Motion of Small Bodies in Classical Field Theory

Abstract

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.

minor typographical changes

References in corpus (11)

• Chameleon Fields: Awaiting Surprises for Tests of Gravity in Space
• Gravitational Radiation Reaction to a Particle Motion
• The motion of point particles in curved spacetime
• Equivalence Principle Implications of Modified Gravity Models
• A Rigorous Derivation of Gravitational Self-force
• A Rigorous Derivation of Electromagnetic Self-force
• Axiomatic approach to radiation reaction of scalar point particles in curved spacetime
• On the Renormalization Group in Curved Spacetime
• The self-consistent gravitational self-force
• A general variational principle for spherically symmetric perturbations in diffeomorphism covariant theories
• Equations of Motion in General Relativity of a Small Charged Black Hole

Cited by in corpus (16)

• Gravitational Wave Tests of General Relativity with Ground-Based Detectors and Pulsar Timing Arrays
• Dark Energy vs. Modified Gravity
• Bumpy Black Holes in Alternate Theories of Gravity
• Effective Gravitational Wave Stress-energy Tensor in Alternative Theories of Gravity
• Conformal equivalence in classical gravity: the example of "veiled" General Relativity
• Mass, Charge and Motion in Covariant Gravity Theories
• Testing general relativity with compact-body orbits: A modified Einstein-Infeld-Hoffmann framework
• Dependence of Self-force on Central Object
• Gravitational self-force in scalar-tensor gravity
• Gravitational self-force in nonvacuum spacetimes
• Compact binary systems in Einstein-Aether gravity: Direct integration of the relaxed field equations to 2.5 post-Newtonian order
• Gravitational self-force in non-vacuum spacetimes: an effective field theory derivation
• QED Plasma and Magnetars
• Equivalence principle in scalar-tensor gravity
• Modified geodesic equations of motion for compact bodies in alternative theories gravity
• Modified Gravity Corrections in Fundamental Orbital Frequencies in Kerr Spacetime