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Quasinormal Modes and GUP-Corrected Hawking Radiation of BTZ Black Holes within Modified Gravity Frameworks

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper claims that f(R) and Ricci-Inverse gravity leave the BTZ black hole geometry unchanged and shift every observable signature—quasinormal-mode frequencies, effective potential barriers, and GUP-corrected Hawking…

desk verdict A parameter-substitution paper whose QNM spectra are not actually quasinormal: the massless case imposes boundary conditions outside the physical domain, and the horizon radius used for thermodynamics is negative for the paper's own cosmological constants. read the letter →

arxiv 2412.04513 v1 pith:5NRV2ZWG submitted 2024-12-04 gr-qc astro-ph.HEhep-th

Faizuddin Ahmed , Ahmad Al-Badawi , İzzet Sakall{i} , Abdelmalek Bouzenadad This is my paper · ORCID
classification gr-qcastro-ph.HEhep-th MSC 83C5783D05 PACS 04.70.-s04.50.Kd04.60.-m
keywords ModifiedgravitytheoriesBTZspace-timeQuasinormalmodesThermodynamicpropertiesGeodesicsequationsEffectivecosmologicalconstantGeneralizeduncertaintyprincipleHawkingradiation
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 tries to establish that two modified theories of gravity—f(R) gravity, where the Ricci scalar in the action becomes a function of itself, and Ricci-Inverse gravity, which adds anti-curvature terms—do not deform the three-dimensional BTZ black hole solution; they only replace the cosmological constant $\Lambda$ with an effective constant $\Lambda_m$ built from the coupling constants $\alpha_1, \alpha_2, \beta_1, \beta_2, \gamma$. From this single substitution the paper derives closed-form quasinormal-mode spectra, most compactly $\omega_{nm} = 2\sqrt{-\Lambda_m}(n+1+|m|/2)$ for the AdS$_3$-type case, plus effective potentials that grow with mass, $\Lambda_m$, and the couplings, and a GUP-corrected Hawking temperature $T_H(1-\alpha\Xi)$ for rotating BTZ black holes. If correct, the result matters because it turns modified-gravity effects on black hole ringing and radiation into one-parameter changes in a known spectrum, giving gravitational-wave or thermodynamic observations a concrete, testable signature that differs from general relativity only through $\Lambda_m$.

What carries the argument

The load-bearing object is the effective cosmological constant $\Lambda_m$, defined in Eqs. (2.10) and (2.15), which packages all modified-gravity couplings into the single parameter that replaces $\Lambda$ inside the standard BTZ metric. The QNM derivations use tortoise coordinates to convert the scalar wave equation into special-function form—a spherical Bessel equation for the massless case and a trigonometric-potential equation for the AdS$_3$-type case—and read off the closed-form frequencies from known eigenvalues. The GUP-corrected thermodynamics uses WKB tunneling through modified Klein-Gordon and Dirac equations to produce the temperature suppression factor $(1-\alpha\Xi)$.

What would settle it

Substitute the BTZ metric with $\Lambda_m^{\rm RI}$ or $\Lambda_m^{f(R)}$ directly into the modified field equations (2.4) and (2.13) for generic nonzero couplings and check whether the equations vanish identically; a nonzero residual would falsify the input. Alternatively, a numerical solution of the scalar wave equation on the claimed background that fails to reproduce $\omega_{nm}=2\sqrt{-\Lambda_m}(n+1+|m|/2)$ at low $n$ would falsify the QNM spectrum.

Watch

Extended reading notes

Core claim

The central claim is that the BTZ line element keeps its standard rotating form in both Ricci-Inverse and f(R) gravity, with the entire modification carried by the effective cosmological constants $\Lambda_m^{\mathrm{RI}} = \Lambda - 6\alpha_1\Lambda^2 - 108\alpha_2\Lambda^3 + \frac{5\beta_1}{4\Lambda} + \frac{15\beta_2}{8\Lambda^2} + \frac{7\gamma}{8\Lambda^2}$ and $\Lambda_m^{f(R)} = \Lambda - 6\alpha_1\Lambda^2 - 108\alpha_2\Lambda^3$. Solving the massless scalar wave equation on these backgrounds gives closed-form quasinormal-mode frequencies—$\omega_{nm} = \sqrt{m^2(-\Lambda_m) + \left(2n+\frac54\right)^2\left(\frac{\pi}{\bar r_*}\right)^2}$ for the massless case and $\omega_{nm} = 2\sqrt{-\Lambda_m}\left(n+1+\frac{|m|}{2}\right)$ for the AdS$_3$-type case—with effective potentials that grow with $m$, $\Lambda_m$, and the coupling constants. For the rotating black hole, the GUP-modified Hawking temperature is $T_H(1-\alpha\Xi)$ for both boson and fermion emission, with $T_H$ itself containing $\Lambda_m$, so the modified-gravity parameters enter the thermodynamics through the same single constant. The paper concludes that these shifts in QNM frequencies and radiation rates constitute observable deviations from general relativity that gravitational-wave observations could in principle test.

Load-bearing premise

The whole parameter dependence collapses if the effective cosmological constants claimed for Ricci-Inverse and f(R) gravity, which the paper imports from an unpublished preprint, are not exact vacuum solutions of the modified field equations for arbitrary coupling constants.

Editorial extensions

If this is right

  • For the AdS$_3$-type BTZ black hole, the quasinormal spectrum is $\omega_{nm} = 2\sqrt{-\Lambda_m}(n+1+|m|/2)$, so any coupling that changes $\Lambda_m$ shifts oscillation frequencies and the overall spectral scale.
  • The effective potentials $V_{\rm eff}(r)$ rise with the perturbation quantum number $m$, with $\Lambda_m$, and with the modified-gravity couplings, implying stronger confinement of scalar perturbations and altered decay rates.
  • The GUP-corrected Hawking temperature $T_H(1-\alpha\Xi)$ is lower than the standard value for both boson and fermion emission, so black hole radiation and evaporation slow down.
  • Setting $\alpha_1=\alpha_2=\beta_1=\beta_2=\gamma=0$ recovers the standard BTZ results of general relativity in every formula.
  • The two modified theories differ only through the extra $\beta_1,\beta_2,\gamma$ terms in $\Lambda_m^{\mathrm{RI}}$, so any test that distinguishes them must be sensitive to those combinations rather than to a changed geometry.

Reading between the lines

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

  • A single QNM frequency measurement constrains only the combination $\Lambda_m$, not the individual couplings; separating $\alpha_1,\alpha_2,\beta_1,\beta_2,\gamma$ would require combining spectra with temperature or geodesic data, an identifiability issue the paper does not address.
  • Because every observable depends on $\Lambda_m$ alone, any modified-gravity theory whose vacuum BTZ solution reduces to a constant shift of $\Lambda$ would inherit the same functional predictions, so the derivation strategy generalizes beyond the two theories studied.
  • Setting $T^{\rm GUP}=0$ in Eq. (3.22) would give a radius where evaporation stops, suggesting a black hole remnant whose mass could be computed as a function of $\alpha$—a direct extension the paper leaves implicit.
  • A numerical WKB solution of the wave equation for low $n$ could check whether the closed-form AdS$_3$ spectrum (2.47) is exact or only an asymptotic large-$n$ formula.
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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

4 major / 4 minor

Summary. The paper studies scalar perturbations and geodesics of BTZ black holes in f(R) and Ricci-Inverse modified gravity, using effective cosmological constants Λ_m taken from the authors' own unpublished preprint [50] to replace Λ in the standard BTZ metric. It claims to derive quasinormal-mode frequencies for massless, AdS3-type, and non-rotating BTZ spacetimes, GUP-corrected Hawking temperatures for rotating BTZ black holes via boson and fermion tunneling, and effective potentials for null and timelike geodesics. The central advertised conclusion is that the modified-gravity coupling constants α1, α2, β1, β2, and γ shift QNM spectra and Hawking temperatures through the renormalized cosmological constant.

Significance. If the effective-cosmological-constant identifications in Sec. 2 were correct, the paper would establish a simple structural result: f(R) and Ricci-Inverse gravity preserve the BTZ form and only renormalize Λ. The paper is clearly organized and gives explicit, falsifiable formulas, which is commendable, but it does not contain machine-checked proofs or reproducible code. The central QNM claims are undermined by a boundary condition imposed outside the physical domain, the horizon-radius formula is negative for the AdS regime used throughout, and the key input Λ_m is imported from an unpublished self-cited preprint without derivation. As presented, the claimed parameter dependence of QNMs and Hawking temperatures is therefore not established.

major comments (4)
  1. [Sec. 2.1, Eqs. (2.22)-(2.36)] For Λ_m<0, the tortoise coordinate defined in Eq. (2.22) is r_* = -1/[(-Λ_m)r], which is strictly negative for all r>0 and tends to 0 from below as r→∞. The quantization condition in Eq. (2.36) imposes R(\bar r_*)=0 at \bar r_*>0, which lies outside the physical domain, and the resulting ω_nm contains the arbitrary coordinate scale \bar r_*. Moreover, the frequency is purely real, so Eq. (2.36) is a normal-mode condition with a free parameter, not a quasinormal spectrum. The same objection applies to Eq. (2.47), which is the standard real normal-mode spectrum of AdS3 with Λ replaced by Λ_m and contains no imaginary part; the paper's claim that modified-gravity couplings shift QNM decay rates is therefore not supported by the derivations presented.
  2. [Sec. 3.1, Eq. (3.3)] The horizon-radius formula r_+^2 = (M/(2Λ_m))[1 + sqrt(1 - J^2 Λ_m/M^2)] is negative whenever Λ_m<0, since the bracket is positive while M/(2Λ_m) is negative. This is the regime used throughout the paper, with Λ=-0.1 and modified Λ_m also negative in all figures and examples. The correct equation for x=r^2 is (-Λ_m)x^2 - Mx + J^2/4 = 0, whose physical outer root for λ=-Λ_m>0 is x = [M + sqrt(M^2 - λ J^2)]/(2λ). Because Eqs. (3.25)-(3.30) and (3.46)-(3.49) are all constructed from Eq. (3.3), the reported Hawking temperatures are not reliable.
  3. [Sec. 2, Eqs. (2.10) and (2.15)] The effective cosmological constants Λ_m^RI and Λ_m^{f(R)} are imported from the authors' own unpublished preprint [50] without derivation or independent verification in this manuscript. Since every subsequent result is obtained by substituting these expressions into standard BTZ or AdS3 formulas, any approximation, sign error, or gauge dependence in that identification propagates directly into the QNM and thermodynamic claims. A referee cannot verify the central input from the material supplied; it should either be derived in this paper or replaced by a peer-reviewed derivation.
  4. [Sec. 3.2, Eqs. (3.19), (3.22), and (3.24)] The GUP correction factor is internally inconsistent. Eq. (3.19) defines §1 = 3µ_0^2 r_+^2 + j^2/(2r_+^2), while Eq. (3.22) approximates 1/(1+α§1) ≈ 1-α§1; together these imply a correction proportional to -α[3µ_0^2 r_+^2 + j^2/(2r_+^2)]. The later boson temperature in Eq. (3.24), however, is printed with a correction of the form -3α(j^2+µ_0^2 r_+^2)/(4r_+^2), which does not match the coefficients obtained from Eqs. (3.19) and (3.22). Since the quantitative GUP temperature shift is a central advertised result, this discrepancy must be fixed.
minor comments (4)
  1. [Figs. 1-6] The effective-potential plots show the horizontal axis extending to negative r, although r is the radial coordinate and is positive by definition; the plots should be restricted to r>0 or the axis should be relabeled.
  2. [Sec. 2.2, Eq. (2.43)] Eq. (2.43) contains the typo cos^2(κ r) in the second denominator; from the coordinate transformation in Eq. (2.40) the argument should be cos^2(κ r_*).
  3. [Sec. 2.3, Eq. (2.51)] The condition labeled |M|<0 is impossible, and the arctangent expression for r_* contains sqrt(Λ_m/M) with Λ_m<0 and M>0, which is imaginary; the massive non-rotating case needs a correct tortoise-coordinate treatment.
  4. [Throughout] Several cross-references to figures are wrong, e.g., in Sec. 2.2 the text refers to 'panel (a) of Figure 2' when discussing Figure 3, and in Sec. 2.3 similar mislabeling occurs for Figures 5 and 6; these should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the modified-gravity dependence enters through the parameter-free effective cosmological constant Λ_m, and the QNM, Hawking-temperature, and geodesic results are derived consequences rather than restatements of that input.

full rationale

The derivation chain is not circular by the paper's own equations. The modified-gravity input is the effective cosmological constant Λ_m, quoted from the authors' prior work [50] and stated as obtained by solving the field equations (2.4) and (2.13) with T_μν = 0. Given Λ_m, the QNM frequencies in Secs. 2.1–2.2 follow from the Klein-Gordon equation on the BTZ-type background, the effective potentials are computed from the wave equations, and the Hawking temperatures in Sec. 3 follow from standard WKB/tunneling calculations with GUP corrections. The dependence of ω_nm and T_H on α_i, β_i, and γ is a direct functional consequence of Λ_m, not an assumption of those frequencies or temperatures. The self-citation [50] is load-bearing, but under the stated rules it counts as independent support: it is a parameter-free vacuum solution with stated assumptions that do not include the target QNM or temperature results. The main substantive weaknesses are physical-correctness issues, not circularity: the massless-sector derivation imposes a boundary condition at positive r* outside the physical domain, yields real frequencies with an arbitrary r*_bar rather than a quasinormal spectrum, and the AdS3-type result is the standard AdS3 formula with Λ replaced by Λ_m. These are correctness risks and should be evaluated separately.

Assumptions & free parameters 7 free parameters · 4 assumptions · 0 invented entities

The paper's quantitative outputs are determined by the effective cosmological constant Lambda_m (Eqs. 2.10, 2.15), which is a function of the hand-chosen coupling constants alpha, beta, gamma imported from the authors' prior preprint [50], plus the GUP parameter alpha. The massless-QNM frequency also depends on the ad hoc node position bar r_*. No parameter is fitted to data; the 'predictions' are restatements of the input parameterization.

free parameters (7)
  • alpha_1 = 0.1 to 0.8 in figures
    Coupling constant of the R^2 term in f(R) and RI actions; chosen by hand in the plots.
  • alpha_2 = 0.001 to 0.004 in figures
    Coupling constant of the R^3 term; chosen by hand.
  • beta_1 = 0.1 to 0.8 in figures (RI only)
    Coupling constant of the A term; chosen by hand.
  • beta_2 = 0.001 to 0.004 in figures (RI only)
    Coupling constant of the A^2 term; chosen by hand.
  • gamma = 0.01 to 0.04 in figures (RI only)
    Coupling constant of the A_mu_nu A^mu_nu term; chosen by hand.
  • GUP_alpha = not specified
    Minimal-length parameter in GUP-modified field equations; no numerical value assigned, but it controls the Hawking-temperature correction.
  • bar_r_star = arbitrary
    Node position in the massless BTZ QNM quantization, Eq. (2.36); the frequency depends on this unphysical free input.
assumptions (4)
  • domain assumption The BTZ metric (2.1)-(2.2) solves the vacuum field equations of f(R) and RI gravity with the effective cosmological constants Lambda_m given in Eqs. (2.10) and (2.15).
    Taken from the authors' own preprint [50]; no derivation is shown in this paper, and the entire subsequent analysis depends on this identification.
  • domain assumption The GUP-modified Klein-Gordon and Dirac equations (3.10) and (3.31) describe scalar and fermion tunneling from the BTZ black hole.
    The GUP equations are quoted from the literature (Refs. [59], [45]) and not derived; the WKB and tunneling-probability interpretation is standard.
  • standard math For the AdS3-type case, the quantization condition leading to Eq. (2.47) assumes the scalar field is regular at the origin and vanishes at the infinite boundary (normal-mode boundary conditions in global AdS3).
    Standard boundary conditions for the Pöschl-Teller-type potential; no black hole horizon exists for M=-1.
  • ad hoc to paper In Sec. 2.1, the radial function is required to vanish at an arbitrary coordinate bar r_* (Eq. 2.36).
    The condition R(bar r_*)=0 with bar r_*>0 is not a physical boundary condition; the tortoise coordinate r_* = -1/((-Lambda_m)r) is non-positive, so bar r_* is chosen by hand and makes the frequency arbitrary.

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Pith. "Pith review of Quasinormal Modes and GUP-Corrected Hawking Radiation of BTZ Black Holes within Modified Gravity Frameworks." pith.science (2026). https://pith.science/paper/5NRV2ZWG

@misc{pith2026241204513,
  author       = {Pith},
  title        = {Pith review of: Quasinormal Modes and GUP-Corrected Hawking Radiation of BTZ Black Holes within Modified Gravity Frameworks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5NRV2ZWG}},
  note         = {Machine review of arXiv:2412.04513}
}
abstract

This paper aims to explore the quasinormal modes (QNMs) and effective potential profiles of massless and rotating BTZ black holes within the frameworks of $f(\mathcal{R})$ and Ricci-Inverse ($\mathcal{RI}$) modified gravity theories, which, while producing similar space-time structures, exhibit variations due to distinct cosmological constants, $\Lambda_m$. We derive wave equations for these black hole perturbations and analyze the behavior of the effective potential $V_{\text{eff}}(r)$ under different values of mass $m$, cosmological constant $\Lambda_m$, and modified gravity parameters $\alpha_1$, $\alpha_2$, $\beta_1$, $\beta_2$, and $\gamma$. The findings indicate that increasing mass and parameter values results in a raised potential barrier, implying stronger confinement of perturbations and impacting black hole stability. Incorporating the generalized uncertainty principle, we also study its effect on the thermodynamics of rotating BTZ black holes, demonstrating how GUP modifies black hole radiation, potentially observable in QNM decay rates. Additionally, we investigate the motion of particles through null and timelike geodesics in static BTZ space-time, observing asymptotic behaviors for null geodesics and parameter-dependent shifts in potential for timelike paths. The study concludes that modified gravity parameters significantly influence QNM frequencies and effective potential profiles, offering insights into black hole stability and suggesting that these theoretical predictions may be tested through gravitational wave observations.

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