Confinement by black holes

The question of whether an observer can escape from a black hole is addressed, using a recent general definition of a black hole in the form of a future outer trapping horizon. An observer on a future outer trapping horizon must enter the neighbouring trapped region. It is possible for the observer to subsequently escape from the trapped region. However, if the horizon separates the space-time into two disjoint components, inside and outside the horizon, then an observer inside a future outer trapping horizon cannot get outside, assuming the null energy condition. A similar confinement property holds for trapped, locally area-preserving cylinders, as suggested by Israel. It is widely believed that anyone falling into a black hole is forever confined within the black hole and cannot escape back to the universe outside. A precise general formulation of this conjecture has been impossible until recently, due to the lack of a general definition of a black hole. The conventional definition, the event horizon, is defined if the space-time is asymptotically flat, but the actual universe is thought not to be asymptotically flat. A definition of black hole in arbitrary space-times has recently been given in the form of a future outer trapping horizon [1]. This article addresses the question of whether an observer inside such a horizon can get outside. This could be stated more geometrically in terms of future-directed temporal curves, along which observers move. Recall the definition of a black hole [1]. The space-time is assumed time-orientable. Given a double-null foliation of spatial 2-surfaces with area form μ, the expansions θ± may be defined by μθ± = L±μ, where L± denotes the Lie derivative in the future-pointing null normal directions. A marginal surface is a spatial 2-surface S on which one expansion vanishes, fixed henceforth as θ+|S = 0. A trapping horizon is the closure H of a 3-surface H foliated by marginal surfaces on which θ−|H 6= 0 and L−θ+|H 6= 0, where the doublenull foliation is adapted to the marginal surfaces. The trapping horizon is said to be outer if L−θ+|H < 0, inner if L−θ+|H > 0, future if θ−|H < 0 and past if θ−|H > 0. For a future outer trapping horizon, the idea is that the outgoing light rays are diverging just outside the horizon and converging just inside, and that the ingoing light rays are converging. This provides the definition of a black hole. Similarly, a past outer trapping horizon provides the definition of a white hole. Inner trapping horizons include cosmological horizons as well as the possible inner boundaries of black or white holes. Recall also that a trapped surface [2] is a compact spatial 2-surface S such that θ+θ−|S > 0, and is said to be future trapped if θ±|S < 0 and past trapped if θ±|S > 0 [1]. A trapped region [1] is a connected subset of space-time for which each point lies on some trapped surface. A basic property of a trapping horizon is that if the foliating marginal