On the volume inside old black holes

TL;DR

This paper explores the interior volume of evaporating black holes using maximal surfaces, revealing that despite shrinking horizons, the interior volume can grow significantly over time, impacting the information paradox.

Contribution

It demonstrates that finding maximal surfaces in spherically symmetric spacetimes reduces to a 1+1 geodesic problem and analyzes how Hawking radiation influences interior volume growth.

Findings

Interior volume grows monotonically with advanced time during evaporation.

Horizon area shrinks to Planckian dimensions while interior volume increases.

Results have implications for the black hole information paradox and remnant theories.

Abstract

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.