Universality on thermodynamic relation with corrections in Einstein-Bel-Robinson gravity Black hole

Hai-Long Zhen; Huai-Fan Li; Yu-Bo Ma

REVIEW 4 major objections 6 minor 48 references

Universality on thermodynamic relation with corrections in Einstein-Bel-Robinson gravity Black hole

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read At extremality, the mass shift of an Einstein-Bel-Robinson black hole under a small perturbation is fixed by the entropy, pressure, and coupling shifts, and the paper conjectures the same identity holds for arbitrary black holes.

desk verdict The central 'universal relation' is just the first law rewritten; the paper overclaims and has algebraic slips, though it is a clean worked example. read the letter →

arxiv 2506.08466 v2 pith:KGFBLKAU submitted 2025-06-10 gr-qc hep-th

Hai-Long Zhen , Huai-Fan Li , Yu-Bo Ma This is my paper

classification gr-qchep-th

keywords blackholethermodynamicsgeneralizedthermodynamicextremumrelationEinstein-Bel-RobinsongravityextendedphasespacefirstlawofperturbationparameterextremalholesWeakConjecture

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

The paper claims that for black holes in Einstein-Bel-Robinson gravity, the derivative of the extremal black hole mass with respect to a small perturbation parameter is exactly fixed by the simultaneous shifts of entropy, pressure, and the theory's coupling. This generalizes the known extremal thermodynamic relation, which previously was derived only when a single state variable varies and the perturbation parameter enters linearly. The identity is obtained without solving for the perturbation parameter as a function of entropy, and it holds for state variables containing higher-order terms in the perturbation. The authors conjecture that the same structure applies to any black hole whose thermodynamic quantities obey the extended first law, which would make the extremal mass shift a universal probe of quantum-gravity corrections.

What carries the argument

The load-bearing mechanism is the full-differential identity for the mass, M = M(S(η), P(η), B(η), η), combined with the extended first law. Differentiating M along a path where S, P, and B all depend on the perturbation parameter η, and then letting the mass approach its extremal value M0, the first law converts the mass differential into the weighted sum of state-variable shifts that appears on the right-hand side of Eq. (3.26). The method matters because it never requires solving for η analytically from the condition of constant mass; instead, the perturbation parameter is treated as an independent variable and all state variables are allowed to move simultaneously.

What would settle it

Evaluate both sides of Eq. (3.26) for an EBR black hole at several values of (η, β, r+) near extremality using the explicit formulas in Eqs. (2.11)–(2.15); the equality must hold algebraically. A sharper test is to apply the relation to a charged or rotating black hole whose first law contains additional terms beyond TδS + ψΛδP + ψβδB: if one of those charges varies with η, the right-hand side of Eq. (3.26) misses its conjugate term, so the two sides will differ, showing that universality requires including every thermodynamic pair in the sum.

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Extended reading notes

Core claim

On the paper's own terms, the central result is Eq. (3.26): at extremality, the derivative of the mass with respect to the perturbation parameter equals the limit, as the mass approaches its extremal value, of minus the sum of temperature times entropy shift, pressure potential times pressure shift, and coupling potential times coupling shift. The derivation starts from the full differential of M(S(η), P(η), B(η), η) and substitutes the extended-phase-space first law δM = TδS + ψΛδP + ψβδB. Because only the first law is used, the final identity does not depend on whether each state variable is linear or higher order in η. The paper verifies the relation by direct calculation, by chain-rule differentiation, and by the full-differential method, and shows that holding any subset of the state variables fixed reduces the formula to the previously known extremal relations.

Load-bearing premise

The derivation assumes the black hole's mass change is fully accounted for by changes in entropy, pressure, and coupling—the same extended first law that holds away from extremality—and that this law stays valid at the extremal limit.

Editorial extensions

If this is right

The previously known extremal mass–entropy relation becomes the special case where pressure and coupling are held fixed; Eqs. (3.27)–(3.32) list all partial reductions.
The method applies directly to black holes whose thermodynamic quantities are higher-order functions of the perturbation parameter, removing the earlier need to restrict to first-order dependence.
For EBR gravity, Eq. (3.26) ties the correction to the extremal mass to the entropy, pressure, and coupling shifts in a single identity, without computing an on-shell free energy.
If the conjecture holds, any proposed black hole solution satisfying the first law must satisfy a corresponding relation, giving a consistency check for new solutions.
The extremal mass shift is the quantity relevant to the Weak Gravity Conjecture, so the relation links corrected black hole thermodynamics to the charge-to-mass bound.

Reading between the lines

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

If the conjecture extends to arbitrary black holes, charged and rotating solutions should obey an analogous relation with one extra term per conserved charge, each of the form conjugate potential times the derivative of the charge with respect to η; the paper's derivation indicates the structure but does not state it.
The identity could be used as a diagnostic: for a numerically constructed black hole, verifying Eq. (3.26) would confirm that the solution satisfies the first law, while a mismatch would locate missing work terms.
One testable extension is to include higher-order perturbations beyond the displayed o(β²) terms and check whether the same first-law structure, and hence the same relation, survives at order β².
A quantum-gravity reading of the result is that the extremal mass shift induced by the perturbation parameter behaves exactly as if each thermodynamic source—entropy, pressure, coupling—contributes independently, which may simplify estimates of Weak Gravity Conjecture corrections.

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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

The main result, Eq. (3.26), is not a new physical relation. It is obtained by differentiating the extended-phase-space first law (3.17) along a constant-mass path and setting the total derivative to zero. That is a tautology. Any black hole satisfying the first law yields the same expression, and the extremal limit plays no essential role. The paper presents this as a universal breakthrough, but it is the first law rearranged.

To its credit, the paper carefully applies the Goon–Penco framework to a specific higher-derivative gravity model, the Einstein–Bel-Robinson black hole, and shows that the relation holds when entropy, pressure, and coupling all vary with the perturbation parameter. The composite-function derivations are algebraically straightforward and internally consistent with the first law. The model choice is somewhat interesting because the thermodynamic quantities contain higher powers of η, but as you noted, that does not alter the relation's form—the GP relation is first-order in variations.

The soft spots are real but secondary. There are clear transcription errors: duplicated terms in Eq. (2.10), and a denominator inconsistency (279 vs 9) in Eqs. (3.2) and (3.8). The notation is sloppy in places, with partial derivatives carrying different fixed variables without careful distinction. None of these are fatal, because the main argument is the identity above.

My main concern is that the authors do not recognize that their 'universal relation' is a mathematical identity rather than a prediction. The conjecture in Sec. 4 that this applies to arbitrary black holes adds nothing beyond what the first law already says. This is not a case of a subtle error buried in a long calculation; the central claim collapses on inspection.

Who is this for? A reader wanting a worked example of the GP relation in a higher-derivative gravity background might find the explicit expressions useful, but the paper does not merit serious refereeing. The core result is vacuous, and the algebraic errors undermine the credibility of the details. I recommend desk rejection, though you could point the authors toward a much shorter note that correctly frames Eq. (3.26) as an identity, if they want to salvage the model-specific computation.

Referee Report

4 major / 6 minor

Summary. The paper claims to derive a generalized Goon-Penco (GP) relation for Einstein-Bel-Robinson (EBR) black holes when several thermodynamic variables (entropy, pressure, coupling) depend simultaneously on a perturbation parameter η, and to conjecture that this relation is universal. The main result is Eq. (3.26), obtained by applying the full-differential chain rule to the extended-phase-space first law (3.17). The paper also presents two alternative methods (direct calculation and composite-function derivation) that are intended to recover the GP relation in single-variable settings.

Significance. If Eq. (3.26) were a nontrivial physical constraint, it would extend the WGC-motivated extremality relations to multiple state variables. However, the derivation is purely algebraic: Eq. (3.26) is a rearrangement of the first law along a constant-M path, and no EBR-specific input is needed. The paper does not verify the imported thermodynamic quantities, and the concrete checks contain algebraic errors. The conjecture in Sec. 4 is not formulated as a falsifiable statement. On the positive side, the paper correctly identifies the chain-rule structure of the GP relation and systematically presents three derivational routes; the full-differential manipulation itself, apart from notation, is formally correct as a differential identity.

major comments (4)

[§3.3, Eqs. (3.24)–(3.26)] The central claim Eq. (3.26) is a tautological rewriting of the first law, not a universal physical relation. In Eq. (3.24) the total derivative of M along the path S(η), P(η), B(η) is expanded by the chain rule; substituting the first-law partial derivatives ∂M/∂S = T, ∂M/∂P = ψΛ, ∂M/∂B = ψβ and imposing dM/dη = 0 yields Eq. (3.26) identically. No property of Einstein-Bel-Robinson gravity enters; any system with a first law of the form δM = Σ Xi δYi satisfies the same identity, and the limit M → M0 is never used (the identity holds at any fixed M). The paper therefore does not generalize the Goon-Penco relation but merely re-expresses the first law along a constant-mass slice. The title's claim to universality is not supported.
[§3.1, Eq. (3.2) and Eq. (3.6)] The direct-method verification contains algebraic inconsistencies with Eq. (2.12). In particular, the coefficient of Λ3 in Eq. (3.2) is written as −56(1+η)2Λ3r+/279, whereas the corresponding term in Eq. (2.12) is −56(1+η)3Λ3r+/27; both the power of (1+η) and the denominator differ. The same erroneous coefficient propagates into Eqs. (3.6) and (3.8) via the definition of A. In addition, Eq. (2.10) contains two identical blocks of terms (the lines with −47185920Λk2m4/r14 and +3749331456Λk2m5/r15 are repeated), which suggests a copy-paste error in the β2 coefficient. These issues undermine the numerical check of the GP relation for the EBR solution.
[§3.2, Eq. (3.19)] The substitution in Eq. (3.19) is not justified. The ratio (∂M/∂r+)/(∂P/∂r+) at fixed η equals (∂M/∂P)η, but the first law (3.17) identifies ψΛ with (∂M/∂P)S,B,η, i.e., at fixed S and B. Since Eq. (3.15) holds S and β fixed but not B, the replacement (∂M/∂r+)/(∂P/∂r+) = ψΛ is not consistent with the fixed variables in Eq. (3.19). The composite-function derivation of the P-dependent GP relation therefore needs an independent justification.
[Sec. 4] The conjecture of a universal relation between displaced thermodynamic quantities is not stated with enough precision to be evaluated or falsified; it appears to be a restatement of Eq. (3.26) in words. A precise conjecture should specify the class of black holes, the first law assumed, and the type of perturbations to which it applies.

minor comments (6)

[Abstract] The phrase 'the black hole state parameter is solely a first-order function of the perturbation parameter' is unclear; presumably the intended meaning is that state parameters depend linearly on the perturbation parameter.
[§3.3, Eq. (3.24)] The partial-derivative notation with subscripts like P(η), B(η), S(η) on ∂/∂η is nonstandard and ambiguous, because one cannot hold a function and its argument fixed simultaneously. Use explicit path or total-derivative notation instead.
[Eq. (3.7)] Eq. (3.7) has unmatched parentheses around the factor +64β(...); the expression as written is ill-defined and should be corrected.
[References] Reference [4] is incomplete (volume/issue pages are missing), and reference [1] lacks publication data; references [31,32] are cited for results that are central to the paper, but they should be listed with full titles to allow verification.
[§2, Eqs. (2.11)–(2.15)] The paper imports the EBR thermodynamic quantities from Refs. [31,32] without rederivation. Given the propagation error noted above, the authors should at least recompute these quantities from f(r) and explicitly display the first law (3.17) to confirm the identification of ψΛ and ψβ.
[Eq. (2.11)] The notation o(β2) in Eq. (2.11) is misleading because the expansion is truncated at first order in β; the text should state that the expansion is kept only to order β.

Circularity Check

1 steps flagged · score 8.0 of 10

Eq. (3.26) is the input first law (3.17) rearranged along a constant-mass slice; the claimed universal GP relation and the Sec. 4 conjecture are algebraic identities forced by the first law by construction.

self definitional [Sec. 3.3, Eqs. (3.17), (3.24)-(3.26); restated as the universality conjecture in Sec. 4]
"It has been demonstrated that the first law of thermodynamics is satisfied by the thermodynamic quantities of the black hole, we obtain δM = T δS+ ψΛδP + ψβδB, with B = β/16π [32, 40, 41]. ... 0 = (∂M/∂η)_{P(η),B(η),S(η)} + T (∂S/∂η)_{P(η),B(η),M(η)} + ψΛ (∂P/∂η)_{S(η),B(η),M(η)} + ψβ (∂B/∂η)_{P(η),S(η),M(η)} , (3.25) namely, (∂M0/∂η)_{P(η),B(η),S(η)} = lim M→M0 − T (∂S(η)/∂η)_{P(η),B(η),M(η)} − lim M→M0 [ψΛ (∂P(η)/∂η)_{S(η),B(η),M(η)} + ψβ (∂B(η)/∂η)_{P(η),S(η),M(η)}]. (3.26)"

(3.26) is obtained from (3.24) and (3.17) alone: the paper substitutes the first-law coefficients ∂M/∂S = T, ∂M/∂P = ψΛ, ∂M/∂B = ψβ into the total-derivative identity (3.24) and sets dM/dη = 0 along M → M0. Hence (3.26) is literally (3.17) divided by dη along a constant-M curve; it is an algebraic identity, not a new physical relation. The derivation never uses the EBR-specific forms of S(η), P(η), B(η), their higher-order η dependence, or any extremality property, so every system whose thermodynamics obey (3.17) satisfies (3.26) identically. The extremal limit M→M0 is decorative; the identity holds at any fixed M. The Sec. 4 conjecture of universality for arbitrary black holes restates the same content, and Secs.

full rationale

The central derivation chain is: (i) import EBR thermodynamic quantities from Refs [31,32] (external, not self-citations); (ii) assume the extended first law (3.17); (iii) write the total derivative (3.24) of M[S(η),P(η),B(η),η]; (iv) impose dM/dη=0 at M=M0 and identify ∂M/∂S=T, ∂M/∂P=ψΛ, ∂M/∂B=ψβ from (3.17); (v) obtain (3.26). Step (iv)-(v) makes (3.26) a rearrangement of the first law, so the claimed universality carries no content beyond the input equation. The direct and composite-function computations (Secs. 3.1-3.2) are consistency checks of the imported EBR functions against the same first law, not independent predictions; the real content, if any, is that the imported quantities satisfy (3.17), and even that is inherited from the external construction of Refs [31,32]. There is no load-bearing self-citation (the author-overlap Ref. [12] is peripheral) and no fitted parameter renamed as a prediction. The algebraic irregularities in Eqs. (2.10), (3.2), and (3.8) are correctness concerns, not circularity. Because the paper's central universal relation is forced by definition to be the first law rewritten, the appropriate score is 8.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central relation rests entirely on the first law and the quoted EBR solution; no numerical fitting or invented entities are used. The only parameters introduced are the coupling β and perturbation η, neither of which is assigned a value.

free parameters (2)

β
Coupling constant of Einstein-Bel-Robinson gravity from the action Eq (2.1); no value is assigned and the derivation is perturbative in β to order β^2.
η
Perturbation parameter inserted into the cosmological constant in Eq (2.6); no value is assigned and the GP relation is a derivative with respect to η, so its value is irrelevant.

assumptions (4)

standard math Chain rule and implicit differentiation are valid for the thermodynamic quantities M, S, P, B as functions of r+, η, and each other.
Used throughout Section 3, e.g., Eqs (3.4), (3.12), (3.20), (3.24).
domain assumption The first law δM = TδS + ψ_ΛδP + ψ_βδB holds for the EBR black hole in extended phase space.
Introduced in Eq (3.17) as the basis for the full differential method; the universal relation is obtained by setting dM/dη=0 in this equation.
domain assumption The EBR metric functions and thermodynamic quantities in Eqs (2.9)-(2.15), taken from Refs [31,32], are correct to order β^2.
The paper does not rederive the solution; all subsequent derivatives use these expressions.
domain assumption The extremal limit M→M0 exists and thermodynamic variables remain differentiable at that limit.
All GP-type relations take lim M→M0 and require the partial derivatives to be well-defined at extremality.

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Pith. "Pith review of Universality on thermodynamic relation with corrections in Einstein-Bel-Robinson gravity Black hole." pith.science (2026). https://pith.science/paper/KGFBLKAU

@misc{pith2026250608466,
  author       = {Pith},
  title        = {Pith review of: Universality on thermodynamic relation with corrections in Einstein-Bel-Robinson gravity Black hole},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KGFBLKAU}},
  note         = {Machine review of arXiv:2506.08466}
}

read the original abstract

The generalized thermodynamic extremum relation, as proposed by Goon and Penco, establishes a novel theoretical framework for the study of spacetime thermodynamics. However, extant investigations generally assume that the black hole state parameter is solely a first-order function of the perturbation parameter when exploring the Goon-Penco relation in diverse spacetime contexts. An analytic expression for the perturbation parameter as a function of the black hole entropy can be expressed by treating the black hole mass as constant. The present study addresses this limitation and provides insight into the universal Goon-Penco relation when multiple thermodynamic state parameters behave as higher order functions of the perturbation parameters. Notably, we have not only established a universal relational formula in the case of multiple state variables, but more importantly, we have put forward an innovative conjecture that reveals the existence of a universal relation between displaced thermodynamic quantities in spacetime in the context of an arbitrary black hole. These theoretical breakthroughs are expected to open up new exploration directions for quantum gravity research.

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Works this paper leans on

48 extracted references · 48 canonical work pages

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INTRODUCTION The Weak Gravity Conjecture (WGC) establishes a fundamental constraint on the theoretical framework of quantum gravity. This conjecture posits that the canonical interactions of specific states with some lighter charged particle or field must exceed the gravitational interactions within any self-consistent quantum gravity theory [1, 2]. Speci...

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4D EINSTEIN-BEL-ROBINSON GRA VITY In 4 dimensions, EBR gravity is determined by the following action [29, 30] I = 1 16πG Z d4x√g R − 2Λ − β P 2 4 − E2 4 . (2.1) Where R is the Ricci scalar, β is the coupling constant of the theory, and Λ = −3/l2, where l refers to the curvature radius of the maximally symmetric AdS solution of the field equations. Where P...

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GOON-PENCO RELA TION The GP relation was constructed using mass, temperature, entropy, and the perturbation parameter η [5, 10, 12]. in addition, each side of the relation includes a partial derivative term of mass and entropy with respect to η, ∂Mext(Q, η) ∂η = lim M →Mext − T ∂S ∂η M,Q , (3.1) where M and Q represent the mass and additional quantities

work page
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∂S ∂η P,β,r + + ∂S ∂r+ P,β,η ∂r+ ∂η P,β,M # . (3.5) 5 Substituting Eqs. (3.2) and (3.3) into Eqs. (3.4) and (3.5), respectively, yields ∂M0 ∂η P,β,S = −

direct method of calculation Under the condition that P and β are constant, when M → M0, the position of the black hole horizon, r+, is a function of perturbation parameter, η, we can obtain 0 = − " Λr3 + 6 − β − 104Λ 3r3 + + 32(1 + η)Λ2 r+ − 56(1 + η)2Λ3r+ 279 − 208(1 + η)3Λ4r3 + 81 !# + 1 2 − (1 + η)Λr2 + 2 + β − 100 r6 + + 104(1 + η)Λ r4 + − 16(1 + η)2...

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(2.12), it can be demonstrated that, under the condition that β and S remain constant, and when M → M0, both the black hole horizon position r+ and P are functions of η

The Derivation of Composite Function 6 In the context of the Eq. (2.12), it can be demonstrated that, under the condition that β and S remain constant, and when M → M0, both the black hole horizon position r+ and P are functions of η. According to the law of derivation of the composite function, we obtain ∂M ∂η P (η),β,S = ∂M ∂η P (η),β,r+ + ∂M ∂r+ P (η),...

work page
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T ∂S (η) ∂η P,B(η),M (η) + ψβ ∂B (η) ∂η P,S(η),M (η) # , (3.30) when B(η) is invariant, Eq. (3.26) reduces to ∂M0 ∂η P (η),B,S (η) = − lim M →M0

full differential method In the context of employing the composite function derivation method, it is imperative to note that the GP relation between each thermodynamic state parameter and the energy can only be calculated separately when the energy is a function of multiple thermodynamic state parameters. In the following, we employ the full differential ...

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(2.12), the thermodynamic quantity and the horizon position of black hole will concomitantly be modified when the perturbation parameter of the spacetime, η, is altered

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

INTRODUCTION The Weak Gravity Conjecture (WGC) establishes a fundamental constraint on the theoretical framework of quantum gravity. This conjecture posits that the canonical interactions of specific states with some lighter charged particle or field must exceed the gravitational interactions within any self-consistent quantum gravity theory [1, 2]. Speci...

work page Pith review arXiv 2025

[2] [2]

4D EINSTEIN-BEL-ROBINSON GRA VITY In 4 dimensions, EBR gravity is determined by the following action [29, 30] I = 1 16πG Z d4x√g R − 2Λ − β P 2 4 − E2 4 . (2.1) Where R is the Ricci scalar, β is the coupling constant of the theory, and Λ = −3/l2, where l refers to the curvature radius of the maximally symmetric AdS solution of the field equations. Where P...

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[3] [3]

GOON-PENCO RELA TION The GP relation was constructed using mass, temperature, entropy, and the perturbation parameter η [5, 10, 12]. in addition, each side of the relation includes a partial derivative term of mass and entropy with respect to η, ∂Mext(Q, η) ∂η = lim M →Mext − T ∂S ∂η M,Q , (3.1) where M and Q represent the mass and additional quantities

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∂S ∂η P,β,r + + ∂S ∂r+ P,β,η ∂r+ ∂η P,β,M # . (3.5) 5 Substituting Eqs. (3.2) and (3.3) into Eqs. (3.4) and (3.5), respectively, yields ∂M0 ∂η P,β,S = −

direct method of calculation Under the condition that P and β are constant, when M → M0, the position of the black hole horizon, r+, is a function of perturbation parameter, η, we can obtain 0 = − " Λr3 + 6 − β − 104Λ 3r3 + + 32(1 + η)Λ2 r+ − 56(1 + η)2Λ3r+ 279 − 208(1 + η)3Λ4r3 + 81 !# + 1 2 − (1 + η)Λr2 + 2 + β − 100 r6 + + 104(1 + η)Λ r4 + − 16(1 + η)2...

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[5] [5]

(2.12), it can be demonstrated that, under the condition that β and S remain constant, and when M → M0, both the black hole horizon position r+ and P are functions of η

The Derivation of Composite Function 6 In the context of the Eq. (2.12), it can be demonstrated that, under the condition that β and S remain constant, and when M → M0, both the black hole horizon position r+ and P are functions of η. According to the law of derivation of the composite function, we obtain ∂M ∂η P (η),β,S = ∂M ∂η P (η),β,r+ + ∂M ∂r+ P (η),...

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T ∂S (η) ∂η P,B(η),M (η) + ψβ ∂B (η) ∂η P,S(η),M (η) # , (3.30) when B(η) is invariant, Eq. (3.26) reduces to ∂M0 ∂η P (η),B,S (η) = − lim M →M0

full differential method In the context of employing the composite function derivation method, it is imperative to note that the GP relation between each thermodynamic state parameter and the energy can only be calculated separately when the energy is a function of multiple thermodynamic state parameters. In the following, we employ the full differential ...

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[7] [7]

(2.12), the thermodynamic quantity and the horizon position of black hole will concomitantly be modified when the perturbation parameter of the spacetime, η, is altered

SUMMAR Y The present study is grounded on the following core idea: Given the thermodynamic quantities of black holes, in accordance with the constraints imposed by Eq. (2.12), the thermodynamic quantity and the horizon position of black hole will concomitantly be modified when the perturbation parameter of the spacetime, η, is altered. This physical mecha...

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[8] [8]

Vafa, The String landscape and the swampland, HUTP-05-A043 9

C. Vafa, The String landscape and the swampland, HUTP-05-A043 9

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