The motion of point particles in curved spacetime

TL;DR

The article develops a complete, self-contained framework for the motion of point particles with scalar, electric, or gravitational mass in a curved spacetime. It builds the theory from first principles, introducing bitensors, world-function, and parallel propagators, and then derives Hadamard-type Green’s functions to cleanly separate singular source effects from the regular fields that drive self-force. The Detweiler-Whiting decomposition is central, enabling a causal, local self-force description (and nonlocal tail terms) across scalar, electromagnetic, and gravitational cases, with careful treatment of gauge issues and the small-body limit via matched asymptotic expansions. The work culminates in practical methods for self-force computations (mode-sum, effective-source, quasilocal), gauges, and physically observable consequences relevant to gravitational-wave astronomy, including ISCO shifts and redshift invariants, and provides extensive coordinate-formalism tools that underpin the field of self-force and its applications to EMRI modeling and beyond.

Abstract

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. What remains after subtraction is a smooth field that is fully responsible for 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. It then applies the theory to the construction of convenient coordinate systems to chart a neighbourhood of the particle's word line. It continues with a thorough discussion of Green's functions in curved spacetime. The review presents a detailed derivation of each of the three equations of motion. Because the notion of a point mass is problematic in general relativity, the review concludes with an alternative derivation of the equations of motion that applies to a small body of arbitrary internal structure.

Figures (11)

• Figure 1: In flat spacetime, the retarded potential at $x$ depends on the particle's state of motion at the retarded point $z(u)$ on the world line; the advanced potential depends on the state of motion at the advanced point $z(v)$.
• Figure 2: In curved spacetime, the retarded potential at $x$ depends on the particle's history before the retarded time $u$; the advanced potential depends on the particle's history after the advanced time $v$.
• Figure 3: In curved spacetime, the singular potential at $x$ depends on the particle's history during the interval $u \leq \tau \leq v$; for the regular potential the relevant interval is $-\infty < \tau \leq v$.
• Figure 4: Retarded coordinates of a point $x$ relative to a world line $\gamma$. The retarded time $u$ selects a particular null cone, the unit vector $\Omega^a := \hat{x}^a/r$ selects a particular generator of this null cone, and the retarded distance $r$ selects a particular point on this generator.
• Figure 5: The base point $x'$, the field point $x$, and the geodesic segment $\beta$ that links them. The geodesic is described by parametric relations $z^\mu(\lambda)$ and $t^\mu = dz^\mu/d\lambda$ is its tangent vector.