The electromagnetic potential of a point charge in arbitrary motion in a weak gravitational field of a mass $M$ is found by a Green's-function approach. This solution is then applied to the study of the geometrical effects on the generation and propagation of electromagnetic waves to first order in the Riemann tensor. The electromagnetic field is considered as a small perturbation in the sense that its gravitational field is negligible. Generally the received radiation is generated by the charge at two retarded times: that for the direct path of a null ray and that representing geometrical scattering off of the central mass. Explicit expressions for the power radiated are found in the small-$\omega {\mathcal{r}}^{\prime }$ and large-$\omega {\mathcal{r}}^{\prime }$ limits, where ${\mathcal{r}}^{\prime }$ is the distance of the charge from the central mass, for both nonrelativistic and relativistic motion. The analysis shows under what conditions one may suitably make an approximation based on ray optics or photon emission. In the small-$\omega {\mathcal{r}}^{\prime }$ limit the power radiated by the charge is found to be smaller than calculated by a red-shift argument, and the total power depends on the orientation of the accelerating system to the mass $M$. For the case of nonrelativistic free-fall acceleration of the charge, geometrical corrections to the power radiated are of order $\frac{v^2}{c^2}$ smaller than the dominant contributions calculated from flat-space electromagnetism. Although the concept of constrained uniform motion is not precisely defined in a gravitational field, any reasonable definition gives electromagnetic radiation for a uniformly moving charge of a calculable amount in a gravitational field, in contrast to the nonexistence of such radiation in empty flat space. In the extreme relativistic limit ($v\simeq c$) the radiation from a uniformly moving charge is orders of magnitude larger than that from a freely falling charge; in either case large amounts of radiation are received by the observer at a time such that the charge, at the retarded time, is far from the mass and unambiguously moving uniformly. The geometrical origin and implications of this radiation are discussed, and related to the corresponding situation in gravitational radiation.

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