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

Towards the solution of the relativistic gravitational radiation reaction problem for binary black holes

Here we present the results of applying the generalized Riemann zeta-function regularization method to the gravitational radiation reaction problem. We analyze in detail the headon collision of two nonspinning black holes with extreme mass ratio. The resulting reaction force on the smaller hole is repulsive. We discuss the possible extensions of these method to generic orbits and spinning black holes. The determination of corrected trajectories allows to add second perturbative corrections with the consequent increase in the accuracy of computed waveforms.Comment: Contribution to the Proceedings of the 3rd LISA Symposiu

Gauge Problem in the Gravitational Self-Force II. First Post Newtonian Force under Regge-Wheeler Gauge

We discuss the gravitational self-force on a particle in a black hole space-time. For a point particle, the full (bare) self-force diverges. It is known that the metric perturbation induced by a particle can be divided into two parts, the direct part (or the S part) and the tail part (or the R part), in the harmonic gauge, and the regularized self-force is derived from the R part which is regular and satisfies the source-free perturbed Einstein equations. In this paper, we consider a gauge transformation from the harmonic gauge to the Regge-Wheeler gauge in which the full metric perturbation can be calculated, and present a method to derive the regularized self-force for a particle in circular orbit around a Schwarzschild black hole in the Regge-Wheeler gauge. As a first application of this method, we then calculate the self-force to first post-Newtonian order. We find the correction to the total mass of the system due to the presence of the particle is correctly reproduced in the force at the Newtonian order.Comment: Revtex4, 43 pages, no figure. Version to be published in PR

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

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

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