The volume of the black hole interior at late times

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.

Gravity factorized

“…This is precisely what one would expect from a system with a discrete spectrum, and it is a non-trivial check to see that our model also works in the case of probe matter. Explicitly, if we again consider an ensemble average as in section 3.4, we find that after averaging the two-point function (5.13), the result is identical to the two-point function of probe matter fields in JT gravity to all orders in e −S 0 [16][17][18]92],…”

“…This is because the presence of the geodesic effectively reduces the mapping class group of the disk with a handle to that of the cylinder [16][17][18]. Similarly, the mapping class group of a surface of genus g and one boundary is effectively reduced by the presence of the geodesic to either that of a connected surface with two boundaries and genus g − 1, or that of two surfaces, each with one boundary, whose genera sum to g [16][17][18]92]. One can then use the same arguments as in section 3 to prove that only (…”

Holographic Complexity of Quantum Black Holes

“…the CFT 3 degrees of freedom below the cutoff yield the proper complexity of the quantum fields C bulk V (|φ ). 14 We have found that in our system their complexity vanishes to leading order in g 2 eff .…”

“…The discussion so far implicitly assumes that the bulk is classical, which describes the CFT to leading order in the limit of large N , or more properly, central charge c → ∞. Beyond this leading order in the large-c expansion, several aspects of holographic complexity have been investigated in two-dimensional dilaton gravity [14,[21][22][23][24], but its more general defining properties still await further elucidation.…”

“…This behavior is the key property that sustains VC and AC as plausible measures of the complexity for the chaotic time-evolution of the black hole. 1 See also [14] for volume and spectral complexity.…”