Stability of saddles and choices of contour in the Euclidean path integral for linearized gravity: dependence on the DeWitt parameter

Due to the conformal factor problem, the definition of the Euclidean gravitational path integral requires a non-trivial choice of contour. The present work examines a generalization of a recently proposed rule-of-thumb [1] for selecting this contour at quadratic order about a saddle. The original proposal depended on the choice of an indefinite-signature metric on the space of perturbations, which was taken to be a DeWitt metric with parameter α = – 1. This choice was made to match previous results, but was otherwise admittedly ad hoc. To begin to investigate the physics associated with the choice of such a metric, we now explore contours defined using analogous prescriptions for α ≠ – 1. We study such contours for Euclidean gravity linearized about AdS-Schwarzschild black holes in reflecting cavities with thermal (canonical ensemble) boundary conditions, and we compare path-integral stability of the associated saddles with thermodynamic stability of the classical spacetimes. While the contour generally depends on the choice of DeWitt parameter α, the precise agreement between these two notions of stability found at α = – 1 continues to hold over the finite interval (– 2, – 2/d), where d is the dimension of the bulk spacetime. This agreement manifestly fails for α > – 2/d when the DeWitt metric becomes positive definite. However, we also find dramatic failures for α < – 2 that correlate with breakdowns of the de Donder-like gauge condition defined by α, and at which the relevant fluctuation operator fails to be diagonalizable. This provides criteria that may be useful in predicting metrics on the space of perturbations that give physically-useful contours in more general settings. Along the way, we also identify an interesting error in [1], though we show this error to be harmless.