Motion of a charge in a gravitational field

The DeWitt-Brehme equation of motion for a charged particle moving in a Riemannian space-time is examined for the case of absence of an external electromagnetic field by decomposing it into four scalar equations. This is done by introducing four orthonormal vectors along the world line of the particle and decomposing every term of the equation along them. It turns out, for example, that the "tail" has at most two nonvanishing components. We discuss the condition for a geodesic motion. We show that the "tail" and the local radiative damping term are not independent; the vanishing of the latter implies the vanishing of one of the two components of the "tail" and determines the other one to be a constant. The DeWitt-Brehme equation is then examined for the case of slow motion in a Schwarzschild gravitational field by expanding every term of it in a power series in 1c (Einstein-Infeld-Hoffmann method). It is shown that the seventh-order (in 1c) force term, which is the first radiative term, is exactly the traditional term (23)e23ξ?t3 in disagreement with a result obtained recently by DeWitt and DeWitt.