Volume inside old black holes

Black holes that have nearly evaporated are often thought of as small objects, due to their tiny exterior area. However, the horizon bounds large spacelike hypersurfaces. A compelling geometric perspective on the evolution of the interior geometry was recently shown to be provided by a generally covariant definition of the volume inside a black hole using maximal surfaces. In this article, we expand on previous results and show that finding the maximal surfaces in an arbitrary spherically symmetric spacetime is equivalent to a $1+1$ geodesic problem. We then study the effect of Hawking radiation on the volume by computing the volume of maximal surfaces inside the apparent horizon of an evaporating black hole as a function of time at infinity: while the area is shrinking, the volume of these surfaces grows monotonically with advanced time, up to when the horizon has reached Planckian dimensions. The physical relevance of these results for the information paradox and the remnant scenarios are discussed.

Figure 1

(Left panel) Maximal surfaces (blue lines) inside a two-sphere in flat Minkowski spacetime. The largest is the $t=\text{const}$. (bold black lines) defining its inertial frame. (Right panel) Maximal surfaces (blue lines) inside a two-sphere on the horizon of a static black hole. Apart from the transient part connecting it to the horizon, the largest surface (bold black lines) lies on the limiting surface $r=3/2m$. The volume difference between the spheres $S_v$ and $S_{v^{\prime }}$ is finite and given by (7).

Figure 2

Penrose diagram illustrating the surface defining the volume (black curve) in the case of a black hole formed by collapse. The details of the surface in the interior of the collapsing object (dotted curve) will depend on the specific metric use to describe the latter. For Oppenheimer-Snyder and null massive shell collapses, this contribution to the volume is of the order $m^3$.

Figure 3

Eddington-Finkelstein diagram of the two families of extremal volume surfaces (blue lines) inside an evaporating black hole formed by a collapsing object. The surface defining the volume is in bold black. Note the close analogy with the static case, compare with Fig. 1.

Figure 4

The surfaces defining the volume enclosed in spheres at the apparent horizon of an evaporating black hole at different times (blue lines). The limiting surface lies close to $r=\alpha m(v)$, with $\alpha$ given by (9) (dashed line). Here $m=10$ in Planck units.

Figure 5

Speculative evolution of maximal surfaces in the case of a long-lived remnant scenario. The volume acquired during the evaporation process (continuous surfaces) deflates back to flat space (dotted surfaces). This is expected to happen in a time of order $m_0^4$, during which all the information stored can be released.