Stability of the microcanonical ensemble in Euclidean Quantum Gravity

TL;DR

The paper resolves a long-standing tension in Euclidean quantum gravity by constructing a microcanonical path integral $Z_{micro}(E)$ as a contour transform of the canonical partition function, thereby fixing an off-shell energy $H$ and enabling a stability analysis of gravitational saddles. Applying a Wick-rotation rule for the conformal mode and analyzing linear perturbations about Euclidean Schwarzschild–AdS black holes in a cavity, the authors show the microcanonical action is positive definite for static, time-independent perturbations in $d=4,5$ with $\Lambda \,\le\,0$, removing the previous worry of negative modes affecting stability. Numerically, the lowest eigenmode has a real, positive eigenvalue and positive DeWitt norm, and while many higher modes are complex, none introduce a microcanonical instability; the result extends to the vanishing cosmological constant limit. The framework clarifies how microcanonical ensembles in gravitating systems can be stable and sets the stage for generalizations to rotating black holes, matter couplings, and Lorentzian‑signature analyses, with open questions about contour choices and conformal mode treatment remaining for future work.

Abstract

This work resolves a longstanding tension between the physically-expected stability of the microcanonical ensemble for gravitating systems and the fact that the known negative mode of the asymptotically flat Schwarzschild black hole decays too rapidly at infinity to affect the ADM energy boundary term at infinity. The key to our study is that we fix an appropriate {\it off-shell} notion of energy, which we obtain by constructing the microcanonical partition function as an integral transform of the canonical partition function. After applying the rule-of-thumb for Wick rotations from our recent companion paper to deal with the conformal mode problem of Euclidean gravity, we find a positive definite action for linear perturbations about any Euclidean Schwarzchild (-AdS) black hole. Most of our work is done in a cavity with reflecting boundary conditions, but the cavity wall can be removed by taking an appropriate limit.

Figures (6)

• Figure 1: An illustration of how we organize the space of bulk metrics ${\mathfrak G}_R$. The horizontal surfaces are codimension-1 slices of constant $\beta$. The red lines are 1-dimensional surfaces that define a notion of what it means to keep 'the rest of the metric' constant while changing $\beta$, and can thus be said to define additional coordinates on the space of all metrics. We may in particular proceed by choosing a reference value $\hat{\beta}$ and lifting any coordinates $\Psi^{(1)}$, $\Psi^{(2)}$ on the corresponding horizontal surface to the full space by taking them to be constant along the red lines. We use $\Psi_{\hat{\beta}}[g]$ to denote the result of inverting this procedure by taking a general metric $g$ and 'projecting' it along the red lines into the horizontal surface at $\hat{\beta}$.
• Figure 2: The lowest lying eigenmode with microcanonical boundary conditions as a function of $y_0$ and $y_+$. On the left panel we have $d=4$, while on the right panel we take $d=5$. To aid visualisation we also plot the plane $\widetilde{\lambda}=0$ in red. This mode turns out to be purely real and, most importantly, it has positive eigevanlue.
• Figure 3: The quantity $\eta$ computed for the lowest lying eigenmode with microcanonical boundary conditions as a function of $y_0$ and $y_+$. On the left panel we have $d=4$, while on the right panel we take $d=5$. In both cases the results indicate that the mode has positive norm. As a result, the rule-of-thumb from CanoPaper does not Wick-rotate either mode. And since we saw above that they have positive eigenvalues, both modes also define stable directions of the original action before any Wick-rotation.
• Figure 4: The real part (left column) and the imaginary part (right column) of the excited modes as a function of $y_0$ for $y_+=0$. The figures in the bottom row show magnified versions of small regions from the figures in the top row. All plots are for $d=4$. The colour coding is as follows: green triangles are non-gauge modes with complex eigenvalues; blue diamonds are non-gauge modes with negative norm under $\hat{\mathcal{G}}$; red squares are non-gauge modes with positive norm under $\hat{\mathcal{G}}$ and the black disks are pure gauge modes. Each green triangle has a two-fold degeneracy since complex modes come in conjugate pairs.
• Figure 5: The real part (left column) and the imaginary part (right column) of the excited modes as a function of $y_0$ for $y_+=0$. The bottom row is a zoom of the top row and all plots have $d=5$. The colour coding is as follows: green triangles are non-gauge modes with complex eigenvalues; blue diamonds are non-gauge modes with negative norm under $\hat{\mathcal{G}}$; red squares are non-gauge modes with positive norm under $\hat{\mathcal{G}}$ and the black disks are pure gauge modes. Each green triangle has a two-fold degeneracy since complex modes come in conjugate pairs.