Tunneling of Massive Vector Particles under the Influence of Quantum Gravity

Ali \"Ovg\"un; Riasat Ali; Rimsha Babar; Wajiha Javed

REVIEW 5 major objections 5 minor 91 references

Tunneling of Massive Vector Particles under the Influence of Quantum Gravity

T0 review · 5 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper derives a quantum-gravity-corrected Hawking temperature for the charged accelerating rotating NUT black hole, $T'_H = T_H[1 - \beta\Xi]$, predicting that evaporation halts at a Planck-scale remnant.

desk verdict Routine GUP tunneling calculation whose new formula may be right, but the determinant reduction that supports it is asserted, not shown. read the letter →

arxiv 1909.02405 v1 pith:RO567CEY submitted 2019-08-19 physics.gen-ph

Wajiha Javed , Riasat Ali , Rimsha Babar , Ali Övgün This is my paper · ORCID

classification physics.gen-ph PACS 04.70.Dy04.70.Bw04.60.-m

keywords GUPHawkingtemperaturevectorparticletunnelingProcaequationWKBapproximationNUTblackholeremnant

open problems Quantum Gravity

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 derives the tunneling rate and Hawking temperature for massive charged vector (spin-1) particles crossing the horizons of a pair of charged, accelerating, rotating black holes with NUT parameter, with quantum-gravity effects encoded through a generalized uncertainty principle (GUP). Using a GUP-modified Proca equation and the WKB (semiclassical) approximation, it obtains the corrected tunneling probability $\Gamma = \exp[-2\pi (E - J\Omega_H - eA_0)(1 + \beta\Xi)/\kappa(r_+)]$ and the corrected Hawking temperature $T'_H = T_H[1 - \beta\Xi]$, where $\Xi = 6(m^2 + (J_\theta^2 + J_\phi^2 \csc^2\theta)/r_+^2)$. The quantum correction lowers the temperature relative to the semiclassical value and drives it to zero at a finite horizon, which the paper interprets as a stable remnant with mass $M_{\rm Res} \gtrsim M_p/\beta_0$. The consequence is that a minimal-length correction changes the endpoint of black-hole evaporation while preserving the known Schwarzschild, Reissner-Nordström, Kerr-Newman, and accelerating-rotating limits.

What carries the argument

The mechanism that carries the argument is the GUP-modified Proca equation for a charged massive spin-1 field, where the correction parameter $\beta$ appears in the field-strength combination $\psi_{\nu\mu} = (1 - \beta\hbar^2\partial_\nu^2)\partial_\nu\psi_\mu - (1 - \beta\hbar^2\partial_\mu^2)\partial_\mu\psi_\nu$ plus charge terms; it is the equation that makes the minimal-length effect feed into the tunneling exponent. Under the WKB ansatz, the four field equations become a homogeneous $4\times4$ linear system in the amplitudes $c_0,\dots,c_3$, and the identity $\det(V)=0$ is the step that converts the system into the single radial action integral. The actual quantum-gravity correction is carried by the positive combination $\Xi = 6(m^2 + (J_\theta^2 + J_\phi^2 \csc^2\theta)/r_+^2)$, which appears in the exponent as $(1 + \beta\Xi)$ and in the temperature as $[1 - \beta\Xi]$. The surface gravity $\kappa(r_+)$ of the outer horizon fixes the semiclassical temperature that the correction multiplies.

What would settle it

Recompute the determinant condition directly in the Schwarzschild limit ($a=l=\alpha=0$, $e=g=0$) and compare the resulting radial integral with the one the paper quotes; if it does not yield $\mathrm{Im}\,R_+ = i\pi E/(2\kappa)$ with $\kappa=1/(4M)$, the correction formula does not follow.

Watch

Extended reading notes

Core claim

Starting from the GUP-modified Proca Lagrangian for a massive charged boson in the spacetime of a pair of charged accelerating rotating NUT black holes, the paper inserts the WKB ansatz $\psi_\nu = c_\nu \exp[(i/\hbar)(-(E - J\Omega_H)t + R(r) + N\chi + \Theta(\theta))]$ and collects the leading-order terms into a $4\times4$ homogeneous system $V(c_0,c_1,c_2,c_3)^t = 0$. The condition $\det(V)=0$ is taken to reduce to the radial integral $R_\pm = \pm \int \sqrt{(E - J\Omega_H - eA_0)^2 + X_2(1 + X_1/X_2 \beta)}/B \, dr$, and a pole integration around $r_+$ gives $\mathrm{Im}\,R_\pm = \pm i\pi (E - \Omega_H J - eA_0)(1 + \beta\Xi)/(2\kappa(r_+))$. Comparing $\Gamma = \exp[-4\,\mathrm{Im}\,R_+]$ with the Boltzmann factor $\exp[-(E - J\Omega_H - eA_0)/T'_H]$ yields $T'_H = T_H[1 - \beta\Xi]$, with $T_H$ the semiclassical temperature built from $\kappa(r_+)$. The same equation, specialized to $\beta = 0$, reproduces the earlier fermion and vector-particle temperatures, and the limit chain $l \to 0$, $\tilde{k}=1$, $\alpha \to 0$, $a \to 0$ reduces it successively to the Kerr-Newman, Reissner-Nordström, and Schwarzschild temperatures. The paper also uses the temperature formula to derive a remnant: applying the stopping condition $(M - dM)(1 + \beta\Xi) \simeq M$ with $\beta = \beta_0/M_p^2$ and $\omega \simeq M_p$ gives $M_{\rm Res} \gtrsim M_p/\beta_0$ and $T_{\rm Res} \lesssim \beta_0/(8\pi)M_p$.

Load-bearing premise

The whole calculation rests on an unstated algebraic step: the claim that one particular $4\times4$ determinant condition is equivalent to the radial integral used for the tunneling rate; if that step has a sign or term error, the tunneling exponent and the final temperature are wrong.

Editorial extensions

If this is right

With $\beta = 0$ the corrected temperature reduces to the earlier semiclassical vector-particle tunneling temperature for these spacetimes, and to the fermion temperature of the same family when vector charge is dropped.
In the special limits $l=0$ with $\tilde{k}=1$, $\alpha=0$, and $a=0$, the formula reproduces, in turn, the accelerating-rotating charged temperature, the non-accelerating temperature, the Kerr-Newman temperature, the Reissner-Nordström temperature, and the Schwarzschild temperature.
Because $\Xi > 0$ for massive or rotating emissions, the corrected temperature is always below the semiclassical value; for the plotted parameters it vanishes at $\beta=100$, and for $\beta>100$ the first-order correction exceeds the semiclassical term and the temperature turns negative, which the paper rejects as non-physical.
Evaporation terminates at a nonzero remnant mass $M_{\rm Res} \gtrsim M_p/\beta_0$ with $T_{\rm Res} \lesssim \beta_0/(8\pi)M_p$, rather than proceeding to zero mass.
The corrected temperature increases with the rotation parameters $a$ and $\omega$, the correction parameter $\beta$, the black-hole acceleration $\alpha$, and the arbitrary parameter $k$, and decreases with the electric and magnetic charges $e$ and $g$.

Reading between the lines

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

If the determinant reduction is correct, the same machinery should apply to higher-spin emissions, with the spin appearing only through the numerical coefficient of $\Xi$; a spin-2 extension would test whether the remnant scale shifts.
The remnant bound is derived from a single stopping condition; a direct computation of the heat capacity or emission spectrum would test whether the remnant is thermodynamically stable rather than merely a zero of the temperature.
Because $\Xi$ is essentially the transverse kinetic energy of the emitted particle at the horizon, the GUP suppression should be stronger for high-mass, high-angular-momentum quanta; the angular-momentum distribution of the final radiation could probe this.
A direct symbolic check of the radial integral from $\det(V)=0$, even in the $\beta=0$ limit, would turn the paper's central algebraic claim into a fully reproducible step and reveal whether the quoted $X_1$ and $X_2$ are unique or gauge-dependent.

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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

First thing to know: this is a routine application of the established GUP-modified Proca tunneling program to one more exotic metric. The genuinely new item is the explicit corrected Hawking temperature T'_H = T_H[1 - beta Xi] for the accelerating rotating charged NUT black hole, with Xi spelled out in Eq (18). If the algebra is right, it extends a known class of results and reduces correctly to the beta=0 temperatures of earlier papers. The authors also state the GUP validity range (beta < 100) and give remnant mass estimates consistent with the earlier literature.

The paper's main problem is that the central derivation is not actually shown. After writing out a 4x4 matrix V whose elements contain terms up to quartic order in \dot R, N, \dot\Theta, and beta, the paper asserts that det(V)=0 leads to the radial integral in Eq (15) with the quoted X1 and X2. No reduction is given. That is a load-bearing step: any sign or term error would propagate into the exponent and the final temperature. The stress-test note is correct that Eq (17) introduces J_theta and J_phi with a factor 6 and a csc^2 theta dependence without explanation, and the action ansatz uses a coordinate chi that never appears in the metric. The final temperature inherits an explicit theta dependence through csc^2 theta, which is odd for a horizon quantity; presumably the angular parts are being treated as constants, but that deserves justification. The parameter k is also treated as independent in the graphs while the metric defines it through a constraint; that is a minor sloppiness, not a fatal flaw.

None of this makes the paper incoherent. The zero-order limit beta=0 does reproduce earlier temperatures, so the semiclassical part is probably right. The open question is whether the GUP correction coefficient Xi is correctly derived. A referee could check the determinant by hand or with a CAS; the authors should be asked to include that computation.

Who is this for? People working in the GUP-tunneling subfield, and anyone cataloguing Hawking temperatures for accelerating NUT families. It is not a paper that changes anyone's view of quantum gravity. I would not cite it, and I would not bring it to reading group, but I would still send it to a referee if the journal wants to publish this genre, on condition that the missing determinant derivation is supplied. Without that, the central claim is unsupported.

Referee Report

5 major / 5 minor

Summary. This paper studies the tunneling of massive charged vector particles through the horizons of a pair of accelerating, rotating, charged NUT black holes in the presence of quantum gravitational effects modeled by the generalized uncertainty principle (GUP). The authors start from the metric (1)-(2), introduce a modified Proca equation (6) containing the GUP parameter beta, apply the WKB ansatz (8), and obtain a 4x4 matrix equation V(c0,c1,c2,c3)^t = 0. Setting det(V)=0, they assert the radial integral (15) and, after a pole integration, the imaginary part (17), leading to the corrected tunneling probability and the corrected Hawking temperature T'_H = T_H(1 - beta Xi) in Eq. (18), where Xi = 6(m^2 + (J_theta^2 + J_phi^2 csc^2 theta)/r_+^2). They also discuss graphical stability and estimate a Planck-scale remnant mass.

Significance. If the derivation were complete, the claimed result would be a useful extension of GUP-corrected vector-particle tunneling to a complicated spacetime with acceleration, rotation, NUT charge, and electromagnetic charges. The parametric form T'_H = T_H(1 - beta Xi) is plausible and consistent with the known qualitative behavior of GUP-corrected Hawking temperatures, and the paper explicitly checks several limiting reductions to previously known results. However, the central derivation is not actually shown: the reduction from det(V)=0 to Eq. (15), the origin of X1 and X2, the factor 6, and the transition to Eq. (17) are all asserted rather than derived. Because any error in these steps would change the tunneling exponent and the final temperature, the central claim is not yet established. The paper also contains an internal contradiction between Eq. (18), which implies T'_H decreases with beta, and the text and figures, which state that T'_H increases with beta.

major comments (5)

[Section II, Eqs. (14)-(15)] The central determinant reduction is not shown. After writing V(c0,c1,c2,c3)^t = 0, the paper only says "we put det(V) = 0 and computing the radial part" and then states the radial integral (15). No derivation of the 4x4 determinant, of the resulting polynomial in \dot R, or of the functions X1 and X2 is provided. This step is load-bearing: any sign or term error in the determinant would propagate into ImR_+, the tunneling exponent, and the corrected temperature in Eq. (18). Please include the explicit computation, or at least the determinant condition reduced to a quadratic equation in \dot R, and verify that X1 and X2 are indeed independent of \dot R so that Eq. (15) is a closed radial integral.
[Section II, Eq. (13)] The action ansatz uses a coordinate \chi that does not appear in the metric (1)-(2), and N = \partial_\chi \tilde I is never defined. Since the azimuthal coordinate in the metric is \phi, the angular momentum separation should use \partial_\phi \tilde I; as written, N cannot be identified with the angular quantum numbers J_\theta and J_\phi that appear later in Eq. (17). Please replace \chi by \phi, or define \chi explicitly and connect it to the spacetime coordinates.
[Section II, Eqs. (15)-(18)] The transition from Eq. (15) to Eq. (17) is unexplained. The factor 6 in \Xi, the appearance of m^2, and the replacement of N^2 and \dot\Theta^2 by (J_\theta^2 + J_\phi^2 \csc^2\theta)/r_+^2 do not follow from the displayed X1 and X2, which depend on N and \dot\Theta rather than on J_\theta and J_\phi. Moreover, the corrected temperature in Eq. (18) retains an explicit \theta-dependence through \csc^2\theta, while a horizon temperature should be a global quantity; the paper should specify how \Xi is evaluated at the horizon or explain why the angular dependence is physical.
[Section II, Eq. (18), and Section III] There is an internal contradiction about the sign of the quantum correction. Since \Xi > 0, Eq. (18) implies that T'_H decreases as \beta increases. However, the abstract, the description of Fig. 1(ii), and the bullet points in the conclusion state that T'_H increases with \beta. These statements cannot both be correct; please reconcile the formula with the written analysis of the figures, or correct the graphical claims.
[Section II, below Eq. (2), and Section III] The abstract and the graphical analysis treat k as an arbitrary parameter, but \tilde k is fixed by a constraint in the metric definitions. Statements such as "T'_H increases with the increase of k" require either a demonstration that the constraint admits independent variation of \tilde k while all other parameters are held fixed, or a rephrasing of the claim in terms of the constrained parameter space.

minor comments (5)

[Section II, Eqs. (5)-(7)] There are index and contraction typos in the Lagrangian: Eq. (5) contains \psi^\mu \psi^\nu where a contraction such as \psi_\mu \psi^\mu is presumably intended, and the gauge-covariant structures in Eq. (6) are written with inconsistent index placements. These should be corrected for readability.
[Section II, Eq. (17)] The sign convention for ImR_\pm is not stated. Since the final tunneling probability is written as exp[-4 ImR_+], please define the branch and signs of the imaginary parts so that the Boltzmann factor comparison is unambiguous.
[Section II, Eq. (18)] The dimensions of \beta are not stated. Since \beta\Xi must be dimensionless in Eq. (18), the later replacement \beta = \beta_0/M_p^2 should be introduced before the main calculation, and the units of \Xi should be fixed consistently.
[Section II, Figure 3] The caption of Fig. 3 says the plots are for "varying a and \omega", but panel (i) varies \alpha; please correct the caption to match the panels.
[Keywords and references] The keyword "Hawking radiation" is listed twice; one duplicate should be removed. Also, reference [83] in the text appears as the source of the modified Proca equation, but the relation to the GUP modification should be stated more explicitly in the main text.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the GUP-modified Proca input and the Boltzmann-factor comparison are independent of the claimed corrected temperature; the main weakness is an unshown determinant reduction, which is a proof gap rather than a self-referential argument.

full rationale

The claimed result T'_H = T_H(1 - beta Xi) is obtained via the tunneling exponent ImR± in Eq. (17), which the paper reaches from det(V)=0 and a pole integration. The GUP correction enters through the modified Proca equation (6) and Lagrangian (5), cited from prior external work [83,84]; it is not fitted to the claimed temperature. Comparing the tunneling probability with the Boltzmann factor Gamma_B = exp[-(E - J Omega_H - e A0)/T'_H] is the standard tunneling method and does not presuppose the corrected temperature. The surface gravity is taken from Ref. [67], which is a self-citation, but it is a concrete, checkable expression for the same metric and is not used to conceal an assumption; the beta=0 limit is also checked against previous results [67,78]. The residual-mass estimate does reuse the external bound beta0 < 10^5 from Refs. [71-73], so the remnant conclusion imports prior constraints, but that is not circular. The real weakness is that the reduction from the 4x4 matrix equation to Eq. (15), and the jump to Eq. (17) with Xi = 6(m^2 + (J_theta^2 + J_phi^2 csc^2 theta)/r_+^2), are asserted without displayed algebra: N, chi, J_theta and J_phi are not carefully connected, and the theta dependence in Xi is unexplained. Those are unverified algebraic/computational gaps and correctness risks, not cases where the output is identical to the input by construction. No fitted parameter is renamed as a prediction, no self-citation chain forces the result, and no external result is merely relabeled. Therefore no significant circularity is established.

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

The central claim rests on the imported GUP-modified Proca equation and the metric from Ref [82]. No new particles or fields are introduced. The free parameters are the quantum gravity strength beta and the inconsistently treated parameter k.

free parameters (2)

beta = beta = beta_0/M_p^2 with beta_0 < 10^5 in residual-mass section
Quantum gravity correction strength; chosen by hand, not fitted to data.
k (k~) = varied as 0.1, 0.2, 0.3, 0.4 in Figure 2(ii)
Described as arbitrary parameter in plots even though the metric defines k~ through a constraint.

assumptions (4)

domain assumption The line element (1) from Ref [82] is a valid pair of accelerating rotating charged NUT black hole spacetime.
All horizon and temperature results use this metric and its charge and rotation parameters without independent derivation.
domain assumption The GUP-modified Proca equation (6) from Refs [83,84] correctly describes spin-1 particles under quantum gravitational effects.
The beta-dependent terms are the source of the claimed correction; if this equation is not the right effective model, T'_H does not follow.
standard math The WKB ansatz and near-horizon Taylor expansion (16) yield the pole integral (17).
This is the standard tunneling method, but the paper does not justify convergence or the dropping of higher-order terms.
domain assumption The tunneling probability can be compared with the Boltzmann factor Gamma_B = exp[-(E - J Omega_H - e A_0)/T'] to read off the temperature.
Standard in the tunneling method; relies on a thermal emission spectrum.

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Cite this review

Pith. "Pith review of Tunneling of Massive Vector Particles under the Influence of Quantum Gravity." pith.science (2026). https://pith.science/paper/RO567CEY

@misc{pith2026190902405,
  author       = {Pith},
  title        = {Pith review of: Tunneling of Massive Vector Particles under the Influence of Quantum Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RO567CEY}},
  note         = {Machine review of arXiv:1909.02405}
}

abstract

This paper is devoted to investigate charged vector particles tunneling via horizons of a pair of accelerating rotating charged NUT black hole under the influence of quantum gravitational effects. For this purpose, we use the modified Proca equation incorporating generalized uncertainty principle. Using the WKB approximation to the field equation, we obtain a modified tunneling rate and the corresponding corrected Hawking temperature for this black hole. Moreover, we analyze the graphical behavior of corrected Hawking temperature $T'_{H}$ with respect to the event horizon for the given black hole. By considering quantum gravitational effects on Hawking temperatures, we discuss the stability analysis of this black hole. For a pair of black holes, the temperature $T'_{H}$ increases with the increase in rotation parameters $a$ and $\omega$, correction parameter $\beta$, black hole acceleration $\alpha$ and arbitrary parameter $k$ and decreases with the increase in electric $e$ and magnetic charges $g$.

Figures

Figures reproduced from arXiv: 1909.02405 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

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