Gravitational thermodynamics without the conformal factor problem: Partition functions and Euclidean saddles from Lorentzian Path Integrals

TL;DR

This paper demonstrates that gravitational thermodynamics and black hole partition functions can be understood through real-time Lorentzian path integrals, avoiding the conformal factor problem inherent in Euclidean approaches.

Contribution

It introduces a Lorentzian path integral framework with codimension-2 singularities to compute gravitational partition functions, providing a new perspective on black hole saddle points.

Findings

Euclidean black hole saddles are recoverable in Lorentzian path integrals.

Black holes with positive specific heat contribute non-trivially to the semiclassical limit.

The approach circumvents the conformal factor problem in Euclidean quantum gravity.

Abstract

Thermal partition functions for gravitational systems have traditionally been studied using Euclidean path integrals. But in Euclidean signature the gravitational action suffers from the conformal factor problem, which renders the action unbounded below. This makes it difficult to take the Euclidean formulation as fundamental. However, despite their familiar association with periodic imaginary time, thermal gravitational partition functions can also be described by real-time path integrals over contours defined by real Lorentzian metrics. The one caveat is that we should allow certain codimension-2 singularities analogous to the familiar Euclidean conical singularities. With this understanding, we show that the usual Euclidean-signature black holes (or their complex rotating analogues) define saddle points for the real-time path integrals that compute our partition functions. Furthermore, when the black holes have positive specific heat, we provide evidence that a codimension-2 subcontour of our real Lorentz-signature contour of integration can be deformed so as to show that these black holes saddles contribute with non-zero weight to the semiclassical limit, and that the same is then true of the remaining two integrals.