Approximate analytical solutions to the initial data problem of black holebinary systems - art. no. 024017

We present approximate analytical solutions to the Hamiltonian and momentum constraint equations, corresponding to systems composed of two black holes with arbitrary linear and angular momentum. The analytical nature of these initial data solutions makes them easier to implement in numerical evoluti ons than the traditional numerical approach of solving the elliptic equatio ns derived from the Einstein constraints. Although in general the problem o f setting up initial conditions for black hole binary simulations is compli cated by the presence of singularities, we show that the methods presented in this work provide initial data with l(1) and l(infinity) norms of violat ion of the constraint equations falling below those of the truncation error (residual error due to discretization) present in finite difference codes for the range of grid resolutions currently used. Thus, these data sets are suitable for use in evolution codes. Detailed results are presented for th e case of a head-on collision of two equal-mass M black holes with specific angular momentum 0.5M at an initial separation of 10M. A straightforward s uperposition method yields data adequate for resolutions of h = M/4, and an "attenuated" superposition yields data usable to resolutions at least as f ine as h = M/8. In addition, the attenuated approximate data may be more tr actable in a full (computational) exact solution to the initial value probl em.