Radiation damping in a gravitational field

Abstract

The validity of the principle of equivalence is examined from the point of view of a charged mass point moving in an externally given gravitational field. The procedure is a covariant generalization of Dirac's work on the classical radiating electron. Just as Dirac's calculation was kept Lorentz invariant throughout, so the present calculation is maintained generally covariant throughout. With the aid of bi-tensors, which are nonlocal generalizations of ordinary local tensors, the manifest general covariance of each step is achieved in an elegant and useful way. The Green's functions for the scalar and vector wave equations in a curved manifold are obtained and applied to the derivation of the covariant Liénard-Wiechert potentials. The computation of energy-momentum balance across a world tube of infinitesimal radius surrounding the particle world-line then leads to the ponderomotive equations including radiation damping.

Because of the nonlocal electromagnetic field which a charged particle carries with itself, its use as a device to distinguish locally between gravitational and inertial fields is really not allowable. One should be prepared to find an explicit occurrence of the Riemann tensor in the ponderomotive equations, leading to the result that acceleration by a “true” gravitational field can produce bremsstrahlung, thereby causing a reactive force in addition to the force of inertia. It is remarkable, however, that such an explicit occurrence does not happen. The particle tries its best to satisfy the equivalenc principle in spite of its charge. It is only prevented from doing so (i.e., from following a geodetic path) because of the fact that, contrary to the case of flat space-time, the electromagnetic Green's function in a curved spacetime does not generally vanish inside the light cone, but gives rise to a “tail” on any initially sharp pulse of radiation. The ponderomotive equations have exactly the same form as Dirac found for the flat-space-time case except for the addition of an integral over the entire past history of the particle, representing the effect of the “tail.”