Complete initial value spacetimes containing black holes in general relativity:Application to black hole-disk systems

We present a new initial data formulation to solve the full set of Einstein equations for spacetimes that contain a black hole under general conditions. The method can be used to construct complete initial data for spacetimes (the full metric) that contain a black hole. For stationary, axisymmetric spacetimes our method yields Kerr-Schild black holes in vacuum and rotating equilibrium neutron stars. We demonstrate the power of our new method by solving for the first time the whole system of Einstein equations for a nonaxisymmetric, self-gravitating torus in the presence of a black hole. The black hole has dimensionless spin 0.99, a rotation axis tilted at a 30 degrees angle with respect to the angular momentum of the disk, and a mass of approximately 1/5 of the disk.