Existence of initial data containing isolated black holes

We present a general construction of initial data for Einstein’s equations containing an arbitrary number of black holes, each of which is instantaneously in equilibrium. Each black hole is taken to be a marginally trapped surface and plays the role of the inner boundary of the Cauchy surface. The black hole is taken to be instantaneously isolated if its outgoing null rays are shear-free. Starting from the choice of a conformal metric and the freely specifiable part of the extrinsic curvature in the bulk, we give a prescription for choosing the shape of the inner boundaries and the boundary conditions that must be imposed there. We show that with these choices, the resulting nonlinear elliptic system always admits solutions.

Figure 1

The inner boundary $S$ of the Cauchy surface $M$. In this figure the inner boundary consists of two disconnected spherical surfaces.

Figure 2

The setup of the Lopatinski-Schapiro condition. $x_0$ is a point on the boundary $\partial \Omega$, $(x_1,x_2)$ span the tangent plane to $\partial \Omega$ at $x_0$, and $x_3$ is orthogonal to $\partial \Omega$; $s^a$ is the unit normal to $\delta \Omega$. The shaded region is $x_3<0$.