Gravitational radiation in black-hole collisions at the speed of light. III. Results and conclusions

This paper summarizes results following from the two preceding papers, I and II, on the gravitational radiation emitted in the head-on collision of two black holes, each with energy μ, at or near the speed of light. The radiation (in the speed-of-light case) near the forward and backward directions ^θ=0, π, where ^θ is the angle from the symmetry axis in the center-of-mass frame, is given by the series ${c}₀({^}{{τ}},{^}{{θ}})={Σ}{n=0}^{{∞}}{a}₂ₙ({{^}{{τ}}}{{μ}}){sin}²ⁿ{^}{{θ}}$ for the news function ${c}₀$ of retarded time ${^}{{τ}}$ and angle ${^}{{θ}}$; successive terms can in principle be found from a perturbation treatment. Here the form of ${a}₂({{^}{{τ}}}{{μ}})$ is presented. Knowledge of ${a}₂$ allows the new mass-loss formula of paper I to be applied, giving a calculation of the mass of the (assumed) final Schwarzschild black hole. Since the "final mass" resulting from the calculation exceeds 2μ, the assumptions of the new mass-loss formula must not all hold. The most likely explanation is that there is a "second burst" of radiation present in the space-time, centered for small angles ^θ on retarded times roughly |8μln^θ| later than the "first burst" described above. A more realistic crude estimate of the energy emitted in gravitational waves is given by the Bondi expression, taking only the first two terms a₀ and a₂ in c₀; this gives an efficiency of 16.4% for gravitational wave generation.