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

Donald Marolf|arXiv (Cornell University)|Mar 14, 2022

Black Holes and Theoretical Physics71 references35 citations

TL;DR

This paper proposes a real-time Lorentzian path integral formulation for gravitational partition functions, resolving the conformal factor problem by allowing codimension-2 conical singularities. It demonstrates that standard Euclidean black hole solutions with positive specific heat contribute as saddle points with non-zero weight in the semiclassical limit, thereby deriving conventional Euclidean results from a fundamental Lorentzian framework without ad hoc contour choices.

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.

Motivation & Objective

• To provide a first-principles foundation for thermal gravitational partition functions without relying on the problematic Euclidean path integral with its unbounded gravitational action.
• To resolve the conformal factor problem in Euclidean gravity by formulating the partition function in Lorentzian signature with controlled singularities.
• To show that standard Euclidean black hole solutions (with positive specific heat) emerge as physical saddle points in a real-time path integral framework.
• To establish a consistent semiclassical description of gravitational thermodynamics using Lorentzian metrics and codimension-2 conical singularities.
• To lay the groundwork for a non-perturbative derivation of Euclidean path integral contours from Lorentzian dynamics, closing a foundational gap in quantum gravity.

Proposed method

• Define the thermal partition function Z(β) as a Fourier-type integral over real Lorentz-signature path integrals ZL(T), with fβ(T) as the Fourier transform of the Boltzmann factor.
• Use real Lorentz-signature metrics as the fundamental integration contour, avoiding the unboundedness of the Euclidean action.
• Introduce codimension-2 conical singularities (analogous to Euclidean conical defects) to allow the formation of black hole-like saddle points.
• Assign an imaginary contribution to the Lorentzian action via a complexified Gauss-Bonnet theorem, enabling non-zero path integral weights.
• Apply saddle-point methods in general dimensions, focusing on fixed-area variations to identify physical saddle points.
• Deform the original Lorentzian contour to access Euclidean black hole geometries, showing they contribute with non-zero weight in the semiclassical limit.

Experimental results

Research questions

• RQ1 Can thermal gravitational partition functions be consistently defined using real Lorentz-signature path integrals instead of Euclidean ones?
• RQ2 How can standard Euclidean black hole solutions emerge as saddle points in a Lorentzian path integral framework?
• RQ3 What is the role of codimension-2 conical singularities in enabling non-zero contributions from black hole saddles?
• RQ4 Can the conformal factor problem be resolved by formulating gravity in Lorentzian signature with controlled singularities?
• RQ5 Is it possible to derive the standard Euclidean path integral contour prescriptions from a fundamental Lorentzian path integral?

Key findings

• The Lorentzian path integral with codimension-2 conical singularities yields a well-defined, oscillatory integrand that can be treated as a distribution.
• The inclusion of conical singularities introduces imaginary contributions to the action, enhancing contributions from high-entropy geometries by a factor of e^S, consistent with thermodynamics.
• Standard Euclidean black holes with positive specific heat are shown to be valid saddle points in the Lorentzian path integral, contributing with non-zero weight in the semiclassical limit.
• A codimension-2 subcontour of the original Lorentzian integration contour can be deformed to access these black hole saddles, and the same holds for the remaining integrals.
• The method provides a first-principles derivation of conventional Euclidean results from a Lorentzian starting point, avoiding ad hoc contour choices.
• The framework suggests that higher-derivative corrections may remain perturbatively small, even near singularities, though a full UV completion requires further study.