Rigorous derivation of electromagnetic self-force

During the past century, there has been considerable discussion and analysis of the motion of a point charge in an external electromagnetic field in special relativity, taking into account “self-force” effects due to the particle’s own electromagnetic field. We analyze the issue of “particle motion” in classical electromagnetism in a rigorous and systematic way by considering a one-parameter family of solutions to the coupled Maxwell and matter equations corresponding to having a body whose charge-current density J^a(λ) and stress-energy tensor T_ab(λ) scale to zero size in an asymptotically self-similar manner about a worldline γ as λ→0. In this limit, the charge, q, and total mass, m, of the body go to zero, and q/m goes to a well-defined limit. The Maxwell field F_ab(λ) is assumed to be the retarded solution associated with J^a(λ) plus a homogeneous solution (the “external field”) that varies smoothly with λ. We prove that the worldline γ must be a solution to the Lorentz force equations of motion in the external field F_ab(λ=0). We then obtain self-force, dipole forces, and spin force as first-order perturbative corrections to the center-of-mass motion of the body. We believe that this is the first rigorous derivation of the complete first-order correction to Lorentz force motion. We also address the issue of obtaining a self-consistent perturbative equation of motion associated with our perturbative result, and argue that the self-force equations of motion that have previously been written down in conjunction with the “reduction of order” procedure should provide accurate equations of motion for a sufficiently small charged body with negligible dipole moments and spin. (There is no corresponding justification for the non-reduced-order equations.) We restrict consideration in this paper to classical electrodynamics in flat spacetime, but there should be no difficulty in extending our results to the motion of a charged body in an arbitrary globally hyperbolic curved spacetime.