Post-Minkowski action for point particles and a helically symmetric binary solution

TL;DR

This paper develops Fokker actions and equations of motion for point particles in a post-Minkowski framework, providing solutions for binary systems with circular orbits and exploring conserved quantities and thermodynamic relations.

Contribution

It introduces new Fokker actions for point particles in post-Minkowski gravity, including a first post-Newtonian correction, and analyzes conserved quantities and thermodynamic relations for binary systems.

Findings

Derived parametrization-invariant and affine Fokker actions for point particles.

Obtained a formal solution for binary systems with circular orbits.

Found conserved energy and angular momentum despite divergent radiation field quantities.

Abstract

Two Fokker actions and corresponding equations of motion are obtained for two point particles in a post-Minkowski framework, in which the field of each particle is given by the half-retarded + half-advanced solution to the linearized Einstein equations. The first action is parametrization invariant, the second a generalization of the affinely parametrized quadratic action for a relativistic particle. Expressions for a conserved 4-momentum and angular momentum tensor are obtained in terms of the particles' trajectories in this post-Minkowski approximation. A formal solution to the equations of motion is found for a binary system with circular orbits. For a bound system of this kind, the post-Minkowski solution is a toy model that omits nonlinear terms of relevant post-Newtonian order; and we also obtain a Fokker action that is accurate to first post-Newtonian order, by adding to the post-Minkowski action a term cubic in the particle masses. Curiously, the conserved energy and angular momentum associated with the Fokker action are each finite and well-defined for this bound 2-particle system despite the fact that the total energy and angular momentum of the radiation field diverge. Corresponding solutions and conserved quantities are found for two scalar charges (for electromagnetic charges we exhibit the solution found by Schild). For a broad class of parametrization-invariant Fokker actions and for the affinely parametrized action, binary systems with circular orbits satisfy the relation $dE = \Omega dL$ (a form of the first law of thermodynamics), relating the energy, angular velocity and angular momentum of nearby equilibrium configurations.