100 research outputs found

### Overspinning a Kerr black hole: the effect of self-force

We study the scenario in which a massive particle is thrown into a rapidly
rotating Kerr black hole in an attempt to spin it up beyond its extremal limit,
challenging weak cosmic censorship. We work in black-hole perturbation theory,
and focus on non-spinning, uncharged particles sent in on equatorial orbits. We
first identify the complete parameter-space region in which overspinning occurs
when back-reaction effects from the particle's self-gravity are ignored. We
find, in particular, that overspinning can be achieved only with particles sent
in from infinity. Gravitational self-force effects may prevent overspinning by
radiating away a sufficient amount of the particle's angular momentum
("dissipative effect"), and/or by increasing the effective centrifugal
repulsion, so that particles with suitable parameters never get captured
("conservative effect"). We analyze the full effect of the self-force, thereby
completing previous studies by Jacobson and Sotiriou (who neglected the
self-force) and by Barausse, Cardoso and Khanna (who considered the dissipative
effect on a subset of orbits). Our main result is an inequality, involving
certain self-force quantities, which describes a necessary and sufficient
condition for the overspinning scenario to be overruled. This "censorship"
condition is formulated on a certain one-parameter family of geodesics in an
extremal Kerr geometry. We find that the censorship condition is insensitive to
the dissipative effect (within the first-order self-force approximation used
here), except for a subset of perfectly fine-tuned orbits, for which a separate
censorship condition is derived. We do not obtain here the self-force input
needed to evaluate either of our two conditions, but discuss the prospects for
producing the necessary data using state-of-the-art numerical codes.Comment: 25 pages, 4 figure

### Beyond the geodesic approximation: conservative effects of the gravitational self-force in eccentric orbits around a Schwarzschild black hole

We study conservative finite-mass corrections to the motion of a particle in
a bound (eccentric) strong-field orbit around a Schwarzschild black hole. We
assume the particle's mass $\mu$ is much smaller than the black hole mass $M$,
and explore post-geodesic corrections of $O(\mu/M)$. Our analysis uses
numerical data from a recently developed code that outputs the Lorenz-gauge
gravitational self-force (GSF) acting on the particle along the eccentric
geodesic. First, we calculate the $O(\mu/M)$ conservative correction to the
periastron advance of the orbit, as a function of the (gauge-dependent)
semilatus rectum and eccentricity. A gauge-invariant description of the GSF
precession effect is made possible in the circular-orbit limit, where we
express the correction to the periastron advance as a function of the invariant
azimuthal frequency. We compare this relation with results from fully nonlinear
numerical-relativistic simulations. In order to obtain a gauge-invariant
measure of the GSF effect for fully eccentric orbits, we introduce a suitable
generalization of Detweiler's circular-orbit "redshift" invariant. We compute
the $O(\mu/M)$ conservative correction to this invariant, expressed as a
function of the two invariant frequencies that parametrize the orbit. Our
results are in good agreement with results from post-Newtonian calculations in
the weak-field regime, as we shall report elsewhere. The results of our study
can inform the development of analytical models for the dynamics of strongly
gravitating binaries. They also provide an accurate benchmark for future
numerical-relativistic simulations.Comment: 29 pages, 4 eps figures, matches PRD versio

### Frequency-domain algorithm for the Lorenz-gauge gravitational self-force

State-of-the-art computations of the gravitational self-force (GSF) on
massive particles in black hole spacetimes involve numerical evolution of the
metric perturbation equations in the time-domain, which is computationally very
costly. We present here a new strategy, based on a frequency-domain treatment
of the perturbation equations, which offers considerable computational saving.
The essential ingredients of our method are (i) a Fourier-harmonic
decomposition of the Lorenz-gauge metric perturbation equations and a numerical
solution of the resulting coupled set of ordinary equations with suitable
boundary conditions; (ii) a generalized version of the method of extended
homogeneous solutions [Phys. Rev. D {\bf 78}, 084021 (2008)] used to circumvent
the Gibbs phenomenon that would otherwise hamper the convergence of the Fourier
mode-sum at the particle's location; and (iii) standard mode-sum
regularization, which finally yields the physical GSF as a sum over regularized
modal contributions. We present a working code that implements this strategy to
calculate the Lorenz-gauge GSF along eccentric geodesic orbits around a
Schwarzschild black hole. The code is far more efficient than existing
time-domain methods; the gain in computation speed (at a given precision) is
about an order of magnitude at an eccentricity of 0.2, and up to three orders
of magnitude for circular or nearly circular orbits. This increased efficiency
was crucial in enabling the recently reported calculation of the long-term
orbital evolution of an extreme mass ratio inspiral [Phys. Rev. D {\bf 85},
061501(R) (2012)]. Here we provide full technical details of our method to
complement the above report.Comment: 27 pages, 4 figure

### Gravitational self-force from radiation-gauge metric perturbations

Calculations of the gravitational self-force (GSF) on a point mass in curved spacetime require as input the metric perturbation in a sufficiently regular gauge. A basic challenge in the program to compute the GSF for orbits around a Kerr black hole is that the standard procedure for reconstructing the metric perturbation is formulated in a class of “radiation” gauges, in which the particle singularity is nonisotropic and extends away from the particle’s location. Here we present two practical schemes for calculating the GSF using a radiation-gauge reconstructed metric as input. The schemes are based on a detailed analysis of the local structure of the particle singularity in the radiation gauges. We show that three types of radiation gauge exist: two containing a radial stringlike singularity emanating from the particle, either in one direction (“half-string” gauges) or both directions (“full-string” gauges); and a third type containing no strings but with a jump discontinuity (and possibly a delta function) across a surface intersecting the particle. Based on a flat-space example, we argue that the standard mode-by-mode reconstruction procedure yields the “regular half” of a half-string solution, or (equivalently) either of the regular halves of a no-string solution. For the half-string case, we formulate the GSF in a locally deformed radiation gauge that removes the string singularity near the particle. We derive a mode-sum formula for the GSF in this gauge, which is analogous to the standard Lorenz-gauge formula but requires a correction to the values of the regularization parameters. For the no-string case, we formulate the GSF directly, without a local deformation, and we derive a mode-sum formula that requires no correction to the regularization parameters but involves a certain averaging procedure. We explain the consistency of our results with Gralla’s invariance theorem for the regularization parameters, and we discuss the correspondence between our method and a related approach by Friedman et al

### Self-force and radiation reaction in general relativity

[Abridged] This review surveys the theory of gravitational self-force in
curved spacetime and its application to the gravitational two-body problem in
the extreme-mass-ratio regime. We first lay the relevant formal foundation,
describing the rigorous derivation of the equation of self-forced motion using
matched asymptotic expansions and other ideas. We then review the progress that
has been achieved in numerically calculating the self-force and its physical
effects in the astrophysical scenario of a compact object inspiralling into a
(rotating) massive black hole. We highlight the way in which, nowadays,
self-force calculations make a fruitful contact with other approaches to the
two-body problem and help inform an accurate universal model of binary black
hole inspirals, valid across all mass ratios. We conclude with a summary of the
state of the art, open problems and prospects. Our review is aimed at
non-specialist readers and is for the most part self-contained and
non-technical; only elementary-level acquaintance with General Relativity is
assumed. Where useful, we draw on analogies with familiar concepts from
Newtonian gravity or classical electrodynamics.Comment: 79 pages, 11 figures; invited by Reports on Progress in Physics. v2
contains minor corrections and it is the published versio

### Critical phenomena at the threshold of immediate merger in binary black hole systems: the extreme mass ratio case

In numerical simulations of black hole binaries, Pretorius and Khurana
[Class. Quant. Grav. {\bf 24}, S83 (2007)] have observed critical behaviour at
the threshold between scattering and immediate merger. The number of orbits
scales as $n\simeq -\gamma\ln|p-p_*|$ along any one-parameter family of initial
data such that the threshold is at $p=p_*$. Hence they conjecture that in
ultrarelavistic collisions almost all the kinetic energy can be converted into
gravitational waves if the impact parameter is fine-tuned to the threshold. As
a toy model for the binary, they consider the geodesic motion of a test
particle in a Kerr black hole spacetime, where the unstable circular geodesics
play the role of critical solutions, and calculate the critical exponent
$\gamma$. Here, we incorporate radiation reaction into this model using the
self-force approximation. The critical solution now evolves adiabatically along
a sequence of unstable circular geodesic orbits under the effect of the
self-force. We confirm that almost all the initial energy and angular momentum
are radiated on the critical solution. Our calculation suggests that, even for
infinite initial energy, this happens over a finite number of orbits given by
$n_\infty\simeq 0.41/\eta$, where $\eta$ is the (small) mass ratio. We derive
expressions for the time spent on the critical solution, number of orbits and
radiated energy as functions of the initial energy and impact parameter.Comment: Version published in PR

### Discontinuous collocation methods and gravitational self-force applications

Numerical simulations of extereme mass ratio inspirals, the mostimportant
sources for the LISA detector, face several computational challenges. We
present a new approach to evolving partial differential equations occurring in
black hole perturbation theory and calculations of the self-force acting on
point particles orbiting supermassive black holes. Such equations are
distributionally sourced, and standard numerical methods, such as
finite-difference or spectral methods, face difficulties associated with
approximating discontinuous functions. However, in the self-force problem we
typically have access to full a-priori information about the local structure of
the discontinuity at the particle. Using this information, we show that
high-order accuracy can be recovered by adding to the Lagrange interpolation
formula a linear combination of certain jump amplitudes. We construct
discontinuous spatial and temporal discretizations by operating on the
corrected Lagrange formula. In a method-of-lines framework, this provides a
simple and efficient method of solving time-dependent partial differential
equations, without loss of accuracy near moving singularities or
discontinuities. This method is well-suited for the problem of time-domain
reconstruction of the metric perturbation via the Teukolsky or
Regge-Wheeler-Zerilli formalisms. Parallel implementations on modern CPU and
GPU architectures are discussed.Comment: 29 pages, 5 figure

### Comparison Between Self-Force and Post-Newtonian Dynamics: Beyond Circular Orbits

The gravitational self-force (GSF) and post-Newtonian (PN) schemes are
complementary approximation methods for modelling the dynamics of compact
binary systems. Comparison of their results in an overlapping domain of
validity provides a crucial test for both methods, and can be used to enhance
their accuracy, e.g. via the determination of previously unknown PN parameters.
Here, for the first time, we extend such comparisons to noncircular
orbits---specifically, to a system of two nonspinning objects in a bound
(eccentric) orbit. To enable the comparison we use a certain orbital-averaged
quantity $\langle U \rangle$ that generalizes Detweiler's redshift invariant.
The functional relationship $\langle U \rangle(\Omega_r,\Omega_\phi)$, where
$\Omega_r$ and $\Omega_\phi$ are the frequencies of the radial and azimuthal
motions, is an invariant characteristic of the conservative dynamics. We
compute $\langle U \rangle(\Omega_r,\Omega_\phi)$ numerically through linear
order in the mass ratio $q$, using a GSF code which is based on a
frequency-domain treatment of the linearized Einstein equations in the Lorenz
gauge. We also derive $\langle U \rangle(\Omega_r,\Omega_\phi)$ analytically
through 3PN order, for an arbitrary $q$, using the known near-zone 3PN metric
and the generalized quasi-Keplerian representation of the motion. We
demonstrate that the $\mathcal{O}(q)$ piece of the analytical PN prediction is
perfectly consistent with the numerical GSF results, and we use the latter to
estimate yet unknown pieces of the 4PN expression at $\mathcal{O}(q)$.Comment: 44 pages, 2 figures, 4 table

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