Just how long can you live in a black hole and what can be done about it?

摘要

We study the problem of how long a journey within a black hole can last.Based on our observations, we make two conjectures. First, for observers thathave entered a black hole from an asymptotic region, we conjecture that thelength of their journey within is bounded by a multiple of the futureasymptotic ``size'' of the black hole, provided the spacetime is globallyhyperbolic and satisfies the dominant-energy and non-negative-pressuresconditions. Second, for spacetimes with ${\Bbb R}^3$ Cauchy surfaces (or anappropriate generalization thereof) and satisfying the dominant energy andnon-negative-pressures conditions, we conjecture that the length of a journeyanywhere within a black hole is again bounded, although here the bound requiresa knowledge of the initial data for the gravitational field on a Cauchysurface. We prove these conjectures in the spherically symmetric case. We alsoprove that there is an upper bound on the lifetimes of observers lying ``deepwithin'' a black hole, provided the spacetime satisfies thetimelike-convergence condition and possesses a maximal Cauchy surface. Further,we investigate whether one can increase the lifetime of an observer that hasentered a black hole, e.g., by throwing additional matter into the hole.Lastly, in an appendix, we prove that the surface area $A$ of the event horizonof a black hole in a spherically symmetric spacetime with ADM mass$M_{\text{ADM}}$ is always bounded by $A \le 16\pi M_{\text{ADM}}^2$, providedthat future null infinity is complete and the spacetime is globally hyperbolicand satisfies the dominant-energy condition.