Thermodynamic relation on rotating charged black strings with arbitrary cosmological constant

Hai-Long Zhen; Huai-Fan Li; Jian-hua Shi; Meng-Sen Ma; Yu-Bo Ma

REVIEW 5 major objections 5 minor 37 references

Thermodynamic relation on rotating charged black strings with arbitrary cosmological constant

T0 review · 5 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The Goon-Penco relation holds for rotating charged black strings in cylindrical coordinates for arbitrary cosmological constant, even when charge, angular momentum, and entropy depend on the perturbation parameter.

desk verdict A clean but essentially tautological unpacking of the extended first law; the cylindrical-coordinate example is routine and the eta/P0 bookkeeping is sloppy. read the letter →

arxiv 2507.18024 v3 pith:OD7GLT5Z submitted 2025-07-24 hep-th

Hai-Long Zhen , Meng-Sen Ma , Huai-Fan Li , Jian-Hua Shi , Yu-Bo Ma This is my paper

classification hep-th PACS 04.70.Dy

keywords Goon-Pencorelationrotatingchargedblackstringsextendedfirstlawperturbationparametercosmologicalconstantholethermodynamicscylindricalcoordinates

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper sets out to show that the Goon-Penco relation---an identity linking the shift of a black hole's mass under a perturbation to the response of its entropy---is not an accident of spherical symmetry. It derives the relation for rotating charged black strings in cylindrical coordinates with an arbitrary cosmological constant, allowing charge, angular momentum, entropy, and pressure to vary with the same perturbation parameter $\eta$. The derivation runs through the extended first law of black hole thermodynamics, which includes a pressure-volume term; holding the black string mass fixed at $M_0$ converts that law into an identity from which the Goon-Penco relation follows under several different assumptions. Establishing the relation for cylindrical horizons extends the known validity of the Goon-Penco result to non-spherically symmetric spacetimes.

What carries the argument

The machinery is the extended first law $$dM=T\,dS+\$\Omega$\,dJ+U\,dQ+V\,dP_0+\left(\frac{\partial M}{\partial \eta}\right)_{S,J,Q,P_0}d\eta,$$ together with the perturbation prescription $\Lambda\to\Lambda(1+\eta)$, which shifts the pressure from $P_0$ to $P_0(1+\eta)$. The mass formula from Ref. [33], written in terms of $S,Q,J,P$, defines all conjugate quantities as partial derivatives. When $M$ is held at $M_0$, differentiating along the perturbation turns the first law into an identity pairing $(\partial M_0/\partial \eta)$ with $T(\partial S/\partial \eta)$, $\Omega(\partial J/\partial \eta)$, $U(\partial Q/\partial \eta)$, and $V(\partial P_0/\partial \eta)$; choosing which state functions depend on $\eta$ selects which Goon-Penco relation emerges.

What would settle it

A direct numerical check: fix $P_0$, $Q$, and $J$, use Eq. (3.8) to compute the temperature and entropy as functions of $M$, and compare both sides of Eq. (3.18) for several values of $M_0$ under a small $\eta$. If the two sides differ beyond numerical precision, the extended first law used in the derivation is not satisfied and the claimed universal relation fails.

Watch

Extended reading notes

Core claim

The central result is Eq. (3.18): with $P_0$, $Q$, and $J$ held fixed and the perturbed mass fixed at $M_0$, $$\left(\frac{\partial M_0}{\partial \eta}\right)_{P_0,S(\eta),Q,J} = -\lim_{M\to M_0} T\left(\frac{\partial S}{\partial \eta}\right)_{P_0,Q,J,M}.$$ The same construction yields companion identities for angular momentum, charge, pressure, and the joint variation of all these state functions, Eqs. (3.19)--(3.22). The proof starts from the mass formula $M(S,P_0,J(\eta),Q(\eta),\eta)$ of Ref. [33], inserts it into the extended first law, and imposes $dM/d\eta=0$ at $M=M_0$. On this basis the paper concludes that the Goon-Penco relation is universal for arbitrary cosmological constant and independent of the coordinate system used to describe the horizon.

Load-bearing premise

The load-bearing assumption is that the extended first law (3.7) is a genuine thermodynamic identity for the mass formula (3.8), so the partial derivatives in Eqs. (3.10)--(3.13) truly give the physical temperature, angular velocity, electric potential, and volume.

Editorial extensions

If this is right

Extremality corrections for rotating charged black strings can be computed from the entropy response alone, without first solving for the extremal configuration.
The earlier restriction to perturbations with $J$ and $Q$ independent of $\eta$ is lifted; variations of charge, spin, and pressure enter the same relation with their own conjugate terms.
The fixed-mass differentiation recipe can be reused for other black objects once an extended first law and a mass formula are known, providing a shortcut to GP-type identities.
Because the cosmological constant enters through $P_0$ and its $\eta$-scaling, the relation is written in a form that does not single out spherical topology.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper's universality claim is demonstrated for one cylindrical-coordinate family; checking toroidal, planar, or hyperbolic horizon topologies would test how far the 'irrespective of coordinate system' conclusion reaches.
A numerical evaluation of both sides of Eq. (3.18) using the explicit mass formula in Eq. (3.8) would independently verify that the extended first law is integrable, something the derivation assumes.
The same 'vary everything with $\eta$, hold $M$ fixed' strategy may carry over to higher-derivative or higher-dimensional black objects, where explicit extremal solutions are harder to obtain.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

Read this one so you know what's being claimed, but don't expect a new physical result. The paper's central 'universal GP relation' for rotating charged black strings is, by the authors' own derivation, just the extended first law with dM=0. That's an identity, not a discovery.

What is new is modest: applying the Goon-Penco trick to a cylindrical/toroidal black string with S(η), J(η), Q(η) all allowed to vary. That hasn't been written down for this exact metric, as far as I know. The review in Sec. 2 of the higher-dimensional RN(A)dS case is correct and could be useful pedagogically. The algebra from Eq. (3.7) to Eqs. (3.18)-(3.22) is internally consistent if you accept the bookkeeping.

The soft spots are proportionate and they land on the load-bearing parts of the claim. The stress-test note is right: the only place η enters is via P = P0(1+η) and through the state parameters. The explicit (∂M/∂η) term in Eq. (3.7) is either redundant (it equals V times dP/dη) or zero if you hold the pressure fixed. The paper writes both V dP0 and (∂M/∂η)dη without defining which pressure is being varied, and that ambiguity propagates into Eqs. (3.16)-(3.17). There's also no check that the mass formula imported from Ref [33] actually satisfies the extended first law; the paper just asserts the partial derivatives give T, Ω, U, V. And the extremal limit that makes the original GP relation interesting for the WGC is replaced by an arbitrary fixed mass M0, so the relation is a generic first-law consequence with no extremality physics.

The cited literature is covered, and the paper is honest about its method (it says the first law is the input). But the 'arbitrary cosmological constant' and 'universal' claims in the abstract are overstated. The reader's verdict of conditional/high is fair; if the η/P0 issue is fixed and the mass formula verified, the remaining content is a routine example. This is for someone tracking GP-relation applications to exotic horizons, not for someone seeking new physics.

A serious referee could handle it, but I'd lean toward asking for major revisions or recommending rejection in its current form, since the main claim is not established. If an editor asked me, I'd say the desk decision to reject is also defensible.

Referee Report

5 major / 5 minor

Summary. The paper claims to extend the Goon-Penco (GP) relation, which connects the shift of the extremal mass under a perturbation to a temperature-weighted entropy derivative, to rotating charged black strings in four-dimensional cylindrical coordinates with an arbitrary cosmological constant. The authors start from an imported mass formula for the black string and differentiate it using an extended first law that contains both a pressure term and an explicit perturbation term. By setting the mass differential to zero, they obtain a set of relations between derivatives of mass and entropy, angular momentum, charge, and pressure with respect to the perturbation parameter. The abstract concludes that the GP relation is universal for arbitrary cosmological constants and independent of the coordinate system.

Significance. If a nontrivial GP relation were established for rotating charged black strings, it would be a useful extension of the original Goon-Penco result beyond spherically symmetric spacetimes and could have implications for weak-gravity-conjecture studies in non-spherical settings. However, the manuscript's central derivation is essentially a rearrangement of the first law of thermodynamics and does not invoke the extremal-limit structure that gives the GP relation its physical content. The claimed universality is therefore not demonstrated, and the significance of the result as a new universal relation is not supported by the presented arguments.

major comments (5)

[Sec. 3, Eqs. (3.7), (3.14), (3.16)] The dependence of the mass function (3.8) on η enters only through the pressure redefinition P = P0(1+η) and through the postulated functions S(η), J(η), Q(η). Therefore the term (∂M/∂η)_{S,J,Q,P0} in the extended first law (3.7) is not an independent derivative: it equals P0 V if P0 is the background pressure, and it double-counts the pressure variation if P0 is meant to be the physical pressure. The subsequent Eq. (3.16) and the fixed-mass constraint (3.17) mix these two dependent terms, so Eqs. (3.18)-(3.22) are not well-defined.
[Sec. 3, Eq. (3.8)] The entire derivation rests on the imported mass formula (3.8) satisfying the extended first law, but the paper never verifies that the partial derivatives in Eqs. (3.10)-(3.13) reproduce the temperature, angular velocity, electric potential, and volume of the black string solution defined by Eqs. (3.1)-(3.4). Without such a check, the replacements T = ∂M/∂S, Ω = ∂M/∂J, U = ∂M/∂Q used in Eq. (3.16) are unjustified, so all subsequent relations may be incorrect.
[Sec. 3, Eqs. (3.18)-(3.22)] The derivation of the GP-type relations is tautological: substituting dM/dη = 0 into the first law and moving terms to the right-hand side yields an identity that is logically equivalent to the input first law. The original GP relation (1.1) is a nontrivial statement about the shift of the extremal mass under a perturbation; it requires taking the extremal limit M → M_ext. The manuscript instead uses an arbitrary fixed mass M0 and never takes an extremal limit, so Eqs. (3.18)-(3.22) are not the GP relation.
[Sec. 2, Eqs. (2.11)-(2.12)] The same criticism applies to the spherically symmetric case: Eq. (2.12) is obtained by setting dM/dη = 0 in the first law (2.11), which is a rearrangement of the first law rather than a derivation of the GP relation. Moreover, the limit M → M0 is taken with M0 a coexistence mass of the black hole and cosmological horizons, not an extremal mass, so the relation obtained is not the GP relation of Eq. (1.1).
[Abstract; Sec. 4] The claim that the GP relation is 'universal for spacetimes with arbitrary cosmological constants, irrespective of the adopted coordinate system' is not supported: the paper analyzes only one specific rotating charged black string solution in cylindrical coordinates and provides no argument that the result extends to other non-spherically symmetric spacetimes.

minor comments (5)

[General] The manuscript contains numerous typographical errors, such as 'universe relation' in the Introduction, 'higer-dimensional' and 'anaysis' in Sec. 2, and 'the relationship of between' in the Discussion; a thorough proofreading is needed.
[Sec. 3, Eq. (3.17)] The subscript notation S(η) in Eq. (3.17) is ambiguous: holding a function S(η) fixed while differentiating with respect to η is not a well-defined partial derivative, and the intended meaning should be stated explicitly.
[Sec. 2] The signs ± and ∓ in Sec. 2 are never defined systematically; since T, S, and the horizon labels + and c carry their own sign conventions, the reader cannot check Eq. (2.8) without reconstructing the conventions from scratch.
[Sec. 3, Eq. (3.8)] The complicated mass formula (3.8) should be accompanied by a check in a simple limit (e.g., J = 0 or Q = 0) to show that it reduces to the known black string mass (3.4), and Ref. [33] should be cited with the exact equation number.
[Sec. 1] The introduction's description of the GP relation and its connection to the weak gravity conjecture is too brief; the manuscript should state more precisely what is new compared with Refs. [13,15,18,19], which already treat η-dependent state parameters in other spacetimes.

Circularity Check

2 steps flagged · score 7.0 of 10

Eq. (3.18) is just dM=0 in the assumed first law Eq. (3.7); with η entering only through P=P0(1+η), the explicit η-derivative is redundant, so the GP 'derivation' is equivalent to its input.

self definitional [Sec. 3, Eq. (3.7) and Eqs. (3.17)-(3.18)]
"By allowing the pressure P0, the first law of black hole thermodynamics should then be modified to [19] dM = T dS+ ΩdJ + U dQ+ V dP0 + (∂M/∂η)_{S,J,Q,P0}dη. ... When the mass M0 of per unit volume of black string is assigned a specific value, Eq. (3.7) is reduced to ... (∂M0/∂η)_{P0,S(η),Q(η),J(η)} + T(∂S/∂η)_{P0,Q(η),J(η),M} + ..."

Eq. (3.18) is obtained by setting dM=0 in Eq. (3.7) and dropping the Q, J, and P0 variations. The quantity (∂M0/∂η) on the left is exactly the explicit (∂M/∂η) term that was put into the first law by hand in Eq. (3.7). No use is made of the explicit mass formula Eq. (3.8) or of any independent computation of how M depends on η; the 'derived' GP relation is an algebraic rearrangement of the assumed first law. The result is therefore equivalent to its input by construction.
other [Sec. 3, Eqs. (3.14)-(3.16)]
"P = 3(1+η)/8πl^2 = P0(1+η) ... Therefore, the state parameter of the energy, denoted by M, in Eq. (3.8) is expressed as M(S,P0,J(η),Q(η),η). When η is subject to perturbation, from Eqs. (3.7) and (3.8), we obtain dM/dη = (∂M/∂η) + (∂M/∂S)(∂S/∂η) + ... + (∂M/∂P0)(∂P0/∂η)."

Since P=P0(1+η), the only η-dependence of M in Eq. (3.8) enters through P, S(η), J(η), and Q(η). There is no separate explicit η-dependence left, so (∂M/∂η)_{S,J,Q,P0} is not an independent first-law coefficient: it equals (∂M/∂P)P0 = V P0. The subsequent separate term V(∂P0/∂η) is either zero (if P0 is the unperturbed constant) or double-counts the pressure dependence (if P0 is read as the perturbed pressure P). Thus Eqs. (3.16)-(3.22) combine an undefined or redundant derivative with the pressure term, so the claimed GP relations are not established independently of the ansatz that introduced η.

full rationale

The paper's central claim—the universal GP relation, Eqs. (3.18)-(3.22)—is not a prediction that goes beyond the input. The derivation starts from the extended first law Eq. (3.7), in which the coefficient of dη is an uncomputed partial derivative (∂M/∂η). Setting M=M0 and rearranging gives Eq. (3.17); dropping terms under constraints gives Eqs. (3.18)-(3.22). This is the chain rule, not a calculation. Moreover, the perturbation is introduced through P=P0(1+η), so the explicit η-derivative either coincides with V P0 or is redundant with the V dP0 term. The paper never evaluates (∂M/∂η) from Eq. (3.8) nor checks the consistency of Eq. (3.8) with the first law. No external benchmark or numerical check is provided. The score reflects that the central result reduces to the assumed first law by construction; this is not a case of harmless self-citation or a minor gap.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

There are no fitted numerical parameters. The central result rests on the extended first law, the imported mass formula, the eta-perturbation prescription, and the replacement of the extremal limit by an arbitrary fixed mass; these are all assumptions rather than derived results.

assumptions (4)

domain assumption Extended first law of black hole thermodynamics (Eq 3.7) with pressure-volume and explicit perturbation terms.
The entire derivation in Sec. 3 is a rearrangement of this equation; its validity for the mass formula is assumed, not proven.
domain assumption Mass formula M(S,P,J,Q) from Ref [33] (Eq 3.8) describes the rotating charged black string and satisfies the thermodynamic derivative relations.
The paper imports this formula and uses it to define T, Omega, U, and V via partial derivatives, but does not verify the first law for it.
ad hoc to paper Perturbation prescription Lambda -> Lambda(1+eta), so P=P0(1+eta), with J, Q, S, and possibly P0 allowed to depend on eta.
This specific eta-dependence is a modeling choice introduced in Sec. 3 (Eqs 3.14-3.15); the claimed universality is contingent on it.
ad hoc to paper Replacement of the extremal limit M->M_ext in the original GP relation by a general fixed mass M0.
Eqs (3.18)-(3.22) are written for an arbitrary fixed M0, whereas Eq (1.1) involves the extremal limit; the paper does not justify this substitution.

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Pith. "Pith review of Thermodynamic relation on rotating charged black strings with arbitrary cosmological constant." pith.science (2026). https://pith.science/paper/OD7GLT5Z

@misc{pith2026250718024,
  author       = {Pith},
  title        = {Pith review of: Thermodynamic relation on rotating charged black strings with arbitrary cosmological constant},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OD7GLT5Z}},
  note         = {Machine review of arXiv:2507.18024}
}

abstract

The Goon-Penco (GP) relation was investigated on rotating charged black strings with an arbitrary cosmological constant. It has been demonstrated that the GP relation retains its form in the context of spacetimes described by cylindrical coordinates. In addition, the GP relation is derived in scenarios where the energy state parameters (including angular momentum $J$ and charge $Q$, etc.) are expressed as functions of the perturbation parameter $\eta$. This finding indicates that the GP relation is not only valid for spacetimes described by spherically symmetric coordinates, but also prevalent for non-spherically symmetric spacetimes, such as cylindrical coordinates. Therefore, the present study demonstrated that the GP relation is universal for spacetimes with arbitrary cosmological constants, irrespective of the adopted coordinate system.

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [33]

R. Zhao, L. C. Zhang, Commun. Theor. Phys. 70 (2018) 5, 578, arXiv:1710.07225 [gr-qc]

work page Pith review arXiv 2018

[2] [1]

INTRODUCTION Black holes represent a distinctive physical system, manifesting characteristics of both strong gravitational fields and thermodynamic systems. As early as the 1970s, prominent physicists such as Hawking and Bekenstein established the four laws of thermodynamics of black holes [1, 2], thereby establishing the theoretical foundation for this f...

work page Pith review arXiv 2025

[3] [2]

HIGHER-DIMENSIONAL RN(A)DS BLACK HOLES: GENERAL CASE To extend to our anaysis to the non-spherically symmetric case, we review our calculation to a more general case, this is, higher-dimensional, spherically spacetime. The action of higer-dimensional Ads black holes is expressed as I = 1 16π Z dn+2x√g(R − FµνF µν − 2(1 + η)Λ), (2.1) Thus,the metric functi...

work page

[4] [3]

(3.1) The metric function f (r) is determined by f (r) = r2 l2 − m r + q2 r2 , Ξ2 = 1 + a l2 , (3.2) 4 and the gauge potential admits the the following form A = − q r (Ξdt − adφ)

ROT A TING CHARGED BLACK STRINGS For a four-dimensional spacetime with cylindrical or toroidal horizons, the line element can be expressed as [31–33]: ds2 = −f (r)(Ξdt − adφ)2 + r2 a l2 dt − Ξdφ 2 + dr2 f (r) + r2 l2 dz2. (3.1) The metric function f (r) is determined by f (r) = r2 l2 − m r + q2 r2 , Ξ2 = 1 + a l2 , (3.2) 4 and the gauge potential admits t...

work page

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Rotating Charged Black Strings described in cylindrical coordinates are analyzed

DISCUSSION In this study, the thermodynamic relationship associated with the thermodynamic quantities of Rotating Charged Black Strings is investigated in the context of the cosmological constant subjected by the perturbation parameter eta. Rotating Charged Black Strings described in cylindrical coordinates are analyzed. The mass of per unit height black ...

work page

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W, Commun

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