The Quasinormal Mode Spectrum of a Kerr Black Hole in the Eikonal Limit

It is well established that the response of a black hole to a generic perturbation is characterized by a spectrum of damped resonances, called quasinormal modes; and that, in the limit of large angular momentum ($l \gg 1$), the quasinormal mode frequency spectrum is related to the properties of unstable null orbits. In this paper we develop an expansion method to explore the link. We obtain new closed-form approximations for the lightly-damped part of the spectrum in the large-$l$ regime. We confirm that, at leading order in $l$, the resonance frequency is linked to the orbital frequency, and the resonance damping to the Lyapunov exponent, of the relevant null orbit. We go somewhat further than previous studies to establish (i) a spin-dependent correction to the frequency at order $1 / l$ for equatorial ($m = \pm l$) modes, and (ii) a new result for polar modes ($m = 0$). We validate the approach by testing the closed-form approximations against frequencies obtained numerically with Leaver's method.Comment: 18 pages, 3 tables, 3 figure

Self-force of a scalar field for circular orbits about a Schwarzschild black hole

The foundations are laid for the numerical computation of the actual worldline for a particle orbiting a black hole and emitting gravitational waves. The essential practicalities of this computation are here illustrated for a scalar particle of infinitesimal size and small but finite scalar charge. This particle deviates from a geodesic because it interacts with its own retarded field \psi^\ret. A recently introduced Green's function G^\SS precisely determines the singular part, \psi^\SS, of the retarded field. This part exerts no force on the particle. The remainder of the field \psi^\R = \psi^\ret - \psi^\SS is a vacuum solution of the field equation and is entirely responsible for the self-force. A particular, locally inertial coordinate system is used to determine an expansion of \psi^\SS in the vicinity of the particle. For a particle in a circular orbit in the Schwarzschild geometry, the mode-sum decomposition of the difference between \psi^\ret and the dominant terms in the expansion of \psi^\SS provide a mode-sum decomposition of an approximation for $\psi^\R$ from which the self-force is obtained. When more terms are included in the expansion, the approximation for $\psi^\R$ is increasingly differentiable, and the mode-sum for the self-force converges more rapidly.Comment: RevTex, 31 pages, 1 figure, modified abstract, more details of numerical method

Canonical Quantization of the Electromagnetic Field on the Kerr Background

We investigate the canonical quantization of the electromagnetic field on the Kerr background. We give new expressions for the expectation value of the electromagnetic stress-energy tensor in various vacua states and give a physical interpretation of the separate terms appearing in them. We numerically calculate the luminosity in these states. We also study the form of the renormalized stress-energy tensor close to the horizon when the electromagnetic field is in the past Boulware state.Comment: 27 zipped, postscript figure file

The Extreme Kerr Throat Geometry: A Vacuum Analog of AdS_2 x S^2

We study the near horizon limit of a four dimensional extreme rotating black hole. The limiting metric is a completely nonsingular vacuum solution, with an enhanced symmetry group SL(2,R) x U(1). We show that many of the properties of this solution are similar to the AdS_2 x S^2 geometry arising in the near horizon limit of extreme charged black holes. In particular, the boundary at infinity is a timelike surface. This suggests the possibility of a dual quantum mechanical description. A five dimensional generalization is also discussed.Comment: 21 page

Spacetime Symmetries and Kepler's Third Law

The curved spacetime geometry of a system of two point masses moving on a circular orbit has a helical symmetry. We show how Kepler's third law for circular motion, and its generalization in post-Newtonian theory, can be recovered from a simple, covariant condition on the norm of the associated helical Killing vector field. This unusual derivation can be used to illustrate some concepts of prime importance in a general relativity course, including those of Killing field, covariance, coordinate dependence, and gravitational redshift.Comment: 11 pages, 3 figures; minor changes and text improvements; matches version to appear in Class. Quant. Gra

The scalar perturbation of the higher-dimensional rotating black holes

The massless scalar field in the higher-dimensional Kerr black hole (Myers- Perry solution with a single rotation axis) has been investigated. It has been shown that the field equation is separable in arbitrary dimensions. The quasi-normal modes of the scalar field have been searched in five dimensions using the continued fraction method. The numerical result shows the evidence for the stability of the scalar perturbation of the five-dimensional Kerr black holes. The time scale of the resonant oscillation in the rapidly rotating black hole, in which case the horizon radius becomes small, is characterized by (black hole mass)^{1/2}(Planck mass)^{-3/2} rather than the light-crossing time of the horizon.Comment: 16 pages, 7 figures, revised versio

Asymptotically Matched Spacetime Metric for Non-Precessing, Spinning Black Hole Binaries

We construct a closed-form, fully analytical 4-metric that approximately represents the spacetime evolution of non-precessing, spinning black hole binaries from infinite separations up to a few orbits prior to merger. We employ the technique of asymptotic matching to join a perturbed Kerr metric in the neighborhood of each spinning black hole to a near-zone, post-Newtonian metric farther out. The latter is already naturally matched to a far-zone, post-Minkowskian metric that accounts for full temporal retardation. The result is a 4-metric that is approximately valid everywhere in space and in a small bundle of spatial hypersurfaces. We here restrict our attention to quasi- circular orbits, but the method is valid for any orbital motion or physical scenario, provided an overlapping region of validity or buffer zone exists. A simple extension of such a metric will allow for future studies of the accretion disk and jet dynamics around spinning back hole binaries

A detailed study of quasinormal frequencies of the Kerr black hole

We compute the quasinormal frequencies of the Kerr black hole using a continued fraction method. The continued fraction method first proposed by Leaver is still the only known method stable and accurate for the numerical determination of the Kerr quasinormal frequencies. We numerically obtain not only the slowly but also the rapidly damped quasinormal frequencies and analyze the peculiar behavior of these frequencies at the Kerr limit. We also calculate the algebraically special frequency first identified by Chandrasekhar and confirm that it coincide with the $n=8$ quasinormal frequency only at the Schwarzschild limit.Comment: REVTEX, 15 pages, 7 eps figure

Brane Decay of a (4+n)-Dimensional Rotating Black Hole. II: spin-1 particles

The present works complements and expands a previous one, focused on the emission of scalar fields by a (4+n)-dimensional rotating black hole on the brane, by studying the emission of gauge fields on the brane from a similar black hole. A comprehensive analysis of the particle, energy and angular momentum emission rates is undertaken, for arbitrary angular momentum of the black hole and dimensionality of spacetime. Our analysis reveals the existence of a number of distinct features associated with the emission of spin-1 fields from a rotating black hole on the brane, such as the behaviour and magnitude of the different emission rates, the angular distribution of particles and energy, the relative enhancement compared to the scalar fields, and the magnitude of the superradiance effect. Apart from their theoretical interest, these features can comprise clear signatures of the emission of Hawking radiation from a brane-world black hole during its spin-down phase upon successful detection of this effect during an experiment.Comment: 35 pages, 19 figures, Latex fil

Analytic approximations, perturbation methods, and their applications

The paper summarizes the parallel session B3 {\em Analytic approximations, perturbation methods, and their applications} of the GR18 conference. The talks in the session reported notably recent advances in black hole perturbations and post-Newtonian approximations as applied to sources of gravitational waves.Comment: Summary of the B3 parallel session of the GR18 conferenc