Holographic pressure and volume for black holes

TL;DR

The paper develops a holographic framework for black-hole thermodynamics by combining York’s quasi-local gravity with boundary (holographic) duality, identifying the surface pressure with the boundary’s thermodynamic pressure P and the boundary area with the thermodynamic volume V. It shows that in static, spherically symmetric spacetimes the quasi-local first law can be written as dE = T dS − P dV, with P and V defined on a finite boundary, leading to a well-defined notion of system size and extensivity that distinguishes between asymptotically flat and AdS cases. Through explicit analyses of Schwarzschild and AdS–Schwarzschild black holes, the work demonstrates that extensivity is representation- and branch-dependent: asymptotically flat black holes are non-extensive in the energy representation but can be extensive for large AdS black holes in the canonical representation, while AdS black holes admit a rich extensive structure tied to their dual CFT, including a Cardy–Verlinde-like entropy and a Killing-volume term signaling finite-volume corrections. The AdS Smarr formula is interpreted in light of quasi-homogeneity, with the finite-volume Killing-volume term acting as a subextensive Casimir contribution in the CFT limit, which vanishes in the large-system limit, restoring the Euler relation and conventional extensivity. Overall, the paper provides a coherent holographic picture where boundary data encode the thermodynamic size and pressure, clarifying the role of subleading corrections and the emergence of extensivity in holographic contexts.

Abstract

We advocate for a holographic definition of thermodynamic pressure and volume for black holes based on quasi-local gravitational thermodynamics. When a black hole is enclosed by a finite timelike boundary, York's quasi-local first law includes a surface pressure conjugate to the boundary area. Assuming the existence of a holographically dual theory living on this boundary, these geometric quantities correspond to the pressure and volume of the dual thermal system. In this work we focus on static, spherically symmetric black holes, for which these quantities reduce to global thermodynamic variables. The holographic volume provides a notion of system size, allowing extensivity to be defined in standard thermodynamic terms, and it yields a definition of the large-system limit. For the asymptotically flat case, we show that, in the canonical thermodynamic representation, small Schwarzschild black holes are non-extensive, whereas large black holes become extensive in the large-system limit. A similar conclusion applies to Anti-de-Sitter Schwarzschild black holes, with the difference that the quasi-local energy of the large black hole also becomes extensive in the large-system limit. Before this limit, the energy decomposes into subextensive and extensive contributions, and we derive an explicit expression for the extensive part as a function of the finite volume and entropy.

Figures (3)

• Figure 1: Quasi-local energy $E$ of an asymptotically flat Schwarzschild black hole in 3+1 dimensions as a function of the boundary radius $r_B$ and the Tolman temperature $T$, seen from two different angles. The red corresponds to the large black hole solution, whereas the blue part corresponds to the small black hole solution.
• Figure 2: Heat capacity $C_V$ as a function of the temperature $T$ for a Schwarzschild black hole in $d=4$ at fixed $r_B=1$, with $G=1$. The small black hole (blue line) has a negative heat capacity, whereas the large black hole (red line) has a positive heat capacity.
• Figure 3: Heat capacity $C_V$ as a function of temperature $T$ at fixed $r_B=1$ for AdS-Schwarzschild in $d=4$, with $G=L=1$. $T_\text{min}$ is the temperature at which the two solutions merge. The small black hole (blue line) has a negative heat capacity, whereas the large black hole (red line) has a positive heat capacity.