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 $\mu$, at or near the speed of light. The radiation (in the speed-of-light case) near the forward and backward directions $\hat{\theta }=0\mathrm{}$, $\pi$, where $\hat{\theta }$ is the angle from the symmetry axis in the center-of-mass frame, is given by the series $c_0(\hat{\tau },\hat{\theta })=\Sigma {n=0\mathrm{}}^{\infty }{a}_{2n}\left(\frac{\hat{\tau }}{\mu }\right){sin}^{2n}\hat{\theta }$ for the news function $c_0$ of retarded time $\hat{\tau }$ and angle $\hat{\theta }$; successive terms can in principle be found from a perturbation treatment. Here the form of $a_2\left(\frac{\hat{\tau }}{\mu }\right)$ 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 $2\mu$, 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 $\hat{\theta }$ on retarded times roughly $\left|8\mathrm{}\mu \ln \hat{\theta }\right|$ 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 $a_0$ and $a_2$ in $c_0$; this gives an efficiency of 16.4% for gravitational wave generation.