Mechanics of extended masses in general relativity

TL;DR

This paper develops a comprehensive framework for understanding the motion of extended bodies in general relativity, incorporating self-force effects and effective metrics, applicable to bodies with arbitrary shape, spin, and internal structure.

Contribution

It introduces exact equations for the evolution of extended bodies in GR using an effective metric, generalizing previous self-force concepts to arbitrary shapes and internal compositions.

Findings

The force and torque on bodies are equivalent to those on test bodies in an effective metric.

Self-field effects only renormalize multipole moments without additional instantaneous forces.

The MiSaTaQuWa self-force expression is derived as a special case.

Abstract

The "external" or "bulk" motion of extended bodies is studied in general relativity. Compact material objects of essentially arbitrary shape, spin, internal composition, and velocity are allowed as long as there is no direct (non-gravitational) contact with other sources of stress-energy. Physically reasonable linear and angular momenta are proposed for such bodies and exact equations describing their evolution are derived. Changes in the momenta depend on a certain "effective metric" that is closely related to a non-perturbative generalization of the Detweiler-Whiting R-field originally introduced in the self-force literature. If the effective metric inside a self-gravitating body can be adequately approximated by an appropriate power series, the instantaneous gravitational force and torque exerted on it is shown to be identical to the force and torque exerted on an appropriate test body moving in the effective metric. This result holds to all multipole orders. The only instantaneous effect of a body's self-field is to finitely renormalize the "bare" multipole moments of its stress-energy tensor. The MiSaTaQuWa expression for the gravitational self-force is recovered as a simple application. A gravitational self-torque is obtained as well. Lastly, it is shown that the effective metric in which objects appear to move is approximately a solution to the vacuum Einstein equation if the physical metric is an approximate solution to Einstein's equation linearized about a vacuum background.