Newton’s second law, radiation reaction and type II Einstein–Maxwell fields

Abstract

Considering perturbations of the Reissner–Nordström metric while keeping the perturbations in the class of type II Einstein–Maxwell metrics, we perform a spherical harmonic expansion of all the variables up to the quadrupole term. This leads to rather surprising results. Referring to the source of the metric as a type II particle (analogous to referring to a Schwarzschild–Reissner–Nordström or Kerr–Newman particle), we see immediately that the Bondi momentum of the particle takes the classical form of mass times velocity plus an electromagnetic radiation reaction term, while the Bondi mass loss equation becomes the classical gravitational and electromagnetic (electric and magnetic) dipole and quadrupole radiation. The Bondi momentum loss equation turns into Newton’s second law of motion containing the Abraham–Lorentz–Dirac radiation reaction force plus a momentum recoil (rocket) force, while the reality condition on the Bondi mass aspect yields the conservation of angular momentum. Two things must be pointed out: (1) these results, (equations of motion, etc) take place, not in the spacetime of the type II metric but in an auxiliary space referred to as -space, whose physical meaning is rather obscure and (2) this analysis of the type II field equations is a very special case of a similar analysis of the general asymptotically flat Einstein–Maxwell equations. Although the final results are similar (though not the same), the analysis uses different equations (specifically, the type II field equations) and is vastly simpler than the general case. Without a great deal of the technical structures needed in the general case, one can see rather easily where the basic results reside in the type II field equations.