Holographic pressure and volume for black holes

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.

💡 Research Summary

The paper tackles two long‑standing puzzles in black‑hole thermodynamics: the apparent absence of a pressure‑volume work term in the first law and the non‑extensive scaling of black‑hole variables. The authors propose that a holographic definition of pressure and volume emerges naturally from York’s quasi‑local gravitational thermodynamics when a black hole is enclosed by a finite timelike boundary (“York boundary”). In this setting the quasi‑local first law reads
 dE = T dS − s dA,
where E is the Brown‑York quasi‑local energy, T the Tolman temperature, S the Bekenstein–Hawking entropy, A the area of the boundary cross‑section, and s the surface pressure conjugate to A. Assuming a holographic dual living on the boundary, the authors identify
 P ≡ s and V ≡ A,
so that the quasi‑local law becomes the standard thermodynamic form dE = T dS − P dV for the dual theory. This identification supplies a genuine notion of system size (the boundary volume) that is independent of the horizon area, allowing the usual definition of extensivity (homogeneity of degree one) to be applied.

Two thermodynamic representations are considered: the internal‑energy representation E(S,V) and the Helmholtz free‑energy representation F(T,V). Extensivity is defined as E(λS,λV)=λE and F(T,λV)=λF for any λ>0, with the large‑system limit taken as V→∞ at fixed entropy density s=S/V (energy representation) or fixed temperature T (canonical representation). The authors emphasize that extensivity can be representation‑ and branch‑dependent.

For asymptotically flat Schwarzschild black holes, the quasi‑local energy E(S,V) remains non‑extensive even in the large‑system limit. In the canonical representation, however, the solution space splits into a small‑black‑hole branch and a large‑black‑hole branch. The small branch stays non‑extensive, while the large branch becomes extensive in the limit V→∞ at fixed T; the Helmholtz free energy scales linearly with V, although the internal energy does not. Thus, in flat space extensivity is not a universal property but appears only for the large‑black‑hole branch in the canonical ensemble.

For asymptotically AdS Schwarzschild black holes the situation is richer. The quasi‑local energy naturally decomposes into a sub‑extensive piece and an extensive piece that dominates as V→∞. The extensive contribution is derived explicitly:  E_ext(S,V)= (d−2)V/(8πGL)