. Our analysis relies on a novel framework to study perturbations of rotating Kerr black holes in the slow-rotation regime, that we developed up to second order in rotation, and that can be extended to other spacetime metrics and other theories. Introduction. The properties of matter making up our universe are mostly unknown. Strong evidence (e.g. from galactic rotation curves and from gravitational lensing) points to the existence of elusive, weakly-interacting matter as the most abundant element in the universe. An interesting possibility is the existence of ultralight bosonic degrees of freedom, such as those appearing in the "string axiverse" scenario [1, 2], or of massive hidden U (1) vector fields, that are also a generic feature of extensions of the standard model [3][4][5][6].Massive fields around rotating black holes (BHs) can trigger a superradiant instability, the so-called "black hole bomb" [7]. This instability is well understood in the case of massive scalar fields [8][9][10][11][12][13][14][15]: it requires the existence of negative energy states in a region around the BH known as the ergoregion. The instability is regulated by the dimensionless parameter M µ (from now on we set G = c = 1), where M is the BH mass and m s = µ is the scalar field mass, and it is most effective when M µ ∼ 1 and for maximally spinning BHs. For a solar mass BH and a field of mass m s ∼ 1 eV the parameter M µ ∼ 10 10 , and therefore in many cases of astrophysical interest the instability timescale is larger than the age of the universe. Superradiant instabilities strong enough to be observationally relevant (M µ ∼ 1) can occur either for light primordial BHs that may have been produced in the early universe [16][17][18] or for ultralight exotic particles found in some extensions of the standard model [1,2]. In the string axiverse scenario, massive scalar fields with

A key result in the proof of black hole uniqueness in 4-dimensions is that a stationary black hole that is "rotating"-i.e., is such that the stationary Killing field is not everywhere normal to the horizon-must be axisymmetric. The proof of this result in 4-dimensions relies on the fact that the orbits of the stationary Killing field on the horizon have the property that they must return to the same null geodesic generator of the horizon after a certain period, P . This latter property follows, in turn, from the fact that the cross-sections of the horizon are two-dimensional spheres. However, in spacetimes of dimension greater than 4, it is no longer true that the orbits of the stationary Killing field on the horizon must return to the same null geodesic generator. In this paper, we prove that, nevertheless, a higher dimensional stationary black hole that is rotating must be axisymmetric. No assumptions are made concerning the topology of the horizon cross-sections other than that they are compact. However, we assume that the horizon is non-degenerate and, as in the 4-dimensional proof, that the spacetime is analytic.

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. * )

We discuss a general method to study linear perturbations of slowly rotating black holes which is valid for any perturbation field, and particularly advantageous when the field equations are not separable. As an illustration of the method we investigate massive vector (Proca) perturbations in the Kerr metric, which do not appear to be separable in the standard Teukolsky formalism. Working in a perturbative scheme, we discuss two important effects induced by rotation: a Zeemanlike shift of nonaxisymmetric quasinormal modes and bound states with different azimuthal number m, and the coupling between axial and polar modes with different multipolar index ℓ. We explicitly compute the perturbation equations up to second order in rotation, but in principle the method can be extended to any order. Working at first order in rotation we show that polar and axial Proca modes can be computed by solving two decoupled sets of equations, and we derive a single master equation describing axial perturbations of spin s = 0 and s = ±1. By extending the calculation to second order we can study the superradiant regime of Proca perturbations in a self-consistent way. For the first time we show that Proca fields around Kerr black holes exhibit a superradiant instability, which is significantly stronger than for massive scalar fields. Because of this instability, astrophysical observations of spinning black holes provide the tightest upper limit on the mass of the photon: mγ 4 × 10 −20 eV under our most conservative assumptions. Spin measurements for the largest black holes could reduce this bound to mγ 10 −22 eV or lower.