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Title: Gravitational self-force on a particle in circular orbit around a Schwarzschild black hole

Abstract

We calculate the gravitational self-force acting on a pointlike particle of mass {mu}, set in a circular geodesic orbit around a Schwarzschild black hole. Our calculation is done in the Lorenz gauge: For given orbital radius, we first solve directly for the Lorenz-gauge metric perturbation using numerical evolution in the time domain; we then compute the (finite) backreaction force from each of the multipole modes of the perturbation; finally, we apply the 'mode-sum' method to obtain the total, physical self-force. The temporal component of the self-force (which is gauge invariant) describes the dissipation of orbital energy through gravitational radiation. Our results for this component are consistent, to within the computational accuracy, with the total flux of gravitational-wave energy radiated to infinity and through the event horizon. The radial component of the self-force (which is gauge dependent) is calculated here for the first time. It describes a conservative shift in the orbital parameters away from their geodesic values. We thus obtain the O({mu}) correction to the specific energy and angular momentum parameters (in the Lorenz gauge), as well as the O({mu}) shift in the orbital frequency (which is gauge invariant)

Authors:
;  [1]
  1. School of Mathematics, University of Southampton, Southampton, SO17 1BJ (United Kingdom)
Publication Date:
OSTI Identifier:
21020167
Resource Type:
Journal Article
Resource Relation:
Journal Name: Physical Review. D, Particles Fields; Journal Volume: 75; Journal Issue: 6; Other Information: DOI: 10.1103/PhysRevD.75.064021; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; ACCURACY; ANGULAR MOMENTUM; BLACK HOLES; CORRECTIONS; COSMOLOGY; DISTURBANCES; GAUGE INVARIANCE; GEODESICS; GRAVITATIONAL INTERACTIONS; GRAVITATIONAL RADIATION; GRAVITATIONAL WAVES; MASS; SCHWARZSCHILD METRIC

Citation Formats

Barack, Leor, and Sago, Norichika. Gravitational self-force on a particle in circular orbit around a Schwarzschild black hole. United States: N. p., 2007. Web. doi:10.1103/PHYSREVD.75.064021.
Barack, Leor, & Sago, Norichika. Gravitational self-force on a particle in circular orbit around a Schwarzschild black hole. United States. doi:10.1103/PHYSREVD.75.064021.
Barack, Leor, and Sago, Norichika. Thu . "Gravitational self-force on a particle in circular orbit around a Schwarzschild black hole". United States. doi:10.1103/PHYSREVD.75.064021.
@article{osti_21020167,
title = {Gravitational self-force on a particle in circular orbit around a Schwarzschild black hole},
author = {Barack, Leor and Sago, Norichika},
abstractNote = {We calculate the gravitational self-force acting on a pointlike particle of mass {mu}, set in a circular geodesic orbit around a Schwarzschild black hole. Our calculation is done in the Lorenz gauge: For given orbital radius, we first solve directly for the Lorenz-gauge metric perturbation using numerical evolution in the time domain; we then compute the (finite) backreaction force from each of the multipole modes of the perturbation; finally, we apply the 'mode-sum' method to obtain the total, physical self-force. The temporal component of the self-force (which is gauge invariant) describes the dissipation of orbital energy through gravitational radiation. Our results for this component are consistent, to within the computational accuracy, with the total flux of gravitational-wave energy radiated to infinity and through the event horizon. The radial component of the self-force (which is gauge dependent) is calculated here for the first time. It describes a conservative shift in the orbital parameters away from their geodesic values. We thus obtain the O({mu}) correction to the specific energy and angular momentum parameters (in the Lorenz gauge), as well as the O({mu}) shift in the orbital frequency (which is gauge invariant)},
doi = {10.1103/PHYSREVD.75.064021},
journal = {Physical Review. D, Particles Fields},
number = 6,
volume = 75,
place = {United States},
year = {Thu Mar 15 00:00:00 EDT 2007},
month = {Thu Mar 15 00:00:00 EDT 2007}
}
  • This is the second of two companion papers on computing the self-force in a radiation gauge; more precisely, the method uses a radiation gauge for the radiative part of the metric perturbation, together with an arbitrarily chosen gauge for the parts of the perturbation associated with changes in black-hole mass and spin and with a shift in the center of mass. In a test of the method delineated in the first paper, we compute the conservative part of the self-force for a particle in circular orbit around a Schwarzschild black hole. The gauge vector relating our radiation gauge to amore » Lorenz gauge is helically symmetric, implying that the quantity h{sub {alpha}{beta}u}{sup {alpha}u{beta}} must have the same value for our radiation gauge as for a Lorenz gauge; and we confirm this numerically to one part in 10{sup 14}. As outlined in the first paper, the perturbed metric is constructed from a Hertz potential that is in a term obtained algebraically from the retarded perturbed spin-2 Weyl scalar, {psi}{sub 0}{sup ret}. We use a mode-sum renormalization and find the renormalization coefficients by matching a series in L=l+1/2 to the large-L behavior of the expression for the self-force in terms of the retarded field h{sub {alpha}{beta}}{sup ret}; we similarly find the leading renormalization coefficients of h{sub {alpha}{beta}u}{sup {alpha}u{beta}} and the related change in the angular velocity of the particle due to its self-force. We show numerically that the singular part of the self-force has the form f{sub {alpha}}{sup S}=<{nabla}{sub {alpha}{rho}}{sup -1}>, the part of {nabla}{sub {alpha}{rho}}{sup -1} that is axisymmetric about a radial line through the particle. This differs only by a constant from its form for a Lorenz gauge. It is because we do not use a radiation gauge to describe the change in black-hole mass that the singular part of the self-force has no singularity along a radial line through the particle and, at least in this example, is spherically symmetric to subleading order in {rho}.« less
  • We present a numerical code for calculating the local gravitational self-force acting on a pointlike particle in a generic (bound) geodesic orbit around a Schwarzschild black hole. The calculation is carried out in the Lorenz gauge: For a given geodesic orbit, we decompose the Lorenz-gauge metric perturbation equations (sourced by the delta-function particle) into tensorial harmonics, and solve for each harmonic using numerical evolution in the time domain (in 1+1 dimensions). The physical self-force along the orbit is then obtained via mode-sum regularization. The total self-force contains a dissipative piece as well as a conservative piece, and we describe amore » simple method for disentangling these two pieces in a time-domain framework. The dissipative component is responsible for the loss of orbital energy and angular momentum through gravitational radiation; as a test of our code we demonstrate that the work done by the dissipative component of the computed force is precisely balanced by the asymptotic fluxes of energy and angular momentum, which we extract independently from the wave-zone numerical solutions. The conservative piece of the self-force does not affect the time-averaged rate of energy and angular-momentum loss, but it influences the evolution of the orbital phases; this piece is calculated here for the first time in eccentric strong-field orbits. As a first concrete application of our code we recently reported the value of the shift in the location and frequency of the innermost stable circular orbit due to the conservative self-force [Phys. Rev. Lett. 102, 191101 (2009)]. Here we provide full details of this analysis, and discuss future applications.« less
  • Recent breakthroughs in numerical relativity enable one to examine the validity of the post-Newtonian expansion in the late stages of inspiral. For the comparison between post-Newtonian (PN) expansion and numerical simulations, the waveforms in terms of the spin-weighted spherical harmonics are more useful than the plus and cross polarizations, which are used for data analysis of gravitational waves. Factorized resummed waveforms achieve better agreement with numerical results than the conventional Taylor expanded post-Newtonian waveforms. In this paper, we revisit the post-Newtonian expansion of gravitational waves for a test particle of mass {mu} in circular orbit of radius r{sub 0} aroundmore » a Schwarzschild black hole of mass M and derive the spherical harmonic components associated with the gravitational wave polarizations up to order v{sup 11} beyond Newtonian. Using the more accurate h{sub lm}'s computed in this work, we provide the more complete set of associated {rho}{sub lm}'s and {delta}{sub lm}'s that form important bricks in the factorized resummation of waveforms with potential applications for the construction of further improved waveforms for prototypical compact binary sources in the future. We also provide ready-to-use expressions of the 5.5PN gravitational waves polarizations h{sub +} and h{sub x} in the test-particle limit for gravitational waves data analysis applications. Additionally, we provide closed analytical expressions for 2.5PN h{sub lm}, 2PN {rho}{sub lm}, and 3PN {delta}{sub lm}, for general multipolar orders l and m in the test-particle limit. Finally, we also examine the implications of the present analysis for compact binary sources in Laser Interferometer Space Antenna.« less
  • The description of the inspiral of a stellar-mass compact object into a massive black hole sitting at a galactic center is a problem of major relevance for the future space-based gravitational-wave observatory Laser Interferometer Space Antenna (LISA), as the signals from these systems will be buried in the data stream and accurate gravitational-wave templates will be needed to extract them. The main difficulty in describing these systems lies in the estimation of the gravitational effects of the stellar-mass compact object on his own trajectory around the massive black hole, which can be modeled as the action of a local force,more » the self-force. In this paper, we present a new time-domain numerical method for the computation of the self-force in a simplified model consisting of a charged scalar particle orbiting a nonrotating black hole. We use a multidomain framework in such a way that the particle is located at the interface between two domains so that the presence of the particle and its physical effects appear only through appropriate boundary conditions. In this way we eliminate completely the presence of a small length scale associated with the need of resolving the particle. This technique also avoids the problems associated with the impact of a low differentiability of the solution in the accuracy of the numerical computations. The spatial discretization of the field equations is done by using the pseudospectral collocation method and the time evolution, based on the method of lines, uses a Runge-Kutta solver. We show how this special framework can provide very efficient and accurate computations in the time domain, which makes the technique amenable for the intensive computations required in the astrophysically relevant scenarios for LISA.« less
  • 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 and Iyer [Classical Quantum Gravity 20, 755 (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, andmore » (2) it accurately approximates the recently calculated shift in the Schwarzschild ISCO frequency due to the conservative-piece of the gravitational self-force [L. Barack and N. Sago, Phys. Rev. Lett. 102, 191101 (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 and Buonanno [Phys. Rev. D 81, 084024 (2010)]. These results serve as a useful point of comparison for future gravitational self-force calculations in the Kerr spacetime.« less