Mass-energy and momentum extraction by gravitational wave emission in the merger of two colliding black holes: The non-head-on case

Abstract

We examine numerically the post-merger regime of two nonspining holes in non-head-on collisions in the realm of nonaxisymmetric Robinson-Trautman spacetimes. Characteristic initial data for the system are constructed and evolved via the Robinson-Trautman equation. The numerical integration is performed using a Galerkin spectral method which is sufficiently stable to reach the final configuration of the remnant black hole, when the gravitational wave emission ceases. The initial data contains three independent parameters, the ratio mass $\ensuremath{\alpha}$ of the individual colliding black holes, their initial premerger infalling velocity and the incidence angle of collision ${\ensuremath{\rho}}_{0}$. The remnant black hole is characterized by its final boost parameter, rest mass and scattering angle. The motion of the remnant black hole is restricted to the plane determined by the directions of the two initial colliding black holes, characterizing a planar collision. The net momentum fluxes carried out by gravitational waves are confined to this plane. We evaluate the efficiency of mass-energy extraction, the total energy and momentum carried out by gravitational waves and the momentum distribution of the remnant black hole for a large domain of initial data parameters. Our analysis is based on the Bondi-Sachs four-momentum conservation laws. The process of mass-energy extraction is shown to be less efficient as the initial data departs from the head-on configuration. Head-on collisions (${\ensuremath{\rho}}_{0}={0}^{o}$) and orthogonal collisions (${\ensuremath{\rho}}_{0}=90\ifmmode^\circ\else\textdegree\fi{}$) constitute, respectively, upper and lower bounds to the power emission and to the efficiency of mass-energy extraction. On the contrary, head-on collisions and orthogonal collisions constitute, respectively, lower and upper bounds for the momentum of the remnant. Distinct regimes of gravitational wave emission (bursts or quiescent emission) are characterized by the analysis of the time behavior of the gravitational wave power as a function of $\ensuremath{\alpha}$. In particular, the net gravitational wave flux is nonzero for equal-mass colliding black holes in non-head-on collisions. The momentum extraction and the patterns of the momentum fluxes, as a function of the incidence angle, are examined. The relation between the incidence angle and the scattering angle closely approximates a relation for the inelastic collision of classical particles in Newtonian dynamics.