The motion of point particles in curved spacetime

TL;DR

The work derives the self-force acting on point particles (scalar charge, electric charge, and mass) moving in a specified curved spacetime by decomposing fields into singular and radiative parts. Using bitensors, Synge's world function, and a Hadamard Green's-function framework, Poisson constructs retarded, singular, and radiative fields, showing that the self-force arises exclusively from the radiative field which satisfies the homogeneous wave equation and includes a curvature-tail term. The MiSaTaQuWa formalism for gravitational self-force emerges from linearized gravity and tail integrals, with Detweiler–Whiting providing a gauge-consistent interpretation: motion is geodesic in the effective metric $g_{ ext{eff}} = g + h^{ ext{R}}$. The review provides comprehensive coordinate systems (Riemann, Fermi, retarded), a full Green's-function toolkit in curved spacetime, and multiple derivations of the self-force, including mode-sum and matched asymptotic approaches, enabling practical calculations in Schwarzschild and Kerr spacetimes. Overall, the framework connects local near-field dynamics to nonlocal tail effects, highlighting the role of curvature in radiation reaction and offering robust, gauge-aware prescriptions for observable predictions in strong-field gravity.

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 -- its only effect is to contribute to the particle's inertia. What remains after subtraction is a smooth 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 (radiative) 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 concludes with a detailed derivation of each of the three equations of motion (part IV).

Figures (10)

• 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 radiative 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 $\Omega^a \equiv \hat{x}^a/r$ 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 $\beta$ that links them. The geodesic is described by parametric relations $z^\mu(\lambda)$ and $t^\mu = dz^\mu/d\lambda$ is its tangent vector.