Self-consistent gravitational self-force

Abstract

I review the problem of motion for small bodies in General Relativity, with an emphasis on developing a self-consistent treatment of the gravitational self-force. An analysis of the various derivations extant in the literature leads me to formulate an asymptotic expansion in which the metric is expanded while a representative worldline is held fixed; I discuss the utility of this expansion for both exact point particles and asymptotically small bodies, contrasting it with a regular expansion in which both the metric and the worldline are expanded. Based on these preliminary analyses, I present a general method of deriving self-consistent equations of motion for arbitrarily structured (sufficiently compact) small bodies. My method utilizes two expansions: an inner expansion that keeps the size of the body fixed, and an outer expansion that lets the body shrink while holding its worldline fixed. By imposing the Lorenz gauge, I express the global solution to the Einstein equation in the outer expansion in terms of an integral over a worldtube of small radius surrounding the body. Appropriate boundary data on the tube are determined from a local-in-space expansion in a buffer region where both the inner and outer expansions are valid. This buffer-region expansion also results in an expression for the self-force in terms of irreducible pieces of the metric perturbation on the worldline. Based on the global solution, these pieces of the perturbation can be written in terms of a tail integral over the body's past history. This approach can be applied at any order to obtain a self-consistent approximation that is valid on long timescales, both near and far from the small body. I conclude by discussing possible extensions of my method and comparing it to alternative approaches.

Sources of Gravitational Waves: Theory and Observations

“…It is important to notice that whereas in the original derivation the MiSaTaQuWa equation appears as the geodesic equation in the metric g µν + h tail µν , in the interpretation by Detweiler and Whiting, it is a geodesic equation in the (physical) metric g µν + h R µν , which is regular on the worldline of the body and satisfies the Einstein equations in vacuum. The derivation in [233,234] is limited to point masses, but Gralla and Wald [236], and Pound [237] demonstrated that the MiSaTaQuWa equation applies to any compact object of arbitrary internal structure. The MiSaTaQuWa equation of motion is not gauge invariant [238] and relies on the Lorenz gauge condition.…”

Gauge and averaging in gravitational self-force

“…This definition would not be sensible within our framework. The mass defined in [19] was found to evolve with time. The conclusion that the perturbed mass evolves with time appears to be at odds with the analysis of [20], where it was found that energy conservation is satisfied under the assumption of no change in mass.…”

“…However, it is easy to see that these effective sources are O(1/r 4 ), tO(1/r 6 ), andt 2 O(1/r 8 ), while the error in the linearized 5 A definition of perturbed mass was given in [19], which appears to correspond to equation (19) applied in the Lorenz gauge. This definition would not be sensible within our framework.…”

Proving the principle: Taking geodesic dynamics too seriously in Einstein's theory

“…48 Signi cant advances have been made in (Mino et al, 1997, Quinn and Wald, 1997, Gralla and Wald, 2008, Pound, 2010, which approximate the consequences of self-force e ects as rst order perturbations in the background metric.…”