Relativistic action-at-a-distance interactions: Lagrangian and Hamiltonian to terms of second order

TL;DR

This paper derives second-order relativistic two-body Lagrangian and Hamiltonian formulations for action-at-a-distance interactions, providing explicit expressions and approximate solutions within a Poincaré invariant framework.

Contribution

It presents the first explicit derivation of single-time Lagrangian and Hamiltonian formulations for relativistic action-at-a-distance interactions up to second order.

Findings

Derived explicit second-order Lagrangian and Hamiltonian formulations.

Presented approximate solutions to the relativistic two-body problem.

Identified static limit components: linear potential, Coulomb-like interaction, and dynamical mass contribution.

Abstract

Relativistic systems of particles interacting pairwise at a distance (interactions not mediated by fields) in flat spacetime are studied. It is assumed that the interactions propagate at the speed of light in vacuum and that all masses are scalars under Poincar\'e transformations. The action functional of the theory depends on multiple times (the proper times of the particles). In the static limit, the theory has three components: a linearly rising potential, a Coulomb-like interaction and a dynamical component to the Poincar\'e invariant mass. In this Letter we obtain explicitly, to terms of second order, the Lagrangian and the Hamiltonian with all the dynamical variables depending on a single time. Approximate solutions of the relativistic two-body problem are presented.