Gravitational radiation from distant encounters and from head-on collisions of black holes: The zero-frequency limit

The zero-frequency limit (ZFL) of the energy spectrum ($\frac{dE}{d\omega d\Omega }$) for the gravitational radiation emitted during the scattering or collision of two particles is investigated. If the asymptotic trajectories have constant velocities, at least one of which is nonzero, then the ZFL of the spectrum is flat and can be easily calculated. These calculations are made for the cases of distant encounters or head-on collisions of two compact objects, and comparisons to previous methods are made. It is found that the ZFL not only gives the exact low-frequency results, but that it provides an estimate of the total energy radiated, its polarization, and its angular distribution. Applied to the high-velocity collision ($V_{\infty }\sim 1$) of two equal-mass black holes it predicts an isotropic angular distribution of gravitational radiation with an efficiency of order unity.