On the gravitational partition function under volume constraints

TL;DR

This work analyzes Einstein gravity under a fixed-volume constraint and constructs volume-constrained Euclidean geometries (VCEGs), including extensions with nonzero mass parameters. Using the covariant phase space formalism, it shows that the on-shell Euclidean action is determined by horizon contributions and, in extended setups, by the sum of horizon areas, mirroring the Wald entropy and the area law. The authors introduce extended VCEGs with two horizons that generically possess conical singularities, identifiable as mass-constrained gravitational instantons, and they reveal a close analogy to Euclidean Schwarzschild–de Sitter systems, suggesting the volume constraint plays a role akin to a cosmological constant in semiclassical quantum gravity. They also develop a constrained-instanton framework for the path integral, discuss ensemble interpretations, and estimate decay rates between massless and massive configurations, highlighting the thermodynamic and topological coherence of these volume-constrained geometries. Overall, the volume constraint shapes the gravitational ensemble similarly to a cosmological constant, with horizon geometry governing the semiclassical partition function and potential implications for holography and quantum gravity thermodynamics.

Abstract

The Euclidean action serves as a bridge between gravitational thermodynamics and the partition function. In this work, we further examine the gravitational partition function under a fixed volume constraint, extending the fixed volume on-shell geometry in the massless case. Moving beyond this massless configuration, we construct solutions with nonzero mass functions, leading to a new class of volume-constrained Euclidean geometries (VCEGs). The VCEG contains both a boundary and a horizon, and its Euclidean action is determined solely by the contribution from the horizon. However, further investigation suggests that this boundary appears to be artificially constructed and can be extended, giving rise to the extended VCEGs. These geometries feature two horizons, each with a conical singularity, and their action is given by one-quarter of the sum of the areas of the two horizons. In general, the conical singularities on both horizons cannot be simultaneously removed, except at a critical mass $m = m^*$, which defines the critical extended VCEG. Configurations with conical singularities are interpreted as constrained gravitational instantons. An analysis of their contributions to the partition function and topology reveals a close analogy between the extended VCEGs and the Euclidean Schwarzschild-de Sitter static patch, suggesting that the volume constraint effectively plays a role akin to that of a cosmological constant in semiclassical quantum gravity.

Figures (11)

• Figure 1: Functional profiles of $N(r)$ at fixed volume $V = 4\pi/3$. The blue, orange, and red curves correspond to $m = 0$ ($r_c = 1$), $m = 0.25$ ($r_c \approx 0.91$), and $m = 0.5$ ($r_c \approx 0.81$), respectively. Throughout the plot, the parameter $\lambda$ is fixed at $-1$.
• Figure 2: Relation between the mass parameter $m$ and $r_c$ at fixed volume. The volume is chosen to be that of a unit ball in three dimensions, $V = 4\pi/3$. The blue curve represents the functional dependence of $m$ on $r_c$, while the red dashed line corresponds to $r_c = m$.
• Figure 3: Relation between the mass parameter $m$ and the Euclidean action $I_E$ at fixed volume $V = 4\pi/3$.
• Figure 4: Profile of the lapse function $N(\chi)$. The curves with different colors correspond to different parameter choices: the blue curve represents the case $\chi_c>\chi_c^*$, the orange curve corresponds to $\chi_c = \chi_c^*$, and the green curve denotes $\chi_c < \chi_c^*$. The critical value $\chi_c^*$ is defined by the condition $H(\chi_c^*) = 0$.
• Figure 5: Relations among the characteristic parameters of the metric, $m$, $\chi_e$, and $\chi_c$. (a) The dependence of $\chi_c$ and $\chi_e$ on $\chi_c$. The blue and red curves correspond to $\chi_c$ and $\chi_e$, respectively, where $\chi_c$ and $\chi_e$ denote the two roots of the lapse function $N(\chi)$. (b) The relation between the mass parameter $m$ and $\chi_e$. The mass parameters reaches its maximal value $m^*$ when $\chi_c = \chi_c^*$.