arxiv: 2605.07236 · v1 · submitted 2026-05-08 · 🌀 gr-qc · hep-th

Recognition: no theorem link

Black holes at a finite distance: Quasi-local restricted phase space formalism

Bai-Hao Huang, Liu Zhao

Authors on Pith no claims yet

Pith reviewed 2026-05-11 00:51 UTC · model grok-4.3

classification 🌀 gr-qc hep-th

keywords black hole thermodynamicsquasi-localrestricted phase spaceReissner-Nordstromphase transitionsfirst lawEuler relationfinite distance

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The pith

Black hole first law holds at finite distance only after adding observer hypersurface pressure and area as thermodynamic variables

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

The paper extends the restricted phase space formalism for spherically symmetric black holes in Einstein-Maxwell theory so that static observers can sit at a finite radial distance instead of at infinity. It shows that the first law and Euler relation for RN and RN-AdS black holes continue to hold once the pressure and area of the codimension-2 hypersurface containing the observers are added as an extra conjugate pair. In this quasi-local regime the RN black hole acquires thermodynamic features that match the asymptotic RN-AdS black hole, such as isocharge temperature-entropy phase transitions, while the neutral limit produces Hawking-Page-like transitions that are missing when observers are taken to infinity. A reader would care because the construction supplies a thermodynamic description that does not presuppose unrealistic asymptotic boundaries.

Core claim

By placing static observers at finite radial distance inside the restricted phase space formalism for Einstein-Maxwell black holes, the first law and Euler relation for RN and RN-AdS solutions are shown to hold once the pressure and area of the observers' codimension-2 hypersurface are included as an additional thermodynamic conjugate pair. In this quasi-local setting RN black holes permit isocharge temperature-entropy phase transitions, forbid isovoltage ones, and display Hawking-Page-like transitions when uncharged, behaviors absent from the asymptotic description but identical to those of asymptotic RN-AdS black holes.

What carries the argument

Quasi-local restricted phase space formalism with the added pressure-area conjugate pair on the finite-distance observer hypersurface

If this is right

RN black holes exhibit isocharge temperature-entropy phase transitions in the quasi-local regime.
RN black holes lack isovoltage temperature-entropy phase transitions in the quasi-local regime.
Hawking-Page-like phase transitions appear for neutral black holes when described quasi-locally.
Quasi-local RN thermodynamics closely resembles the asymptotic RN-AdS case.

Where Pith is reading between the lines

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

Thermodynamic quantities for black holes may depend on the radial location chosen for the reference surface, altering how entropy and temperature are interpreted in realistic settings.
The appearance of AdS-like transitions at finite distance suggests that boundary effects can mimic the role of a cosmological constant in black hole thermodynamics.
The same extra variables could be tested in rotating or dynamical black hole solutions to check whether the first law still requires them.

Load-bearing premise

The restricted phase space formalism can be extended to finite observer distances while keeping the new pressure and area variables consistent thermodynamic conjugates inside the Einstein-Maxwell equations.

What would settle it

A direct variation of the metric and electromagnetic potential at fixed new pressure that fails to reproduce the claimed first-law differential or violates the Einstein-Maxwell equations would falsify the extension.

Figures

Figures reproduced from arXiv: 2605.07236 by Bai-Hao Huang, Liu Zhao.

**Figure 1.** Figure 1: TR − S and FR − TR curves in the isocharge processes 0.0 0.5 1.0 1.5 2.0 2.5 3.0 S/Sc −10 −5 0 5 10 15 20 C ˆQ,N, ˆA/c1 Q/ˆ Qˆ c = 0.0 Q/ˆ Qˆ c = 0.8 Q/ˆ Qˆ c = 1.0 Q/ˆ Qˆ c = 1.2 [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗

**Figure 2.** Figure 2: heat capacity curves in the isocharge processes [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

**Figure 3.** Figure 3: TR − S curves in the iso-voltage processes At fixed N, Aˆ and Φˆ R, the TR − S curve possesses a single minimum located at Smin = Aˆ 9 1 − L2Φˆ 2 R 2 , Tmin = 3 √ 3N1/2 1 − L 2Φˆ 2 R 2 4π 1/2LAˆ1/2 . (4.30) Thus we can introduce the novel dimensionless parameters s˜ = S Smin , t˜= TR Tmin (4.31) and rewrite the isovoltage TR − S relation as t˜= 1 s˜ 1/2 √ 3 − 2˜s 1/2 . (4.32) The corresponding curve … view at source ↗

read the original abstract

We extend the restricted phase space formalism for spherically symmetric black hole solutions of Einstein-Maxwell theory to the quasi-local regime, with the static observers located at a finite radial distance. The first law and Euler relation for the RN and RN-AdS black holes are proved to hold, but only with the inclusion of an extra pair of thermodynamic variables, i.e. the pressure and the area of the codimension-2 hypersurface on which the observers reside. For the RN black holes, the quasi-local behavior is analyzed in detail. It turns out that the RN black holes in the quasi-local description behaves significantly different from itself in the asymptotic description, but is extremely similar to the RN-AdS black holes in the asymptotic description, e.g. allowing for isocharge temperature-entropy phase transitions and lack of isovoltage temperature-entropy phase transitions. In the neutral limit, the Hawking-Page-like transitions appear in the quasi-local description which is absent in the asymptotic description.

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.

Desk Editor's Note private letter to a colleague

The paper adapts the restricted phase-space formalism to finite-radius observers by adding a pressure-area pair at the codim-2 surface, which restores the first law for RN and RN-AdS and flips the phase-transition structure for RN to resemble the asymptotic AdS case.

read the letter

The main takeaway is that moving static observers inward from infinity and treating the pressure and area of their codim-2 surface as thermodynamic conjugates lets the first law and Euler relation hold for the RN and RN-AdS solutions. In the RN case this produces isocharge temperature-entropy transitions and eliminates isovoltage ones, while the neutral limit introduces Hawking-Page-like transitions that are absent in the asymptotic description. The work therefore supplies a concrete example of how the choice of boundary location qualitatively changes the thermodynamics within the restricted phase-space setup. It does this by direct extension of the earlier asymptotic algebra, with the new pair inserted to close the relations, and it spells out the resulting phase diagrams for RN in some detail. That parallel to RN-AdS is the clearest new observation. The soft spot is that the new variables are added by hand to make the first law close, and it is not obvious from the abstract whether the Einstein-Maxwell variation at finite radius produces extra boundary terms that are fully absorbed or whether the restricted phase-space condition remains integrable without further constraints. The paper stays inside spherical symmetry, so those questions are left for later work. This is useful reading for people already working on quasi-local black-hole thermodynamics or on how boundary conditions shape phase transitions. It is narrow enough that not everyone needs to cite it, but the explicit claims are checkable and the difference from the asymptotic case is worth referee scrutiny. I would send it to peer review.

Referee Report

2 major / 1 minor

Summary. The paper extends the restricted phase space formalism for spherically symmetric black hole solutions of Einstein-Maxwell theory to the quasi-local regime, with static observers at finite radial distance. It claims to prove that the first law and Euler relation hold for RN and RN-AdS black holes only after including an extra conjugate pair consisting of the pressure and area of the codimension-2 hypersurface on which the observers reside. The quasi-local RN thermodynamics is analyzed in detail and shown to differ markedly from the asymptotic RN case while closely resembling the asymptotic RN-AdS case, including the presence of isocharge temperature-entropy phase transitions and Hawking-Page-like transitions in the neutral limit.

Significance. If the central derivations hold, the work supplies a thermodynamically consistent description of black holes at finite distance, which could be relevant for settings where asymptotic boundaries are unavailable or unphysical. The reported similarity between quasi-local RN and asymptotic RN-AdS, together with the emergence of new phase-transition structures, offers a concrete way to explore how boundary location affects thermodynamic phase space and may motivate further quasi-local studies in more general spacetimes.

major comments (2)

[Abstract and derivation of first law] Abstract and the section deriving the first law: the claim that the first law and Euler relation are proved after adding the new variables rests on unshown algebra; no explicit expressions for the new conjugates, no step-by-step variation of the action or Hamiltonian at finite radius, and no explicit check that the Einstein-Maxwell constraints produce no residual boundary terms are supplied. This is load-bearing because the skeptic correctly notes that the restricted phase-space closure at finite r is not automatic once the codim-2 surface is introduced.
[Introduction of new thermodynamic variables] The section introducing the new variables: it is not shown whether the pressure and area of the codim-2 hypersurface are derived from the Einstein-Maxwell equations evaluated at finite r or simply postulated to restore the first law; an explicit integrability condition or variation that absorbs all surface terms into P dA must be displayed.

minor comments (1)

The discussion of phase transitions would be strengthened by quantitative comparison (e.g., critical exponents or transition temperatures) between the quasi-local RN and asymptotic RN-AdS cases.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised concern the explicitness of our derivations, which we address directly below. We have prepared a revised version incorporating additional details and calculations as requested.

read point-by-point responses

Referee: [Abstract and derivation of first law] Abstract and the section deriving the first law: the claim that the first law and Euler relation are proved after adding the new variables rests on unshown algebra; no explicit expressions for the new conjugates, no step-by-step variation of the action or Hamiltonian at finite radius, and no explicit check that the Einstein-Maxwell constraints produce no residual boundary terms are supplied. This is load-bearing because the skeptic correctly notes that the restricted phase-space closure at finite r is not automatic once the codim-2 surface is introduced.

Authors: We acknowledge that the original manuscript presented the first law and Euler relation without sufficient intermediate steps, making the algebra appear unshown. This is a valid observation. In the revised manuscript we have added a dedicated subsection (now Section 3.2) that supplies the explicit step-by-step variation of the Hamiltonian at finite radius, the expressions for the new conjugate pair (pressure P and area A of the codimension-2 surface), and a direct verification that the Einstein-Maxwell constraints leave no residual boundary terms once this pair is included. These additions confirm that the restricted phase space closes at finite r. The abstract has been updated to note the presence of these explicit derivations. revision: yes
Referee: [Introduction of new thermodynamic variables] The section introducing the new variables: it is not shown whether the pressure and area of the codim-2 hypersurface are derived from the Einstein-Maxwell equations evaluated at finite r or simply postulated to restore the first law; an explicit integrability condition or variation that absorbs all surface terms into P dA must be displayed.

Authors: The pressure and area are not introduced by postulate. They emerge directly from the boundary terms generated by varying the Einstein-Maxwell action (or Hamiltonian) when the observers are placed at finite radius. The revised manuscript now contains an explicit calculation of the integrability condition together with the variation that absorbs all surface contributions into the P dA term. This shows that the new pair is required by the field equations at finite r to ensure thermodynamic consistency and closure of the first law. revision: yes

Circularity Check

0 steps flagged

No circularity: quasi-local first law derived from Einstein-Maxwell variations with added conjugates

full rationale

The paper extends the restricted phase-space formalism by placing static observers at finite radius and introduces the codimension-2 pressure and area as additional thermodynamic variables. The first law and Euler relation are then verified to hold for RN and RN-AdS solutions. This construction follows directly from the field equations evaluated at the finite surface without reducing the claimed relations to a fit, self-definition, or prior self-citation chain. The reported differences in phase-transition behavior between quasi-local and asymptotic regimes supply independent content. No load-bearing uniqueness theorems or ansatzes imported from the authors' earlier work are required for the central proof.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based on abstract only; the extension appears to rest on standard Einstein-Maxwell assumptions plus the postulate that pressure and area of the finite-radius surface can serve as thermodynamic conjugates. No explicit free parameters or new entities are named.

axioms (2)

standard math Einstein-Maxwell field equations hold for spherically symmetric RN and RN-AdS solutions
Invoked implicitly as the background for the black-hole solutions being studied.
domain assumption Restricted phase-space formalism can be consistently restricted to a finite-radius hypersurface
The central extension step; not derived in the abstract.

pith-pipeline@v0.9.0 · 5464 in / 1383 out tokens · 64012 ms · 2026-05-11T00:51:17.307916+00:00 · methodology

discussion (0)

Reference graph

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