Charged accelerating black hole in $f(R)$ gravity

Ming Zhang; Robert B. Mann

REVIEW 4 major objections 5 minor 1 cited by

Charged accelerating black hole in $f(R)$ gravity

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper constructs a charged accelerating AdS black hole in f(R) gravity, proves a first law for it, and shows that a generalized reverse isoperimetric inequality makes the η>1 case super-entropic.

desk verdict A useful extension of accelerating black hole thermodynamics to f(R) gravity, but the central inequality rests on a mass formula that is asserted rather than derived; worth refereeing with a request for more detail. read the letter →

arxiv 1908.05118 v2 pith:GA7OA24Q submitted 2019-08-14 hep-th gr-qc

Ming Zhang , Robert B. Mann This is my paper · ORCID

classification hep-thgr-qc MSC 83C5783D05 PACS 04.70.-s04.50.Kd

keywords f(R)gravityacceleratingblackholeC-metricthermodynamicsfirstlawreverseisoperimetricinequalitysuper-entropicAdS

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 constructs a charged, slowly accelerating anti-de Sitter black hole in f(R) gravity and shows that its thermodynamics is well-defined: the first law $dM = T\,dS + \Phi\,dQ + \lambda_+\,d\mu_+ + \lambda_-\,d\mu_- + V\,dP$ holds, including the tension variables $\mu_\pm$ and their conjugates $\lambda_\pm$. The key new quantity is $\eta = 1 + f'(R_0)$, which enters linearly in the mass and Wald entropy, cannot be scaled away by a charge redefinition, and enlarges the allowed parameter space of solutions when $\eta > 1$. The paper derives a generalized reverse isoperimetric inequality, $(3V/4\pi)^2 \ge (1/\eta\Delta)(S/\pi)^3$, and concludes that for $\eta > 1$ the black holes are super-entropic relative to their Einstein-gravity counterparts, while larger conical deficits act in the opposite direction.

What carries the argument

The load-bearing object is the parameter $\eta = 1 + f'(R_0)$, the derivative of the f(R) action evaluated at the constant Ricci scalar $R_0$ of the solution. In the field equations it appears through $\eta R_{ab} - (\eta/4) R_0 g_{ab} = 2 T_{ab}$, so the metric has the same form as the charged accelerating C-metric with $q^2$ shifted to $q^2/\eta$; superficially this shift can be absorbed into the charge, but the mass and entropy both carry an explicit linear factor of $\eta$, so $\eta$ cannot be scaled out of the thermodynamics. The derivation of the first law and of the inequality (47) proceeds through writing the mass entirely in terms of the extrinsic thermodynamic variables (41), from which the conjugates are obtained by differentiation and the thermodynamic volume is fixed by the Smarr relation.

What would settle it

Compute the thermodynamic volume geometrically, for example by varying the action with respect to the AdS scale l at fixed S, Q and μ±, and compare the result with Eq. (33). Any discrepancy would shift the inequality (47) and overturn the super-entropic conclusion.

Watch

Extended reading notes

Core claim

On its own terms, the paper claims the first accelerating black hole solution in f(R) gravity, with a full first law of thermodynamics. The mass $M = \eta m(1 - A^2 l^2 \Xi)/(K\alpha)$, the Wald entropy $S = \eta \pi r_+^2/(K(1 - A^2 r_+^2))$, the temperature, charge, potentials, and tension variables are computed, and the mass is rewritten in Eq. (41) in terms of the extrinsic parameters $S$, $Q$, $P$, $\mu_\pm$. From that rewriting the tension conjugates $\lambda_\pm$ follow, the first law is checked, and the reverse isoperimetric inequality (47) is derived. The central result is that $\eta = 1 + f'(R_0)$ controls entropy-versus-volume behavior: at fixed volume, larger $\eta$ means more entropy, so solutions with $\eta > 1$ are super-entropic.

Load-bearing premise

The thermodynamic volume is not measured from the geometry but is fixed by requiring the Smarr relation to hold, and the reverse isoperimetric inequality depends on that choice.

Editorial extensions

If this is right

The f(R) accelerating black hole inherits the standard phase structure of the Einstein-gravity case, including van der Waals transitions, snapping swallowtails, and Hawking-Page-like curves, but with η-dependent terminal temperatures and pressures.
For η>1 the enlarged parameter space and the super-entropic character make these black holes a new setting for studying thermodynamic instabilities of super-entropic objects in modified gravity.
The generalized inequality (47) gives a concrete observational test: a measurement of the entropy-volume ratio of an accelerating black hole would constrain η and hence the form of f(R).
If η is promoted to an independent thermodynamic variable, the first law gains a new work term, which would turn the inequality into a sharper constraint and introduce a chemical-like potential for modified gravity.

Reading between the lines

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

The Smarr-fixed volume is a choice; a direct geometric definition of the thermodynamic volume could shift the inequality, so the super-entropic conclusion should be checked against an independent computation of V.
Since both mass and entropy scale linearly with η, the specific heat and evaporation rate of these black holes are nearly η-independent, but the charge-to-entropy ratio is not, which could yield observable differences in charged black hole evaporation within f(R) gravity.
The same construction should extend to rotating accelerating f(R) black holes; checking whether the pressure 'splitting' of the reentrant transition found for the rotating Einstein case survives for η≠1 would be a concrete test.

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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

This paper does what it says: it writes down a charged accelerating AdS black hole in constant-curvature f(R) gravity and works out its extended thermodynamics. The metric itself is basically the Einstein C-metric with the charge rescaled by η, as they admit, so the new content is in the thermodynamic treatment. That treatment is mostly careful: Wald entropy, conformal mass, temperature from the blackening factor, and a first law with tensions. The η=1 limit recovers the known Einstein-gravity results, which is a helpful anchor.

The genuinely new things are the generalized reverse isoperimetric inequality (47) and the η-dependent phase structure they map out. The RII looks plausible and reduces properly. The phase diagrams show real work—parameter space bounds, snapping swallowtail behavior, and Hawking-Page curves in the grand canonical ensemble.

Now the soft spots. The big one is the master mass formula (41). It is introduced with a hand-wave ('rather cumbersome' direct route) and then everything—T, Φ, V, the first law, and the RII—follows from it. The paper does claim that T and Φ obtained from (41) agree with the geometric expressions, which is a decent consistency check, but the derivation is not shown. That makes the central results hard to audit. The thermodynamic volume is fixed by the Smarr relation rather than measured from the geometry, and the RII is sensitive to that choice. They also leave open whether η itself should be a thermodynamic variable; if it is, the first law and the RII would change. They flag this themselves in the summary, so it is an acknowledged caveat, not an oversight. Still, it means the super-entropic conclusion for η>1 is conditional.

A smaller issue: the solution (12) is asserted, not verified against the f(R) field equations. I would have liked a line stating that direct substitution works, even if routine. And their disagreement with the charge definitions in Refs. [27,30] is stated without detail; a referee will want that spelled out.

None of these are fatal. The consistency checks they do provide—the η=1 limit, the T/Φ match, the reduction of the RII—give me reasonable confidence that the core is right. I would send this to peer review. The right referee will push for a derivation of (41) or at least a more explicit reconstruction from the first law, and for a discussion of the η-as-thermodynamic-variable possibility. If those checks survive, this will be a citable contribution to accelerating black hole thermodynamics in modified gravity.

For my part: I would bring it to the reading group, and I would cite it if I worked on accelerating black holes or f(R) thermodynamics. The verdict is 'conditionally accept'.

Referee Report

4 major / 5 minor

Summary. The paper constructs a charged, slowly accelerating AdS black hole in f(R) gravity with constant Ricci scalar R0, parametrized by η=1+f'(R0). It computes the conformal mass, Wald entropy, temperature, electric charge, electric potential, and conical-defect tensions, and claims a first law dM = T dS + Φ dQ + λ+ dμ+ + λ− dμ− + V dP (Eq. 46) built on a master mass formula (41). It also derives a generalized reverse isoperimetric inequality (47) with an η-dependent coefficient, which for η>1 indicates super-entropic behavior relative to Einstein gravity. The paper further maps the allowed parameter space and studies canonical and grand canonical phase behaviour, including van der Waals transitions, snapping swallowtails, and Hawking-Page-like curves.

Significance. If correct, this work extends accelerating black hole thermodynamics beyond Einstein gravity and gives a concrete η-dependent modification of the reverse isoperimetric inequality that can alter the sub/super-entropic classification. The paper contains useful internal checks: the η=1 limit reproduces the known Einstein results of Refs. [18,50]; the A=0 limit reduces to the charged f(R) AdS thermodynamics; the Smarr relation used to define V is consistent with the mass formula; and the authors make a clear, falsifiable claim disagreeing with earlier f(R) charge and potential definitions [27,30]. The phase-transition phenomenology is a valuable extension. However, the central thermodynamic claims rest on Eqs. (41) and (44), which are asserted rather than derived, and the solution (12) is not shown to satisfy the field equations.

major comments (4)

[II.A, Eqs. (12)-(15)] The accelerating f(R) solution is announced without a verification that the metric satisfies the f(R)-Maxwell equations (4) and (5). Equation (10) reduces the task to checking that (12) obeys η(R_ab − R0/4 g_ab) = 2T_ab with R0 = −12/l² and F from (18)-(19), but no substitution, Ricci-component check, or reference is provided. Since every thermodynamic quantity in the paper is computed from this line element, a direct verification should be included.
[II.B, Eq. (41)] The master mass formula (41) is load-bearing: equations (42)-(45), the first law (46), and the generalized reverse isoperimetric inequality (47) are all algebraic consequences of it. The text says the direct route is 'rather cumbersome' and simply writes (41) without deriving it from the mass (25), the horizon condition N(r+)=0, and the definitions of S,Q,P,Δ,C. The later claim that the derivatives (42)-(44) agree with (27), (30), and (33) is also only asserted. An appendix should derive (41) and explicitly verify the agreement; without this, the central thermodynamic claims are unsubstantiated.
[II.B, Eqs. (33)-(34) and (44)] The thermodynamic volume V is fixed by imposing the Smarr relation (34), and then Eq. (44), obtained as a derivative of (41), is said to agree with (33). Because (34) was used to define V, this agreement is a self-consistency condition rather than an independent check. The logical status should be clarified: the independent content resides in (41), and if (41) were modified, both V and the inequality (47) would change. Please show the explicit algebra for (44)=(33) rather than asserting it.
[II.B, Eqs. (48)-(50)] The boundary action used to obtain the free energy G is stated without derivation. In f(R) gravity the Gibbons-Hawking surface term is not generally ηK when f(R) contains higher derivatives; boundary terms involving ∂_n R and δR must be addressed even if R is constant in the bulk. The counterterm (49) and its η-dependence therefore need a calculation or a clear reference. Since G underlies the phase-transition analysis in Section III.B, this is a load-bearing issue for that part of the paper.

minor comments (5)

[II.A, after Eq. (20)] The Einstein-gravity limit f'(R0)=0 should also specify f(R0)=6/l², since Eq. (9) then requires R0+2f(R0)=0.
[II.B, Eq. (27)] The equality between the two displayed forms of T uses the horizon condition in a nontrivial way; one intermediate line of algebra would make the expression much easier to follow.
[II.B, Eq. (50)] The mass term is split as 'ηm(1−A²l²)/(αK) − mA⁴l²q²/(αK)'; it would be clearer to keep the single compact form ηm(1−A²l²Ξ)/(αK).
[III.B, after Eq. (71)] The assertion that T'(S)=0 always has a solution for S>0 in the charged accelerating case is not demonstrated; an explicit expression for T'(S) beyond the C=0 case should be provided.
[Figs. 2 and 3] The captions should define P1, P2, P3 and explain what the terminal points in the G−T diagrams represent.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the f(R) accelerating black hole thermodynamics are a consistency-checked extension, not a reduction to inputs.

full rationale

The central derivation chain is self-contained rather than circular. The mass, entropy, temperature, charge, and electric potential are computed from independent geometric definitions: the conformal mass (25), Wald entropy (26), surface gravity temperature (27), Gauss-law charge (29), and gauge-potential integral (30). The thermodynamic volume is fixed by the Smarr relation (34), a standard extended-phase-space convention, and then re-derived independently as the pressure derivative of the mass formula (41); the agreement of (42)-(44) with the geometric expressions (27), (30), and (33) is a consistency check, not a fit, because those geometric quantities were obtained before the mass formula was introduced. The reverse isoperimetric inequality (47) follows algebraically from (41) and (44) and reduces to the known Einstein-gravity result when η=1, providing an external benchmark. The paper explicitly flags that 'the possibility that η itself could be a thermodynamic parameter merits investigation', which is an acknowledged completeness/risk issue rather than a circular step: no predicted quantity is used to set a free constant, and no target result is assumed in defining the inputs. The master mass formula (41) is asserted rather than derived in detail from the metric, which is a derivation gap and a correctness risk, but the paper's own cross-checks against geometric quantities make the chain a self-consistent derivation rather than a definitional equivalence. Overall, no circular step of the kind defined in the rubric is present.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The derivation introduces no fitted numbers. The only new model parameter is η, which is an input of f(R) gravity rather than a fit. All other inputs are standard black-hole thermodynamic definitions from the cited literature. The most fragile input is the Smarr-defined volume, which is also the weakest assumption of the paper.

assumptions (6)

domain assumption The f(R) action is restricted to constant Ricci scalar R0 satisfying Eq. (9).
The field equations reduce to η R_ab - (η/4) R0 g_ab = 2 T_ab only for constant R0; the solution and all thermodynamics depend on this restriction.
domain assumption η = 1 + f'(R0) > 0.
The paper follows [30] in requiring η>0 to have inner and outer horizons; the parameter space plots and phase analysis all take η>0.
domain assumption The slow-acceleration regime has no acceleration horizon, enforced by conditions such as 1 - A^2 l^2 Ξ > 0 (Eq. 61).
The single-horizon thermodynamics and first law (46) apply only in this regime.
standard math The conformal mass prescription (25) and Wald entropy (26) are the correct charge/entropy definitions for this solution.
Adopted from [32,33] and [38] without re-derivation; they carry the thermodynamic interpretation.
domain assumption The conical-deficit tensions μ± are dimensionless and do not enter the Smarr relation (34).
Following [14,50]; this closes the first law and underlies the mass formula (41) and the RII.
domain assumption Extended thermodynamics with P=3/(8π l^2) applies, and V is defined to satisfy the Smarr relation.
This is the standard extended phase space convention; the RII depends on this V.

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Pith. "Pith review of Charged accelerating black hole in $f(R)$ gravity." pith.science (2026). https://pith.science/paper/GA7OA24Q

@misc{pith2026190805118,
  author       = {Pith},
  title        = {Pith review of: Charged accelerating black hole in $f(R)$ gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GA7OA24Q}},
  note         = {Machine review of arXiv:1908.05118}
}

abstract

We obtain a charged accelerating AdS black hole solution in $f(R)$ gravity and investigate its thermodynamic behaviour. We consider low-acceleration black holes that do not have an acceleration horizon and obtain the first law of thermodynamics for them. We further study the parameter space of charged slowly accelerating $f(R)$ AdS black holes before investigating the behaviour of the free energy in both the canonical and grand canonical ensembles. We find a generalization of the reverse isoperimetric inequality, applicable to black holes in $f(R)$ gravity, that indicates these black holes can become super-entropic relative to their counterparts in Einstein gravity.

Figures

Figures reproduced from arXiv: 1908.05118 by the authors.

**Figure 2.** FIG. 2. The free energy in terms of the temperature in the [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** FIG. 3. The Gibbs free energy [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

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Reviewed August 14, 2026 · model on record in the stance chip above.

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