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 self-force contains both conservative and dissipative terms, and the latter are responsible for the radiation reaction. The work done by the self-force matches the energy radiated away by the particle.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 regular 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 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 presents a detailed derivation of each of the three equations of motion (Part IV). Because the notion of a point mass is problematic in general relativity, the review concludes (Part V) with an alternative derivation of the equations of motion that applies to a small body of arbitrary internal structure.
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.
The detection of gravitational waves from binary black-hole mergers by the LIGO-Virgo Collaboration marks the dawn of an era when generalrelativistic dynamics in its most extreme manifestation is directly accessible to observation. In the future, planned (space-based) observatories operating in the millihertz band will detect the intricate gravitational-wave signals from the inspiral of compact objects into massive black holes residing in galactic centers. Such inspiral events are extremely effective probes of black-hole geometries, offering unparalleled precision tests of General Relativity in its most extreme regime. This prospect has in the past two decades motivated a programme to obtain an accurate theoretical model of the strong-field radiative dynamics in a two-body system with a small mass ratio. The problem naturally lends itself to a perturbative treatment based on a systematic expansion of the field equations in the small mass ratio. At leading order one has a pointlike particle moving in a geodesic orbit around the large black hole. At subsequent orders, interaction of the particle with its own gravitational perturbation gives rise to an effective "self-force", which drives the radiative evolution of the orbit, and whose effects can be accounted for order by order in the mass ratio.This review surveys the theory of gravitational self-force in curved spacetime and its application to the astrophysical inspiral problem. We first lay the relevant formal foundation, describing the rigorous derivation of the equation of self-forced motion using matched asymptotic expansions and other ideas. We then review the progress that has been achieved in numerically calculating the self-force and its physical effects in astrophysically realistic inspiral scenarios. We highlight the way in which, nowadays, self-force calculations make a fruitful contact with other approaches to the two-body problem and help inform an accurate universal model of binary black hole inspirals, valid across all mass ratios. We conclude with a summary of the state of the art, open problems and prospects.Our review is aimed at non-specialist readers and is for the most part selfcontained and non-technical; only elementary-level acquaintance with General Relativity is assumed. Where useful, we draw on analogies with familiar concepts from Newtonian gravity or classical electrodynamics. 7
Using a rigorous method of matched asymptotic expansions, I derive the equation of motion of a small, compact body in an external vacuum spacetime through second order in the body's mass (neglecting effects of internal structure). The motion is found to be geodesic in a certain locally defined regular geometry satisfying Einstein's equation at second order. I outline a method of numerically obtaining both the metric of that regular geometry and the complete second-order metric perturbation produced by the body.PACS numbers: 04.25.Nx, 04.30.Db Introduction. The governing equation of general relativity, the Einstein field equation (EFE), describes how bodies influence spacetime curvature and move within the resultant curved geometry. Yet, since the seminal work of Einstein, Infeld, and Hoffman in 1938 [1], study of this nonlinear problem of motion has largely focused on the post-Newtonian limit of slow motion and weak fields. In the strong-field regime, bodies have instead typically been approximated as test bodies moving in a spacetime that is unaffected by them. Only within the last fifteen years [2,3] has there arisen an analytical description of the gravitational backreaction: the body's perturbative effect on spacetime geometry and that perturbation's effect on the body's motion. In the case of a small mass m, this backreaction is called the gravitational self-force, and it is now well understood at linear order in m [4][5][6][7]. Beyond its foundational role, the self-force is also potentially of great astrophysical importance, as it describes the evolution of extreme-mass-ratio inspirals (EMRIs), in which a stellar black hole or neutron star spirals into a supermassive black hole. Such systems are predicted to be key sources for the planned gravitational wave detector LISA [8], and they will afford both a unique probe of strong-field dynamics and a map of the spacetime near black holes. The self-force also provides an essential point of comparison with other treatments of the problem of motion: it complements post-Newtonian theory [9,10] and fully nonlinear numerical simulations [10,11], both of which are ill-suited to extreme mass ratios; and it fixes mass-dependent parameters in Effective One Body (EOB) theory [12][13][14].
A small extended body moving through an external spacetime g αβ creates a metric perturbation h αβ , which forces the body away from geodesic motion in g αβ . The foundations of this effect, called the gravitational self-force, are now well established, but concrete results have mostly been limited to linear order. Accurately modeling the dynamics of compact binaries requires proceeding to nonlinear orders. To that end, I show how to obtain the metric perturbation outside the body at all orders in a class of generalized wave gauges. In a small buffer region surrounding the body, the form of the perturbation can be found analytically as an expansion for small distances r from a representative worldline. Given only a specification of the body's multipole moments, the field obtained in the buffer region suffices to find the metric everywhere outside the body via a numerical puncture scheme. Following this procedure at first and second order, I calculate the field in the buffer region around an arbitrarily structured compact body at sufficiently high order in r to numerically implement a second-order puncture scheme, including effects of the body's spin. I also define nthorder (local) generalizations of the Detweiler-Whiting singular and regular fields and show that in a certain sense, the body can be viewed as a skeleton of multipole moments.
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