On the existence of initial data containing isolated black holes

TL;DR

This paper develops a general method for constructing initial data for Einstein's equations that include multiple black holes in equilibrium, ensuring the existence of solutions under specified boundary conditions.

Contribution

It provides a novel, rigorous construction of initial data with multiple isolated black holes using boundary conditions on marginally trapped surfaces.

Findings

Existence of solutions for the constructed initial data system.

A prescription for boundary shape and conditions for black holes.

Framework applicable to arbitrary numbers of black holes.

Abstract

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 rigorously that with these choices, the resulting non-linear elliptic system always admits solutions.