Analytically Approximate Black Hole Solution to Higher Curvature Gravity

Seyed Naseh Sajadi

REVIEW 4 major objections 4 minor 23 references

Analytically Approximate Black Hole Solution to Higher Curvature Gravity

T0 review · 4 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read A black hole metric can be reconstructed from its thermodynamics, without solving the field equations, using a continued-fraction expansion.

desk verdict Paper advertises higher-curvature black hole solutions but only reconstructs known RN-AdS by plugging in its own thermodynamics; no alpha term is ever computed. read the letter →

arxiv 2608.01869 v1 pith:7UPGKNCL submitted 2026-08-03 gr-qc

Seyed Naseh Sajadi This is my paper · ORCID

classification gr-qc

keywords blackholeshigher-curvaturegravitycontinuedfractionexpansionSmarrrelationfirstlawofthermodynamicsReissner-Nordström-AdSapproximateanalyticalsolutionsmodified

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 tries to show that black hole thermodynamics can substitute for solving the field equations when constructing approximate analytical black hole solutions in higher-curvature gravity. The proposed route is to parameterize the metric with a continued-fraction expansion matched at the horizon and at infinity, then use the first law dM = TdS + Σψdα and the Smarr relation M = 2TS + Σαψ to fix the previously free near-horizon slope. In the worked Einstein-Maxwell-AdS example, this determines f1 = -r+Λ - Q²/r+³ + C1/r+, and choosing C1 = 1 reproduces the exact RN-AdS metric and temperature with a four-term truncation. The broader claim is that the same framework applies to any modified-gravity theory, including f(Q) and f(T) gravity.

What carries the argument

The key mechanism is the continued-fraction parametrization h(r) = xA(x), h(r)/f(r) = B²(x), with x = 1 - r+/r, which keeps the metric accurate from the horizon to spatial infinity using only a handful of coefficients. The coefficients are fixed by matching asymptotic and near-horizon expansions; the two remaining unknowns, f1 and h1, are fixed by imposing the Smarr relation and the first law as differential equations for f1. This substitution of thermodynamic identities for the equations of motion is what carries the argument.

What would settle it

Apply the thermodynamic-plus-continued-fraction procedure to a genuine higher-curvature theory where the exact solution is unknown, such as Einsteinian cubic gravity, and compare the resulting f1 and metric against numerical integration of the field equations. If the metric does not match the numerics, or if the required inputs S and ψi can only be computed after solving the field equations, the central claim fails.

Watch

Extended reading notes

Core claim

The central claim is that the first law of thermodynamics and the Smarr relation are sufficient to determine black hole solutions and their thermodynamic properties without directly solving the field equations. Concretely, for a static spherically symmetric charged AdS black hole, the Smarr relation yields a mass formula M = 2TS - 2PV + Qφ, and combining it with the first law produces differential equations for the near-horizon expansion coefficient f1. Solving those equations gives f1 = -r+Λ - Q²/r+³ + C1/r+, and setting C1 = 1 exactly reproduces the RN-AdS temperature and mass. The continued-fraction metric built from this f1 matches the exact solution to within about 10⁻² outside the hori

Load-bearing premise

The load-bearing premise is that the Smarr relation and the first law can be imposed on the trial metric before the solution is known, with S, V, φ, and any conjugate potentials as available inputs; in the worked example those inputs are borrowed from the exact RN-AdS solution the method is meant to construct, and C1 is set to 1 by hand.

Editorial extensions

If this is right

If the method works in genuine higher-curvature theories, one can obtain closed-form approximations to black hole metrics and temperatures without solving fourth-order field equations.
The same continued-fraction structure should apply to non-curvature-based modified gravity theories such as f(Q) and f(T), with only the coefficients changed.
Low-order truncation is sufficient: four continued-fraction terms reproduce the RN-AdS metric function with deviation at or below 10⁻² outside the horizon.
The framework yields geometry and thermodynamics simultaneously, so thermodynamic consistency is built into the approximate solution by construction.

Reading between the lines

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

Editorial extension: The worked example is not yet evidence for higher-curvature theories because the thermodynamic inputs S = πr+², V = 4πr+³/3, φ = Q/r+, and the setting C1 = 1 are all taken from the exact RN-AdS solution the method is meant to construct.
Editorial extension: A decisive test would be to run the same algorithm in a theory where the Wald entropy and higher-curvature conjugate potentials are not known in advance, such as Einsteinian cubic gravity or quadratic gravity, and compare against numerical integration of the full field equations.
Editorial extension: The paper mentions but does not work out the f1 ≠ h1 branch; applying the thermodynamic fixing to that case would extend the method to theories with two independent metric functions and could reveal branches missed by numerical scans.
Editorial extension: If the method generalizes, black hole thermodynamics becomes a solution-generating technique, making it possible to compute quasinormal-mode or lensing observables from approximate metrics without solving the equations of motion for each coupling.

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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,

Short version: the title oversells. The paper never turns on the higher-curvature coupling. The one worked example is Einstein–Maxwell–AdS, and the "construction" is a calibration: S, V, φ, and the Smarr mass formula are taken from the known RN–AdS solution, and the integration constant C1 is set to 1 so the temperature comes out matching. That is not a derivation of a higher-curvature metric.

For what it is worth, the machinery is standard and the algebra is mostly clean. The continued-fraction parametrization comes from Rezzolla–Zhidenko, and the thermodynamic fixing of the near-horizon slope follows Bonanno–Silveravalle. The paper lays out the differential equations (22)-(24) sensibly, and the final f1 expression (37) is correct for RN–AdS when C1 = 1. The reconstructed metric agrees with the exact solution to about 10^-2 in the figure, which is a reasonable sanity check. The conclusions are also honest: they admit only Einstein gravity is treated and the higher-curvature extension is left for future work.

The problems are not minor. First, there is no α ≠ 0 calculation anywhere, so the central claim in the abstract and title is unsupported. Second, the method as demonstrated is circular for the only example: the Smarr mass formula (36) with S = πr+², V = 4πr+³/3, φ = Q/r+ is an identity of the exact RN–AdS solution. For a genuine higher-curvature theory, the conjugate potential ψ_α in Eq. (17) is an integral of horizon curvature that depends on the unknown metric, so you cannot impose Eq. (23) before solving the field equations. Third, there are internal arithmetic inconsistencies: Eq. (31) with (29) implies a near-horizon slope with a factor of 2 different from Eq. (37), and the f2, f3 coefficients in Eq. (33) do not match the Taylor expansion of the claimed solution. The paper's own equations are not mutually consistent.

Who would get value from this? Someone teaching continued-fraction methods might use it as a worked exercise once the arithmetic is corrected. But it is not a new result, and it does not establish the advertised method for higher-curvature gravity.

I would not send this to a referee in its current form. The fix is substantial: compute a genuine α ≠ 0 example with a well-defined ψ_α, or change the title and claims to describe a consistency check on RN–AdS. With that, the piece could be a useful technical note; as it stands, it is not.

Referee Report

4 major / 4 minor

Summary. The paper proposes a method to construct approximate static, spherically symmetric black hole solutions by combining a continued-fraction metric parametrization with the first law of black hole thermodynamics and the Smarr relation, thereby avoiding direct solution of the field equations. The general formalism is set up for a Lagrangian L = R - 2Λ + α L1, with generic conjugate potentials ψ_i and Wald entropy. The only explicit calculation, however, is for Einstein–Maxwell–AdS gravity (α = 0). The near-horizon coefficient f1 is obtained from thermodynamic differential equations, an integration constant is set to unity to reproduce the RN–AdS temperature, and the reconstructed metric is compared with the exact RN solution.

Significance. If the advertised program worked, it would be genuinely useful: it would allow approximate metrics and thermodynamics in higher-curvature theories without solving fourth-order field equations, and it would extend a well-established continued-fraction parametrization. The formal first-law consistency equations (22)–(24) are reasonable, and the continued-fraction framework is a known and valuable tool. However, the paper does not carry out any higher-curvature calculation; its only worked example reconstructs RN–AdS by inserting RN–AdS thermodynamic identities. Moreover, the example contains a factor-of-two inconsistency between the displayed field equation and the thermodynamic result. The potential significance is therefore not realized in the present manuscript.

major comments (4)

[Title, Abstract, §3 Conclusions] The central claim is not demonstrated. The generic action (2) contains α L1, but §2.1 only treats the Einstein–Maxwell action (25) and never sets α ≠ 0. No ψ_α is computed for any higher-curvature theory, and no metric-dependent Wald entropy or thermodynamic volume for α ≠ 0 is provided. The conclusion that the framework is demonstrated for higher-curvature gravity, and speculatively for f(Q)/f(T) theories, is unsupported by the content of the paper.
[§2.1, Eqs. (31) and (37)] Solving Eq. (31) as written gives f(r) = 1 + C/r - Λ r^2/3 + Q^2/(2r^2). After imposing f(r_+)=0, the near-horizon slope is f1 = 1/r_+ - Λ r_+ - Q^2/(2 r_+^3). Eq. (37) with C1 = 1 gives f1 = 1/r_+ - Λ r_+ - Q^2/r_+^3. The Q^2-dependent terms differ by a factor of 2. Thus the thermodynamic f1 is not the near-horizon slope of the solution of the displayed field equation. This is an internal inconsistency, not a convention choice, because it is the same metric ansatz and the same stress-energy tensor.
[§2.1, Eq. (33)] The near-horizon Taylor coefficients in Eq. (33) are not those of the exact RN–AdS metric. For f = 1 - 2M/r + Q^2/r^2 - Λ r^2/3 with f(r_+)=0, the coefficient of (r - r_+)^2 is f''(r_+)/2 = (2Q^2 - r_+^2)/r_+^4, not (Q^2 - r_+^2)/r_+^4; the coefficient of (r - r_+)^3 also differs. Eq. (33) appears instead to be the Taylor expansion of the solution of Eq. (31), which has Q^2/(2r^2) rather than the RN Q^2/r^2 term. The comparison in Fig. 1 is therefore made with a metric that does not correspond to the solution whose temperature is quoted.
[§2.1, Eqs. (35)–(38)] The procedure in the only worked example is circular. The inputs S = π r_+^2, V = 4π r_+^3/3, φ = Q/r_+, and the Smarr form M = 2TS - 2PV + Qφ are identities of the known exact RN–AdS solution, not consequences of the continued-fraction ansatz. The first law then determines f1 up to an integration constant, and C1 = 1 is chosen in Eq. (38) by matching the known RN–AdS temperature. In a genuine higher-curvature theory, S and ψ_i are metric-dependent and unknown before solving the field equations, so the claimed 'without directly solving the field equations' feature is not established.

minor comments (4)

[Eq. (17)] The definition ψ_i = ∫_H ξ_H · A_i is vague: for the cosmological constant and for higher-curvature couplings the conjugate potentials are not generally gauge potentials in this sense. A concrete definition, or a reference, is needed.
[Eq. (32)] The asymptotic expansion writes f(r) = Σ F_i/r^i but then includes terms Q^2/r^2 and -Λ r^2/3, which are not of the form F_i/r^i. Separate the known asymptotic RN–AdS terms from the continued-fraction residual.
[Eq. (34)] The text states that the continued-fraction coefficients are obtained '(Λ = 0)', but Λ appears later in f1 and in the thermodynamic relations. It should be clarified whether the adopted expansion is valid for AdS or only for Λ = 0.
[General] There are several typographical and grammatical issues (e.g., 'diﬃculty', 'but they are subject to several constraints', 'with multiple horizons'), and the notation f = g/h = 1 in Eq. (27) is not introduced consistently with the general ansatz having two independent functions f(r) and h(r).

Circularity Check

2 steps flagged · score 6.0 of 10

Smarr/first-law framework reconstructs RN-AdS only by inputting its own thermodynamic identities and fixing C1=1 to the known temperature.

fitted input called prediction [Section 2.1, Eqs. (37)-(39)]
"For C1 = 1, this exactly reproduces the temperature of the RN–AdS black hole. Therefore, f1(r+,Λ,Q) = −r+Λ − Q²/r³+ + 1/r+, F1 = −2M = −r+[1 + Q²/r²+ − Λr²+/3]."

The first-law differential equations determine f1 only up to an integration constant C1. The paper then sets C1 = 1 because that value reproduces the already-known RN–AdS temperature. No boundary condition or independent input fixes C1 within the thermodynamic framework, so the subsequent agreement with the exact solution is imposed by hand. The temperature in Eq. (38) is not a prediction; it is the target used to select the free constant.
self definitional [Section 2.1, Eqs. (35)-(36)]
"The thermodynamical quantities are given as T = f1/4π, S = πr²+, P = −Λ/8π, V = 4/3πr³+, φ = Q/r+. Therefore, the Smarr formula yields M = 2TS − 2PV + Qφ = r²+/2 f1(r+,Λ,Q) + Λ/3 r³+ + Q²/r+."

The inputs S, V, φ and the Smarr combination M = 2TS − 2PV + Qφ are identities of the exact RN-AdS solution that the method is supposed to construct. They are inserted as given inputs, and the first law is then solved for f1 (and hence F1 = −2M). For the advertised higher-curvature case, the conjugate ψ_α in Eq. (17) is an integral over horizon curvature that depends on the unknown metric, so the procedure cannot even start without already knowing the target solution. The worked example is thus self-referential: the solution's own thermodynamic identities are used to produce the solution.

full rationale

The paper's central Einstein-Maxwell-AdS demonstration is partially circular. The thermodynamic identities S = πr+², V = 4πr+³/3, φ = Q/r+, and the Smarr form M = 2TS − 2PV + Qφ are taken from the exact solution and used as inputs to the first law. The first-law equations then fix f1 only up to C1, and the paper explicitly sets C1 = 1 to match the known RN-AdS temperature. Thus the claimed agreement with the exact solution is enforced by construction rather than derived from the method. The continued-fraction parametrization itself is a legitimate and independently motivated approximation tool, tested against the exact RN geometry, but it does not supply the missing predictive content. No load-bearing self-citation chain is present: the continued-fraction references are external, and the author's own references are not used to justify the circular step. The higher-curvature extension is asserted without a single α ≠ 0 example and without explicit formulas for the Smarr conjugate ψ_α, so that part of the claim is unsupported rather than independently verified. Overall, the key reduction fits the pattern of a fitted input called a prediction, giving a circularity score of 6.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

Everything the example imports: the RN-AdS thermodynamic identities (S, V, phi, Smarr form), the f = h metric ansatz, the truncation choice, and the field equation as displayed. None of these is derived in the paper; most are properties of the exact solution the method is supposed to construct.

free parameters (2)

C1 (integration constant in f1(r+, Lambda, Q)) = 1
General solution (37) of the first-law differential equations; set to C1 = 1 to reproduce the known RN-AdS temperature in Eq. (38). The thermodynamics alone determines the functional form of f1 only up to this constant.
Continued fraction truncation order = four terms
The expansion is truncated at order four by hand (Section 2). Accuracy is checked for one charge value in Fig. 1; no error analysis in Q, Lambda, or r.

assumptions (5)

domain assumption Static, spherically symmetric ansatz with f(r) = h(r) for the charged example (line element (27)), while the general line element (6) allows f != h.
Restricts the solution space before any field equation or thermodynamic input; in the Einstein-Maxwell case this is standard, but the generic higher-curvature framework keeps f and h independent.
domain assumption Smarr relation (19) and first law (18) hold with conjugate potentials psi_i defined by (17).
These are identities of the exact solution family; the paper imposes them as inputs to fix the near-horizon coefficients in Section 2, Eqs. (18)-(24).
domain assumption For the example: S = pi r+^2, V = 4 pi r+^3 / 3, phi = Q/r+, T = f1/(4 pi) (Eq. (35)).
These are standard thermodynamic data of the known RN-AdS solution, imported as inputs; they already encode the target result.
domain assumption A four-term continued fraction truncation accurately represents the metric from the horizon to infinity.
Assumed throughout; supported only by the single comparison in Fig. 1 for q = 0.4 r+.
ad hoc to paper The field equation (31) with stress tensor (30) is the correct Einstein-Maxwell equation for the metric (27) as displayed.
As displayed, Eq. (31) implies f1 = 1/r+ - Lambda r+ - Q^2/(2 r+^3), a factor 2 from the thermodynamic value (37)-(38); the near-horizon coefficients (33) inherit the same discrepancy against the Taylor expansion of (32)/(39).

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Pith. "Pith review of Analytically Approximate Black Hole Solution to Higher Curvature Gravity." pith.science (2026). https://pith.science/paper/7UPGKNCL

@misc{pith2026260801869,
  author       = {Pith},
  title        = {Pith review of: Analytically Approximate Black Hole Solution to Higher Curvature Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7UPGKNCL}},
  note         = {Machine review of arXiv:2608.01869}
}

read the original abstract

Higher-curvature gravity usually leads to very complicated field equations, which makes it hard to find analytical solutions. In this work, we obtain analytical charged black hole solutions in higher-curvature gravity by using black hole thermodynamics and the continued fraction expansion. We study the thermodynamics of static black holes using the first law of thermodynamics and the Smarr formula. We show that our results agree with those obtained by directly solving the field equations in Einstein gravity.

Figures

Figures reproduced from arXiv: 2608.01869 by the authors.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

H. Lu, A. Perkins, C. N. Pope and K. S. Stelle, Phys. Rev. Let t. 114, no.17, 171601 (2015)

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[2] [2]

R. C. Myers, doi:10.1007/978-94-017-0934-7 8

work page doi:10.1007/978-94-017-0934-7

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Capozziello, R

S. Capozziello, R. D’Agostino, A. Lapponi and O. Luongo, E ur. Phys. J. C 83, no.2, 175 (2023)

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Luongo and H

O. Luongo and H. Quevedo, Phys. Rev. D 90, no.8, 084032 (2014)

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work page doi:10.1142/9789814374552

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S. N. Sajadi, S. Ponglertsakul and O. Luongo, Phys. Dark Un iv. 48, 101938 (2025)

2025