It is shown that a charge, accelerated by a gravitational field, radiates as if it were accelerated by any other external force. General relativistic effects appear only as corrections, which can be neglected for slow motions in weak fields. The bearing of this fact upon the equivalence principle is discussed in relation to a recent paper of Dewitt and Brehme.
According to classical electrodynamics (i.e., within the framework of the special theory of relativity) an accelerated charge radiates energy, whatever may be the cause of its acceleration. It seems, however, that the general relativity theory would make a distinction between gravitational forces, and other ones: according to the principle of equivalence, a gravitational field can be locally transformed away by choosing a suitable coordinate system, and thus it seems that a freely gravitating charge should not radiate.
In fact, the principle of equivalence holds strictly for uniform gravitational fields, and it is well known that a uniformly accelerated charge is a very controversial subject (I). On the other hand, in the case of a nonuniform gravitational field, one cannot rely on the local correctness of the equivalence principle, because a charged particle carries with it a nonlocal electromagnetic field, and thus may be "able to distinguish" a true gravitational field (defined by a non-Euclidean metric) from an inertial one (2). One should thus be prepared to find an explicit occurrence of the Riemann tensor in the ponderomotive equations.
Quite surprisingly, a recent investigation by DeWitt and Brehme (2) showed no such occurrence. Their final formula, however, is not explicit, and it seems extremely difficult to get from it any clear information as to the value of the radiated energy.
We here intend to attack this problem by a method similar to that of Einstein, Infeld, and Hoffman in pure gravitation theory. We shall assume that all motions are slow, and consider velocities as small quantities of the first order, accelerations of the second order, and so on.
The Maxwell-Lorentz equations in curved space are (3) * Partly supported by the U. S. Air Force, through the European Office of the Air Research and Development Command.