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REVIEW 5 major objections 5 minor 54 references

Quantum Black Holes: Perihelion Advance, Quasi Normal Modes and Classical/ Topological Thermodynamics

T0 review · 5 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Quantum black hole modes decay; topology matches Reissner-Nordström

desk verdict Routine but honest extension of QNM/thermo methods to a known LQG-inspired metric; the topological thermodynamics numbers are internally inconsistent and need recomputation before the W=0 claim can be trusted. read the letter →

arxiv 2507.22945 v1 pith:LZNOXUO4 submitted 2025-07-26 gr-qc

Grigoris Panotopoulos , Francisco Tello-Ortiz This is my paper
classification gr-qc
keywords quantumblackholequasinormalmodesWKBapproximationperihelionadvancethermodynamicsthermodynamictopologyloopgravityReissner-Nordström
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 studies a Schwarzschild-like black hole from loop quantum cosmology, whose only free parameter $\alpha$ adds a short-range $M^2/r^4$ term to the Newtonian potential. Mercury's perihelion advance puts an upper bound on $\alpha$, and the paper then fixes $\alpha=0.02M^2$ and computes quasinormal modes for massless and massive scalar fields and for electromagnetic perturbations with the WKB method. Every mode in the tables has a positive real part and a negative imaginary part, so the black hole is stable under those perturbations, and both parts grow with the test-field mass. The paper also regularizes the first law by defining the energy through $dE=W\,dM$ with $W=\sqrt{r_+^2-\alpha}/r_+$ while keeping $S=\pi r_+^2$, and shows that the classical Gibbs free energy and the topological $\phi$-mapping both put this black hole in the same thermodynamic class as Reissner-Nordström: stable small and unstable large branches, one generating point, and total topological charge $W=0$.

What carries the argument

The argument runs on the one-parameter metric $f(r)=1-2M/r+\alpha M^2/r^4$ with $\alpha$ the only free parameter and the physical horizon branch $r_+\ge\sqrt{\alpha}$. Perturbations are reduced to Schrödinger-like equations in the tortoise coordinate, with effective potentials $V_{sc}=f\,[m^2+\ell(\ell+1)/r+f'/r]$ and $V_{EM}=f\,\ell(\ell+1)/r^2$, and the WKB method supplies the complex frequencies. Thermodynamics is carried by the regularized first law $dE=W\,dM$ with $W=\sqrt{r_+^2-\alpha}/r_+$, giving $E(r_+)=[2\alpha r_+-r_+^3+(r_+^2-\alpha)^{3/2}]/\alpha$, and by the topological $\phi$-mapping construction, in which the off-shell free energy $F=E-S/\tau$ defines a vector field whose zero-point winding numbers sum to $W=0$.

What would settle it

A direct time-domain integration of the scalar and electromagnetic perturbation equations at $\alpha=0.02M^2$ that produced any mode with $\mathrm{Im}\,\omega>0$ would overturn the mechanical-stability claim, as would a first-law derivation from an action, or from the original $dM\neq T\,dS$ form, that gave a total topological charge different from $W=0$.

Watch

Extended reading notes

Core claim

The central claim, stated on the paper's own terms, is that the quantum-corrected metric $f(r)=1-2M/r+\alpha M^2/r^4$ is observationally viable and, at the representative coupling $\alpha=0.02M^2$, mechanically stable: the WKB quasinormal frequencies for massless scalar, massive scalar ($m=0.1M$ and $0.2M$), and electromagnetic perturbations all satisfy $\mathrm{Re}\,\omega>0$ and $\mathrm{Im}\,\omega<0$ over the computed $\ell$ and $n$ range. The perihelion analysis gives a quantum correction $\Delta\theta_p^{(QG)}=-3\alpha\pi M(4+e^2)/2L^3$, about ten orders of magnitude below the Einstein term at Mercury's orbit, so the observationally allowed $\alpha$ can exceed the loop-quantum-gravity motivated value $\alpha\approx1.1663$. After the first law is regularized so that the Hawking temperature equals the first-law temperature, the thermodynamic analysis yields a Gibbs free energy with one stable small-black-hole branch and one unstable large-black-hole branch, and the topological $\phi$-mapping produces a generating point with winding numbers $+1$ and $-1$ summing to $W=0$, the same pattern as Reissner-Nordström. The paper's conclusion is that the quantum correction leaves the Reissner-Nordström thermodynamic class intact while mechanical stability holds at the chosen coupling.

Load-bearing premise

The thermodynamic conclusion rests on the paper's imposed regularization of the first law—taking $S=\pi r_+^2$ and defining the energy by $dE=W\,dM$ with $W=\sqrt{r_+^2-\alpha}/r_+$—so the stable/unstable branches and $W=0$ follow only if that prescription is accepted.

Editorial extensions

If this is right

  • For $\alpha=0.02M^2$, the black hole is stable against massless scalar, massive scalar ($m=0.1M$ and $0.2M$), and electromagnetic perturbations across the tabulated $\ell$ and $n$ values.
  • The quantum coupling is compatible with Mercury's perihelion precession, with a correction to the Newtonian potential of order $10^{-10}$ at Mercury's orbit, so solar-system tests do not rule out this quantum-corrected black hole.
  • Thermodynamically the quantum black hole falls into the same local and global topological class as Reissner-Nordström, with one stable and one unstable branch, one generating point, and total charge $W=0$.
  • Increasing the scalar-field mass increases both the oscillation frequency and the damping rate of the modes, a trend that could be compared with ringdown data if such fields are present.

Reading between the lines

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

  • Beyond the paper: applying the same topological machinery to the unregularized first law, with $M$ still treated as the energy and $dM\neq T\,dS$, would not necessarily return $W=0$, so the Reissner-Nordström-like thermodynamics is best read as a property of the regularized framework rather than of the bare metric.
  • Beyond the paper: the perihelion bound admits values of $\alpha$ many orders of magnitude larger than the $\alpha=0.02M^2$ used here; recomputing the quasinormal modes near the observational bound would test whether stability persists across the allowed parameter space.
  • Beyond the paper: the WKB tables cover one coupling value and low overtones; a time-domain or higher-order WKB survey over the allowed $\alpha$ range would show whether the stability finding is generic rather than specific to the chosen point.
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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

5 major / 5 minor

Summary. The paper studies the loop-quantum-gravity-inspired Schwarzschild-like metric f(r)=1-2M/r+αM^2/r^4 of Lewandowski et al. It aims (i) to constrain the coupling α using Mercury's perihelion advance, (ii) to compute quasinormal modes for massless and massive scalar fields and electromagnetic perturbations using the WKB method, and (iii) to analyze black hole thermodynamics both classically and via Duan's topological method. The authors report that all computed modes have positive real parts and negative imaginary parts, indicating stability, and that the thermodynamics shows stable/unstable branches, one generating point, and total topological charge W=0, so the quantum black hole behaves like Reissner-Nordström.

Significance. If the results are correct, the paper provides a useful stability analysis and thermodynamic classification for a specific quantum-corrected black hole metric. Its strengths include the use of standard, easily reproducible perturbation equations; extensive tables of QNM frequencies; and a clear statement of the regularization needed to obtain a consistent first law. The paper is transparent that the thermodynamic framework is constructed, not derived from a Lagrangian, and it correctly discards the unphysical upper branch of the horizon mass. However, the perihelion constraint is so weak that it does not actually constrain α, the QNM results lack numerical validation and order estimates, and the topological thermodynamic section contains a concrete internal inconsistency in the quoted zero points and critical temperature.

major comments (5)
  1. [V.B, Eq. (53), Figs. 3-5] The quoted zero points and critical point in the topological thermodynamics do not satisfy Eq. (53). For α=0.02, Eq. (53) gives τ(0.5672)≈7.4 and τ(1.9049)≈26, so these radii cannot both be on-shell points for τ=60 as stated in the text and Fig. 4. The claimed generating point (r+=0.7870, τ_c=13.95) is not a stationary point of τ(r): evaluating Eq. (53) gives τ(0.787)≈9.9 and dτ/dr≠0. The true minimum of τ(r) for α=0.02 occurs near r≈0.27 with τ_c≈4.3. Consequently, the winding numbers reported in Figs. 4 and 5 were computed at locations that are not the true on-shell solutions for the stated τ. The qualitative conclusion W=0 may survive because τ(r) still has a single minimum and two branches, but all quantitative results—critical temperature, generating point, zero-point positions, and the associated contour/winding-number assignments—must be recomputed and the figure captions corrected (Fig. 5 states τ=25 while the text refers to τ=60).
  2. [III, Sec. VI, Abstract] The perihelion bound does not meaningfully constrain the coupling α. Eq. (10) and Fig. 1 show that the observational uncertainty allows α up to ≈10^87 l_pl^2, which includes the theoretical value α≈1.17 l_pl^2 and essentially any plausible value. The paper itself states that the theoretical value falls within the bound, and then selects α=0.02 'to see how this small correction is affecting' the background. Thus the abstract's claim that the coupling is constrained by perihelion advance is overstated, and the QNM and thermodynamic results are obtained for an unmotivated, ad hoc parameter value. The authors should explicitly state that α=0.02 is a toy value and that the perihelion test provides no practical restriction.
  3. [IV, Tables I-IV] The WKB computations lack the numerical details needed for independent verification. The paper cites the WKB method [49,50] but does not specify the WKB order used (e.g., 3rd, 6th, or 13th), provides no error estimate from comparing successive WKB orders, and does not validate the code against the known Schwarzschild limit α→0. For a stability claim based on tables of complex frequencies, such checks are essential; without them the reader cannot assess the reliability of the quoted values. The authors should add this information or provide a benchmark table for the Schwarzschild case.
  4. [V.A, Eqs. (30)-(37)] The thermodynamic results are largely a consequence of the imposed regularization rather than an independent prediction of the metric. By defining the energy through dE=W dM with W=√(r+²-α)/r+ and fixing the area entropy S=πr+², the first-law temperature is forced to match the Hawking temperature by construction. The subsequent stable/unstable branches and total topological charge W=0 therefore depend on this choice of energy and entropy prescriptions; a different first-law formulation would not automatically give the same result. The authors acknowledge the issue, but the physical significance of the claimed Reissner-Nordström-like behavior is accordingly limited and should be framed more cautiously.
  5. [V.A, Eq. (38)] Equation (38) writes T(r+) = f(r+)/(4π), but f(r+)=0 by definition of the horizon. The intended expression is T(r+) = f'(r+)/(4π), and the algebraic form given in Eq. (38) is indeed consistent with f'(r+)/(4π). This typo should be corrected.
minor comments (5)
  1. [Overall] There are several typographical and language issues: 'and small value' should be 'a small value' in Sec. VI; 'ı.e.' should be 'i.e.' in Sec. V.B; 're' in 'detailed derivation of the above formula can be referred to' should be removed; and Spanish 'y' appears in Eqs. (45) and (47).
  2. [Fig. 3 caption] The caption 'we have used M 1[km] and α = 0.02 [km²]' is unclear; it should read M=1 km (or explicitly dimensionless in units of the Planck mass) so that the reader knows the units of the axes and parameters.
  3. [V.A, Eq. (33)] The integral expression for W(M,r+) is introduced but not evaluated; the subsequent Eq. (36) is simply stated. Showing the intermediate step that leads from T^0_0 = -3M^2α/r^6 to W=√(r+²-α)/r+ would improve transparency.
  4. [V.B, Eqs. (40) and (51)] The off-shell free energy is first written as F = M - S/τ in Eq. (40), then replaced by E - S/τ in Eq. (51). The paper should explicitly state that in the topological construction the horizon mass M is replaced by the regularized energy E, otherwise the reader may confuse the two quantities.
  5. [Introduction] The sentence 'The main a major difference between the present study and these antecedents' contains a grammar error ('The main a major difference') and should be rewritten.

Circularity Check

1 steps flagged · score 4.0 of 10

Thermodynamic topology result W=0 is largely a consequence of the first-law regularization that defines E via dE=W dM so that T=TH; the perihelion and QNM analyses are independent and not circular.

  1. self definitional [Sec. V.A, Eqs. (31)-(37); used in Sec. V.B, Eqs. (51)-(53)]
    "One way to reconcile at least the standard definitions for the entropy S and temperature T , is by modifying the thermodynamics first law (30) introduction a correction term W(M, r+) as follows ... In this way, T = TH , then TH = WTh, and S = πr2 + ... dE = W(M, r+)dM."

    The first-law temperature T is not independently derived; it is defined by WdM=T dS with W (Eq. 36) chosen so that T equals the Hawking temperature TH, and S is fixed to the area law. The off-shell topological analysis then uses F=E−S/τ with E from dE=W dM, and its on-shell condition is τ=1/T. Therefore the critical temperature, generation point, stable/unstable branches, and total charge W=0 (the paper's RN-like conclusion) are consequences of this imposed (E,S) prescription rather than an independent prediction of the metric. The paper itself flags this: it says the first law was 'regularized' to match Hawking. This does not affect the perihelion bound or QNM stability, which are independent.

full rationale

The perihelion-advance constraint and the WKB quasinormal-mode computation are self-contained checks: Eq. (10) is a direct integral of the perturbing potential, and Tables I-IV are standard WKB outputs for the given f(r). No circularity there. The only circularity-adjacent step is the thermodynamic construction: the paper explicitly 'regularizes' the first law (WdM=T dS, S=πr+², dE=W dM) so that the first-law temperature matches the Hawking temperature by construction, and then builds the thermodynamic topology on that E. The claimed RN-like W=0 is thus a property of the chosen first-law frame, not a model-independent prediction. A separate numerical inconsistency in the reported critical point and zero points (Fig. 3/4 values do not satisfy Eq. (53)) is a correctness issue, not circularity. Overall score 4 reflects one constructed central step alongside independent mechanical-stability content.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The paper contributes a scan over a known metric with one free coupling α, plus a first-law regularization. The QNM calculation rests on standard field theory and WKB. The thermodynamic topology result depends on the ad hoc energy and entropy prescription. No new particles, forces, or dimensions are introduced.

free parameters (2)
  • α (quantum coupling) = 0.02 M² for QNMs and thermodynamics; theoretical value α≈1.166 l_pl² from γ≈0.2375
    Section III gives only a very loose upper bound (α up to roughly 10^87 l_pl²); the value 0.02 is chosen by hand in Sections IV and VI. The QNM and thermodynamic results depend on this choice.
  • test scalar field mass m = 0, 0.1, 0.2 (in units of 1/M)
    Probe values chosen in Tables I-III; the claim that both real and imaginary parts increase with mass only covers this short range.
assumptions (4)
  • domain assumption The LQG-inspired metric f(r)=1-2M/r+αM²/r⁴ from [29] is the correct quantum-corrected black hole spacetime.
    Section II adopts the line element (1)-(2) without derivation; all later results inherit this background.
  • domain assumption Standard minimally coupled Klein-Gordon and Maxwell equations are valid on the quantum-corrected background with no additional quantum modifications.
    Section IV justifies this by the minimal-coupling replacement η→g, ∂→D; if quantum corrections alter the field equations, the QNM tables change.
  • ad hoc to paper The first law can be regularized by defining energy E through dE=W dM with W=√(r+²-α)/r+ and fixing area entropy S=πr+².
    Section V.A introduces W specifically so that T=TH; this is not derived from LQG and drives the RN-like thermodynamic topology.
  • standard math The WKB approximation [49,50] gives accurate quasinormal frequencies for the potentials at hand.
    Section IV uses WKB without stating the order or error estimates; accuracy is assumed from the literature.

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Pith. "Pith review of Quantum Black Holes: Perihelion Advance, Quasi Normal Modes and Classical/ Topological Thermodynamics." pith.science (2026). https://pith.science/paper/LZNOXUO4

@misc{pith2026250722945,
  author       = {Pith},
  title        = {Pith review of: Quantum Black Holes: Perihelion Advance, Quasi Normal Modes and Classical/ Topological Thermodynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LZNOXUO4}},
  note         = {Machine review of arXiv:2507.22945}
}
abstract

We report on some properties of a quantum black hole obtained recently. The correction to the Newtonian gravitational potential is proportional to a coupling $\alpha$, which is the only free parameter of the theory. We constrain the coupling using the perihelion advance, we compute the quasi-normal modes for scalar (both massless and massive) and electromagnetic perturbations. We find that all modes computed here are complex numbers characterized by a positive real part and a negative imaginary part, while both parts increase with the mass of the test scalar field. Also thermodynamics properties are investigated from the classical and topological point of view. In this regard, the quantum black hole exhibits the same behavior as the classical Reissner-Nordstr\"om space-time, that is, it presents stable/unstable branches in the Gibbs potential, one generating point and a topological charge $W=0$.

Figures

Figures reproduced from arXiv: 2507.22945 by the authors.

Figure 1
Figure 1. FIG. 1. Observational bound on the parameter [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The trend of the BH temperature versus the BH horizon (left panel). The behavior of the Gibbs free energy against [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Zero points of the vector [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The black arrows represent the unit vector field [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The Ω vs [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.