Gravitational Partition Function Admits New Solutions under Volume Constraints and Horizon Conditions
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The nature of gravity at a fundamental level remains a central challenge in physics, and understanding its connection to quantum mechanics requires exploring concepts like the gravitational partition function. Shan-Ping Wu, alongside Peng Cheng from Tianjin University and Shao-Wen Wei from Lanzhou University, investigate this function under specific volume constraints, extending previous work on massless configurations to include solutions with mass. Their research constructs a new class of geometries featuring both boundaries and horizons, revealing that the gravitational action is determined by the horizon’s properties. Importantly, the team demonstrates that these constrained geometries exhibit a surprising connection to semiclassical gravity, behaving similarly to the well-known Schwarzschild-de Sitter static patch and suggesting that imposing a volume constraint effectively mimics the influence of a cosmological constant. Scientists constructed a new class of volume-constrained Euclidean geometries (VCEGs) that possess both a boundary and a horizon, initially finding that the Euclidean action is determined solely by the contribution from the horizon. Further investigation revealed that these geometries can be extended, resulting in configurations with two horizons, each exhibiting a conical singularity, and an action proportional to the combined area of both horizons.
Volume Partition Function for Black Hole Entropy This research explores the relationship between black hole thermodynamics, gravitational constraints, and the concept of a partition function for a defined volume of space. It investigates methods for calculating black hole entropy and seeks to establish a consistent framework for understanding the thermodynamic properties of black holes and de Sitter space, a model of an expanding universe. A central idea is to define a partition function not just for a system on the boundary of spacetime, but within a defined volume itself, offering a new perspective on fundamental physics. Researchers investigated different methods for calculating black hole entropy, including the use of conserved quantities derived from symmetries in spacetime and calculations based on the Euclidean action.
The team also explored the idea of constrained instantons, solutions to the equations of gravity that satisfy specific constraints related to the volume of space being considered, linking this to the definition of a partition function for a fixed volume. These concepts are also applied to de Sitter space, suggesting that the same thermodynamic principles governing black holes may also apply to cosmology. The research incorporates an ensemble-averaged theory to describe black hole thermodynamics, involving averaging over different configurations of the black hole.
The team carefully compared and contrasted different methods for calculating entropy, highlighting their strengths and weaknesses. The study touches upon the role of asymptotic symmetries in the calculation of conserved charges and entropy, including Wald’s formula and Tolman’s solutions. This work contributes to a deeper understanding of the fundamental nature of gravity and its connection to thermodynamics, with implications for the development of a theory of quantum gravity. The application of these concepts to de Sitter space has implications for our understanding of the universe’s expansion and its ultimate fate. The work may also shed light on the black hole information paradox and proposes new approaches to calculating entropy that may be more accurate and reliable. Extended Geometries, Critical Mass, and Horizon Area This work investigates Euclidean geometries with a fixed volume constraint, extending previous massless configurations to include solutions with non-zero mass. Scientists constructed a new class of volume-constrained Euclidean geometries (VCEGs) possessing both a boundary and a horizon, initially finding that the Euclidean action is determined solely by the contribution from the horizon. Further analysis revealed that these initial geometries can be extended to include two horizons, each with conical singularities, and the action becomes proportional to the sum of the areas of these horizons. Researchers discovered a critical mass at which these conical singularities simultaneously vanish, defining a critical extended VCEG. These constrained geometries, possessing conical singularities, are interpreted as constrained gravitational instantons, and their contributions to the partition function reveal a striking analogy to the Euclidean Schwarzschild-de Sitter static patch, suggesting the volume constraint acts similarly to a cosmological constant in semiclassical gravity.
The team found that the Euclidean action takes a universal form proportional to the total horizon area.
Horizon Area Defines Critical Extended Geometry This work extends the established connection between the Euclidean gravitational action and the gravitational partition function, specifically by investigating volume-constrained Euclidean geometries. Researchers successfully constructed a new class of these geometries, featuring both boundaries and horizons, where the Euclidean action is determined by the contribution from the horizon itself. Further analysis revealed that these initial geometries can be extended to include two horizons, each with conical singularities, and the action becomes proportional to the sum of the areas of these horizons. Notably, the team found that simultaneously removing these conical singularities is generally impossible, except at a specific critical mass, defining a critical extended geometry. These constrained geometries, possessing conical singularities, are interpreted as constrained gravitational instantons, and their contribution to the partition function demonstrates a strong analogy with the Euclidean Schwarzschild-de Sitter static patch. This suggests that imposing a volume constraint effectively mimics the role of a cosmological constant in semiclassical gravity. 👉 More information 🗞 On the gravitational partition function under volume constraints 🧠 ArXiv: https://arxiv.org/abs/2512.12138 Tags: