Relativistic Gravitational Bremsstrahlung

Abstract

Gravitational radiation is calculated for the situation of a small mass passing a large mass in an unbound trajectory, where the velocity of the small mass can be relativistic. This allows one to study gravitational radiation for cases in which the slow-motion approximation is not valid. The gravitational potentials, or perturbations in the metric, arising from the small mass, are determined explicitly by solving the perturbed field equations of general relativity, which are obtained by expanding the metric about a metric representing the geometry of the large mass. From these the energy flux of the emitted waves is calculated. In the nonrelativistic limit, the results agree with those of the slow-motion approximation. The qualitative behavior of the radiation at extreme relativistic velocities is discussed, and is found to disagree with what one would expect from the fast-motion approximation in that same limit. Numerical results are presented for the total energy, power, and angular distribution of energy radiated for a range of velocities from $0.01c$ to $0.9999c$. Significant features in the extreme relativistic limit are the peaking of the radiation in the forward direction and the peaking also in time, which both occur in electromagnetic radiation, and the fact that the total energy radiated in one transit is proportional to ${(1\ensuremath{-}\frac{{v}^{2}}{{c}^{2}})}^{\ensuremath{-}\frac{3}{2}}$.