Motion of Small Bodies in Classical Field Theory
  I show how prior work with R. Wald on geodesic motion in general relativity
can be generalized to classical field theories of a metric and other tensor
fields on four-dimensional spacetime that 1) are second-order and 2) follow
from a diffeomorphism-covariant Lagrangian. The approach is to consider a
one-parameter-family of solutions to the field equations satisfying certain
assumptions designed to reflect the existence of a body whose size, mass, and
various charges are simultaneously scaled to zero. (That such solutions exist
places a further restriction on the class of theories to which our results
apply.) Assumptions are made only on the spacetime region outside of the body,
so that the results apply to ordinary bodies as well as black holes. The
worldline "left behind" by the shrinking, disappearing body is interpreted as
its lowest-order motion. An equation for this worldline follows from the
"Bianchi identity" for the theory, without use of any properties of the field
equations beyond their being second-order. The form of the force law for a
theory therefore depends only on the ranks of its various tensor fields, with
the field equations relevant only for determining the charges of a particular
body from its exterior fields. I explicitly derive the force law (and
mass-evolution law) in the case of scalar and vector fields, and give the
recipe in the higher-rank case. Note that the vector force law is quite
complicated, simplifying to the Lorentz force law only in the presence of the
Maxwell gauge symmetry. Example applications of the results are the motion of
"chameleon" bodies beyond the Newtonian limit, and the motion of bodies in
(classical) non-Abelian gauge theory. I also make some comments on the role
that scaling plays in the appearance of universality in the motion of bodies.
