Abstract
An idealized ``test'' object in general relativity moves along a geodesic. However, if the object has a finite mass, this will create additional curvature in the spacetime, causing it to deviate from geodesic motion. If the mass is nonetheless sufficiently small, such an effect is usually treated perturbatively and is known as the gravitational self-force due to the object. This issue is still an open problem in gravitational physics today, motivated not only by basic foundational interest, but also by the need for its direct application in gravitational wave astronomy. In particular, the observation of extreme-mass-ratio inspirals by the future space-based detector LISA will rely crucially on an accurate modeling of the self-force driving the orbital evolution and gravitational wave emission of such systems. In this paper, we present a novel derivation, based on conservation laws, of the basic equations of motion for this problem. They are formulated with the use of a quasilocal (rather than matter) stress-energy-momentum tensor---in particular, the Brown-York tensor---so as to capture gravitational effects in the momentum flux of the object, including the self-force. Our formulation and resulting equations of motion are independent of the choice of the perturbative gauge. We show that, in addition to the usual gravitational self-force term, they also lead to an additional ``self-pressure'' force not found in previous analyses, and also that our results correctly recover known formulas under appropriate conditions. Our approach thus offers a fresh geometrical picture from which to understand the self-force fundamentally, and potentially useful new avenues for computing it practically.
Highlights
We have used quasilocal conservation laws to develop a novel formulation of self-force effects in general relativity, one that is independent of the choice of the perturbative gauge and applicable to any perturbative scheme designed to describe the correction to the motion of a localized object
We have shown that the correction to the motion of any finite spatial region, due to any perturbation of any spacetime metric, is dominated when that region is small by an extended gravitational self-force; this is the standard gravitational self-force term known up to now plus a new term, not found in previous analyses and attributable to a gravitational pressure effect with no analog in Newtonian gravity, which we have dubbed the gravitational self-pressure force
We have found that the total change in momentum ΔpðφÞ 1⁄4 pðfiφnÞal − pðinφitÞial between an initial and final time of any system subject to any metric perturbation h is given, in a direction determined by a conformal Killing vector φ
Summary
Average estimates indicate that LISA will be able to see on the order of hundreds of EMRI events per year [10], with an expectation of observing, for each, thousands of orbital cycles over a period on the order of one year before the final plunge [11] The trajectories defining these cycles and the gravitational wave signals produced by them will generally look much more complex than the relatively generic signals from mergers of stellar-mass black holes of comparable masses as observed, for example, by LIGO/Virgo. 1. Sketch of an EMRI, a two-body system consisting of a stellar-mass compact object (SCO), usually a stellar-mass black hole, of mass m ∼ 100−2M⊙, orbiting and eventually spiralling into a (super-) massive black hole (MBH), of mass M ∼ 104−7M⊙, and emitting gravitational waves in the process. Problem which can be posed in any (not just gravitational) classical field theory: the so-called self-force problem
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