1,163 research outputs found
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
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 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
Can the post-Newtonian gravitational waveform of an inspiraling binary be improved by solving the energy balance equation numerically?
The detection of gravitational waves from inspiraling compact binaries using
matched filtering depends crucially on the availability of accurate template
waveforms. We determine whether the accuracy of the templates' phasing can be
improved by solving the post-Newtonian energy balance equation numerically,
rather than (as is normally done) analytically within the post-Newtonian
perturbative expansion. By specializing to the limit of a small mass ratio, we
find evidence that there is no gain in accuracy.Comment: 13 pages, RevTeX, 5 figures included via eps
Nonsingular Black Hole Evaporation and ``Stable'' Remnants
We examine the evaporation of two--dimensional black holes, the classical
space--times of which are extended geometries, like for example the
two--dimensional section of the extremal Reissner--Nordstrom black hole. We
find that the evaporation in two particular models proceeds to a stable
end--point. This should represent the generic behavior of a certain class of
two--dimensional dilaton--gravity models. There are two distinct regimes
depending on whether the back--reaction is weak or strong in a certain sense.
When the back--reaction is weak, evaporation proceeds via an adiabatic
evolution, whereas for strong back--reaction, the decay proceeds in a somewhat
surprising manner. Although information loss is inevitable in these models at
the semi--classical level, it is rather benign, in that the information is
stored in another asymptotic region.Comment: 23 pages, 6 figures, harvmac and epsf, RU-93-12, PUPT-1399,
NSF-ITP-93-5
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.
Gravitational waveforms from a point particle orbiting a Schwarzschild black hole
We numerically solve the inhomogeneous Zerilli-Moncrief and Regge-Wheeler
equations in the time domain. We obtain the gravitational waveforms produced by
a point-particle of mass traveling around a Schwarzschild black hole of
mass M on arbitrary bound and unbound orbits. Fluxes of energy and angular
momentum at infinity and the event horizon are also calculated. Results for
circular orbits, selected cases of eccentric orbits, and parabolic orbits are
presented. The numerical results from the time-domain code indicate that, for
all three types of orbital motion, black hole absorption contributes less than
1% of the total flux, so long as the orbital radius r_p(t) satisfies r_p(t)> 5M
at all times.Comment: revtex4, 24 pages, 23 figures, 3 tables, submitted to PR
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
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