The volume of the black hole interior at late times

TL;DR

The paper defines a nonperturbative, path-integral notion of the interior volume (the ER bridge length) for 2D dilaton gravity and computes its time dependence across all genera. It uncovers a universal cancellation between semiclassical and nonperturbative spectral contributions, causing the interior length to grow linearly at early times and saturate at late times at an exponentially large scale in the entropy, offering a nonperturbative realization of complexity equals volume. By deriving all-genus two-point functions in JT gravity and related dilaton theories, and introducing a spectral-complexity observable, the work connects interior geometry to spectral data and boundary quantities, with validation in JT/dilaton models and SYK. The study also analyzes fluctuations and discusses tensions and interpretations for complexity in gravity, highlighting the role of nonperturbative effects and proposing avenues for generalization to other ensembles and higher-dimensional black holes.

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.

Figures (8)

• Figure 1: We show the numerical evaluation of the Einstein-Rosen bridge length (see \ref{['Ct3']}) for $\beta=15$ in orange, while the early and late time asymptotics \ref{['Asymp-ell']} are shown by the blue dashed and gray dotted curves, respectively.
• Figure 2: An example (with genus 2) of the type of hyperbolic surfaces which the path integral \ref{['general-formula']} sums over. The green curves represent the wiggly asymptotic boundaries, while the red curve represents the boundary to boundary geodesic. While there are an infinite number of non-self-intersecting boundary-to-boundary geodesics on such surfaces, above, we have drawn an example of all possible surface topologies that can result by cutting a genus two surface along the geodesics. The purple curves represent the closed geodesics which we use to glue the trumpet wavefunctions in \ref{['two-point-function-vol-decomp']} to different bordered Riemann surfaces.
• Figure 3: We show $\langle\ell(t)\rangle_\text{micro}$ (blue) and $\text{SFF}_\text{micro}(t)$ (orange) for JT gravity with $s=1$ (or $E=1/2$). We are plotting the expressions in \ref{['micro2']}, so we have dropped inessential prefactors. $\langle\ell(t)\rangle$ is a cubic polynomial in time, while the SFF is its second time derivative (up to the addition of a constant). We have normalized $\langle\ell(t)\rangle_\text{micro}$ and $\text{SFF}_\text{micro}(t)$ in such a way that their plateau values are the same.
• Figure 4: An example (with genus 3) of the type of hyperbolic surfaces which the path integral \ref{['general-formula-variance']} sums over. The figures represent the cuts ${\mathcal{M}}_{g, 2} \to {\mathcal{M}}_{h_1, 1}\oplus {\mathcal{M}}_{h_2, 1}\oplus {\mathcal{M}}_{g-h_1 - h_2, 2}$ with $h_{1,2}\geq 0$ and $h_{1}+ h_2 \leq g$ (top left), ${\mathcal{M}}_{g, 2} \to {\mathcal{M}}_{h, 1} \oplus {\mathcal{M}}_{g-h-1, 3}$ (top right), ${\mathcal{M}}_{g, 2} \to {\mathcal{M}}_{h, 2} \oplus {\mathcal{M}}_{g-h-1, 2}$ (bottom left), and ${\mathcal{M}}_{g, 2} \to {\mathcal{M}}_{g-2,\, 4}$ (bottom right).
• Figure 5: We show $\langle \ell \rangle$ and the band $\langle \ell \rangle \pm \sigma_{\ell}$, which at long times goes as $e^{S_0}\pm \sqrt{t}$ giving the graph above on a log-log plot. We put $\beta=15$ and $S_0=10$ on this plot. The opening time of the band in $\hat{t}=e^{-S_0}t$ can be made arbitrarily late by increasing $S_0$.