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

Static spherically symmetric scalar field spacetimes with C^0 matching

All the classes of static massless scalar field models available currently in the Einstein theory of gravity necessarily contain a strong curvature naked singularity. We obtain here a family of solutions for static massless scalar fields coupled to gravity, which does not have any strong curvature singularity. This class of models contain a thin shell of singular matter, which has a physical interpretation. The central curvature singularity is, however, avoided which is common to all static massless scalar field spacetimes models known so far. Our result thus points out that the full class of solutions in this case may contain non-singular models, which is an intriguing possibility.Comment: revised version, 10 pages, no figures, accepted in Mod. Phys. Let.

On the fate of singularities and horizons in higher derivative gravity

We study static spherically symmetric solutions of high derivative gravity theories, with 4, 6, 8 and even 10 derivatives. Except for isolated points in the space of theories with more than 4 derivatives, only solutions that are nonsingular near the origin are found. But these solutions cannot smooth out the Schwarzschild singularity without the appearance of a second horizon. This conundrum, and the possibility of singularities at finite r, leads us to study numerical solutions of theories truncated at four derivatives. Rather than two horizons we are led to the suggestion that the original horizon is replaced by a rapid nonsingular transition from weak to strong gravity. We also consider this possibility for the de Sitter horizon.Comment: 15 pages, 3 figures, improvements and references added, to appear in PR

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

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

Upper limits of particle emission from high-energy collision and reaction near a maximally rotating Kerr black hole

The center-of-mass energy of two particles colliding near the horizon of a maximally rotating black hole can be arbitrarily high if the angular momentum of either of the incident particles is fine-tuned, which we call a critical particle. We study particle emission from such high-energy collision and reaction in the equatorial plane fully analytically. We show that the unconditional upper limit of the energy of the emitted particle is given by 218.6% of that of the injected critical particle, irrespective of the details of the reaction and this upper limit can be realized for massless particle emission. The upper limit of the energy extraction efficiency for this emission as a collisional Penrose process is given by 146.6%, which can be realized in the collision of two massive particles with optimized mass ratio. Moreover, we analyze perfectly elastic collision, Compton scattering, and pair annihilation and show that net positive energy extraction is really possible for these three reactions. The Compton scattering is most efficient among them and the efficiency can reach 137.2%. On the other hand, our result is qualitatively consistent with the earlier claim that the mass and energy of the emitted particle are at most of order the total energy of the injected particles and hence we can observe neither super-heavy nor super-energetic particles.Comment: 22 pages, 3 figures, typos corrected, reference updated, accepted for publication in Physical Review D, typos correcte

Dirty rotating black holes: regularity conditions on stationary horizons

We consider generic, or "dirty" (surrounded by matter), stationary rotating black holes with axial symmetry. The restrictions are found on the asymptotic form of metric in the vicinity of non-extremal, extremal and ultra-extremal horizons, imposed by the conditions of regularity of increasing strength: boundedness on the horizon of the Ricci scalar, of scalar quadratic curvature invariants, and of the components of the curvature tensor in the tetrad attached to a falling observer. We show, in particular, that boundedness of the Ricci scalar implies the "rigidity" of the horizon's rotation in all cases, while the finiteness of quadratic invariants leads to the constancy of the surface gravity. We discuss the role of quasiglobal coordinate r that is emphasized by the conditions of regularity. Further restrictions on the metric are formulated in terms of subsequent coefficients of expansion of metric functions by r. The boundedness of the tetrad components of curvature tensor for an observer crossing the horizon is shown to lead in the horizon limit to diagonalization of Einstein tensor in the frame of zero angular momentum observer on a circular orbit (ZAMO frame) for horizons of all degrees of extremality.Comment: 31 pages. Misprints correcte

Gravitational waves from inspiraling compact binaries: Second post-Newtonian waveforms as search templates

We ascertain the effectiveness of the second post-Newtonian approximation to the gravitational waves emitted during the adiabatic inspiral of a compact binary system as templates for signal searches with kilometer-scale interferometric detectors. The reference signal is obtained by solving the Teukolsky equation for a small mass moving on a circular orbit around a large nonrotating black hole. Fitting factors computed from this signal and these templates, for various types of binary systems, are all above the 90% mark. According to Apostolatos' criterion, second post-Newtonian waveforms should make acceptably effective search templates.Comment: LaTeX, one eps figure. Hires and color versions are available from http://jovian.physics.uoguelph.ca/~droz/uni/papers/search.htm

Quadrupole moments of rotating neutron stars

Numerical models of rotating neutron stars are constructed for four equations of state using the computer code RNS written by Stergioulas. For five selected values of the star's gravitational mass (in the interval between 1.0 and 1.8 solar masses) and for each equation of state, the star's angular momentum is varied from J=0 to the Keplerian limit J=J_{max}. For each neutron-star configuration we compute Q, the quadrupole moment of the mass distribution. We show that for given values of M and J, |Q| increases with the stiffness of the equation of state. For fixed mass and equation of state, the dependence on J is well reproduced with a simple quadratic fit, Q \simeq - aJ^2/M c^2, where c is the speed of light, and a is a parameter of order unity depending on the mass and the equation of state.Comment: ReVTeX, 7 pages, 5 figures, additional material, and references adde