Gravitational thermodynamics without the conformal factor problem: partition functions and Euclidean saddles from Lorentzian path integrals

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