1,696 research outputs found
A light-cone gauge for black-hole perturbation theory
The geometrical meaning of the Eddington-Finkelstein coordinates of
Schwarzschild spacetime is well understood: (i) the advanced-time coordinate v
is constant on incoming light cones that converge toward r=0, (ii) the angles
theta and phi are constant on the null generators of each light cone, (iii) the
radial coordinate r is an affine-parameter distance along each generator, and
(iv) r is an areal radius, in the sense that 4 pi r^2 is the area of each
two-surface (v,r) = constant. The light-cone gauge of black-hole perturbation
theory, which is formulated in this paper, places conditions on a perturbation
of the Schwarzschild metric that ensure that properties (i)--(iii) of the
coordinates are preserved in the perturbed spacetime. Property (iv) is lost in
general, but it is retained in exceptional situations that are identified in
this paper. Unlike other popular choices of gauge, the light-cone gauge
produces a perturbed metric that is expressed in a meaningful coordinate
system; this is a considerable asset that greatly facilitates the task of
extracting physical consequences. We illustrate the use of the light-cone gauge
by calculating the metric of a black hole immersed in a uniform magnetic field.
We construct a three-parameter family of solutions to the perturbative
Einstein-Maxwell equations and argue that it is applicable to a broader range
of physical situations than the exact, two-parameter Schwarzschild-Melvin
family.Comment: 12 page
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
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
Regularization of the second-order gravitational perturbations produced by a compact object
The equations for the second-order gravitational perturbations produced by a
compact-object have highly singular source terms at the point particle limit.
At this limit the standard retarded solutions to these equations are
ill-defined. Here we construct well-defined and physically meaningful solutions
to these equations. These solutions are important for practical calculations:
the planned gravitational-wave detector LISA requires preparation of waveform
templates for the potential gravitational-waves. Construction of templates with
desired accuracy for extreme mass ratio binaries, in which a compact-object
inspirals towards a supermassive black-hole, requires calculation of the
second-order gravitational perturbations produced by the compact-object.Comment: 12 pages, discussion expanded, to be published in Phys. Rev. D Rapid
Communicatio
Construction of the second-order gravitational perturbations produced by a compact object
Accurate calculation of the gradual inspiral motion in an extreme mass-ratio
binary system, in which a compact-object inspirals towards a supermassive
black-hole requires calculation of the interaction between the compact-object
and the gravitational perturbations that it induces. These metric perturbations
satisfy linear partial differential equations on a curved background spacetime
induced by the supermassive black-hole. At the point particle limit the
second-order perturbations equations have source terms that diverge as
, where is the distance from the particle. This singular behavior
renders the standard retarded solutions of these equations ill-defined. Here we
resolve this problem and construct well-defined and physically meaningful
solutions to these equations. We recently presented an outline of this
resolution [E. Rosenthal, Phys. Rev. D 72, 121503 (2005)]. Here we provide the
full details of this analysis. These second-order solutions are important for
practical calculations: the planned gravitational-wave detector LISA requires
preparation of waveform templates for the expected gravitational-waves.
Construction of templates with desired accuracy for extreme mass-ratio binaries
requires accurate calculation of the inspiral motion including the interaction
with the second-order gravitational perturbations.Comment: 30 page
Gauge and Averaging in Gravitational Self-force
A difficulty with previous treatments of the gravitational self-force is that
an explicit formula for the force is available only in a particular gauge
(Lorenz gauge), where the force in other gauges must be found through a
transformation law once the Lorenz gauge force is known. For a class of gauges
satisfying a ``parity condition'' ensuring that the Hamiltonian center of mass
of the particle is well-defined, I show that the gravitational self-force is
always given by the angle-average of the bare gravitational force. To derive
this result I replace the computational strategy of previous work with a new
approach, wherein the form of the force is first fixed up to a gauge-invariant
piece by simple manipulations, and then that piece is determined by working in
a gauge designed specifically to simplify the computation. This offers
significant computational savings over the Lorenz gauge, since the Hadamard
expansion is avoided entirely and the metric perturbation takes a very simple
form. I also show that the rest mass of the particle does not evolve due to
first-order self-force effects. Finally, I consider the ``mode sum
regularization'' scheme for computing the self-force in black hole background
spacetimes, and use the angle-average form of the force to show that the same
mode-by-mode subtraction may be performed in all parity-regular gauges. It
appears plausible that suitably modified versions of the Regge-Wheeler and
radiation gauges (convenient to Schwarzschild and Kerr, respectively) are in
this class
Retarded Green's Functions In Perturbed Spacetimes For Cosmology and Gravitational Physics
Electromagnetic and gravitational radiation do not propagate solely on the
null cone in a generic curved spacetime. They develop "tails," traveling at all
speeds equal to and less than unity. If sizeable, this off-the-null-cone effect
could mean objects at cosmological distances, such as supernovae, appear dimmer
than they really are. Their light curves may be distorted relative to their
flat spacetime counterparts. These in turn could affect how we infer the
properties and evolution of the universe or the objects it contains. Within the
gravitational context, the tail effect induces a self-force that causes a
compact object orbiting a massive black hole to deviate from an otherwise
geodesic path. This needs to be taken into account when modeling the
gravitational waves expected from such sources. Motivated by these
considerations, we develop perturbation theory for solving the massless scalar,
photon and graviton retarded Green's functions in perturbed spacetimes,
assuming these Green's functions are known in the background spacetime. In
particular, we elaborate on the theory in perturbed Minkowski spacetime in
significant detail; and apply our techniques to compute the retarded Green's
functions in the weak field limit of the Kerr spacetime to first order in the
black hole's mass and angular momentum. Our methods build on and generalizes
work appearing in the literature on this topic to date, and lays the foundation
for a thorough, first principles based, investigation of how light propagates
over cosmological distances, within a spatially flat inhomogeneous
Friedmann-Lema\^{i}tre-Robertson-Walker (FLRW) universe. This perturbative
scheme applied to the graviton Green's function, when pushed to higher orders,
may provide approximate analytic (or semi-analytic) results for the self-force
problem in the weak field limits of the Schwarzschild and Kerr black hole
geometries.Comment: 23 pages, 5 figures. Significant updates in v2: Scalar, photon and
graviton Green's functions calculated explicitly in Kerr black hole spacetime
up to first order in mass and angular momentum (Sec. V); Visser's van Vleck
determinant result shown to be equivalent to ours in Sec. II. v3: JWKB
discussion moved to introduction; to be published in PR
Absorption of mass and angular momentum by a black hole: Time-domain formalisms for gravitational perturbations, and the small-hole/slow-motion approximation
The first objective of this work is to obtain practical prescriptions to
calculate the absorption of mass and angular momentum by a black hole when
external processes produce gravitational radiation. These prescriptions are
formulated in the time domain within the framework of black-hole perturbation
theory. Two such prescriptions are presented. The first is based on the
Teukolsky equation and it applies to general (rotating) black holes. The second
is based on the Regge-Wheeler and Zerilli equations and it applies to
nonrotating black holes. The second objective of this work is to apply the
time-domain absorption formalisms to situations in which the black hole is
either small or slowly moving. In the context of this small-hole/slow-motion
approximation, the equations of black-hole perturbation theory can be solved
analytically, and explicit expressions can be obtained for the absorption of
mass and angular momentum. The changes in the black-hole parameters can then be
understood in terms of an interaction between the tidal gravitational fields
supplied by the external universe and the hole's tidally-induced mass and
current quadrupole moments. For a nonrotating black hole the quadrupole moments
are proportional to the rate of change of the tidal fields on the hole's world
line. For a rotating black hole they are proportional to the tidal fields
themselves.Comment: 36 pages, revtex4, no figures, final published versio
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
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