Inner Horizon Saddles and a Spectral KSW Criterion

TL;DR

This paper explores complex saddle geometries related to inner horizons in black holes, proposing a spectral KSW criterion to assess the physical validity of one-loop corrections in such complex gravitational configurations.

Contribution

It introduces the concept of inner horizon saddles with negative boundary length, analyzes their role in black hole entropy corrections, and proposes a spectral KSW criterion for complex metrics.

Findings

Inner horizon saddle geometries contribute to entropy corrections.

A weaker KSW criterion can handle unphysical divergences in complex path integrals.

The analysis connects complex saddles with the vanishing density of states near extremality.

Abstract

The Bekenstein-Hawking entropy formula $\rho = e^{A/4G}$ receives significant corrections for charged black holes near extremality. Using standard results in JT gravity, the correction term can semiclassically be expressed as minus the exponential of the inner horizon area, $e^{A_{\text{inner}}/4G}$, and the cancellation between these two exponentials enforces a vanishing density of states towards extremality, when the two horizons collide. Building on arXiv:2402.10162, we argue that the correction term corresponds to a complex saddle geometry of the bulk gravitational path integral. The proposed geometry has a negative boundary length and caps off at the inner horizon; we refer to it as the inner horizon saddle. We discuss how the saddle, and its accompanying minus sign, contribute to the density of states through a Picard-Lefschetz analysis of the inverse Laplace contour, together with a stability analysis of the saddle. We also address the inner horizon saddle's violation of the Kontsevich-Segal-Witten (KSW) allowability criterion for the inclusion of complex metrics. Despite this violation, which is believed to cause unphysical divergences in path integral computations, one can describe one-loop effects on the inner horizon saddle by carefully treating wrong-sign modes. Motivated by this observation, we propose a weaker version of the KSW criterion, which we call the spectral KSW criterion. Its purpose is to characterize when one-loop corrections around complex gravitational saddles are well defined.