UNDERSTANDING INITIAL DATA FOR BLACK-HOLE COLLISIONS

Numerical relativity, applied to collisions of black holes, starts wit h initial data for black holes already in each other's strong field. F or the initial data to be astrophysically meaningful, it must approxim ately represent conditions that evolved from holes originally at large separation. The initial hypersurface data typically used for computat ion is based on mathematically simplifying prescriptions, such as conf ormal flatness of the 3-geometry and longitudinality of the extrinsic curvature. In the case of head-on collisions of equal-mass holes, ther e 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 hea d on collisions of nonspinning holes, i.e., the limit of an extreme ra tio of hole masses. The mass of the small hole is considered to be a p erturbation of the Schwarzschild spacetime of the larger hole, and Ein stein's equations are linearized in this perturbation and described by a single gauge-invariant spacetime function psi for each multipole. T he resulting quadrupole equations have been solved by numerical evolut ion 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, psi and its time derivative, on s urfaces of constant background Schwarzschild time. These evolved data can then be compared with ''prescribed'' data, evolved data can be rep laced by prescribed data on any hypersurface and evolved further forwa rd in time, a gauge-invariant measure of devia tion from conformal fla tness can be evaluated, and other comparisons can be made. The main fi ndings of this study are (i) for holes of unequal mass the use of pres cribed 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 extrinsi c curvature is more important than in the 3-geometry, and (iv) the use of the more general conformally flat longitudinal data does not notab ly improve this picture. [S0556-28u (97)03320-1].