Scalar, Vector Perturbations and Effective Hawking Radiation of Cylindrical Black Holes in $f(\mathcal{R})$ and Ricci-Inverse Gravity

Abdelmalek Bouzenada; Ahmad Al-Badawi; Faizuddin Ahmed; \.Izzet Sakall{\i}

REVIEW 4 major objections 3 minor 110 references

Scalar, Vector Perturbations and Effective Hawking Radiation of Cylindrical Black Holes in $f(\mathcal{R})$ and Ricci-Inverse Gravity

T0 review · 4 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read For a cylindrical black hole, modified gravity changes the effective cosmological constant, and all perturbation and thermodynamic results follow from that replacement.

desk verdict The perturbation potentials are standard cylindrical black hole results with Λ replaced by Λ_m from the authors' own preprint, and the Hawking-temperature derivation has a square-root error that propagates into all thermodynamic formulas. read the letter →

arxiv 2501.00060 v1 pith:E6HWOYJ3 submitted 2024-12-29 gr-qc astro-ph.HEhep-thphysics.space-phquant-ph

Faizuddin Ahmed , İzzet Sakall{i} , Ahmad Al-Badawi , Abdelmalek Bouzenada This is my paper · ORCID

classification gr-qcastro-ph.HEhep-thphysics.space-phquant-ph PACS 04.70.Dy04.50.Kd

keywords cylindricalblackholesf(R)gravityRicci-inverseeffectivecosmologicalconstantquasinormalmodesHawkingradiationgeneralizeduncertaintyprincipleholeremnants

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 claims that, for a cylindrical black hole, switching from general relativity to f(R) or Ricci-inverse gravity changes only one input: the cosmological constant Λ is replaced by an effective constant Λ_m built from the coupling constants of the modified action. Starting from that replacement, the authors derive closed-form effective potentials for scalar and vector perturbations, identify the quasinormal-mode equation they feed into, and then compute GUP-corrected Hawking temperatures and entropies. The payoff is a concrete dictionary: each coupling constant of the modified theory controls how much the potential, the evaporation temperature, and the entropy deviate from their general-relativity values. In the Ricci-inverse model the temperature stays finite as Λ→0, which the authors read as a sign of black-hole remnant formation. If the formulas are right, they give a direct way to test these modified gravities through ringdown or radiation signatures.

What carries the argument

The load-bearing object is the effective cosmological constant Λ_m, obtained in an earlier paper by the same authors [77]: $Λ_m^{{f(R)}}$=Λ+α_k(1−k)4^$kΛ^{{k+1}}$ for the polynomial f(R), and $Λ_m^{{RI}}$=Λ−16Λ³α2+3β1/Λ+16β2/Λ²+4γ/Λ² for the Ricci-inverse model. Every result in the paper is this constant inserted into the GR cylindrical black hole metric and into the perturbation and thermodynamic equations; all coupling constants of the modified actions enter only through Λ_m. The second essential piece is the GUP-deformed Klein-Gordon equation, which modifies the radial momentum by the factor (1−2α_GUP(...)) and ultimately multiplies the Hawking temperature and entropy by √(1−2m_p²α_GUP). The mechanism that carries the argument is therefore substitution: solve the modified field equations once to get Λ_m, then recycle the standard GR computation with Λ_m in place of Λ.

What would settle it

Take the cylindrical metric (2.1) with f(r)=α²r²−4M/(α r), plug it together with f(R)=R+α1R²+...+$α4R^{5}$ into the f(R) field equations (2.5) with zero stress-energy, and solve for the constant that makes the equations hold; if the resulting effective cosmological constant does not match Eq. (2.8), the paper's central substitution claim collapses.

Watch

Extended reading notes

Core claim

The paper's central claim is that the cylindrical black hole solutions of f(R)-gravity and Ricci-inverse gravity are the same GR metric with Λ replaced by $Λ_m^{{f(R)}}$ or $Λ_m^{{RI}}$, and that every derived quantity follows from that single substitution. For f(R) with f=R+Σ α_i $R^{{i+1}}$, the effective constant is Λ + α_k(1−k)4^k $Λ^{{k+1}}$; for the Ricci-inverse Class III model f=R+α1R²+α2R³+β1A+β2A²+γA_{μν}$A^{{μν}}$, it is Λ−16Λ³α2+3β1/Λ+16β2/Λ²+4γ/Λ². Writing the metric with Λ_m, the authors reduce the Klein-Gordon equation for a massless scalar to a Schrödinger-type equation with effective potential V=(ι²/r²+f′/r)f, ι²=m²−3k²/Λ_m, and the Maxwell equation to a similar potential V_e=f ι²/r²; these potentials, with the modified constants, are the paper's predictions for scalar and electromagnetic quasinormal modes. For thermodynamics, solving the GUP-modified Klein-Gordon equation near the horizon gives a tunneling rate whose Boltzmann comparison yields T_GUP=T_H√(1−2m_p²α_GUP), with T_H=(−3Λ_m)^{1/4}(4M)^{1/6}/(4π), and the first law then gives S_GUP=(πz/4)√(1−2m_p²α_GUP) r_h². The coupling constants of both modified actions therefore control the deviations from GR in all these quantities.

Load-bearing premise

The argument stands on the effective cosmological constants quoted from reference [77]; if those constants are not the correct solutions of the f(R) and Ricci-inverse field equations for this metric, then every potential, temperature, and entropy in the paper is wrong.

Editorial extensions

If this is right

For both modified theories, the scalar and vector perturbation potentials reduce exactly to the GR form when the coupling constants vanish, so any observational difference from GR ringdown is controlled by the strengths of the higher-curvature and anti-curvature terms.
The GUP correction factor √(1−2m_p²α_GUP) lowers the Hawking temperature and entropy for every model, slowing evaporation, and the same factor appears in the f(R) and RI cases because the derivation is metric-independent once Λ_m is substituted.
In the Ricci-inverse model the Hawking temperature remains nonzero as Λ→0, implying that evaporation stops at a remnant, whereas the f(R) model in the GR limit instead shows complete evaporation as Λ→0.
Because the effective potentials V and V_e are explicit functions of the coupling constants, quasinormal frequencies computed from them would shift with α2, α3, α4 (f(R)) and α2, β1, β2, γ (RI), providing a parameter-dependent ringdown signature.

Reading between the lines

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

Since the paper's derivation stops at the potentials, a direct numerical computation of the quasinormal frequencies would be the obvious next step; if the potentials take negative values in the plotted ranges, the stability conclusion may not follow without checking the sign of the imaginary part.
The same Λ_m substitution should apply to other observables of this metric family, such as photon orbits, shadows, and geodesic precession, so the two modified theories predict a full parametric family of deviations from GR that could be tested with horizon-scale imaging.
The temperature formula T_GUP=T_H√(1−2m_p²α_GUP) is real only for 2m_p²α_GUP≤1; the paper does not discuss this bound, but it implies a maximum particle mass or GUP parameter beyond which the semiclassical temperature is undefined.
The remnant behavior in RI-gravity could be sharpened by computing the heat capacity: a change of sign in C=dM/dT near the minimum temperature would tell whether the remnant is thermodynamically stable.

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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 honest headline: this is a substitution exercise with a broken thermodynamics section. The scalar and vector potentials are the standard cylindrical black hole potentials with Λ swapped for Λ_m, imported from ref. [77] by two of the present authors. No QNM frequencies or damping rates are computed, despite the abstract and conclusion advertising a QNM analysis.

What the paper does well is modest but real. The scalar perturbation derivation in Sec. 3 is algebraically consistent, and the potential (3.9) plus its f(R)/RI special cases reduce properly to the GR limit when the coupling constants vanish. The plots comparing potentials for different coupling strengths are clear and might be useful to someone working on these specific models. The paper also honestly states that the effective cosmological constants come from ref. [77], which is a preprint by two of the authors; that is not by itself a flaw, but it is an unchecked input on which every modified-gravity result depends.

The soft spots are serious. First, the advertised QNM analysis is absent: the paper stops at effective potentials and never computes ω, so the stability claims are unsupported. Second, the GUP tunneling derivation in Sec. 5 has a load-bearing error. The Klein-Gordon/WKB equation should use g^{rr}(W')^2, not g_{rr}(W')^2; the paper uses the wrong component, which leads to Eq. (5.23) asserting T_H = sqrt(Δ'(r_h))/(4π). The correct surface-gravity relation is T_H = Δ'(r_h)/(4π). This square-root mistake propagates into Eqs. (5.24)-(5.28) and into the GUP temperatures. The entropy section is internally inconsistent on its own terms: Eq. (5.33) sets A = 2π z r_h, but Eq. (5.34) integrates 2π z r_h dr_h, yielding S ∝ r_h^2 instead of S ∝ r_h. These are not minor typos; they invalidate the central thermodynamic claims.

The stress-test note holds up on reading. I would not send this paper to a referee: the advertised QNM result is missing and the thermodynamics needs to be redone. The effective-potential part could become a short, useful note if the authors compute actual QNM frequencies and redo the GUP calculation properly, but as submitted it is not a reliable source for modified black hole thermodynamics.

Referee Report

4 major / 3 minor

Summary. The paper studies scalar and vector perturbations of Lemos-type cylindrical black holes embedded in f(R) and Ricci-Inverse gravity, and derives GUP-corrected Hawking temperatures and entropies via a tunneling calculation. The modified-gravity dependence enters only through effective cosmological constants Lambda_m^{f(R)} and Lambda_m^{RI} imported from the authors' earlier preprint [77]. The main results are the effective potentials (3.9) and (4.5) and the thermodynamic formulas (5.25)-(5.26) and (5.38)-(5.39), all obtained by substituting Lambda -> Lambda_m into known expressions for the Lemos metric.

Significance. If correct, the paper would provide explicit modified-gravity corrections to the perturbation potentials and to the Hawking radiation of cylindrical black holes, with the f(R) and Ricci-Inverse coupling constants controlling the deviations. The manuscript is transparent about the substitution structure and presents clear plots of the potentials and thermodynamic quantities. However, the advertised quasinormal-mode analysis is not actually performed, and the Hawking-temperature and entropy derivations contain internal, load-bearing errors. Because the central formulas of Section 5 are wrong as written, the contribution in its present form does not constitute a reliable quantitative result.

major comments (4)

[Sec. 5, Eqs. (5.33)-(5.39)] The tunneling calculation uses the wrong metric components. For the metric (2.22), the Hamilton-Jacobi equation from the Klein-Gordon equation is g^{tt}E^2 + g^{rr}(W')^2 + ... = 0, i.e. -E^2/f + f(W')^2 + ... = 0. Equation (5.4) instead writes 1/g_{tt}E^2 = g_{rr}(W')^2 + ..., which effectively uses g_{rr} in place of g^{rr}. This leads to W' ~ 1/sqrt(f) near the horizon rather than W' ~ 1/f, and consequently Eq. (5.23) gives T_H = sqrt(Delta'(r_h))/(4 pi) instead of the standard T_H = Delta'(r_h)/(4 pi) following from kappa = Delta'/2. The error propagates into the explicit temperatures (5.24)-(5.28) and into the entropy formulas (5.38)-(5.39).
[Sec. 5, Eqs. (5.33)-(5.39)] The entropy derivation is internally inconsistent. Equation (5.33) sets A = 2 pi z r_h, which implies dA = 2 pi z dr_h, but Eq. (5.34) integrates 2 pi z r_h dr_h and yields S proportional to r_h^2. Moreover, for the metric (2.22) with g_phi phi = r^2 and g_zz = alpha^2 r^2, the horizon area for an axial length z is A = 2 pi alpha z r_h^2, not 2 pi z r_h. Thus both the area expression and its integration in Eqs. (5.34)-(5.36) are wrong, and the final GUP entropies (5.38)-(5.39) do not follow from the stated first-law integration.
[Secs. 3 and 4] The abstract and section titles promise an analysis of quasinormal modes, but no quasinormal frequencies or damping times are computed anywhere. Sections 3 and 4 reduce the perturbation equations to the Schrodinger-type form (3.8) and (4.3) and plot the effective potentials, but they never impose QNM boundary conditions, solve the eigenvalue problem, or list any omega values. The claim that QNMs are analyzed is therefore unsupported by the content of the paper.
[Sec. 2, Eqs. (2.8), (2.12), (2.21)] Every modified-gravity result in the paper depends on the effective cosmological constants Lambda_m^{f(R)} and Lambda_m^{RI} taken from ref. [77], a preprint by two of the present authors. The manuscript does not re-derive these constants or check them against an independent calculation, so the potentials, temperatures, and entropies inherit any error in [77]. Since this dependence is the sole channel through which the modified-gravity parameters enter the final formulas, the authors should either verify these constants within the present manuscript or state explicitly that the results are conditional on the correctness of [77].

minor comments (3)

[Eq. (3.4)] The displayed determinant is sqrt(-g) = r sqrt(-Lambda_m/3), but for the metric (3.3) the correct value is sqrt(-g) = r^2 sqrt(-Lambda_m/3). The subsequent radial equation (3.5) appears consistent with the r^2 factor, so this is likely a typographical error that should be corrected.
[Eqs. (2.13), (2.22), (2.23), (3.3)] The metric functions are typeset ambiguously, e.g. expressions like '4 M q - Lambda/3 r' should clearly indicate whether the square-root factor is in the numerator or denominator. Please use explicit notation such as 4M/sqrt(-Lambda/3) r or 4M sqrt(-3/Lambda)/r.
[Sec. 5] There are several typographical and referencing issues: 'Feyman' should be 'Feynman'; the factor-of-two discussion around Eqs. (5.17)-(5.20) is described but the cited method in refs. [103,104] is not applied explicitly; and Eq. (5.10) contains the square root of a negative quantity without specifying that it is sqrt(-3/Lambda) for Lambda < 0.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central derivations are standard substitutions and WKB calculations; reliance on the self-cited effective cosmological constants is a verification gap, not a tautology.

full rationale

We find no step in which a claimed prediction is equivalent to an input by construction. The scalar and vector effective potentials, Eqs. (3.9) and (4.5), are obtained by the standard separation of the Klein-Gordon and Maxwell equations in the line element (3.3), followed by the tortoise-coordinate transformation; the GUP-corrected temperatures, Eqs. (5.22)-(5.26), are solved from the GUP-modified Klein-Gordon equation (5.1)-(5.5), not read off from the input. The only modified-gravity input is the effective cosmological constant Lambda_m taken from ref. [77], a preprint by two of the present authors. That reliance is load-bearing, but it is not circular: the paper cites [77] as a prior derivation with stated assumptions (the Lagrangians (2.6) and (2.19) and zero energy-momentum tensor), and the subsequent algebra does not presuppose the potentials, temperatures, or entropies that it derives. The GUP correction factor sqrt(1 - 2 m_p^2 alpha_GUP) is obtained by solving the WKB equations, not assumed in the final form. We do note internal algebraic inconsistencies, e.g. Eq. (5.23) writes T_H = sqrt(Delta'(r_h))/(4 pi) even though the paper's own Delta'(r_h) = 2 kappa implies T_H = Delta'(r_h)/(4 pi), and Eqs. (5.33)-(5.36) set A = 2 pi z r_h but integrate 2 pi z r_h dr_h to obtain S proportional to r_h^2; these are correctness errors, not circularity. The lack of an independent check of the self-cited [77] result is a reproducibility concern, but it does not make the derivation circular.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The paper adds no new entities. Its load-bearing input is the effective cosmological constant imported from the authors' own earlier preprint, plus the free coupling constants of the two modified gravity actions and the GUP model. The entropy derivation further relies on an inconsistent horizon-area assumption.

free parameters (3)

f(R) coupling constants α2, α3, α4 = varied in figures, e.g., α2=0.1 to 0.4
Free constants in the action (2.6); all modified results are parameterized by them. Not fitted to data, but chosen by hand in the plots.
RI coupling constants α2, β1, β2, γ = varied in figures, e.g., β1=0.5 to 1.0
Free constants in the action (2.19); not fitted to data. The effective cosmological constant and all downstream quantities depend on them.
GUP parameter α_GUP and scalar particle mass m_p = α_GUP = 0 or 10, m_p = 0.1 in figures
Free parameters of the GUP model; the correction factor sqrt(1−2m_p²α_GUP) depends on them and is central to the claimed temperature and entropy corrections.

assumptions (5)

domain assumption The Lemos cylindrical vacuum solution (2.1) satisfies R_μν = Λ g_μν with Λ = −3α².
Used as the seed metric for both modified gravity embeddings; entered in Sec. 2.
ad hoc to paper Effective cosmological constants Λ_m^{f(R)} and Λ_m^{RI} from ref. [77] are correct.
Imported from a self-cited preprint and not re-derived here; all subsequent results depend on them. This is the load-bearing input from outside the present paper.
domain assumption The GUP-modified Klein-Gordon equation (5.1) from ref. [100] correctly describes quantum gravitational corrections.
Adopted without modification; the Hawking temperature result follows from it.
domain assumption Horizon area is A = 2π z r_h, Eq. (5.33).
Used to derive entropy; inconsistent with g_zz = −Λ/3 r² = α²r², so this assumption is internally contradicted by the metric.
domain assumption Tunneling probability normalization P_− = 1 and the factor-2 correction (Sec. 5) are valid.
Standard in the GUP tunneling literature; adopted to obtain the quoted temperature.

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Pith. "Pith review of Scalar, Vector Perturbations and Effective Hawking Radiation of Cylindrical Black Holes in $f(\mathcal{R})$ and Ricci-Inverse Gravity." pith.science (2026). https://pith.science/paper/E6HWOYJ3

@misc{pith2026250100060,
  author       = {Pith},
  title        = {Pith review of: Scalar, Vector Perturbations and Effective Hawking Radiation of Cylindrical Black Holes in $f(\mathcalR)$ and Ricci-Inverse Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E6HWOYJ3}},
  note         = {Machine review of arXiv:2501.00060}
}

abstract

This paper investigates scalar perturbations and quasinormal modes (QNMs) associated with cylindrical black holes constructed within the frameworks of $f(\mathcal{R})$-gravity and Ricci-Inverse ($\mathcal{RI}$) gravity. Moreover, we study the modified Hawking radiation in these black hole solutions and analyze the effects of coupling constants. These modified theories, which extend general relativity by introducing higher-order curvature corrections and additional geometric terms, provide a rich platform for exploring deviations from standard gravitational physics. The study begins by revisiting the cylindrical black holes in these modified gravity theories, where the effective cosmological constants respectively, are represented by $\Lambda_m^{f(\mathcal{R})}$ and $\Lambda_m^{\mathcal{RI}}$ related to the coupling constants unique to each framework. Afterwards, the QNMs, intrinsic damped oscillations of the black hole space-time, are analyzed to probe the stability of the system, with the effective potential $V$ revealing the impact of the modified gravity parameters. Additionally, the thermodynamic properties of the black holes are examined through the lens of the Generalized Uncertainty Principle (GUP), which introduces quantum corrections to Hawking radiation. The GUP-modified Hawking temperature and entropy are derived, demonstrating significant deviations from classical results and highlighting the quantum gravitational effects in these modified frameworks. By linking QNMs, thermodynamics, and quantum corrections, this work not only deepens the understanding of modified gravity theories but also offers potential observational pathways to test their validity.

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