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We present a new initial data formulation to solve the full set of Einsteinequations for spacetimes that contain a black hole under general conditions.The method can be used to construct complete initial data for spacetimes (thefull metric) that contain a black hole. Contrary to most current studies theformulation requires minimal assumptions. For example, rather than imposing theform of the spatial conformal metric we impose 3 gauge conditions adapted tothe coordinates describing the system under consideration. For stationary,axisymmetric spacetimes our method yields Kerr-Schild black holes in vacuum androtating equilibrium neutron stars. We demonstrate the power of our new methodby solving for the first time the whole system of Einstein equations for anonaxisymmetric, self-gravitating torus in the presence of a black hole. Theblack hole has dimensionless spin $J_{\rm bh}/M_{\rm bh}^2=0.9918$, a rotationaxis tilted at a $30^\circ$ angle with respect to the angular momentum of thedisk, and a mass of $\sim 1/5$ of the disk.

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