{"id":"9e0954d0-4cb8-44d4-afd9-c1319e321951","arxiv_id":"2412.04513","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":2.0,"correctness_risk":"high","formal_verification":"none","parameter_count":7,"one_line_summary":"QNM frequencies and GUP-corrected Hawking temperatures of BTZ black holes in f(R) and Ricci-Inverse gravity are expressed in terms of an effective cosmological constant, but key derivations contain errors.","lead":"This paper computes oscillation frequencies and quantum radiation temperatures of BTZ black holes in two modified gravity frameworks, showing that both depend on a modified cosmological constant. The work is a 2+1-dimensional theoretical exercise that does not connect to observable astrophysics.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Massless BTZ QNM derivation in Sec. 2.1 imposes a boundary condition at positive r*, outside the physical domain, and yields real, not quasinormal, frequencies.","rationale":"The reader's verdict of REJECT is correct, but the weakest assumption I would stress is not (only) the imported Λ_m from reference [50]; it is the internal invalidity of the massless-BTZ QNM derivation. Equation (2.36) is built on a boundary condition at positive r*, while Eq. (2.22) shows r* is negative for all physical radii when Λ_m < 0. This is a self-contained error: no external input can repair it. Additionally, the formula is real, so it cannot describe damped quasinormal oscillations. The AdS3-type spectrum (2.47) is also a known normal-mode spectrum, so the paper's QNM claims reduce to relabeling normal modes. I therefore agree with the REJECT verdict, but my primary reason is the invalid QNM boundary condition rather than the unproven effective cosmological constant. The imported Λ_m remains a secondary concern: because it is taken verbatim from an unpublished self-cited preprint and all results are functions of it, an independent derivation of Eqs. (2.10) and (2.15) would also be needed for any positive claim. My proposed test is decisive for the QNM concern and should be performed first.","tokens_in":175,"tokens_out":5187,"duration_ms":62978,"concrete_test":"Solve Eq. (2.27) on the correct physical domain r* ∈ (-∞, 0) with standard QNM boundary conditions: purely ingoing as r* → -∞ (horizon) and Dirichlet at r* → 0 (AdS boundary). Compare the resulting complex frequencies with Eq. (2.36) for a fixed Λ_m (e.g., -0.1) and m. The frequencies will differ and will have nonzero imaginary parts, confirming that Eq. (2.36) is not a QNM condition.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's claim that QNM frequencies depend on modified-gravity couplings rests on Eq. (2.36) for massless BTZ. With Λ_m < 0, the tortoise coordinate (2.22), r* = -1/((-Λ_m) r), is strictly negative for all r > 0, with r* → 0^- as r → ∞ and r* → -∞ as r → 0. Equation (2.36) is obtained by requiring R(r*_bar) = 0 at an arbitrary r*_bar > 0, a point not in the physical domain. Moreover, the resulting ω_nm is purely real, so it is a normal-mode frequency with an undetermined parameter r*_bar, not a quasinormal frequency (which should have Im ω ≠ 0). Even if the imported effective cosmological constant Λ_m were correct, this invalidates the advertised QNM spectrum and the conclusion that modified-gravity parameters shift QNM decay rates. The AdS3-type result in Eq. (2.47) is likewise the standard normal-mode formula for AdS3 with Λ replaced by Λ_m, not a quasinormal spectrum. Thus the central claim of parameter-dependent QNMs is not established by the derivations presented.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":23387,"tokens_out":6881,"duration_ms":66892,"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":[{"comment":"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.","section":"Sec. 2.1, Eqs. (2.22)-(2.36)"},{"comment":"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.","section":"Sec. 3.1, Eq. (3.3)"},{"comment":"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.","section":"Sec. 2, Eqs. (2.10) and (2.15)"},{"comment":"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.","section":"Sec. 3.2, Eqs. (3.19), (3.22), and (3.24)"}],"minor_comments":[{"comment":"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.","section":"Figs. 1-6"},{"comment":"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_*).","section":"Sec. 2.2, Eq. (2.43)"},{"comment":"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.","section":"Sec. 2.3, Eq. (2.51)"},{"comment":"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.","section":"Throughout"}],"recommendation":"reject","confidential_remarks":"The central input of the paper, the effective cosmological constant, is taken from the authors' own unpublished preprint [50], and the main results are substitutions of that input into known formulas. The QNM derivation in Sec. 2.1 and the horizon-radius formula in Sec. 3.1 contain errors that cannot be repaired by local editing; the manuscript would need a substantially revised derivation and validation of the modified-gravity background before its central claims could be assessed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this paper is a substitution exercise built on an input it never derives, and its two central QNM results do not survive contact with the authors' own definitions. For the massless BTZ case, the tortoise coordinate (2.22) is strictly negative on the physical domain r>0 when Λ_m<0, with r*→0^- as r→∞. The quantization condition (2.36) imposes R(r*_bar)=0 at a positive r*_bar that is not in the domain. The resulting ω_nm is real, so it is a normal-mode formula with a free parameter, not a quasinormal frequency. The AdS3-type result (2.47) is the standard AdS3 normal-mode formula with Λ replaced by the effective Λ_m; it is not a QNM spectrum. The advertised parameter dependence of QNM frequencies is therefore not established.\n\nThere is some honest work here. The observation that both f(R) and Ricci-Inverse gravity with the chosen Lagrangians leave the BTZ metric unchanged up to a renormalized cosmological constant is correct. The effective potentials follow from the wave equations, and the geodesic section is standard and mostly correctly executed. The GUP tunneling calculation is long but mechanically follows the established Gecim-Sucu template, so it is at least reproducible.\n\nThe soft spots are load-bearing, not cosmetic. Section 3 uses the event horizon radius (3.3), which gives negative r_+^2 for the Λ_m<0 values used in the paper, making the Hawking temperatures imaginary. That is not a GUP issue; the base temperature is already unphysical. Section 2.3 states QNM frequencies without derivation. And the effective cosmological constants (2.10) and (2.15) are imported from the authors' own unpublished preprint [50]; nothing in this paper checks that identification or its domain of validity. The heavy self-citation is not the problem per se—the problem is that the cited result is doing all the work and is never verified here.\n\nWho gets value? A reader collecting examples of how one might plug modified-gravity couplings into known BTZ formulas, and maybe the geodesic solutions. But the paper's conclusions about observable QNM shifts and gravitational-wave tests are not supported. I would desk reject rather than send to referees; the Sec. 2.1 domain error and Sec. 3 sign problem are enough to sink the central claims.","headline":"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.","tokens_in":24006,"tokens_out":2860,"would_cite":false,"duration_ms":27949,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["83C57","83D05"],"pacs":["04.70.-s","04.50.Kd","04.60.-m"],"model":"deepseek-v4-flash","headline":"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…","keywords":["Modified gravity theories","BTZ space-time","Quasinormal modes","Thermodynamic properties","Geodesics equations","Effective cosmological constant","Generalized uncertainty principle","Hawking radiation"],"falsifier":"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.","tokens_in":22883,"feed_emoji":"🕳️","tokens_out":12616,"duration_ms":105652,"temperature":0.7,"pith_summary":"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 $\\Lambda_m$.","feed_headline":"One constant governs how modified gravity changes black hole ringing","feed_subtitle":"In f(R) and Ricci-Inverse gravity, black hole signals shift only through a redefined cosmological constant.","key_machinery":"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 $(1-\\alpha\\Xi)$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the effective cosmological constants $\\Lambda_m^{\\rm RI}$ and $\\Lambda_m^{f(R)}$ that the whole parameter-dependence argument assumes.","marker":"[50]"},{"why":"Provides the rotating BTZ line element whose form the paper keeps with $\\Lambda$ replaced by $\\Lambda_m$.","marker":"[48, 49]"},{"why":"Sets up the Ricci-Inverse gravity action and field equations from which $\\Lambda_m^{\\rm RI}$ follows.","marker":"[25-27]"},{"why":"Sets up f(R) gravity and its field equations from which $\\Lambda_m^{f(R)}$ follows.","marker":"[28, 29]"},{"why":"Define quasinormal modes as ringing frequencies whose real part is oscillation and imaginary part is decay.","marker":"[12, 13]"},{"why":"Provide the GUP-modified Klein-Gordon and Dirac equations used for tunneling and temperature corrections.","marker":"[59, 60]"},{"why":"Supplies the WKB ansatz that reduces the modified wave equations to Hamilton-Jacobi form.","marker":"[61]"}],"fun_headline_variants":["One constant governs modified gravity's effect on black hole ringing","Black hole signals in f(R) and Ricci-Inverse gravity shift with one effective constant","Modified gravity changes BTZ black hole ringing via a single cosmological constant","Both modified gravity theories reduce to one constant for BTZ black hole modes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["One constant governs modified gravity's effect on black hole ringing","Black hole signals in f(R) and Ricci-Inverse gravity shift with one effective constant","Modified gravity changes BTZ black hole ringing via a single cosmological constant","Both modified gravity theories reduce to one constant for BTZ black hole modes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000387,"raw_usage":{"total_tokens":2147,"prompt_tokens":1154,"completion_tokens":993,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":770,"completion_tokens_details":{"reasoning_tokens":915}},"tokens_in":770,"tokens_out":993,"duration_ms":8292,"temperature":1.0,"reasoning_tokens":915,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T22:13:24.453523+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Gecim and Y","cited_arxiv_id":null,"evidence_quote":"Supplies the WKB ansatz that reduces the modified wave equations to Hamilton-Jacobi form."}],"review_version":1}