The volume of the black hole interior at late times

TL;DR

This paper defines and analyzes the volume of black hole interiors at late times in two-dimensional gravity models, revealing a saturation phenomenon that supports the complexity equals volume conjecture in quantum gravity.

Contribution

It introduces a path integral approach to measure black hole interior volume and demonstrates its late-time saturation, confirming a key aspect of the complexity equals volume hypothesis.

Findings

Interior volume grows linearly at early times

Volume saturates at late times exponentially large in entropy

Supports the complexity equals volume conjecture

Abstract

Understanding the fate of semi-classical black hole solutions at very late times is one of the most important open questions in quantum gravity. In this paper, we provide a path integral definition of the volume of the black hole interior and study it at arbitrarily late times for black holes in various models of two-dimensional gravity. Because of a novel universal cancellation between the contributions of the semi-classical black hole spectrum and some of its non-perturbative corrections, we find that, after a linear growth at early times, the length of the interior saturates at a time, and towards a value, that is exponentially large in the entropy of the black hole. This provides a non-perturbative confirmation of the complexity equals volume proposal since complexity is also expected to plateau at the same value and at the same time.