Numerical analysis of quasinormal modes in nearly extremal Schwarzschild-de Sitter spacetimes

We calculate high-order quasinormal modes with large imaginary frequencies for electromagnetic and gravitational perturbations in nearly extremal Schwarzschild-de Sitter spacetimes. Our results show that for low-order quasinormal modes, the analytical approximation formula in the extremal limit derived by Cardoso and Lemos is a quite good approximation for the quasinormal frequencies as long as the model parameter $r_1\kappa_1$ is small enough, where $r_1$ and $\kappa_1$ are the black hole horizon radius and the surface gravity, respectively. For high-order quasinormal modes, to which corresponds quasinormal frequencies with large imaginary parts, on the other hand, this formula becomes inaccurate even for small values of $r_1\kappa_1$. We also find that the real parts of the quasinormal frequencies have oscillating behaviors in the limit of highly damped modes, which are similar to those observed in the case of a Reissner-Nordstr{\" o}m black hole. The amplitude of oscillating ${\rm Re(\omega)}$ as a function of ${\rm Im}(\omega)$ approaches a non-zero constant value for gravitational perturbations and zero for electromagnetic perturbations in the limit of highly damped modes, where $\omega$ denotes the quasinormal frequency. This means that for gravitational perturbations, the real part of quasinormal modes of the nearly extremal Schwarzschild-de Sitter spacetime appears not to approach any constant value in the limit of highly damped modes. On the other hand, for electromagnetic perturbations, the real part of frequency seems to go to zero in the limit.Comment: 9 pages, 7 figures, to appear in Physical Review

Late-Time Tails of Wave Propagation in Higher Dimensional Spacetimes

We study the late-time tails appearing in the propagation of massless fields (scalar, electromagnetic and gravitational) in the vicinities of a D-dimensional Schwarzschild black hole. We find that at late times the fields always exhibit a power-law falloff, but the power-law is highly sensitive to the dimensionality of the spacetime. Accordingly, for odd D>3 we find that the field behaves as t^[-(2l+D-2)] at late times, where l is the angular index determining the angular dependence of the field. This behavior is entirely due to D being odd, it does not depend on the presence of a black hole in the spacetime. Indeed this tails is already present in the flat space Green's function. On the other hand, for even D>4 the field decays as t^[-(2l+3D-8)], and this time there is no contribution from the flat background. This power-law is entirely due to the presence of the black hole. The D=4 case is special and exhibits, as is well known, the t^[-(2l+3)] behavior. In the extra dimensional scenario for our Universe, our results are strictly correct if the extra dimensions are infinite, but also give a good description of the late time behaviour of any field if the large extra dimensions are large enough.Comment: 6 pages, 3 figures, RevTeX4. Version to appear in Rapid Communications of Physical Review

Highly Damped Quasinormal Modes of Kerr Black Holes: A Complete Numerical Investigation

We compute for the first time very highly damped quasinormal modes of the (rotating) Kerr black hole. Our numerical technique is based on a decoupling of the radial and angular equations, performed using a large-frequency expansion for the angular separation constant_{s}A_{l m}. This allows us to go much further in overtone number than ever before. We find that the real part of the quasinormal frequencies approaches a non-zero constant value which does not depend on the spin s of the perturbing field and on the angular index l: \omega_R=m\varpi(a). We numerically compute \varpi(a). Leading-order corrections to the asymptotic frequency are likely to be of order 1/\omega_I. The imaginary part grows without bound, the spacing between consecutive modes being a monotonic function of a.Comment: 5 pages, 3 figure

Field propagation in de Sitter black holes

We present an exhaustive analysis of scalar, electromagnetic and gravitational perturbations in the background of Schwarzchild-de Sitter and Reissner-Nordstrom-de Sitter spacetimes. The field propagation is considered by means of a semi-analytical (WKB) approach and two numerical schemes: the characteristic and general initial value integrations. The results are compared near the extreme cosmological constant regime, where analytical results are presented. A unifying picture is established for the dynamics of different spin fields.Comment: 15 pages, 16 figures, published versio

Quasinormal modes of Schwarzschild black holes in four and higher dimensions

We make a thorough investigation of the asymptotic quasinormal modes of the four and five-dimensional Schwarzschild black hole for scalar, electromagnetic and gravitational perturbations. Our numerical results give full support to all the analytical predictions by Motl and Neitzke, for the leading term. We also compute the first order corrections analytically, by extending to higher dimensions, previous work of Musiri and Siopsis, and find excellent agreement with the numerical results. For generic spacetime dimension number D the first-order corrections go as $\frac{1}{n^{(D-3)/(D-2)}}$. This means that there is a more rapid convergence to the asymptotic value for the five dimensional case than for the four dimensional case, as we also show numerically.Comment: 12 pages, 5 figures, RevTeX4. v2. Typos corrected, references adde

Dirty black holes: Quasinormal modes

In this paper, we investigate the asymptotic nature of the quasinormal modes for "dirty" black holes -- generic static and spherically symmetric spacetimes for which a central black hole is surrounded by arbitrary "matter" fields. We demonstrate that, to the leading asymptotic order, the [imaginary] spacing between modes is precisely equal to the surface gravity, independent of the specifics of the black hole system. Our analytical method is based on locating the complex poles in the first Born approximation for the scattering amplitude. We first verify that our formalism agrees, asymptotically, with previous studies on the Schwarzschild black hole. The analysis is then generalized to more exotic black hole geometries. We also extend considerations to spacetimes with two horizons and briefly discuss the degenerate-horizon scenario.Comment: 15 pages; uses iopart.cls setstack.sty; V2: one additional reference added, no physics changes; V3: two extra references, minor changes in response to referee comment

Dirty black holes: Quasinormal modes for "squeezed" horizons

We consider the quasinormal modes for a class of black hole spacetimes that, informally speaking, contain a closely ``squeezed'' pair of horizons. (This scenario, where the relevant observer is presumed to be ``trapped'' between the horizons, is operationally distinct from near-extremal black holes with an external observer.) It is shown, by analytical means, that the spacing of the quasinormal frequencies equals the surface gravity at the squeezed horizons. Moreover, we can calculate the real part of these frequencies provided that the horizons are sufficiently close together (but not necessarily degenerate or even ``nearly degenerate''). The novelty of our analysis (which extends a model-specific treatment by Cardoso and Lemos) is that we consider ``dirty'' black holes; that is, the observable portion of the (static and spherically symmetric) spacetime is allowed to contain an arbitrary distribution of matter.Comment: 15 pages, uses iopart.cls and setstack.sty V2: Two references added. Also, the appendix now relates our computation of the Regge-Wheeler potential for gravity in a generic "dirty" black hole to the results of Karlovini [gr-qc/0111066

Comparison of area spectra in loop quantum gravity

We compare two area spectra proposed in loop quantum gravity in different approaches to compute the entropy of the Schwarzschild black hole. We describe the black hole in general microcanonical and canonical area ensembles for these spectra. We show that in the canonical ensemble, the results for all statistical quantities for any spectrum can be reproduced by a heuristic picture of Bekenstein up to second order. For one of these spectra - the equally-spaced spectrum - in light of a proposed connection of the black hole area spectrum to the quasinormal mode spectrum and following hep-th/0304135, we present explicit calculations to argue that this spectrum is completely consistent with this connection. This follows without requiring a change in the gauge group of the spin degrees of freedom in this formalism from SU(2) to SO(3). We also show that independent of the area spectrum, the degeneracy of the area observable is bounded by $C A\exp(A/4)$, where $A$ is measured in Planck units and $C$ is a constant of order unity.Comment: 8 pages, Revtex 4, version to appear in Classical and Quantum Gravit

Dynamical Boson Stars

The idea of stable, localized bundles of energy has strong appeal as a model for particles. In the 1950s John Wheeler envisioned such bundles as smooth configurations of electromagnetic energy that he called {\em geons}, but none were found. Instead, particle-like solutions were found in the late 1960s with the addition of a scalar field, and these were given the name {\em boson stars}. Since then, boson stars find use in a wide variety of models as sources of dark matter, as black hole mimickers, in simple models of binary systems, and as a tool in finding black holes in higher dimensions with only a single killing vector. We discuss important varieties of boson stars, their dynamic properties, and some of their uses, concentrating on recent efforts.Comment: 79 pages, 25 figures, invited review for Living Reviews in Relativity; major revision in 201