Spherically Symmetric Event Horizons and Trapped Surfaces Developing {}from Innociuous Data

Abstract

In this paper we show the existence of a large class of spherically symmetric data $d$ (on a spacelike 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 {\it innociuous}, i.e. the data--surface $S$ does not contain any point of the horizon or of the trapped surface area. 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 spacelike 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 $S$ it develops (into the future) the event horizon $H$. 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 $\rho$ and $R:=[\text{curvature}]^{-1/2}$. Our data fulfil positivity of $\rho$, $2m/R=1$ (at the star boundary) and other properties. This is archieved by an algorithm, which for given $\rho $ yields $R$ (from an input parameter function $h \in C^1( \left[0,1\right],\left]0,-\infty \right[)$).