375 research outputs found
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 from which the
self-force is obtained. When more terms are included in the expansion, the
approximation for 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
Gravitational self-force and the effective-one-body formalism between the innermost stable circular orbit and the light ring
We compute the conservative piece of the gravitational self-force (GSF)
acting on a particle of mass m_1 as it moves along an (unstable) circular
geodesic orbit between the innermost stable circular orbit (ISCO) and the light
ring of a Schwarzschild black hole of mass m_2>> m_1. More precisely, we
construct the function h_{uu}(x) = h_{\mu\nu} u^{\mu} u^{\nu} (related to
Detweiler's gauge-invariant "redshift" variable), where h_{\mu\nu} is the
regularized metric perturbation in the Lorenz gauge, u^{\mu} is the
four-velocity of m_1, and x= [Gc^{-3}(m_1+m_2)\Omega]^{2/3} is an invariant
coordinate constructed from the orbital frequency \Omega. In particular, we
explore the behavior of h_{uu} just outside the "light ring" at x=1/3, where
the circular orbit becomes null. Using the recently discovered link between
h_{uu} and the piece a(u), linear in the symmetric mass ratio \nu, of the main
radial potential A(u,\nu) of the Effective One Body (EOB) formalism, we compute
a(u) over the entire domain 0<u<1/3. We find that a(u) diverges at the
light-ring as ~0.25 (1-3u)^{-1/2}, explain the physical origin of this
divergence, and discuss its consequences for the EOB formalism. We construct
accurate global analytic fits for a(u), valid on the entire domain 0<u<1/3 (and
possibly beyond), and give accurate numerical estimates of the values of a(u)
and its first 3 derivatives at the ISCO, as well as the O(\nu) shift in the
ISCO frequency. In previous work we used GSF data on slightly eccentric orbits
to compute a certain linear combination of a(u) and its first two derivatives,
involving also the O(\nu) piece \bar d(u) of a second EOB radial potential
{\bar D}(u,\nu). Combining these results with our present global analytic
representation of a(u), we numerically compute {\bar d}(u)$ on the interval
0<u\leq 1/6.Comment: 44 pages, 8 figures. Extended discussion in Section V and minor
typographical corrections throughout. Version to be published in PR
Orbital evolution of a test particle around a black hole: Indirect determination of the self force in the post Newtonian approximation
Comparing the corrections to Kepler's law with orbital evolution under a self
force, we extract the finite, already regularized part of the latter in a
specific gauge. We apply this method to a quasi-circular orbit around a
Schwarzschild black hole of an extreme mass ratio binary, and determine the
first- and second-order conservative gravitational self force in a post
Newtonian expansion. We use these results in the construction of the
gravitational waveform, and revisit the question of the relative contribution
of the self force and spin-orbit coupling.Comment: 5 pages, 2 figure
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 (), 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- regime. We confirm that, at leading order in
, 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 for equatorial ()
modes, and (ii) a new result for polar modes (). 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
Regularization of the Teukolsky Equation for Rotating Black Holes
We show that the radial Teukolsky equation (in the frequency domain) with
sources that extend to infinity has well-behaved solutions. To prove that, we
follow Poisson approach to regularize the non-rotating hole, and extend it to
the rotating case. To do so we use the Chandrasekhar transformation among the
Teukolsky and Regge-Wheeler-like equations, and express the integrals over the
source in terms of solutions to the homogeneous Regge-Wheeler-like equation, to
finally regularize the resulting integral. We then discuss the applicability of
these results.Comment: 14 pages, 1 Table, REVTE
Orbital evolution of a particle around a black hole: II. Comparison of contributions of spin-orbit coupling and the self force
We consider the evolution of the orbit of a spinning compact object in a
quasi-circular, planar orbit around a Schwarzschild black hole in the extreme
mass ratio limit. We compare the contributions to the orbital evolution of both
spin-orbit coupling and the local self force. Making assumptions on the
behavior of the forces, we suggest that the decay of the orbit is dominated by
radiation reaction, and that the conservative effect is typically dominated by
the spin force. We propose that a reasonable approximation for the
gravitational waveform can be obtained by ignoring the local self force, for
adjusted values of the parameters of the system. We argue that this
approximation will only introduce small errors in the astronomical
determination of these parameters.Comment: 11 pages, 7 figure
Gravitational Self-Force Correction to the Binding Energy of Compact Binary Systems
Using the first law of binary black-hole mechanics, we compute the binding
energy E and total angular momentum J of two non-spinning compact objects
moving on circular orbits with frequency Omega, at leading order beyond the
test-particle approximation. By minimizing E(Omega) we recover the exact
frequency shift of the Schwarzschild innermost stable circular orbit induced by
the conservative piece of the gravitational self-force. Comparing our results
for the coordinate invariant relation E(J) to those recently obtained from
numerical simulations of comparable-mass non-spinning black-hole binaries, we
find a remarkably good agreement, even in the strong-field regime. Our findings
confirm that the domain of validity of perturbative calculations may extend
well beyond the extreme mass-ratio limit.Comment: 5 pages, 1 figure; matches the published versio
Universal Bound on Dynamical Relaxation Times and Black-Hole Quasinormal Ringing
From information theory and thermodynamic considerations a universal bound on
the relaxation time of a perturbed system is inferred, , where is the system's temperature. We prove that black holes
comply with the bound; in fact they actually {\it saturate} it. Thus, when
judged by their relaxation properties, black holes are the most extreme objects
in nature, having the maximum relaxation rate which is allowed by quantum
theory.Comment: 4 page
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
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
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