Mass-energy radiative transfer and momentum extraction by gravitational wave emission in the collision of two black holes

Abstract

We examine numerically the head-on collision of two boosted Schwarzschild black holes, in the realm of Robinson-Trautman spacetimes. Characteristic initial data for the system are constructed and the Robinson-Trautman equation is integrated for these data using a numerical code based on the Galerkin-collocation method. The initial data already have a common horizon so that the evolution covers the post-merger regime up to the final configuration, when the gravitational wave emission ceases. In the nonlinear regime gravitational waves are emitted, extracting mass and linear momentum from the system. The final configuration is a boosted Schwarzschild black hole with rest mass larger than the masses of the two individual initial black holes, and with a smaller final boost parameter characterizing the recoil velocity of the remnant. The efficiency $\ensuremath{\Delta}$ of the mass-energy extraction by gravitational waves is evaluated. The points $(\ensuremath{\Delta},y)$, where $y$ is the (normalized) rest mass of the remnant black hole, satisfy a nonextensive Tsallis distribution with entropic index $q\ensuremath{\simeq}1/2$ for $y\ensuremath{\lesssim}12$. Beyond $y\ensuremath{\sim}12$ the experimental points deviate from the distribution function and the efficiency presents an absolute maximum for the case of equally massive individual colliding black holes; the remnant has no recoil in this case. By using the Bondi mass formula we also evaluate the total energy ${E}_{W}$ carried out by gravitational waves as well as the radiative corrections to the efficiency. ${E}_{W}$ increases monotonically with $y$ and the experimental points $({E}_{W},y)$ also satisfy a nonextensive Tsallis distribution but with $q\ensuremath{\simeq}2/3$, up to $y\ensuremath{\sim}14.2$. Beyond this value the experimental points increase faster than the distribution function. For any initial infalling velocity $v$, the distribution of momentum of the remnant exhibits a maximum at ${\ensuremath{\alpha}}_{1}={\ensuremath{\alpha}}_{m}\ensuremath{\simeq}0.667$, where ${\ensuremath{\alpha}}_{1}$ is related to the ratio of pre-merger rest masses, and has a one-to-one correspondence with $y$ for fixed $v$. Two distinct regimes of gravitational wave emission can be characterized according to (i) ${\ensuremath{\alpha}}_{1}<{\ensuremath{\alpha}}_{m}$: bursts of gravitational bremsstrahlung; (ii) ${\ensuremath{\alpha}}_{1}>{\ensuremath{\alpha}}_{m}$: quiescent long time emission of gravitational waves. This picture is also sustained by the analysis of the time behavior of the power emitted ($d{E}_{W}/du$).