Christodoulou and Rovelli have shown that black holes have large interiors that grow asymptotically linearly in advanced time, and speculated that this may be relevant to the information loss paradox. We show that there is no simple relation between the interior volume of an arbitrary black hole and its horizon area. That is, the volume enclosed is not necessarily a monotonically increasing function of the surface area.
Black Holes and Their Large InteriorsAn asymptotically flat Schwarzschild black hole has a spherical event horizon. The usual metric of this geometry in 4-dimensions, in the units G = c = 1, iswhere M is the ADM mass, and r is the areal radius. That is, the area of the event horizon r h is the area of the 2-sphere: 4πr 2 h . Unlike its surface area, the concept of "the volume" of a black hole is a tricky one. The reason is that volume depends on the choice of 3-dimensional spacelike hypersurface. As such, no unique volume can be prescribed to a black hole. Furthermore, the interior of a static black hole is nevertheless dynamical, so one should definitely not think of a black hole as a black box that bounds a certain amount of volume that can be easily estimated from knowing the size of its area. This is well-known: a maximally extended Schwarzschild [Kruskal-Szekeres] geometry has an infinitely large asymptotically flat region on the "other side", connected via the Einstein-Rosen bridge. Similarly, one could attach a closed FLRW universe to the interior of a black hole via the Einstein-Rosen bridge, resulting in the socalled Wheeler's "bag-of-gold" geometry [1]. Even non-black hole configurations can have arbitrarily large interiors than their areas might suggest [2]. What about black holes that were formed from gravitational collapse and have no second asymptotic region?One important motivation to study the interiors of generic black holes is of course the information loss paradox. As matter falls into a black hole and the black hole gradually Hawking evaporates away, it seems that when the black hole disappears, the information of the in-fallen matter will be lost forever, thereby threatening the unitarity of quantum