1,496 research outputs found
The singular field used to calculate the self-force on non-spinning and spinning particles
The singular field of a point charge has recently been described in terms of
a new Green's function of curved spacetime. This singular field plays an
important role in the calculation of the self-force acting upon the particle.
We provide a method for calculating the singular field and a catalog of
expansions of the singular field associated with the geodesic motion of
monopole and dipole sources for scalar, electromagnetic and gravitational
fields. These results can be used, for example, to calculate the effects of the
self-force acting on a particle as it moves through spacetime.Comment: 14 pages; addressed referee's comments; published in PhysRev
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
Osculating orbits in Schwarzschild spacetime, with an application to extreme mass-ratio inspirals
We present a method to integrate the equations of motion that govern bound,
accelerated orbits in Schwarzschild spacetime. At each instant the true
worldline is assumed to lie tangent to a reference geodesic, called an
osculating orbit, such that the worldline evolves smoothly from one such
geodesic to the next. Because a geodesic is uniquely identified by a set of
constant orbital elements, the transition between osculating orbits corresponds
to an evolution of the elements. In this paper we derive the evolution
equations for a convenient set of orbital elements, assuming that the force
acts only within the orbital plane; this is the only restriction that we impose
on the formalism, and we do not assume that the force must be small. As an
application of our method, we analyze the relative motion of two massive
bodies, assuming that one body is much smaller than the other. Using the hybrid
Schwarzschild/post-Newtonian equations of motion formulated by Kidder, Will,
and Wiseman, we treat the unperturbed motion as geodesic in a Schwarzschild
spacetime whose mass parameter is equal to the system's total mass. The force
then consists of terms that depend on the system's reduced mass. We highlight
the importance of conservative terms in this force, which cause significant
long-term changes in the time-dependence and phase of the relative orbit. From
our results we infer some general limitations of the radiative approximation to
the gravitational self-force, which uses only the dissipative terms in the
force.Comment: 18 pages, 6 figures, final version to be published in Physical Review
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
Gravitational waves from binary systems in circular orbits: Convergence of a dressed multipole truncation
The gravitational radiation originating from a compact binary system in
circular orbit is usually expressed as an infinite sum over radiative multipole
moments. In a slow-motion approximation, each multipole moment is then
expressed as a post-Newtonian expansion in powers of v/c, the ratio of the
orbital velocity to the speed of light. The bare multipole truncation of the
radiation consists in keeping only the leading-order term in the post-Newtonian
expansion of each moment, but summing over all the multipole moments. In the
case of binary systems with small mass ratios, the bare multipole series was
shown in a previous paper to converge for all values v/c < 2/e, where e is the
base of natural logarithms. In this paper, we extend the analysis to a dressed
multipole truncation of the radiation, in which the leading-order moments are
corrected with terms of relative order (v/c)^2 and (v/c)^3. We find that the
dressed multipole series converges also for all values v/c < 2/e, and that it
coincides (within 1%) with the numerically ``exact'' results for v/c < 0.2.Comment: 9 pages, ReVTeX, 1 postscript figur
Intrinsic and extrinsic geometries of a tidally deformed black hole
A description of the event horizon of a perturbed Schwarzschild black hole is
provided in terms of the intrinsic and extrinsic geometries of the null
hypersurface. This description relies on a Gauss-Codazzi theory of null
hypersurfaces embedded in spacetime, which extends the standard theory of
spacelike and timelike hypersurfaces involving the first and second fundamental
forms. We show that the intrinsic geometry of the event horizon is invariant
under a reparameterization of the null generators, and that the extrinsic
geometry depends on the parameterization. Stated differently, we show that
while the extrinsic geometry depends on the choice of gauge, the intrinsic
geometry is gauge invariant. We apply the formalism to solutions to the vacuum
field equations that describe a tidally deformed black hole. In a first
instance we consider a slowly-varying, quadrupolar tidal field imposed on the
black hole, and in a second instance we examine the tide raised during a close
parabolic encounter between the black hole and a small orbiting body.Comment: 27 pages, 4 figure
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
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.
Misconceptions About General Relativity in Theoretical Black Hole Astrophysics
The fundamental role played by black holes in our study of microquasars,
gamma ray bursts, and the outflows from active galactic nuclei requires an
appreciation for, and at times some in-depth analysis of, curved spacetime. We
highlight misconceptions surrounding the notion of coordinate transformation in
general relativity as applied to metrics for rotating black holes that are
beginning to increasingly appear in the literature. We emphasize that there is
no coordinate transformation that can turn the metric of a rotating spacetime
into that for a Schwarzschild spacetime, or more generally, that no coordinate
transformation exists that can diagonalize the metric for a rotating spacetime.
We caution against the notion of "local" coordinate transformation, which is
often incorrectly associated with a global analysis of the spacetime.Comment: MNRAS accepte
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