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

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

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

Conservative corrections to the innermost stable circular orbit (ISCO) of a Kerr black hole: a new gauge-invariant post-Newtonian ISCO condition, and the ISCO shift due to test-particle spin and the gravitational self-force

The innermost stable circular orbit (ISCO) delimits the transition from circular orbits to those that plunge into a black hole. In the test-mass limit, well-defined ISCO conditions exist for the Kerr and Schwarzschild spacetimes. In the finite-mass case, there are a large variety of ways to define an ISCO in a post-Newtonian (PN) context. Here I generalize the gauge-invariant ISCO condition of Blanchet & Iyer (2003) to the case of spinning (nonprecessing) binaries. The Blanchet-Iyer ISCO condition has two desirable and unexpected properties: (1) it exactly reproduces the Schwarzschild ISCO in the test-mass limit, and (2) it accurately approximates the recently-calculated shift in the Schwarzschild ISCO frequency due to the conservative-piece of the gravitational self-force [Barack & Sago (2009)]. The generalization of this ISCO condition to spinning binaries has the property that it also exactly reproduces the Kerr ISCO in the test-mass limit (up to the order at which PN spin corrections are currently known). The shift in the ISCO due to the spin of the test-particle is also calculated. Remarkably, the gauge-invariant PN ISCO condition exactly reproduces the ISCO shift predicted by the Papapetrou equations for a fully-relativistic spinning particle. It is surprising that an analysis of the stability of the standard PN equations of motion is able (without any form of "resummation") to accurately describe strong-field effects of the Kerr spacetime. The ISCO frequency shift due to the conservative self-force in Kerr is also calculated from this new ISCO condition, as well as from the effective-one-body Hamiltonian of Barausse & Buonanno (2010). These results serve as a useful point-of-comparison for future gravitational self-force calculations in the Kerr spacetime.Comment: 17 pages, 2 figures, 1 table. v2: references added; minor changes to match published versio

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

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 radiation from a particle in circular orbit around a black hole. VI. Accuracy of the post-Newtonian expansion

A particle of mass $\mu$ moves on a circular orbit around a nonrotating black hole of mass $M$. Under the assumption $\mu \ll M$ 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

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

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.

Misconceptions About General Relativity in Theoretical Black Hole Astrophysics

The fundamental role played by black holes in our study of microquasars, gamma ray bursts, and the outflows from active galactic nuclei requires an appreciation for, and at times some in-depth analysis of, curved spacetime. We highlight misconceptions surrounding the notion of coordinate transformation in general relativity as applied to metrics for rotating black holes that are beginning to increasingly appear in the literature. We emphasize that there is no coordinate transformation that can turn the metric of a rotating spacetime into that for a Schwarzschild spacetime, or more generally, that no coordinate transformation exists that can diagonalize the metric for a rotating spacetime. We caution against the notion of "local" coordinate transformation, which is often incorrectly associated with a global analysis of the spacetime.Comment: MNRAS accepte