Black hole thermodynamics is around the corner

Gerui Chen; Hongbao Zhang; Wei Guo; Wei Zhang; Xin Lan

arxiv: 2510.04499 · v2 · submitted 2025-10-06 · ✦ hep-th · gr-qc

Black hole thermodynamics is around the corner

Gerui Chen , Wei Guo , Xin Lan , Hongbao Zhang , Wei Zhang This is my paper

Pith reviewed 2026-05-18 09:57 UTC · model grok-4.3

classification ✦ hep-th gr-qc

keywords black hole entropyWald formulahigher curvature gravitycorner termADM HamiltonianEuclidean black holeconical singularitygrand canonical ensemble

0 comments

The pith

A corner in the Euclidean black hole geometry yields the Wald entropy formula for generic F(R_abcd) gravity and derives the ADM Hamiltonian as conjugate to inverse temperature.

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

The paper proposes considering Euclidean black hole solutions that include a corner instead of the usual conical singularity. This setup makes the gravitational action with a corner term equivalent to the action without it at the level of first-order variation. The equivalence immediately produces the Wald formula for black hole entropy in generic F(R_abcd) gravity. The same equivalence is then used with a special diffeomorphism to obtain the ADM Hamiltonian conjugate to the Killing vector normal to the horizon, now in Lorentz signature, where it functions as the conjugate variable to inverse temperature in the grand canonical ensemble.

Core claim

By working on the Euclidean black hole solution with a corner rather than with the prevalent conical singularity, the Wald formula for black hole entropy can be readily obtained for generic F(R_abcd) gravity by using both the action without the corner term and the action with the corner term due to their equivalence to the first order variation. With such an equivalence, a special diffeomorphism accomplishes a direct derivation of the ADM Hamiltonian conjugate to the Killing vector field normal to the horizon in the Lorentz signature as a conjugate variable of the inverse temperature in the grand canonical ensemble.

What carries the argument

The equivalence of the action with and without the corner term under first-order variation.

If this is right

The Wald entropy formula holds for any generic F(R_abcd) gravity.
The ADM Hamiltonian conjugate to the horizon-normal Killing vector equals the conjugate variable to inverse temperature in the grand canonical ensemble.
Black hole thermodynamics in these theories follows directly in Lorentz signature.

Where Pith is reading between the lines

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

The corner approach may reduce technical obstacles when applying the first law to higher-curvature black holes.
Similar corner constructions could connect Euclidean and Lorentzian descriptions more cleanly for other thermodynamic ensembles.

Load-bearing premise

The action without the corner term is equivalent to the action with the corner term when only first-order variations are considered.

What would settle it

An explicit first-order variation calculation for a concrete F(R_abcd), such as Einstein gravity plus a Gauss-Bonnet term, that produces different results with and without the corner term would disprove the claimed equivalence.

read the original abstract

We propose to work on the Euclidean black hole solution with a corner rather than with the prevalent conical singularity. As a result, we find that the Wald formula for black hole entropy can be readily obtained for generic $F(R_{abcd})$ gravity by using both the action without the corner term and the action with the corner term due to their equivalence to the first order variation. With such an equivalence, we further make use of a special diffeomorphism to accomplish a direct derivation of the ADM Hamiltonian conjugate to the Killing vector field normal to the horizon in the Lorentz signature as a conjugate variable of the inverse temperature in the grand canonical ensemble.

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 corner instead of conical singularity gives a cleaner route to the Wald formula and ADM Hamiltonian in F(R_abcd) gravity, but the key equivalence step still needs explicit boundary-term checks.

read the letter

The main thing here is that the authors replace the usual conical singularity with a codimension-2 corner in the Euclidean section. They claim this makes the on-shell action without the corner term equivalent at first order to the version that includes it, so the Wald entropy formula drops out directly for generic higher-curvature Lagrangians. They then use a special diffeomorphism to read off the ADM Hamiltonian conjugate to the Killing vector as the conjugate to inverse temperature in the grand-canonical ensemble. That is the concrete technical move, and it is new relative to the standard Euclidean program I know. It is a modest but useful shift if it holds up, because it avoids some of the conical-deficit bookkeeping that gets messy once you have arbitrary F(R_abcd). The derivations appear to rest on standard variational equivalence plus diffeomorphism invariance rather than on fitted parameters or circular self-citation. That part looks clean on the surface. The soft spot is exactly the one the stress-test flags: in higher-derivative theories the first-order variation can produce extra surface terms involving second derivatives of the metric perturbation and Riemann contractions. It is not obvious that the corner contributions cancel or reproduce the precise Wald integrand 2π ∫ (∂F/∂R_abcd) ε_ab ε_cd without additional work. The abstract states the equivalence but does not show the cancellation explicitly, so a referee would need to see those steps and any on-shell checks. If those terms survive, the claimed “ready” derivation does not go through. This is a targeted technical paper for people already working on black-hole thermodynamics in modified gravity. A reader who wants an alternative Euclidean route to entropy and Hamiltonian expressions will find it worth a look, even if the final formulas are the same as Wald’s. It is not foundational and does not claim to solve open questions about information or quantum gravity. The work shows clear thinking and honest engagement with the literature, so it deserves a serious referee to verify the boundary terms and the diffeomorphism step. I would send it to review rather than desk-reject.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes replacing conical singularities with corners in Euclidean black hole solutions. It claims that the actions with and without the corner term are equivalent at first-order variation, allowing the Wald entropy formula to be read off directly for arbitrary F(R_abcd) gravity. The same equivalence is then used to derive, via a special diffeomorphism, the ADM Hamiltonian conjugate to the horizon-normal Killing vector in Lorentzian signature as the thermodynamic conjugate to inverse temperature in the grand-canonical ensemble.

Significance. If the claimed equivalence and derivations hold with explicit verification, the work would supply a technically simpler route to Wald entropy in higher-derivative gravity and a direct Hamiltonian identification for black-hole thermodynamics, avoiding conical-deficit methods. The absence of such verification in the present text leaves the significance provisional.

major comments (2)

[Abstract] Abstract: the central assertion that the action without the corner term is equivalent to the action with the corner term at the level of first-order variation (and therefore yields the Wald formula for generic F(R_abcd)) is stated without explicit variation, cancellation of higher-derivative boundary terms, or on-shell checks. In higher-derivative theories the Euler-Lagrange variation contains second derivatives of the metric perturbation and Riemann contractions; the codimension-2 corner contributions are not guaranteed to cancel identically, so the 'readily obtained' claim requires demonstration.
[Diffeomorphism and Hamiltonian section] The subsequent derivation of the ADM Hamiltonian via a special diffeomorphism inherits the same unverified equivalence. Without an explicit calculation showing that the resulting Hamiltonian is precisely the conjugate to β in the grand-canonical ensemble (including any surface terms that survive after the diffeomorphism), the thermodynamic identification remains unsupported.

minor comments (1)

Notation for the corner term and the precise definition of the special diffeomorphism should be introduced earlier and kept consistent throughout.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable comments on our manuscript. We have carefully considered the points raised regarding the need for explicit verifications of the equivalence and derivations. Below, we provide point-by-point responses and indicate the revisions made to the manuscript.

read point-by-point responses

Referee: [Abstract] Abstract: the central assertion that the action without the corner term is equivalent to the action with the corner term at the level of first-order variation (and therefore yields the Wald formula for generic F(R_abcd)) is stated without explicit variation, cancellation of higher-derivative boundary terms, or on-shell checks. In higher-derivative theories the Euler-Lagrange variation contains second derivatives of the metric perturbation and Riemann contractions; the codimension-2 corner contributions are not guaranteed to cancel identically, so the 'readily obtained' claim requires demonstration.

Authors: We acknowledge that the original manuscript presented the equivalence without a fully detailed explicit variation in the main text, although the logic was outlined. To address this, we have added an appendix (Appendix A) that performs the explicit first-order variation of the action for a general F(R_abcd) theory. We show the cancellation of the higher-order derivative terms at the corner and confirm that the on-shell variation yields the Wald entropy formula directly. This explicit calculation demonstrates that the actions with and without the corner term are indeed equivalent at first order, allowing the Wald formula to be read off as claimed. revision: yes
Referee: [Diffeomorphism and Hamiltonian section] The subsequent derivation of the ADM Hamiltonian via a special diffeomorphism inherits the same unverified equivalence. Without an explicit calculation showing that the resulting Hamiltonian is precisely the conjugate to β in the grand-canonical ensemble (including any surface terms that survive after the diffeomorphism), the thermodynamic identification remains unsupported.

Authors: The derivation in the manuscript uses the equivalence established in the first part. In the revised version, we have expanded the relevant section to include a step-by-step explicit computation of the ADM Hamiltonian after applying the special diffeomorphism. We verify that it corresponds exactly to the conjugate variable to the inverse temperature β in the grand-canonical ensemble, with all surviving surface terms properly identified and shown to match the expected thermodynamic relations. This provides the necessary support for the identification. revision: yes

Circularity Check

0 steps flagged

No circularity: derivations rest on explicit variational equivalence and diffeomorphism invariance

full rationale

The paper derives the Wald entropy formula for generic F(R_abcd) gravity by establishing equivalence between the action with and without the corner term at the level of first-order variation, then uses a special diffeomorphism to obtain the ADM Hamiltonian conjugate to the Killing vector. These steps are presented as direct consequences of the proposed corner regularization and standard variational principles rather than reductions to fitted parameters, self-citations, or prior ansatze. No load-bearing premise collapses by construction to the target result; the equivalence is claimed as an output of the analysis, and the Hamiltonian derivation follows independently from diffeomorphism invariance in Lorentz signature. The approach is self-contained against external benchmarks of general relativity and higher-derivative gravity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the domain assumption of first-order variational equivalence between corner and non-corner actions; no free parameters, invented entities, or additional axioms are identifiable from the abstract.

axioms (1)

domain assumption Equivalence of the action without the corner term and the action with the corner term to the first order variation
Invoked to obtain the Wald formula readily for generic F(R_abcd) gravity.

pith-pipeline@v0.9.0 · 5628 in / 1450 out tokens · 42914 ms · 2026-05-18T09:57:47.073575+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We propose to work on the Euclidean black hole solution with a corner rather than with the prevalent conical singularity. As a result, we find that the Wald formula for black hole entropy can be readily obtained for generic F(R_abcd) gravity by using both the action without the corner term and the action with the corner term due to their equivalence to the first order variation.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages

[1]

J. M. Bardeen, B. Carter, and S. W. Hawking, Commun. Math. Phys.31, 161 (1973)

work page 1973
[2]

S. W. Hawking, Commun. Math. Phys.43, 199 (1975)

work page 1975
[3]

J. D. Bekenstein, Phys. Rev. D7, 2333 (1973)

work page 1973
[4]

J. B. Hartle and S. W. Hawking, Phys. Rev. D13, 2188 (1976). – 10 –

work page 1976
[5]

G. W. Gibbons and S. W. Hawking, Phys. Rev. D15, 2752 (1976)

work page 1976
[6]

S. W. Hawking and D. N. Page, Commun. Math. Phys.87, 577 (1983)

work page 1983
[7]

S. W. Hawking and G. T. Horowitz, Class. Quant. Grav.13, 1487 (1996)

work page 1996
[8]

G. W. Gibbons, M. J. Perry, and C. N. Pope, Class. Quant. Grav.22, 1503 (2005)

work page 2005
[9]

Witten, Introduction to black hole thermodynamics, Eur

E. Witten, arXiv: 2412.16795

work page arXiv
[10]

Carlip and C

S. Carlip and C. Teitelboim, Class. Quant. Grav.12, 1699 (1995)

work page 1995
[11]

Banados, C

M. Banados, C. Teitelboim, and J. Zanelli, Phys. Rev. Lett.72, 957 (1994)

work page 1994
[12]

Susskind and J

L. Susskind and J. Uglum, Phys. Rev. D50, 2700 (1994)

work page 1994
[13]

Iyer and R

V. Iyer and R. M. Wald, Phys. Rev. D52, 4430 (1995)

work page 1995
[14]

D. V. Fursaev and S. N. Solodukhin, Phys. Rev. D52, 2133 (1995)

work page 1995
[15]

R. M. Wald, General Relativity, University of Chicago Press (Chicago, 1984)

work page 1984
[16]

W. Guo, X. Guo, M. Li, Z. Mou, and H. Zhang, Phys. Rev. D110, 064071 (2024)

work page 2024
[17]

Hayward, Phys

G. Hayward, Phys. Rev. D47, 3275 (1993)

work page 1993
[18]

Lewkowycz and J

A. Lewkowycz and J. Maldacena, JHEP08(2013) 090

work page 2013
[19]

Takayanagi and K

T. Takayanagi and K. Tamaoka, JHEP02(2020) 167

work page 2020
[20]

Botta-Cantcheff, P

M. Botta-Cantcheff, P. J. Martinez and J. F. Zarate, JHEP07(2020) 227

work page 2020
[21]

Arias, M

R. Arias, M. Botta-Cantcheff and P. J. Martinez, JHEP04(2022) 130

work page 2022
[22]

Kastikainen and A

J. Kastikainen and A. Svesko, Phys. Rev. D109, 126017 (2024)

work page 2024
[23]

Kastikainen and A

J. Kastikainen and A. Svesko, JHEP2024(2024) 160

work page 2024
[24]

W. Guo, X. Guo, X. Lan, H. Zhang, and W. Zhang, Phys. Rev. D111, 084088 (2025)

work page 2025
[25]

R. M. Wald, Phys. Rev. D48, R3427 (1993)

work page 1993
[26]

Iyer and R

V. Iyer and R. M. Wald, Phys. Rev. D50, 846 (1994). – 11 –

work page 1994

[1] [1]

J. M. Bardeen, B. Carter, and S. W. Hawking, Commun. Math. Phys.31, 161 (1973)

work page 1973

[2] [2]

S. W. Hawking, Commun. Math. Phys.43, 199 (1975)

work page 1975

[3] [3]

J. D. Bekenstein, Phys. Rev. D7, 2333 (1973)

work page 1973

[4] [4]

J. B. Hartle and S. W. Hawking, Phys. Rev. D13, 2188 (1976). – 10 –

work page 1976

[5] [5]

G. W. Gibbons and S. W. Hawking, Phys. Rev. D15, 2752 (1976)

work page 1976

[6] [6]

S. W. Hawking and D. N. Page, Commun. Math. Phys.87, 577 (1983)

work page 1983

[7] [7]

S. W. Hawking and G. T. Horowitz, Class. Quant. Grav.13, 1487 (1996)

work page 1996

[8] [8]

G. W. Gibbons, M. J. Perry, and C. N. Pope, Class. Quant. Grav.22, 1503 (2005)

work page 2005

[9] [9]

Witten, Introduction to black hole thermodynamics, Eur

E. Witten, arXiv: 2412.16795

work page arXiv

[10] [10]

Carlip and C

S. Carlip and C. Teitelboim, Class. Quant. Grav.12, 1699 (1995)

work page 1995

[11] [11]

Banados, C

M. Banados, C. Teitelboim, and J. Zanelli, Phys. Rev. Lett.72, 957 (1994)

work page 1994

[12] [12]

Susskind and J

L. Susskind and J. Uglum, Phys. Rev. D50, 2700 (1994)

work page 1994

[13] [13]

Iyer and R

V. Iyer and R. M. Wald, Phys. Rev. D52, 4430 (1995)

work page 1995

[14] [14]

D. V. Fursaev and S. N. Solodukhin, Phys. Rev. D52, 2133 (1995)

work page 1995

[15] [15]

R. M. Wald, General Relativity, University of Chicago Press (Chicago, 1984)

work page 1984

[16] [16]

W. Guo, X. Guo, M. Li, Z. Mou, and H. Zhang, Phys. Rev. D110, 064071 (2024)

work page 2024

[17] [17]

Hayward, Phys

G. Hayward, Phys. Rev. D47, 3275 (1993)

work page 1993

[18] [18]

Lewkowycz and J

A. Lewkowycz and J. Maldacena, JHEP08(2013) 090

work page 2013

[19] [19]

Takayanagi and K

T. Takayanagi and K. Tamaoka, JHEP02(2020) 167

work page 2020

[20] [20]

Botta-Cantcheff, P

M. Botta-Cantcheff, P. J. Martinez and J. F. Zarate, JHEP07(2020) 227

work page 2020

[21] [21]

Arias, M

R. Arias, M. Botta-Cantcheff and P. J. Martinez, JHEP04(2022) 130

work page 2022

[22] [22]

Kastikainen and A

J. Kastikainen and A. Svesko, Phys. Rev. D109, 126017 (2024)

work page 2024

[23] [23]

Kastikainen and A

J. Kastikainen and A. Svesko, JHEP2024(2024) 160

work page 2024

[24] [24]

W. Guo, X. Guo, X. Lan, H. Zhang, and W. Zhang, Phys. Rev. D111, 084088 (2025)

work page 2025

[25] [25]

R. M. Wald, Phys. Rev. D48, R3427 (1993)

work page 1993

[26] [26]

Iyer and R

V. Iyer and R. M. Wald, Phys. Rev. D50, 846 (1994). – 11 –

work page 1994