Head-on collisions of black holes: The particle limit

We compute gravitational radiation waveforms, spectra, and energies for a point particle of mass m₀ falling from rest at radius r₀ into a Schwarzschild hole of mass M. This radiation is found to lowest order in (m₀/M) with the use of a Laplace transform. In contrast with numerical relativity results for head-on collisions of equal-mass holes, the radiated energy is found not to be a monotonically increasing function of initial separation; there is a local radiated-energy maximum at r₀≈4.5M. The present results, along with results for infall from infinity, provide a complete catalog of waveforms and spectra for particle infall. We give a representative sample from that catalog and an interesting observation: Unlike the simple spectra for other head-on collisions (either of particle and hole, or of equal mass holes) the spectra for ∞>r₀>~5M show a series of evenly spaced bumps. A simple explanation is given for this. Lastly, our energy versus r₀ results are compared with approximation methods used elsewhere, for small and for large initial separation.