Spherically symmetric event horizons and trapped surfaces developing from innocuous data

Abstract

In this paper we show the existence of a large class of spherically symmetric data d (on a space-like hypersurface S), from which a perfect fluid spacetime (surrounded by vacuum) develops. This spacetime contains an event horizon (with trapped surfaces behind it). The data d are regular and innocuous, i.e. the data surface S does not contain any point of the horizon or of the trapped surface area. The occurrence of the horizon (and trapped surfaces) is stable under small spherically symmetric variations of the data d. We give auxiliary data on an auxiliary hypersurface H and also on the star boundary; then we solve Einstein's equations for perfect fluid in the future and past of H. Our solution induces the above-mentioned data d on some chosen space-like hypersurface S in the past of H. By construction H turns out to be the matter part of the horizon, once we attach a vacuum to our matter spacetime. Obviously, from these data d on S the event horizon H can be developed (into the future). We solve the constraint equations for the auxiliary data posed on the null-surface H. This reduces the choice of these data to the choice of the density and . Our data fulfil positivity of , 2m/R=1 (at the star boundary) and other properties. This is archieved by an algorithm, which for given yields R (from an input parameter function ).