Understanding initial data for black hole collisions

Numerical relativity, applied to collisions of black holes, starts with initial data for black holes already in each other’s strong field. For the initial data to be astrophysically meaningful, it must approximately represent conditions that evolved from holes originally at large separation. The initial hypersurface data typically used for computation is based on mathematically simplifying prescriptions, such as conformal flatness of the 3-geometry and longitudinality of the extrinsic curvature. In the case of head-on collisions of equal-mass holes, there is evidence that such prescriptions work reasonably well, but it is not clear why, or whether, this success is more generally valid. Here we study these questions by considering the “particle limit” for head on collisions of nonspinning holes, i.e., the limit of an extreme ratio of hole masses. The mass of the small hole is considered to be a perturbation of the Schwarzschild spacetime of the larger hole, and Einstein’s equations are linearized in this perturbation and described by a single gauge-invariant spacetime function ψ for each multipole. The resulting quadrupole equations have been solved by numerical evolution for collisions starting from various initial separations, and the evolution is studied on a sequence of hypersurfaces. In particular, we extract hypersurface data, that is, ψ and its time derivative, on surfaces of constant background Schwarzschild time. These evolved data can then be compared with “prescribed” data, evolved data can be replaced by prescribed data on any hypersurface and evolved further forward in time, a gauge-invariant measure of deviation from conformal flatness can be evaluated, and other comparisons can be made. The main findings of this study are (i) for holes of unequal mass the use of prescribed data on late hypersurfaces is not successful, (ii) the failure is likely due to the inability of the prescribed data to represent the near field of the smaller hole, (iii) the discrepancy in the extrinsic curvature is more important than in the 3-geometry, and (iv) the use of the more general conformally flat longitudinal data does not notably improve this picture.