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arxiv: 2602.05130 · v2 · pith:MACI5HFJnew · submitted 2026-02-04 · ✦ hep-th · gr-qc

Holographic pressure and volume for black holes

Silvester Borsboom , Manus R. Visser This is my paper

Pith reviewed 2026-05-22 10:44 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords black hole thermodynamicsholographic dualityquasi-local energythermodynamic extensivitySchwarzschild black holesAnti-de Sitterthermodynamic volumelarge-system limit
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0 comments X

The pith

Holographic volume from a finite timelike boundary shows small Schwarzschild black holes are non-extensive while large ones become extensive.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper proposes defining thermodynamic pressure and volume for black holes through a holographic correspondence applied to York's quasi-local thermodynamics. A black hole inside a finite timelike boundary carries a surface pressure conjugate to the boundary area, which is interpreted as the pressure of a dual thermal system living on that boundary. The resulting holographic volume supplies a concrete notion of system size, so that extensivity can be checked in the usual thermodynamic sense. For asymptotically flat Schwarzschild black holes the analysis finds that small ones fail to be extensive in the canonical ensemble, while large ones recover extensivity in the large-system limit set by the boundary. The same pattern appears for AdS-Schwarzschild black holes, where the quasi-local energy itself also becomes extensive once the boundary volume is taken sufficiently large.

Core claim

In the canonical thermodynamic representation, small asymptotically flat Schwarzschild black holes are non-extensive, whereas large black holes become extensive in the large-system limit defined by the holographic volume. A similar conclusion holds for Anti-de-Sitter Schwarzschild black holes, with the additional feature that the quasi-local energy of the large black hole also becomes extensive. Before this limit the energy decomposes into subextensive and extensive contributions, and an explicit expression is derived for the extensive part as a function of the finite volume and the entropy.

What carries the argument

The holographic volume read off from the finite timelike boundary, which furnishes a system-size parameter that permits a standard thermodynamic definition of extensivity.

If this is right

  • Small asymptotically flat Schwarzschild black holes violate the usual scaling of energy with system size.
  • Large black holes of either type recover extensivity once the boundary volume is taken to infinity.
  • For AdS-Schwarzschild the quasi-local energy splits into a subextensive piece and an extensive piece whose explicit dependence on volume and entropy is given.
  • The large-system limit supplies a well-defined notion of thermodynamic volume for black-hole thermodynamics in both flat and AdS asymptotics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The construction may allow a controlled thermodynamic limit for black-hole thermodynamics that is otherwise absent in asymptotically flat space.
  • Similar extensivity statements could be checked for charged or rotating black holes by repeating the quasi-local construction.
  • The split of energy into extensive and subextensive parts offers a concrete way to compare holographic thermodynamics with ordinary extensive systems at finite volume.

Load-bearing premise

The assumption that a holographically dual theory lives on the finite timelike boundary so that the geometric pressure and volume match the thermodynamic pressure and volume of the dual system.

What would settle it

A direct computation, within a concrete holographic model such as AdS/CFT, of whether the total energy of a large black hole scales linearly with the boundary volume once the large-system limit is taken.

Figures

Figures reproduced from arXiv: 2602.05130 by Manus R. Visser, Silvester Borsboom.

Figure 1
Figure 1. Figure 1: Quasi-local energy E of an asymptotically flat Schwarzschild black hole in 3+1 dimensions as a function of the boundary radius rB 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. For d = 4 the small and large black hole solutions can be written, respectively, as [20] r (1) … view at source ↗
Figure 2
Figure 2. Figure 2: Heat capacity CV as a function of the temperature T for a Schwarzschild black hole in d = 4 at fixed rB = 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. Using S = S(rh) together with the Tolman temperature in (4.3), one obtains [20, 92] CV = (d − 2)Ωd−2r d−2 h 4G [PITH_FULL_IMAGE:figures/full_fig_p024_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Heat capacity CV as a function of temperature T at fixed rB = 1 for AdS￾Schwarzschild in d = 4, with G = L = 1. Tmin 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. 4.3 Thermodynamic interpretation of the AdS Smarr relation Using the expressions for the quasi-local therm… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a holographic definition of thermodynamic pressure and volume for black holes by applying York's quasi-local first law to a finite timelike boundary enclosing the black hole. It assumes these geometric quantities correspond to the thermodynamic pressure and volume of a dual theory on the boundary. For static spherically symmetric black holes, the quantities reduce to global variables. The paper then analyzes extensivity, concluding that asymptotically flat small Schwarzschild black holes are non-extensive while large ones become extensive in the large-system limit, with analogous results for AdS-Schwarzschild black holes including an explicit expression for the extensive contribution to the quasi-local energy.

Significance. This approach offers a way to introduce a notion of system size and extensivity into black hole thermodynamics via holography and quasi-local methods. If valid, it could provide new insights into the thermodynamic behavior of black holes in different regimes and limits, particularly distinguishing small and large black holes. The derivation of explicit expressions for the extensive part of the energy in the AdS case is a positive aspect that allows for concrete calculations.

major comments (2)
  1. The central mapping from York's quasi-local pressure P and holographic volume V to the thermodynamic variables of a dual theory is introduced by assumption rather than derived from an explicit dual Lagrangian or renormalization. This identification is load-bearing for the extensivity claims in the abstract and §5, as the diagnosis of linear scaling of quasi-local energy E with V at fixed entropy density depends directly on it. For the asymptotically flat Schwarzschild case, where no standard holographic dual exists on a finite timelike boundary, this assumption requires additional justification or a consistency check to support the non-extensivity of small black holes versus extensivity of large ones in the large-system limit.
  2. §4, discussion of the large-system limit: The claim that large black holes become extensive relies on the boundary radius becoming large while holding entropy density fixed. It is unclear whether the thermodynamic identity dE = T dS - P dV continues to hold without further renormalization in this limit, which could affect the reported distinction between small and large black holes and the decomposition into subextensive and extensive energy contributions.
minor comments (2)
  1. The abstract states that explicit expressions are derived for the extensive part of the energy; ensure these are prominently labeled with equation numbers in the main text and cross-referenced in the discussion of the AdS case.
  2. Consider adding a brief comparison in the introduction to prior literature on quasi-local thermodynamics (e.g., York’s original work) and holographic volume definitions in AdS to better contextualize the novelty.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment below and have revised the text to clarify assumptions and limits where appropriate.

read point-by-point responses

  1. Referee: The central mapping from York's quasi-local pressure P and holographic volume V to the thermodynamic variables of a dual theory is introduced by assumption rather than derived from an explicit dual Lagrangian or renormalization. This identification is load-bearing for the extensivity claims in the abstract and §5, as the diagnosis of linear scaling of quasi-local energy E with V at fixed entropy density depends directly on it. For the asymptotically flat Schwarzschild case, where no standard holographic dual exists on a finite timelike boundary, this assumption requires additional justification or a consistency check to support the non-extensivity of small black holes versus extensivity of large ones in the large-system limit.

    Authors: The identification is indeed presented as an assumption grounded in the holographic principle and the structure of York's quasi-local first law, as stated explicitly in the abstract and Section 2. For the asymptotically flat case we acknowledge that no standard dual exists, and the analysis is offered as a proposal for defining extensivity via quasi-local quantities. We have added a dedicated paragraph in the revised Section 2 that motivates the assumption from the geometric first law and includes a consistency check: the thermodynamic relations follow identically from the Einstein equations and boundary terms without invoking a specific dual Lagrangian. This supports the scaling analysis for both small and large black holes in the large-system limit. revision: partial

  2. Referee: §4, discussion of the large-system limit: The claim that large black holes become extensive relies on the boundary radius becoming large while holding entropy density fixed. It is unclear whether the thermodynamic identity dE = T dS - P dV continues to hold without further renormalization in this limit, which could affect the reported distinction between small and large black holes and the decomposition into subextensive and extensive energy contributions.

    Authors: York's quasi-local first law dE = T dS - P dV holds exactly for any finite boundary radius by direct derivation from the Einstein-Hilbert action with the appropriate boundary terms; no additional renormalization is required at finite radius. In the large-system limit (boundary radius R → ∞ at fixed entropy density s = S / (4π R²)), the identity remains valid because the subleading corrections vanish in a controlled manner. We have expanded the discussion in the revised Section 4 with an explicit asymptotic expansion of the quasi-local quantities, confirming that the identity persists and that the decomposition into subextensive and extensive contributions is well-defined, thereby preserving the distinction between small and large black holes. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained under explicit assumptions

full rationale

The paper states its central assumption explicitly: a holographically dual theory on the finite timelike boundary is assumed to exist so that York's quasi-local pressure and volume can be identified with the dual system's thermodynamic pressure and volume. All subsequent results, including the distinction between non-extensive small Schwarzschild black holes and extensive large ones in the large-system limit, follow from direct evaluation of the quasi-local energy E against the holographic volume V at fixed entropy density, using the first law and explicit expressions derived for the asymptotically flat and AdS cases. No step reduces a claimed result to its input by construction, renames a fitted quantity as a prediction, or relies on a load-bearing self-citation chain. The framework is proposed and then applied via standard quasi-local thermodynamics; the extensivity conclusions are computed outputs rather than tautological redefinitions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The construction relies on the existence of a dual theory and on the validity of York's quasi-local first law; no free parameters are introduced in the abstract, but the dual-theory assumption is an ad-hoc domain premise for the holographic step.

axioms (2)
  • domain assumption Existence of a holographically dual theory living on the finite timelike boundary
    Invoked to equate geometric pressure/volume with thermodynamic pressure/volume of the dual system.
  • standard math York's quasi-local first law applies to the enclosed black hole
    Used as the starting point for defining surface pressure conjugate to boundary area.
invented entities (1)
  • Holographic volume no independent evidence
    purpose: To provide a notion of system size for defining extensivity of black-hole thermodynamics
    Defined from the area of the finite timelike boundary; no independent falsifiable prediction given in abstract.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Holographic Stirling engines and the route to Carnot efficiency

    hep-th 2026-04 unverdicted novelty 6.0

    Stirling efficiency reaches Carnot when fixed-volume heat capacity is volume-independent, true for classical gases but not quantum or CFTs; holographic CFTs approach Carnot at large potentials with faster convergence ...

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