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REVIEW 3 major objections 6 minor 1 cited by

Black hole thermodynamics in modified gravity

T0 review · 3 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A small modification to Einstein gravity changes the horizon, entropy, temperature, heat capacity, and free energy of rotating black holes, so modified gravity cannot be ignored near black holes.

desk verdict Plausible qualitative claim, but the quantitative derivation has load-bearing algebra errors; a corrected version could be worth publishing. read the letter →

arxiv 2411.18693 v1 pith:I46QQGVK submitted 2024-11-27 gr-qc

Mriganka Dutta , Banibrata Mukhopadhyay This is my paper · ORCID
classification gr-qc MSC 83C5783D05 PACS 04.70.Dy04.50.Kd
keywords blackholethermodynamicsmodifiedgravityf(R)rotatinghorizonradiusentropyheatcapacityfreeenergy
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 argues that even a small deviation from Einstein's general relativity alters the thermodynamic properties of a rotating black hole. Using a rotating black hole metric from an f(R) modified gravity theory, parametrized by beta = B(B-6)/2 with small negative B, it derives the horizon radius, entropy, temperature, heat capacity, and free energy. All of these quantities depend on beta, so they differ from the Kerr values. The paper concludes that modified gravity cannot be ignored in the strong-field regime near black holes, and that these differences grow as B becomes more negative.

What carries the argument

The central object is the rotating black hole metric in f(R) gravity, Eq. (2), with f(R) chosen so that df/dR tends to 1 + B/r at large r. The parameter $\beta$ = B(B-6)/2, with B negative and |B| < 1, is the effective modification strength. With the approximation X(r,$\theta$) approximately 1, the metric function $\Delta$ simplifies to $r^{2}$ - 2Mr + $a^{2}$ - $\beta$, which reduces the horizon and thermodynamic quantities to algebraic functions of $\beta$. The work this machinery does is to turn a modified-gravity theory into a one-parameter deformation of Kerr, so every classic black-hole thermodynamic relation (area law, mass formula, heat capacity, free energy) can be evaluated analytically.

What would settle it

Evaluate the exact X(r,$\theta$) and $\Delta$ at B = -0.6 and a in the range 0 to 3 with M = 1, and check whether $\Delta$ = 0 still gives r_+ = M + $\sqrt$($M^{2}$ - $a^{2}$ + $\beta$). If the exact horizon differs by more than a few percent, the paper's analytical thermodynamics does not actually describe the metric it starts from. A more direct test is to solve the f(R) field equations numerically for this B and compare the resulting metric with Eq. (2), or to check the validity of the small-|B| expansion at B = -0.6.

Watch

Extended reading notes

Core claim

The paper derives analytical expressions for the thermodynamic variables of a rotating black hole in a specific f(R) gravity, using the metric of Eq. (2). With |B| < 1 and X(r,$\theta$) approximately 1, the horizon condition reduces to $r^{2}$ - 2Mr + $a^{2}$ - $\beta$ = 0, giving r_+ = M + $\sqrt$($M^{2}$ - $a^{2}$ + $\beta$). From the horizon area A = 4*pi*(r_+^2 + $a^{2}$), entropy S = A/4, temperature T = (r_+ - r_-)/(4*pi*(r_+^2 + $a^{2}$)), heat capacity C_L = M*T*S / (1/(4*pi) - 2*M*T - $T^{2}$*S), and free energy F = M - T*S. For negative B (positive $\beta$), the outer horizon is larger, entropy is larger, temperature at a given spin is lower, the heat-capacity phase transition shifts to higher spin, and free energy is lower. This is the central claim: small modifications to Einstein gravity produce non-negligible changes in black hole thermodynamics.

Load-bearing premise

The paper assumes the rotating metric of Eq. (2) is the correct vacuum solution of the f(R) theory and that the approximation X(r,theta) approximately 1 remains valid at the plotted values of B, which go as low as -0.6 and give beta around 1.98. If the metric is not a solution or the small-B expansion fails, all the thermodynamic quantities change.

Editorial extensions

If this is right

  • Near a rotating black hole, a small f(R) modification increases the horizon radius and entropy relative to Kerr at the same mass and spin.
  • The Hawking temperature is lowered by the modification, so black holes in this modified gravity radiate more slowly at a given mass and spin.
  • The heat capacity switches from negative to positive at a spin value that grows as B becomes more negative, so the modified gravity resists the angular-momentum-driven instability.
  • Free energy drops with more negative B, meaning the strong-curvature region is thermodynamically more stable than in Einstein gravity.
  • Observable quantities tied to horizon size, such as shadow radius and quasinormal modes, would carry a modified-gravity signature.

Reading between the lines

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

  • Going beyond the paper: one could compute the exact X(r,theta) correction at the plotted values, e.g. B = -0.6, and check whether the horizon condition r^2 - 2Mr + a^2 - beta = 0 still holds to the accuracy claimed; if not, the thermodynamic formulas need corrections at large |B|.
  • Going beyond the paper: the metric of Eq. (2) is assumed to be the true vacuum solution; this could be checked by numerically solving the f(R) field equations and comparing the metric components for |B| up to 0.6.
  • Going beyond the paper: the shift in the heat-capacity phase transition suggests a modified-gravity signature in black hole spin measurements, which could be searched for in gravitational wave ringdown data.
  • Going beyond the paper: the same beta-deformed thermodynamics implies a longer evaporation time at fixed mass and spin, a testable extension the authors flag as future work.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper studies the thermodynamics of a rotating black hole in an f(R) modified-gravity theory. Using the rotating metric of Ref. [6], parametrized by B (with beta = B(B-6)/2), the authors write the horizon radius r_+, the entropy S = A/4, the temperature T_BH, the heat capacity C_L, and the free energy F = M - T_BH S. They plot these quantities for B = 0, -0.2, -0.4, -0.6 and conclude that even a small modification of Einstein gravity substantially changes black hole thermodynamic properties, so modified gravity cannot be ignored near black holes.

Significance. If the derivation were correct, the paper would provide a concrete worked example showing how an f(R) modification shifts standard Kerr thermodynamic quantities. A strength is that the metric is taken from an earlier independent publication rather than being fitted to the thermodynamic conclusions, and the paper makes explicit quantitative predictions for the location of the heat-capacity phase transition and for the free energy. However, the central quantitative derivation contains internal inconsistencies: Eq. (10) does not invert Eq. (14), Eq. (11) does not follow from the first law and contradicts the Kerr limit of Eq. (8), and the X(r,theta) = 1 approximation is used at parameter values where it is not valid. The qualitative direction of the conclusion may survive a corrected derivation, but the paper as printed does not establish its quantitative claims. The paper is a short proceedings contribution; its value would be as a reproducible example, not as a fundamental new result.

major comments (3)
  1. [§3.1, Eq. (4)] The horizon condition is reduced to r^2 - 2Mr + a^2 - beta = 0 by setting X(r,theta) = 1, but the paper then plots values as large as B = -0.6, for which beta = 1.98. For these parameters the approximation is not valid. For example, at theta = 0 and a = 0, X = [(r^2 + a^2 cos^2 theta)^2] / [(r + B/2)^2 + a^2 cos^2 theta]^2, and with M = 1, B = -0.6, r = r_+ from Eq. (4), one obtains X of order 1.6, not 1. Thus the simplified horizon radius and all subsequent formulas are uncontrolled at the plotted parameter values. The paper should either restrict itself to |B| much smaller than 1, where X = 1 can be quantitatively justified, or retain the X dependence in the horizon calculation.
  2. [§3.2, Eq. (10)] Equation (10) is not the inverse of Eq. (14). Solving the horizon condition together with S = pi(r_+^2 + a^2) and L = aM gives M^2 = S/(4pi) - beta/2 + pi beta^2/(4S) + pi L^2/S, whereas Eq. (10) contains 4pi L^2/S. The factor 4 in the angular-momentum term is incorrect; in the Kerr limit beta = 0, Eq. (10) contradicts the standard relation M^2 = S/(4pi) + pi L^2/S. Since the later algebra uses Eq. (10), this inconsistency undermines the derivation of the temperature and heat capacity.
  3. [§3.2, Eq. (11)] Equation (11) is not the temperature obtained from the first law. From the corrected mass-entropy relation, T = (partial M / partial S)_L = (1/(2M)) [1/(4pi) - pi beta^2/(4S^2) - pi L^2/S^2]. The printed Eq. (11) has beta^2/4 without the 1/S^2 factor and has +L^2/S^2 instead of -pi L^2/S^2. It is also numerically inconsistent with Eq. (8) in the Kerr limit: for M = 1, a = 0.5, beta = 0, Eq. (8) gives T about 0.0369, while Eq. (11) gives about 0.0407. Because the heat capacity in Eq. (12) and the free energy in Eq. (13) are evaluated with this temperature, the phase-transition locations in Fig. 4 and the free-energy shifts in Fig. 5 are not supported as printed.
minor comments (6)
  1. [§3.1] The statement that the temperature computed from the surface gravity in Eq. (6) equals the temperature in Eq. (8) is asserted but not demonstrated. Since the metric is not the Kerr metric, this equality is a nontrivial check and should be shown explicitly or at least sketched.
  2. [§3.2, Eq. (12)] The derivation of Eq. (12) is omitted. Even if Eq. (12) can be obtained as an algebraic identity from the mass formula, the authors should present the steps, especially because the preceding equations contain errors.
  3. [§3.3, Eq. (14)] The entropy as a function of M, L, and beta is stated without derivation. It would be helpful to show that Eq. (14) follows from the horizon condition and S = pi(r_+^2 + a^2).
  4. [§4] The sentence 'The lower value of the free energy makes the curvature of spacetime less affected' is not physically justified. Free energy is a thermodynamic potential; relating it directly to spacetime curvature requires an argument that the paper does not provide.
  5. [References] Reference [7] contains a typographical error in the URL: 'https:://doi.org' should be 'https://doi.org'.
  6. [Notation] The paper uses both B and beta, with beta = B(B-6)/2, and the figures are labeled by B. Since beta is what actually enters the metric and thermodynamics, the authors should explicitly state the sign and magnitude of beta for each plotted curve, especially because B = -0.6 gives beta = 1.98, which is not small.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper derives thermodynamic quantities from a previously published metric via standard area–entropy and surface-gravity relations; no fitted inputs are relabeled as predictions.

full rationale

The thermodynamic quantities are obtained by applying standard black-hole thermodynamics to the rotating metric of Eq. (2), which is taken from the earlier work cited as [6]. The paper does not fit any parameter to the target thermodynamic claims; entropy is set by S=A/4, temperature by the Killing-vector surface gravity and by the first-law differential, heat capacity by the definition C_L=T(∂S/∂T)_L, and free energy by the explicit definition F=M-TS. The metric itself is an external prior result, not derived from thermodynamics, and the paper does not use thermodynamics to justify the metric; hence relying on it is a normal use of a prior solution rather than a circular step. The algebraic inconsistencies between Eq. (10) and Eq. (14), and between Eq. (11) and the first law, are correctness risks, not circularity, because they do not make any derived quantity equivalent to an input by construction. No part of the derivation reduces to a fitted parameter disguised as a prediction, and no load-bearing claim is justified solely by a self-citation chain whose content is the target result itself. Therefore no significant circularity is found.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The paper introduces no fitted constants; the only free parameter is B from the modified gravity model, chosen by hand. The main load-bearing axioms are the validity of the borrowed metric, the small-B approximation X is approximately 1, and the unmodified area-entropy law in f(R) gravity.

free parameters (1)
  • B (modification parameter, via beta = B(B-6)/2) = negative, |B| < 1; plotted values: -0.2, -0.4, -0.6
    B controls the deviation from Einstein gravity in the metric (Eq. 2) and is chosen by hand to be negative and small; the paper does not fit it to data, but all results depend on its value.
assumptions (5)
  • domain assumption The rotating metric of Eq. (2) is a valid vacuum solution of f(R) gravity with f'(R) = 1 + B/r.
    Section 2, 'Following a previous work [6], one can consider the metric...' The solution's validity is taken from prior literature.
  • domain assumption X(r,theta) is approximately 1 for |B| < 1, so Delta reduces to r^2 - 2Mr + a^2 - beta.
    Section 2, 'Considering the fact that |B| is small; we can take X(r,theta) is approximately 1.' This justifies all subsequent horizon formulas; no validity range check is given.
  • domain assumption Bekenstein-Hawking area-entropy law S = A/4 and Hawking temperature from surface gravity hold unchanged in f(R) gravity.
    Section 3.1, 'Following the earlier works of Bekenstein and Hawking, S = A/4.' In f(R) gravity, entropy may receive Wald corrections; this is not discussed.
  • domain assumption The first law dM = T dS + Omega dL applies with L = aM as the total angular momentum.
    Section 3.1, Eq. (7); the Smarr relation is assumed to carry over to the modified metric.
  • standard math Standard thermodynamic definitions of heat capacity (C = T(dS/dT)_L) and free energy (F = M - TS) apply to black holes in this setting.
    Section 3.2 and 3.3; these are conventional definitions.

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Cite this review

Pith. "Pith review of Black hole thermodynamics in modified gravity." pith.science (2026). https://pith.science/paper/I46QQGVK

@misc{pith2026241118693,
  author       = {Pith},
  title        = {Pith review of: Black hole thermodynamics in modified gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I46QQGVK}},
  note         = {Machine review of arXiv:2411.18693}
}
read the original abstract

The theory of general relativity is often considered under the framework of modified Einstein gravity to explain different phenomena under strong curvature. The strong curvature effect plays a main role near black holes, where the gravitational field is strongest. The idea of black hole thermodynamics is to describe the strong field curvature properties of a black hole in the effective thermodynamical framework, e.g. entropy, temperature, heat capacity etc. In this paper, our aim is to explore how the effect of modified gravity changes the thermodynamic properties of black hole. We show that even a small modification to Einstein gravity affects the thermodynamical properties of a black hole.

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Forward citations

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