105 research outputs found
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
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
The imposition of Cauchy data to the Teukolsky equation II: Numerical comparison with the Zerilli-Moncrief approach to black hole perturbations
We revisit the question of the imposition of initial data representing
astrophysical gravitational perturbations of black holes. We study their
dynamics for the case of nonrotating black holes by numerically evolving the
Teukolsky equation in the time domain. In order to express the Teukolsky
function Psi explicitly in terms of hypersurface quantities, we relate it to
the Moncrief waveform phi_M through a Chandrasekhar transformation in the case
of a nonrotating black hole. This relation between Psi and phi_M holds for any
constant time hypersurface and allows us to compare the computation of the
evolution of Schwarzschild perturbations by the Teukolsky and by the Zerilli
and Regge-Wheeler equations. We explicitly perform this comparison for the
Misner initial data in the close limit approach. We evolve numerically both,
the Teukolsky (with the recent code of Ref. [1]) and the Zerilli equations,
finding complete agreement in resulting waveforms within numerical error. The
consistency of these results further supports the correctness of the numerical
code for evolving the Teukolsky equation as well as the analytic expressions
for Psi in terms only of the three-metric and the extrinsic curvature.Comment: 14 pages, 7 Postscript figures, to appear in Phys. Rev.
Perspective on gravitational self-force analyses
A point particle of mass moving on a geodesic creates a perturbation
, of the spacetime metric , that diverges at the particle.
Simple expressions are given for the singular part of and its
distortion caused by the spacetime. This singular part h^\SS_{ab} is
described in different coordinate systems and in different gauges. Subtracting
h^\SS_{ab} from leaves a regular remainder . The
self-force on the particle from its own gravitational field adjusts the world
line at \Or(\mu) to be a geodesic of ; this adjustment
includes all of the effects of radiation reaction. For the case that the
particle is a small non-rotating black hole, we give a uniformly valid
approximation to a solution of the Einstein equations, with a remainder of
\Or(\mu^2) as .
An example presents the actual steps involved in a self-force calculation.
Gauge freedom introduces ambiguity in perturbation analysis. However,
physically interesting problems avoid this ambiguity.Comment: 40 pages, to appear in a special issue of CQG on radiation reaction,
contains additional references, improved notation for tensor harmonic
The radial infall of a highly relativistic point particle into a Kerr black hole along the symmetry axis
In this Letter we consider the radial infall along the symmetry axis of an
ultra-relativistic point particle into a rotating Kerr black hole. We use the
Sasaki-Nakamura formalism to compute the waveform, energy spectra and total
energy radiated during this process. We discuss possible connections between
these results and the black hole-black hole collision at the speed of light
process.Comment: 1 figur
Quasinormal Modes of AdS Black Holes and the Approach to Thermal Equilibrium
We investigate the decay of a scalar field outside a Schwarzschild anti de
Sitter black hole. This is determined by computing the complex frequencies
associated with quasinormal modes. There are qualitative differences from the
asymptotically flat case, even in the limit of small black holes. In
particular, for a given angular dependence, the decay is always exponential -
there are no power law tails at late times. In terms of the AdS/CFT
correspondence, a large black hole corresponds to an approximately thermal
state in the field theory, and the decay of the scalar field corresponds to the
decay of a perturbation of this state. Thus one obtains the timescale for the
approach to thermal equilibrium. We compute these timescales for the strongly
coupled field theories in three, four, and six dimensions which are dual to
string theory in asymptotically AdS spacetimes.Comment: 25 pages, 9 figures extended discussion of horizon boundary
conditions, added note on higher l mode
Constraint propagation in the family of ADM systems
The current important issue in numerical relativity is to determine which
formulation of the Einstein equations provides us with stable and accurate
simulations. Based on our previous work on "asymptotically constrained"
systems, we here present constraint propagation equations and their eigenvalues
for the Arnowitt-Deser-Misner (ADM) evolution equations with additional
constraint terms (adjusted terms) on the right hand side. We conjecture that
the system is robust against violation of constraints if the amplification
factors (eigenvalues of Fourier-component of the constraint propagation
equations) are negative or pure-imaginary. We show such a system can be
obtained by choosing multipliers of adjusted terms. Our discussion covers
Detweiler's proposal (1987) and Frittelli's analysis (1997), and we also
mention the so-called conformal-traceless ADM systems.Comment: 11 pages, RevTeX, 2 eps figure
A Toy Model for Testing Finite Element Methods to Simulate Extreme-Mass-Ratio Binary Systems
Extreme mass ratio binary systems, binaries involving stellar mass objects
orbiting massive black holes, are considered to be a primary source of
gravitational radiation to be detected by the space-based interferometer LISA.
The numerical modelling of these binary systems is extremely challenging
because the scales involved expand over several orders of magnitude. One needs
to handle large wavelength scales comparable to the size of the massive black
hole and, at the same time, to resolve the scales in the vicinity of the small
companion where radiation reaction effects play a crucial role. Adaptive finite
element methods, in which quantitative control of errors is achieved
automatically by finite element mesh adaptivity based on posteriori error
estimation, are a natural choice that has great potential for achieving the
high level of adaptivity required in these simulations. To demonstrate this, we
present the results of simulations of a toy model, consisting of a point-like
source orbiting a black hole under the action of a scalar gravitational field.Comment: 29 pages, 37 figures. RevTeX 4.0. Minor changes to match the
published versio
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
A nonlinear scalar model of extreme mass ratio inspirals in effective field theory I. Self force through third order
The motion of a small compact object in a background spacetime is
investigated in the context of a model nonlinear scalar field theory. This
model is constructed to have a perturbative structure analogous to the General
Relativistic description of extreme mass ratio inspirals (EMRIs). We apply the
effective field theory approach to this model and calculate the finite part of
the self force on the small compact object through third order in the ratio of
the size of the compact object to the curvature scale of the background (e.g.,
black hole) spacetime. We use well-known renormalization methods and
demonstrate the consistency of the formalism in rendering the self force finite
at higher orders within a point particle prescription for the small compact
object. This nonlinear scalar model should be useful for studying various
aspects of higher-order self force effects in EMRIs but within a comparatively
simpler context than the full gravitational case. These aspects include
developing practical schemes for higher order self force numerical
computations, quantifying the effects of transient resonances on EMRI waveforms
and accurately modeling the small compact object's motion for precise
determinations of the parameters of detected EMRI sources.Comment: 30 pages, 8 figure
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