Black holes in closed universes

Abstract

BLACK holes are now the subject matter of at least half the papers in general relativity. These papers rest on a foundation of sand, for black holes have been successfully defined only in asymptotically flat spacetimes, and no one really believes that the Universe is asymptotically flat. I shall provide a firm basis for black hole physics in the present paper; I shall extend the concept of ‘black hole’ to arbitrary stably causal spacetimes by essentially defining a black hole to be that object which contains all the ‘small’ trapped surfaces. As for most astrophysical applications black hole surfaces are located approximately by the outermost trapped surface boundary, this new definition allows results which depend only on the local behaviour of black holes in asymptotically flat spacetimes to be extended (approximately) to closed universes. Results which depend on the global behaviour of black holes cannot in general be extended to closed universes. For example, it is shown that the black hole area theorem—the statement that black holes never decrease their cross-sectional area—cannot be extended to closed universes. The new concept of black hole yields a purely geometrical definition of time direction in closed universes. My notation will be that of ref. 1.