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

TL;DR

The paper demonstrates that gravitational thermal partition functions can be defined from real Lorentz-signature path integrals by including a controlled class of codimension-2 conical defects, which contribute imaginary terms to the action and allow saddles corresponding to Euclidean black holes to weigh into the semiclassical result. By analyzing saddle structure with Morse theory and fixing the area of the conical defect, the authors derive canonical and microcanonical partition functions that reproduce familiar black-hole thermodynamics, and extend the framework to rotating and charged black holes via a grand-canonical formulation with $E_\xi=E-\Omega J- Q\Phi$. They further show that higher-derivative corrections can be incorporated perturbatively by replacing $A/4G$ with Wald entropy $\sigma$, preserving the overall structure of the semiclassical contributions. The work provides a principled route to recover Euclidean gravity results from a Lorentzian starting point and suggests future directions for nonperturbative extensions and a deeper understanding of contour choices in gravity. The approach offers potential insights into resolving the conformal factor problem and connects Lorentzian saddles, fixed-area geometries, and standard black-hole thermodynamics in a unified real-time framework.

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.

Figures (3)

• Figure 1: An example of our cut-and-paste construction, where we cut pieces $\widetilde{\cal M}_i$ from four smooth spacetimes (left) and then paste them together cyclicly (right) to form a new spacetime which may include a conical singularity. Each piece $\widetilde{\cal M}_i$ is bounded by two hypersurfaces $\partial \Sigma_{i\pm}$ (colored curves) with common boundary $\gamma_i$ (red dots). The two surfaces $\partial \Sigma_{i\pm}$ are not allowed to intersect away from $\gamma_i$. The gluing is then done in a way that identifies all $\gamma_i$ and which cyclicly identifies $\Sigma_{i+}$ with $\Sigma_{(i+1)-}$. In the figure, we have given $\Sigma_{i+}$ and $\Sigma_{(i+1)-}$ the same color to make the cut-and-paste visually clear.
• Figure 2: In a smooth Lorentz-signature spacetime, every codimension-2 surface (red dot) is approached by four orthogonal null congruences. These approach from future-left and past right (both blue), and from future-right and past left (both green).
• Figure 3: We take our black hole exterior ${\cal M}_A$ to include the bifurcation surface $\gamma$ (red dot), but not the past horizon $H^-$ (dashed blue) or the future horizon $H^+$ (dashed green). As a result, the quotient by $e^{\xi T}$ may also be described by introducing two slices $\Sigma_{\pm}$ (each at a constant Killing time $\pm T/2$), focussing on the region $\widetilde{\cal M}_A$ between them, and identifying $\Sigma_+$ with $\Sigma_i$. This is a special case of the cut-and-paste construction of section \ref{['sec:prelim']} using only a single spacetime ${\cal M}_1 = {\cal M}_A$. Note that any geodesic that remains in the region $\widetilde{\cal M}_A$ and approaches the bifurcation surface must do so in a spacelike manner. As a result, the quotient ${\cal M}_{A,T}$ contains no null congruences that approach the image of the bifurcation surface. This means that ${\cal M}_{A,T}$ has ${\cal N}=0$ in the notation of section \ref{['sec:prelim']}. Recall also that the image of $\gamma$ in ${\cal M}_{A,T}$ will be called $\widetilde{\gamma}$.