1,161 research outputs found
Conservative corrections to the innermost stable circular orbit (ISCO) of a Kerr black hole: a new gauge-invariant post-Newtonian ISCO condition, and the ISCO shift due to test-particle spin and the gravitational self-force
The innermost stable circular orbit (ISCO) delimits the transition from
circular orbits to those that plunge into a black hole. In the test-mass limit,
well-defined ISCO conditions exist for the Kerr and Schwarzschild spacetimes.
In the finite-mass case, there are a large variety of ways to define an ISCO in
a post-Newtonian (PN) context. Here I generalize the gauge-invariant ISCO
condition of Blanchet & Iyer (2003) to the case of spinning (nonprecessing)
binaries. The Blanchet-Iyer ISCO condition has two desirable and unexpected
properties: (1) it exactly reproduces the Schwarzschild ISCO in the test-mass
limit, and (2) it accurately approximates the recently-calculated shift in the
Schwarzschild ISCO frequency due to the conservative-piece of the gravitational
self-force [Barack & Sago (2009)]. The generalization of this ISCO condition to
spinning binaries has the property that it also exactly reproduces the Kerr
ISCO in the test-mass limit (up to the order at which PN spin corrections are
currently known). The shift in the ISCO due to the spin of the test-particle is
also calculated. Remarkably, the gauge-invariant PN ISCO condition exactly
reproduces the ISCO shift predicted by the Papapetrou equations for a
fully-relativistic spinning particle. It is surprising that an analysis of the
stability of the standard PN equations of motion is able (without any form of
"resummation") to accurately describe strong-field effects of the Kerr
spacetime. The ISCO frequency shift due to the conservative self-force in Kerr
is also calculated from this new ISCO condition, as well as from the
effective-one-body Hamiltonian of Barausse & Buonanno (2010). These results
serve as a useful point-of-comparison for future gravitational self-force
calculations in the Kerr spacetime.Comment: 17 pages, 2 figures, 1 table. v2: references added; minor changes to
match published versio
Mode-sum regularization of the scalar self-force: Formulation in terms of a tetrad decomposition of the singular field
We examine the motion in Schwarzschild spacetime of a point particle endowed
with a scalar charge. The particle produces a retarded scalar field which
interacts with the particle and influences its motion via the action of a
self-force. We exploit the spherical symmetry of the Schwarzschild spacetime
and decompose the scalar field in spherical-harmonic modes. Although each mode
is bounded at the position of the particle, a mode-sum evaluation of the
self-force requires regularization because the sum does not converge: the
retarded field is infinite at the position of the particle. The regularization
procedure involves the computation of regularization parameters, which are
obtained from a mode decomposition of the Detweiler-Whiting singular field;
these are subtracted from the modes of the retarded field, and the result is a
mode-sum that converges to the actual self-force. We present such a computation
in this paper. There are two main aspects of our work that are new. First, we
define the regularization parameters as scalar quantities by referring them to
a tetrad decomposition of the singular field. Second, we calculate four sets of
regularization parameters (denoted schematically by A, B, C, and D) instead of
the usual three (A, B, and C). As proof of principle that our methods are
reliable, we calculate the self-force acting on a scalar charge in circular
motion around a Schwarzschild black hole, and compare our answers with those
recorded in the literature.Comment: 38 pages, 2 figure
Self force in 2+1 electrodynamics
The radiation reaction problem for an electric charge moving in flat
space-time of three dimensions is discussed. The divergences stemming from the
pointness of the particle are studied. A consistent regularization procedure is
proposed, which exploits the Poincar\'e invariance of the theory. Effective
equation of motion of radiating charge in an external electromagnetic field is
obtained via the consideration of energy-momentum and angular momentum
conservation. This equation includes the effect of the particle's own field.
The radiation reaction is determined by the Lorentz force of point-like charge
acting upon itself plus a non-local term which provides finiteness of the
self-action.Comment: 20 pages, 3 figure
Emergence of thin shell structure during collapse in isotropic coordinates
Numerical studies of gravitational collapse in isotropic coordinates have
recently shown an interesting connection between the gravitational Lagrangian
and black hole thermodynamics. A study of the actual spacetime was not the main
focus of this work and in particular, the rich and interesting structure of the
interior has not been investigated in much detail and remains largely unknown.
We elucidate its features by performing a numerical study of the spacetime in
isotropic coordinates during gravitational collapse of a massless scalar field.
The most salient feature to emerge is the formation of a thin shell of matter
just inside the apparent horizon. The energy density and Ricci scalar peak at
the shell and there is a jump discontinuity in the extrinsic curvature across
the apparent horizon, the hallmark that a thin shell is present in its
vicinity. At late stages of the collapse, the spacetime consists of two vacuum
regions separated by the thin shell. The interior is described by an
interesting collapsing isotropic universe. It tends towards a vacuum (never
reaches a perfect vacuum) and there is a slight inhomogeneity in the interior
that plays a crucial role in the collapse process as the areal radius tends to
zero. The spacetime evolves towards a curvature (physical) singularity in the
interior, both a Weyl and Ricci singularity. In the exterior, our numerical
results match closely the analytical form of the Schwarzschild metric in
isotropic coordinates, providing a strong test of our numerical code.Comment: 24 pages, 10 figures. version to appear in Phys. Rev.
A matched expansion approach to practical self-force calculations
We discuss a practical method to compute the self-force on a particle moving
through a curved spacetime. This method involves two expansions to calculate
the self-force, one arising from the particle's immediate past and the other
from the more distant past. The expansion in the immediate past is a covariant
Taylor series and can be carried out for all geometries. The more distant
expansion is a mode sum, and may be carried out in those cases where the wave
equation for the field mediating the self-force admits a mode expansion of the
solution. In particular, this method can be used to calculate the gravitational
self-force for a particle of mass mu orbiting a black hole of mass M to order
mu^2, provided mu/M << 1. We discuss how to use these two expansions to
construct a full self-force, and in particular investigate criteria for
matching the two expansions. As with all methods of computing self-forces for
particles moving in black hole spacetimes, one encounters considerable
technical difficulty in applying this method; nevertheless, it appears that the
convergence of each series is good enough that a practical implementation may
be plausible.Comment: IOP style, 8 eps figures, accepted for publication in a special issue
of Classical and Quantum Gravit
Gravitational radiation from infall into a black hole: Regularization of the Teukolsky equation
The Teukolsky equation has long been known to lead to divergent integrals
when it is used to calculate the gravitational radiation emitted when a test
mass falls into a black hole from infinity. Two methods have been used in the
past to remove those divergent integrals. In the first, integrations by parts
are carried out, and the infinite boundary terms are simply discarded. In the
second, the Teukolsky equation is transformed into another equation which does
not lead to divergent integrals. The purpose of this paper is to show that
there is nothing intrinsically wrong with the Teukolsky equation when dealing
with non-compact source terms, and that the divergent integrals result simply
from an incorrect choice of Green's function. In this paper, regularization of
the Teukolsky equation is carried out in an entirely natural way which does not
involve modifying the equation.Comment: ReVTeX, 23 page
Gravitational radiation from a particle in circular orbit around a black hole. VI. Accuracy of the post-Newtonian expansion
A particle of mass moves on a circular orbit around a nonrotating black
hole of mass . Under the assumption the gravitational waves
emitted by such a binary system can be calculated exactly numerically using
black-hole perturbation theory. If, further, the particle is slowly moving,
then the waves can be calculated approximately analytically, and expressed in
the form of a post-Newtonian expansion. We determine the accuracy of this
expansion in a quantitative way by calculating the reduction in signal-to-noise
ratio incurred when matched filtering the exact signal with a nonoptimal,
post-Newtonian filter.Comment: 5 pages, ReVTeX, 1 figure. A typographical error was discovered in
the computer code used to generate the results presented in the paper. The
corrected results are presented in an Erratum, which also incorporates new
results, obtained using the recently improved post-Newtonian calculations of
Tanaka, Tagoshi, and Sasak
Perturbative evolution of particle orbits around Kerr black holes: time domain calculation
Treating the Teukolsky perturbation equation numerically as a 2+1 PDE and
smearing the singularities in the particle source term by the use of narrow
Gaussian distributions, we have been able to reproduce earlier results for
equatorial circular orbits that were computed using the frequency domain
formalism. A time domain prescription for a more general evolution of nearly
geodesic orbits under the effects of radiation reaction is presented. This
approach can be useful when tackling the more realistic problem of a
stellar-mass black hole moving on a generic orbit around a supermassive black
hole under the influence of radiation reaction forces.Comment: 8 pages, 5 figures, problems with references and double-printing
fixe
LISA detections of massive black hole inspirals: parameter extraction errors due to inaccurate template waveforms
The planned Laser Interferometer Space Antenna (LISA) is expected to detect
the inspiral and merger of massive black hole binaries (MBHBs) at z <~ 5 with
signal-to-noise ratios (SNRs) of hundreds to thousands. Because of these high
SNRs, and because these SNRs accrete over periods of weeks to months, it should
be possible to extract the physical parameters of these systems with high
accuracy; for instance, for a ~ 10^6 Msun MBHBs at z = 1 it should be possible
to determine the two masses to ~ 0.1% and the sky location to ~ 1 degree.
However, those are just the errors due to noise: there will be additional
"theoretical" errors due to inaccuracies in our best model waveforms, which are
still only approximate. The goal of this paper is to estimate the typical
magnitude of these theoretical errors. We develop mathematical tools for this
purpose, and apply them to a somewhat simplified version of the MBHB problem,
in which we consider just the inspiral part of the waveform and neglect
spin-induced precession, eccentricity, and PN amplitude corrections. For this
simplified version, we estimate that theoretical uncertainties in sky position
will typically be ~ 1 degree, i.e., comparable to the statistical uncertainty.
For the mass and spin parameters, our results suggest that while theoretical
errors will be rather small absolutely, they could still dominate over
statistical errors (by roughly an order of magnitude) for the strongest
sources. The tools developed here should be useful for estimating the magnitude
of theoretical errors in many other problems in gravitational-wave astronomy.Comment: RevTeX4, 16 pages, 2 EPS figures. Corrected typos, clarified
statement
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
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