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

TL;DR

This work probes how the contour choice in Euclidean quantum gravity, used to tame the conformal-factor problem, depends on the DeWitt parameter $\alpha$ by generalizing the Wick-rotation rule-of-thumb to $L_\alpha$-fluctuation operators around ESAdS saddles in a reflecting cavity. It shows that path-integral stability agrees with thermodynamic stability for $\alpha\in(-2,-2/d)$ (e.g., $\alpha\in(-2,-0.5)$ in $d=4$) but fails when the DeWitt metric becomes positive definite ($\alpha> -2/d$) or when it becomes problematic ($\alpha\le -2$) due to gauge-fixing breakdown and non-diagonalizability. The study identifies complex-bubble transitions where diagonalizability breaks down, but demonstrates that contours on either side are related by smooth deformations, preserving the same path integral in the Gaussian setting. These results yield concrete criteria for selecting perturbation metrics that yield physically meaningful contours in linearized gravity and offer guidance for extending contour prescriptions to more general gravitational systems and boundary conditions.

Abstract

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 \cite{Marolf:2022ntb} 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 {\it 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 $α\neq -1$. 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 \cite{Marolf:2022ntb}, though we show this error to be harmless.

Figures (13)

• Figure 1: Orthogonal pairs of negative- and positive-norm vectors are shown in a fixed 1+1 Minkowski space. The vectors are normalized with respect to a $1+1$ Minkowski metric, so the vectors diverge in the limit where they become null. In cases of interest the metric will turn out to vary at the same order at which the vectors fail to be null, and thus at which the two eigenvalues differ. As a result, this picture is not an accurate depiction of the general case, though normalized vectors will always diverge in the null limit.
• Figure 2: The lowest dimensionless eigenvalue $\tilde{\lambda}$ is plotted in panel (a) for $y_+=0$ as a function of $\alpha$. The dashed line indicates $\tilde{\lambda}=0$, and the data agrees well with the known change in thermodynamic stability at $y_0=1.5$. Panel (b) shows the same data, but with the vertical axis now taken to be the difference between the dimensionless eigenvalue at each given $\alpha$ and the value at $\alpha=-1$. Panel (b) also uses an enlarged vertical scale. Panel (c) shows the analogue of (b) for $y_+ = \sqrt{3/11}$ (with an even more enlarged scale). For this value of $y_+$ the transition in thermodynamic stability occurs at $y_0=3$, which agrees well with the data shown. In both cases we take $d=4$ so that we focus on the range of $\alpha$ between $-2/d=-1/2$ and $-2$.
• Figure 3: The real part (left column) and the magnitude of the imaginary part (right column) of the first $20$ excited-mode eigenvalues are shown as functions of $y_0$ in the case $\Lambda=0$. The eigenvalues of pure gauge modes (black) are always real and their eigenvectors have positive norm. Positive-norm (negative-norm) physical modes with real eigenvalues are shown in red (blue). Physical modes with complex eigenvalues are shown in green, with each green data point representing a pair of complex-conjugate modes with complex-conjugate eigenvalues. The only complex modes from panel (d) that ccan be seen in figure (c) are in the family that runs to $y_0=80$ and beyond. The others have $\rm{Re}\,\tilde{\lambda} >1$ and so do not appear in figure (c).
• Figure 4: The functions $a(y)$ for merging modes at values of $y_0$ close to a bubble wall. Top: The left/right panel shows a real physical mode with positive/negative norm. Bottom: The left/right panel shows the real/imaginary part of the complex mode. Here $\alpha=-1$ and $y_+=0$. In the upper panels, from light to dark, the values of $y_0$ are $6.028489,\ 6.028490,\ 6.028491,\ 6.028492,\ 6.028493$. In the lower panel, from dark to light, the values of $y_0$ are $6.028495,\ 6.028496,\ 6.028497,\ 6.028498,\ 6.028499$. Darker colors are closer to the wall (at $y_0 \sim 6.028494$) and show larger amplitudes due to the eigenvectors becoming null. Our parameter values are close to being symmetric about the edge of the bubble so that the eigenfunctions on opposite sides agree to a good precision.
• Figure 5: The left panel shows the real part of a pair of eigenvalues that merge at a bubble wall. The red/blue data points are real physical eigenvalues with positive/negative eigenvectors. Each green dot represents a complex-conjugate pair. The right panel shows the absolute value of the sine of the angle (defined by the Cartesian metric \ref{['eq:CartesianMetric']}) between the relevant two modes (the two real eigenvectors for $y_0<y_0^*$ and the real and imaginary parts of the complex eigenvector for $y_0>y_0^*$). The red dashed horizontal line indicates $|\sin (v_1,v_2)|=0$. The vertical red dashed line indicates the bubble wall. Here $\alpha=-1$ and $y_0=0$.