Mass, charge, and motion in covariant gravity 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 WillNordvedt 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.While general relativity (together with a cosmological constant) appears to provide a successful macroscopic description of all known gravitational phenomena, it is of interest to explore alternative theories that may provide a more fundamentally appealing description or suggest new experiments leading to the discovery of new phenomena. A basic framework for inventing and analyzing such theories is provided by Lagrangian field theory of a Lorentz metric and other tensor fields, and indeed some version of this framework is applied in essentially all modern attempts. In the present work we will employ this framework and restrict further to theories that are diffeomorphism covariant (no background structure), have second-order field equations, and live in four spacetime dimensions. This class contains a majority of the theories commonly considered, and many theories that apparently violate these criteria may be rewritten so as to satisfy them. For example, it is well known that f (R) theories (with fourth-order field equations) may be rewritten as scalar-tensor theories with second-order field equations, and many higher-dimensional theories are analyzed with a four-dimensional effective Lagrangian.Previous work [1] (hereafter paper I) considered the problem of the motion of small bodies in such theories. Here, a "small body" is defined in terms of its external fields: if a region surrounding the putative body may be found such that each field approximately splits into a 1/r-type "body field" plus a smoothly varying "external field", th...

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