arXiv subjects

Eric Poisson

Publications and source records attributed to Eric Poisson.

At least 73 records · Page 4Linked to original sources

arXiv2006

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.

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arXiv2005

Limitations of the adiabatic approximation to the gravitational self-force

A small body moving in the field of a much larger black hole and subjected to its own gravity moves on an accelerated world line in the background spacetime of the large black hole. The acceleration is produced by the body's gravitational self-force, which is constructed from the body's retarded gravitational field. The adiabatic approximation to the gravitational self-force is obtained instead from the half-retarded minus half-advanced field; it is known to produce the same dissipative effects as the true self-force. We argue that the adiabatic approximation is limited, because it discards important conservative terms which lead to the secular evolution of some orbital elements. We argue further that this secular evolution has measurable consequences; in particular, it affects the phasing of the orbit and the phasing of the associated gravitational wave. Our argument rests on a simple toy model involving a point electric charge moving slowly in the weak gravitational field of a central mass; the charge is also subjected to its electromagnetic self-force. In this simple context the true self-force is known explicitly and it can cleanly be separated into conservative and radiation-reaction pieces. We observe that the conservative part of the self-force produces a secular regression of the orbit's periapsis. We explain how the conclusions reached on the basis of the toy model can be extended to the gravitational self-force, and to fast motions and strong fields. While the limitations of the adiabatic approximation are quite severe in a post-Newtonian context in which the motion is slow and the gravitational field weak, they appear to be less so for rapid motions and strong fields.

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arXiv2005

Metric of a tidally distorted, nonrotating black hole

The metric of a tidally distorted, nonrotating black hole is presented in a light-cone coordinate system that penetrates the event horizon and possesses a clear geometrical meaning. The metric is expressed as an expansion in powers of r/R << 1, where r is a measure of distance from the black hole and R is the local radius of curvature of the external spacetime; this is assumed to be much larger than M, the mass of the black hole. The metric is calculated up to a remainder of order (r/R)^4, and it depends on a family of tidal gravitational fields which characterize the hole's local environment. The coordinate system allows an easy identification of the event horizon, and expressions are derived for its surface gravity and the rates at which the tidal interaction transfers mass and angular momentum to the black hole.

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arXiv2005

Gravitational perturbations of the Schwarzschild spacetime: A practical covariant and gauge-invariant formalism

We present a formalism to study the metric perturbations of the Schwarzschild spacetime. The formalism is gauge invariant, and it is also covariant under two-dimensional coordinate transformations that leave the angular coordinates unchanged. The formalism is applied to the typical problem of calculating the gravitational waves produced by material sources moving in the Schwarzschild spacetime. We examine the radiation escaping to future null infinity as well as the radiation crossing the event horizon. The waveforms, the energy radiated, and the angular-momentum radiated can all be expressed in terms of two gauge-invariant scalar functions that satisfy one-dimensional wave equations. The first is the Zerilli-Moncrief function, which satisfies the Zerilli equation, and which represents the even-parity sector of the perturbation. The second is the Cunningham-Price-Moncrief function, which satisfies the Regge-Wheeler equation, and which represents the odd-parity sector of the perturbation. The covariant forms of these wave equations are presented here, complete with covariant source terms that are derived from the stress-energy tensor of the matter responsible for the perturbation. Our presentation of the formalism is concluded with a separate examination of the monopole and dipole components of the metric perturbation.

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arXiv2004

Mass change and motion of a scalar charge in cosmological spacetimes

Continuing previous work reported in an earlier paper [L.M. Burko, A.I. Harte, and E. Poisson, Phys. Rev. D 65, 124006 (2002)] we calculate the self-force acting on a point scalar charge in a wide class of cosmological spacetimes. The self-force produces two types of effect. The first is a time-changing inertial mass, and this is calculated exactly for a particle at rest relative to the cosmological fluid. We show that for certain cosmological models, the mass decreases and then increases back to its original value. For all other models except de Sitter spacetime, the mass is restored only to a fraction of its original value. For de Sitter spacetime the mass steadily decreases. The second effect is a deviation relative to geodesic motion, and we calculate this for a charge that moves slowly relative to the dust in a matter-dominated cosmology. We show that the net effect of the self-force is to push on the particle. We show that this is not an artifact of the scalar theory: The electromagnetic self-force acting on an electrically charged particle also pushes on the particle. The paper concludes with a demonstration that the pushing effect can also occur in the context of slow-motion electrodynamics in flat spacetime.

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arXiv2004

The gravitational self-force

The self-force describes the effect of a particle's own gravitational field on its motion. While the motion is geodesic in the test-mass limit, it is accelerated to first order in the particle's mass. In this contribution I review the foundations of the self-force, and show how the motion of a small black hole can be determined by matched asymptotic expansions of a perturbed metric. I next consider the case of a point mass, and show that while the retarded field is singular on the world line, it can be unambiguously decomposed into a singular piece that exerts no force, and a smooth remainder that is responsible for the acceleration. I also describe the recent efforts, by a number of workers, to compute the self-force in the case of a small body moving in the field of a much more massive black hole. The motivation for this work is provided in part by the Laser Interferometer Space Antenna, which will be sensitive to low-frequency gravitational waves. Among the sources for this detector is the motion of small compact objects around massive (galactic) black holes. To calculate the waves emitted by such systems requires a detailed understanding of the motion, beyond the test-mass approximation.

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arXiv2004

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.

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arXiv2004

The motion of point particles in curved spacetime

This review is concerned with the motion of a point scalar charge, a point electric charge, and a point mass in a specified background spacetime. In each of the three cases the particle produces a field that behaves as outgoing radiation in the wave zone, and therefore removes energy from the particle. In the near zone the field acts on the particle and gives rise to a self-force that prevents the particle from moving on a geodesic of the background spacetime. The field's action on the particle is difficult to calculate because of its singular nature: the field diverges at the position of the particle. But it is possible to isolate the field's singular part and show that it exerts no force on the particle -- its only effect is to contribute to the particle's inertia. What remains after subtraction is a smooth field that is fully responsible for the self-force. Because this field satisfies a homogeneous wave equation, it can be thought of as a free (radiative) field that interacts with the particle; it is this interaction that gives rise to the self-force. The mathematical tools required to derive the equations of motion of a point scalar charge, a point electric charge, and a point mass in a specified background spacetime are developed here from scratch. The review begins with a discussion of the basic theory of bitensors (part I). It then applies the theory to the construction of convenient coordinate systems to chart a neighbourhood of the particle's word line (part II). It continues with a thorough discussion of Green's functions in curved spacetime (part III). The review concludes with a detailed derivation of each of the three equations of motion (part IV).

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arXiv2003

Low multipole contributions to the gravitational self-force

We calculate the unregularized monopole and dipole contributions to the self-force acting on a particle of small mass in a circular orbit around a Schwarzschild black hole. From a self-force point of view, these non-radiating modes are as important as the radiating modes with l greater than 2. In fact, we demonstrate how the dipole self-force contributes to the dynamics even at the Newtonian level. The self-acceleration of a particle is an inherently gauge-dependent concept, but the Lorenz gauge is often preferred because of its hyperbolic wave operator. Our results are in the Lorenz gauge and are also obtained in closed form, except for the even-parity dipole case where we formulate and implement a numerical approach.

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arXiv2003

Retarded coordinates based at a world line, and the motion of a small black hole in an external universe

In the first part of this article I present a system of retarded coordinates based at an arbitrary world line of an arbitrary curved spacetime. The retarded-time coordinate labels forward light cones that are centered on the world line, the radial coordinate is an affine parameter on the null generators of these light cones, and the angular coordinates are constant on each of these generators. The spacetime metric in the retarded coordinates is displayed as an expansion in powers of the radial coordinate and expressed in terms of the world line's acceleration vector and the spacetime's Riemann tensor evaluated at the world line. The formalism is illustrated in two examples, the first involving a comoving world line of a spatially-flat cosmology, the other featuring an observer in circular motion in the Schwarzschild spacetime. The main application of the formalism is presented in the second part of the article, in which I consider the motion of a small black hole in an empty external universe. I use the retarded coordinates to construct the metric of the small black hole perturbed by the tidal field of the external universe, and the metric of the external universe perturbed by the presence of the black hole. Matching these metrics produces the MiSaTaQuWa equations of motion for the small black hole.

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arXiv2002

A One-Parameter Family of Time-Symmetric Initial Data for the Radial Infall of a Particle into a Schwarzschild Black Hole

A one-parameter family of time-symmetric initial data for the radial infall of a particle into a Schwarzschild black hole is constructed within the framework of black-hole perturbation theory. The parameter measures the amount of gravitational radiation present on the initial spacelike surface. These initial data sets are then evolved by integrating the Zerilli-Moncrief wave equation in the presence of the particle. Numerical results for the gravitational waveforms and their power spectra are presented; we show that the choice of initial data strongly influences the waveforms, both in their shapes and their frequency content. We also calculate the total energy radiated by the particle-black-hole system, as a function of the initial separation between the particle and the black hole, and as a function of the choice of initial data. Our results confirm that for large initial separations, a conformally-flat initial three-geometry minimizes the initial gravitational-wave content, so that the total energy radiated is also minimized. For small initial separations, however, we show that the conformally-flat solution no longer minimizes the energy radiated.

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arXiv2002

A reformulation of the Barrabes-Israel null-shell formalism

We consider a situation in which two metrics are joined at a null hypersurface. It often occurs that the union of the two metrics gives rise to a Ricci tensor that contains a term proportional to a Dirac delta-function supported on the hypersurface. This singularity is associated with a thin distribution of matter on the hypersurface, and following Barrabes and Israel, we seek to determine its stress-energy tensor in terms of the geometric properties of the null hypersurface. While our treatment here does not deviate strongly from their previous work, it offers a simplification of the computational operations involved in a typical application of the formalism, and it gives rise to a stress-energy tensor that possesses a more recognizable phenomenology. Our reformulation of the null-shell formalism makes systematic use of the null generators of the singular hypersurface, which define a preferred flow to which the flow of matter can be compared. This construction provides the stress-energy tensor with a simple characterization in terms of a mass density, a mass current, and an isotropic pressure. Our reformulation also involves a family of freely-moving observers that intersect the surface layer and perform measurements on it. This construction gives operational meaning to the stress-energy tensor by fixing the argument of the delta-function to be proper time as measured by these observers.

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arXiv2002

Radiative falloff of a scalar field in a weakly curved spacetime without symmetries

We consider a massless scalar field propagating in a weakly curved spacetime whose metric is a solution to the linearized Einstein field equations. The spacetime is assumed to be stationary and asymptotically flat, but no other symmetries are imposed -- the spacetime can rotate and deviate strongly from spherical symmetry. We prove that the late-time behavior of the scalar field is identical to what it would be in a spherically-symmetric spacetime: it decays in time according to an inverse power-law, with a power determined by the angular profile of the initial wave packet (Price falloff theorem). The field's late-time dynamics is insensitive to the nonspherical aspects of the metric, and it is governed entirely by the spacetime's total gravitational mass; other multipole moments, and in particular the spacetime's total angular momentum, do not enter in the description of the field's late-time behavior. This extended formulation of Price's falloff theorem appears to be at odds with previous studies of radiative decay in the spacetime of a Kerr black hole. We show, however, that the contradiction is only apparent, and that it is largely an artifact of the Boyer-Lindquist coordinates adopted in these studies.

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arXiv2002

Mass loss by a scalar charge in an expanding universe

We study the phenomenon of mass loss by a scalar charge -- a point particle that acts a source for a noninteracting scalar field -- in an expanding universe. The charge is placed on comoving world lines of two cosmological spacetimes: a de Sitter universe, and a spatially-flat, matter-dominated universe. In both cases, we find that the particle's rest mass is not a constant, but that it changes in response to the emission of monopole scalar radiation by the particle. In de Sitter spacetime, the particle radiates all of its mass within a finite proper time. In the matter-dominated cosmology, this happens only if the charge of the particle is sufficiently large; for smaller charges the particle first loses some of its mass, but then regains it all eventually.

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arXiv2001

Scalar, electromagnetic, and gravitational self-forces in weakly curved spacetimes

We calculate the self-force experienced by a point scalar charge, a point electric charge, and a point mass moving in a weakly curved spacetime characterized by a time-independent Newtonian potential. The self-forces are calculated by first computing the retarded Green's functions for scalar, electromagnetic, and (linearized) gravitational fields in the weakly curved spacetime, and then evaluating an integral over the particle's past world line. In all three cases the self-force contains both a conservative and a nonconservative (radiation-reaction) part. The conservative part of the self-force is directly related to the presence of matter in the spacetime. The radiation-reaction part of the self-force, on the other hand, is insensitive to the presence of matter. Our result for the gravitational self-force is disturbing: a radiation-reaction force should not appear in the equations of motion at this level of approximation, and it should certainly not give rise to radiation antidamping. The last section of this paper attempts to make sense of this result by placing it in the context of the post-Newtonian N-body problem.

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arXiv2001

Radiative falloff in Einstein-Straus spacetime

The Einstein-Straus spacetime describes a nonrotating black hole immersed in a matter-dominated cosmology. It is constructed by scooping out a spherical ball of the dust and replacing it with a vacuum region containing a black hole of the same mass. The metric is smooth at the boundary, which is comoving with the rest of the universe. We study the evolution of a massless scalar field in the Einstein-Straus spacetime, with a special emphasis on its late-time behavior. This is done by numerically integrating the scalar wave equation in a double-null coordinate system that covers both portions (vacuum and dust) of the spacetime. We show that the field's evolution is governed mostly by the strong concentration of curvature near the black hole, and the discontinuity in the dust's mass density at the boundary; these give rise to a rather complex behavior at late times. Contrary to what it would do in an asymptotically-flat spacetime, the field does not decay in time according to an inverse power-law.

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arXiv2000

Regular coordinate systems for Schwarzschild and other spherical spacetimes

The continuation of the Schwarzschild metric across the event horizon is almost always (in textbooks) carried out using the Kruskal-Szekeres coordinates, in terms of which the areal radius r is defined only implicitly. We argue that from a pedagogical point of view, using these coordinates comes with several drawbacks, and we advocate the use of simpler, but equally effective, coordinate systems. One such system, introduced by Painleve and Gullstrand in the 1920's, is especially simple and pedagogically powerful; it is, however, still poorly known today. One of our purposes here is therefore to popularize these coordinates. Our other purpose is to provide generalizations to the Painleve-Gullstrand coordinates, first within the specific context of Schwarzschild spacetime, and then in the context of more general spherical spacetimes.

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arXiv1999

Can the post-Newtonian gravitational waveform of an inspiraling binary be improved by solving the energy balance equation numerically?

The detection of gravitational waves from inspiraling compact binaries using matched filtering depends crucially on the availability of accurate template waveforms. We determine whether the accuracy of the templates' phasing can be improved by solving the post-Newtonian energy balance equation numerically, rather than (as is normally done) analytically within the post-Newtonian perturbative expansion. By specializing to the limit of a small mass ratio, we find evidence that there is no gain in accuracy.

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