Mass, Charge and Motion in Covariant Gravity Theories

TL;DR

This paper derives explicit, universal surface integral formulas for the mass and charges of small bodies in covariant gravity theories, simplifying the calculation of their motion across various theories.

Contribution

It provides a universal, explicit method to compute mass and charges of bodies in covariant gravity theories using surface integrals, applicable to multiple field types.

Findings

Derived explicit surface integral expressions for mass and charges.

Unified the computation of body motion across different gravity theories.

Clarified the meaning of sensitivities in scalar-tensor theories.

Abstract

Previous work established a universal form for the equation of motion of small bodies in theories of a metric and other tensor fields that have second-order field equations following from a covariant Lagrangian in four spacetime dimensions. Differences in the motion of the "same" body in two different theories are entirely accounted for by differences in the body's effective mass and charges in those different theories. Previously the process of computing the mass and charges for a particular body was left implicit, to be determined in each particular theory as the need arises. I now obtain explicit expressions for the mass and charges of a body as surface integrals of the fields it generates, where the integrand is constructed from the symplectic current for the theory. This allows the entire prescription for computing the motion of a small body to be written down in a few lines, in a manner universal across bodies and theories. For simplicity I restrict to scalar and vector fields (in addition to the metric), but there is no obstacle to treating higher-rank tensor fields. I explicitly apply the prescription to work out specific equations for various body types in Einstein gravity, generalized Brans-Dicke theory (in both Jordan and Einstein frames), Einstein-Maxwell theory and the Will-Nordvedt vector-tensor theory. In the scalar-tensor case, this clarifies the origin and meaning of the "sensitivities" defined by Eardley and others, and provides explicit formulae for their evaluation.