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How does non-metricity affect particle creation and evaporation in bumblebee gravity?

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

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Pith's one-line read This paper claims that non-metricity in bumblebee gravity changes black-hole emission: boson densities and most greybody factors rise, fermion densities fall, and evaporation accelerates.

desk verdict A competent set of semiclassical calculations, let down by unsupported constraints in Section VI; the fermionic-density contradiction the reader saw is not real. read the letter →

arxiv 2501.00927 v2 pith:CN7SJJXO submitted 2025-01-01 gr-qc hep-th

A. A. Araújo Filho This is my paper · ORCID
classification gr-qchep-th MSC 83C5783C4783D05 PACS 04.70.Dy04.62.+v11.30.Cp
keywords bumblebeegravityLorentzsymmetryviolationnon-metricityHawkingradiationblackholeevaporationgreybodyfactorstunnelingmethodmetric-affine
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 asks what non-metricity does to black-hole particle creation and evaporation in bumblebee gravity, by comparing two exact black-hole solutions: one in the metric formalism, parametrised by $\ell$, and one in the metric-affine formalism, parametrised by $X$. The author derives, for each background, the Hawking temperature from Bogoliubov coefficients, particle densities from the Parikh-Wilczek tunneling picture, greybody bounds, emission rates, and evaporation timescales. The paper's central answer is that non-metricity raises the created boson density, increases most greybody factors (tensor perturbations are the exception), enlarges the emission rate, and shortens the evaporation time, while slightly suppressing fermion production. It closes by converting astrophysical black-hole-lifetime data into bounds on the Lorentz-violating parameters, $\ell, X \lesssim 10^{-25}$ to $10^{-38}$.

What carries the argument

The load-bearing object is the controlled pair of spacetimes: the metric-formalism bumblebee black hole with Lorentz-violating parameter $\ell$ and the metric-affine bumblebee black hole with parameter $X$; non-metricity here means the metric-affine connection's failure to be metric-compatible, which is what distinguishes the $X$ background from the $\ell$ background. Each line element is fed through the same chain of tools: Bogoliubov-coefficient derivation of the Hawking temperature from scalar-field modes; the Parikh-Wilczek tunneling prescription, where the imaginary part of the action is evaluated by contour integration around the shifted horizon; the bound $T_b \ge \mathrm{sech}^2\left(\int G\,dr_*\right)$ for greybody factors; and Stefan-Boltzmann integration for the evaporation time. The comparison isolates what the non-metricity of the metric-affine connection adds to each observable.

What would settle it

Take Eq. (104) or Eq. (134) and compute $dM/(M\,dt)$ for a $10\,M_\odot$ black hole with $\ell = X = 10^{-25}$; if the result is orders of magnitude below $10\,$Hz (the gravitational-wave bound in Eq. (156)), the quoted constraints do not follow from the paper's evaporation equations, and a re-derivation of the bounds would be needed.

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Extended reading notes

Core claim

In its own terms, the paper shows that replacing the metric-formalism bumblebee line element $$$ds^{2}$ = -\left(1-\frac{2M}{r}\right)$dt^{2}$ + (1+\ell)\left(1-\frac{2M}{r}\right)^{-1}$dr^{2}$ + $r^{2}$d\$\Omega$^2$$ with the metric-affine solution changes the quantum emission in a definite direction. Solving the Klein-Gordon equation on each background gives the Bogoliubov coefficients and the temperatures $T_{\rm metric}=1/(8\pi\sqrt{1+\ell}\,M)$ and $T_{\rm metric\text{-}affine}\approx 1/(8\pi M)-X/(16\pi M)$; the tunneling calculation then yields boson densities $n=1/(e^{8\pi\sqrt{1+\ell}\,\omega(M-\omega/2)}-1)$ and fermion densities $n_\psi=1/(e^{8\pi\sqrt{1+\ell}\,M\omega}+1)$, with metric-affine analogues. The central finding is that the metric-affine background produces a larger boson density, larger greybody bounds for scalar, vector, and fermion perturbations, and a larger emission rate than the metric background, while the tensor greybody factor runs the other way; the evaporation times obey $t_{\rm metric}>t_{\rm metric\text{-}affine}$, so non-metricity accelerates evaporation. The same comparison is used to quote bounds $\ell, X \lesssim 10^{-25}$-$10^{-38}$ from black-hole lifetime observations.

Load-bearing premise

The Section VI constraints assume that the astrophysical upper limits on $dM/(M\,dt)$ translate directly into the quoted upper bounds on $\ell$ and $X$, even though the paper never writes that translation and its own evaporation equations predict mass-loss rates far too small to saturate those limits for stellar-mass and supermassive black holes.

Editorial extensions

If this is right

  • If the central comparison is right, Hawking spectra carry a non-metricity fingerprint: boson counts and most greybody factors sit higher in the metric-affine solution, fermion counts lower, so the boson-to-fermion ratio distinguishes the two formulations.
  • The evaporation hierarchy $t_{\rm KR} < t_{\rm Schw} < t_{\rm metric\text{-}affine} < t_{\rm metric}$ implies that measured black-hole lifetimes could, in principle, indicate which Lorentz-violating gravity is realized.
  • The quoted constraints $\ell, X \lesssim 10^{-25}$-$10^{-38}$ would push bumblebee Lorentz violation far beyond current laboratory reach, making astrophysical evaporation the primary probe of these parameters.
  • The greybody-quasinormal-mode correspondence predicts a frequency-dependent reversal for scalar metric-affine perturbations, with the effect of $X$ changing sign near $\omega\approx0.336$; that inflection is a concrete ringdown feature to look for.

Reading between the lines

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

  • A direct consistency check the paper leaves implicit: inserting the derived evaporation rates into the cited astrophysical bounds may show that $\dot M/M$ is far too small to saturate them, in which case the Section VI limits are an upper bound on observability, not on $\ell$ and $X$.
  • The same comparison could be run for rotating Kerr-like bumblebee solutions; if the boson/fermion asymmetry persists there, gravitational-wave ringdown from spinning remnants would be a sharper discriminator than the spherically symmetric emission considered here.
  • Because the metric-affine temperature is lower than the Schwarzschild value for $X>0$ yet the lifetime is shorter, the faster evaporation must be driven by the modified cross-section and greybody factors; separating those contributions would show which quantity actually controls the lifetime.
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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

3 major / 3 minor

Summary. This paper compares Hawking radiation, particle creation, greybody factors, emission rates, and evaporation times for two bumblebee-gravity black-hole solutions: the metric-formalism solution of Ref. [1] and the metric-affine solution of Ref. [2]. It derives Hawking temperatures from Bogoliubov coefficients and from surface gravity, computes bosonic and fermionic particle densities via the tunneling method, obtains greybody bounds, estimates evaporation lifetimes, and relates greybody factors to quasinormal modes. The paper's advertised conclusions are that non-metricity raises bosonic particle density, lowers fermionic particle density, increases greybody factors except for tensor perturbations, accelerates evaporation, and that astrophysical lifetime data constrain the Lorentz-violating parameters to levels of about 10^{-25} to 10^{-38}.

Significance. If the results held, the analytic comparison between metric and metric-affine bumblebee gravity would be useful: the paper provides closed-form expressions for temperatures, densities, greybody bounds, and lifetimes with no free parameters introduced at the level of the derivations from the published metrics. However, two load-bearing claims are not supported by the manuscript's own equations. First, the Section VI constraints on ℓ and X do not follow from the paper's evaporation rates. Second, the abstract and conclusion state that non-metricity reduces fermionic particle density, while Eqs. (54) and (123) show the opposite, and Eq. (123) also disagrees with the thermal exponent implied by Eq. (116). These issues must be resolved before the central message can be accepted.

major comments (3)
  1. [Section VI, Eqs. (104), (134), (155), and Table I] The claimed bounds ℓ, X ≲ 10^{-25}, 10^{-36}, 10^{-35}, and 10^{-38} do not follow from the paper's own evaporation equations. The paper never writes the relation between the astrophysical limits on dM/(M dt) and ℓ or X. Using Eq. (104), dM/dt = -27ξ/[4096π^3(1+ℓ)^2 M^2], so dM/(M dt) ≈ -C/[(1+ℓ)^2 M^3]. For a 10 M_⊙ black hole with ℓ=0, M ≈ 10^{38} in Planck units, giving a fractional mass-loss rate of order 10^{-118} Planck^{-1} ≈ 10^{-75} s^{-1}. This is decades of orders of magnitude below all four limits quoted in Section VI (10 Hz, 2×10^{-10} Hz, 10^{-9} Hz, and 6×10^{-12} Hz). Therefore the inequalities in Eqs. (153)-(162) are already satisfied for any ℓ or X and impose no upper bound on these parameters. The sentence "which yields the following constraint" after Eq. (155) is a gap; the numbers in Table I do not appear in any equation of the paper. The section should either be removed or replaced by a genuine derivation of bounds from the derived lifetimes.
  2. [Abstract, Section IV B, Eqs. (54) and (123)] The abstract and conclusion claim that non-metricity reduces the fermionic particle density. This is contradicted by the manuscript's own formulas. For ℓ = X = 0.1 and fixed M and ω, the exponent in Eq. (54) is 8πMω√(1+ℓ) = 8πMω × 1.049, while the exponent in Eq. (123) as written is 8πMω(4-X)/(3X+4) = 8πMω × 0.907. Since the Fermi-Dirac density decreases with increasing exponent, n^ψ_met-aff is larger than n^ψ_metric, not smaller. In addition, the exponent in Eq. (123) does not match the Hawking temperature T of Eq. (116): the correct thermal factor from Eq. (116) would be exp(8πMω√(3X+4)/√(4-X)), which for X = 0.1 gives 8πMω × 1.054, whereas the written Eq. (123) gives 8πMω × 0.907. The ratio of the two exponents is ((4-X)/(3X+4))^{3/2}, so this is not a typo in prefactor alone. The fermionic sector must be recalculated or the statements in the abstract and conclusion corrected accordingly.
  3. [Section IV F, Eq. (139)] There is a contradictory sentence immediately after Eq. (139). The equation gives t_metric = 1.00899 × t_met-aff, which means t_metric > t_met-aff, i.e., the metric-formalism black hole takes longer to evaporate. The text states that this "confirms that t_metric corresponds to a faster evaporation process compared to t_met-aff". The word "faster" should be "slower"; otherwise the sentence disagrees with both Eq. (138) and the ordering t_KR < t_Schw < t_met-aff < t_metric stated in the conclusion.
minor comments (3)
  1. [Section V B 3] The text says the tensor-perturbation calculation relies on Eq. (23), but the relevant effective potential is given in Eq. (128); Eq. (23) is the Bogoliubov relation. Please fix the cross-reference.
  2. [General presentation] There are numerous typographical errors, including "greybody facotrs" in Section III C 3, "ir reads" before Eq. (121), and repeated "the the top panel" in several figure captions. A careful proofreading pass is needed.
  3. [Section III C 2 and Section IV A 1] In the metric case the vector effective potential is said to be unchanged from Schwarzschild, but in the metric-affine case the vector potential is claimed to depend on X. This asymmetry in treatment is not discussed, and it would help to clarify why the tetrad procedure yields a nontrivial X-dependence in one formalism but not the other.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the Section VI bounds are an unsupported non sequitur, not a circular reduction.

full rationale

The paper's central derivations (Hawking temperatures, tunneling rates, particle densities, greybody bounds, emission rates, evaporation times) are obtained by standard QFT-in-curved-spacetime, tunneling, and WKB calculations applied directly to the two input metrics, Eq. (1) from Ref. [1] and Eq. (2) from Ref. [2]. Taking a previously published spacetime as input is not circular even when the present author is also an author of the solution paper, because the target results are not assumed anywhere in the derivation; they are computed from the metric. The self-citations to Refs. [2], [111], and [114] supply background solutions and cross-sections, but the paper re-derives the Hawking temperature (Eqs. (27) and (116)) and the evaporation equations (Eqs. (104) and (134)) from the metrics, so those citations are not load-bearing in a circular sense. The one serious weakness is Section VI: after Eq. (155), the phrase 'which yields the following constraint on the parameters ℓ and X' introduces no equation connecting the astrophysical dM/(M dt) bounds to ℓ or X. The paper's own fractional mass-loss rates from Eqs. (104) and (134) are many orders of magnitude below the quoted upper limits (for a 10-solar-mass Schwarzschild black hole the fractional rate is roughly 10^-75 s^-1, while the weakest limit used is about 10 Hz), so the numbers 10^-25, 10^-36, 10^-35, and 10^-38 in Table I do not follow from any displayed relation. This is an omitted derivation or non sequitur, not a circular reduction: the constraints are not equivalent to the inputs by construction, they are simply unsubstantiated. Under the hard rule that circularity requires a specific equation-to-equation reduction or a fitted parameter renamed as a prediction, no such step is present.

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

The central results use standard semi-classical techniques on two pre-existing bumblebee gravity solutions. No new entities are introduced, and no parameters are fitted to data. The main unstated burden is the unjustified translation of astrophysical mass-loss limits into constraints on ℓ and X in Section VI.

assumptions (5)
  • domain assumption The bumblebee black hole metrics (Eqs. (1) and (2)) are exact vacuum solutions of bumblebee gravity, as claimed in Refs. [1] and [2].
    The paper adopts these metrics without re-deriving them; if the solutions are wrong or the parameter ranges invalid, all derived quantities shift.
  • domain assumption Hawking radiation can be computed via the tunneling method (Parikh-Wilczek) and the Bogoliubov coefficient approach as presented.
    The paper relies on the standard semi-classical framework; it does not justify its applicability beyond the cited literature.
  • domain assumption The greybody factor bound T_b ≥ sech^2(∫ G dr*) with ξ = ω is a valid lower bound for the transmission probability.
    Used to produce closed-form greybody bounds; the bound is only an inequality, not the exact greybody factor.
  • domain assumption The Stefan-Boltzmann law with geometric optics cross-section σ = π(3√3 M)^2 governs the black hole evaporation rate.
    Used to derive evaporation lifetimes; assumes the black hole radiates as a blackbody with the computed temperature.
  • ad hoc to paper The astrophysical ˙M/M limits from Ref. [168] can be directly applied to constrain ℓ and X.
    The paper asserts the resulting bounds (ℓ, X ≲ 10^-25 to 10^-38) without showing the explicit relation between the observed limits and the Lorentz-violating parameters; the evaporation rates for the masses considered are many orders of magnitude below the quoted limits.

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Pith. "Pith review of How does non-metricity affect particle creation and evaporation in bumblebee gravity?." pith.science (2026). https://pith.science/paper/CN7SJJXO

@misc{pith2026250100927,
  author       = {Pith},
  title        = {Pith review of: How does non-metricity affect particle creation and evaporation in bumblebee gravity?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CN7SJJXO}},
  note         = {Machine review of arXiv:2501.00927}
}
abstract

In this work, we analyze the impact of non-metricity on particle creation and the evaporation process of black holes within the framework of bumblebee gravity. In general lines, we compare black holes in the metric formalism [1] and the metric-affine approach [2]. Initially, we focus on bosonic particle modes to investigate Hawking radiation. Using the Klein-Gordon equation, we compute the Bogoliubov coefficients and derive the Hawking temperature. Subsequently, we examine Hawking radiation as a tunneling process, resolving divergent integrals through the residue method. The analysis is then extended to fermionic particle modes, also within the tunneling framework. Particle creation densities are calculated for both bosonic and fermionic cases. Additionally, greybody bounds are estimated for bosonic and fermionic particles. Furthermore, we explore the evaporation process, considering the final state of the black holes and we also investigate the correlation between the greybody factors and the quasinormal modes. Finally, constraints on the Lorentz-violating parameters $\ell$ (for the metric case) and $X$ (for the metric-affine case) are established using recent astrophysical data on black hole lifetimes. In a general panorama, non-metricity (except for the tensor perturbations) in bumblebee gravity raises particle density for bosons while reducing it for fermions, increases greybody factors (for both bosons and fermions), amplifies the emission rate, and accelerates the evaporation process.

Figures

Figures reproduced from arXiv: 2501.00927 by the authors.

Figure 1
Figure 1. The Hawking temperature Tmetric as a function of mass M for various values of ℓ, being compared with the Schwarzschild and Kalb–Ramond cases. An important point to highlight is that comparing the expression above with the Planck distribution reveals that P(ω, ℓ) = dω 2π 1 e ω T − 1 . (26) In this manner, we can properly obtain Tmetric = 1 8π √ 1 + ℓM . (27) As we shall see in the evaporation subsection, the Hawking … view at source ↗
Figure 2
Figure 2. The particle density nmetric is shown for different values of ℓ for the metric case. The Schwarzschild and the Kalb–Ramond cases are also compared. spin–1 bosons has revealed that the Hawking temperature remains unaffected, even when higher–order quantum corrections are considered [130, 131]. The action for fermions is typically associated with the phase of the spinor wave function, which satisfies the Hamilton–Jaco… view at source ↗
Figure 3
Figure 3. The particle density nψmetric is shown for various values of ℓ. The Schwarzschild and the Kalb–Ramond cases are compared. It is important to note that, based on the dominant energy condition and the Einstein field equations, the functions A(r) and B(r) have identical zeros. Therefore, near r = rh, we can approximate these functions to first order as: A(r)B(r) = A ′ (rh)B ′ (rh)(r − rh) 2 + . . . . (51) This reveals … view at source ↗
Figures from the paper (34 more)
Figure 4
Figure 4. Figure 4: The effective potential V s metric is shown for different values of ℓ for l = 1. Also, the Schwarzschild case is compared in this analysis. with G = q (ξ ′ ) 2 + (ω2 − V s metric − ξ 2 ) 2 2ξ . (60) The function ξ is positive and fulfills the conditions ξ(+∞) = ξ(−∞) =…
Figure 5
Figure 5. Figure 5: The greybody factors T s bmetric is displayed for different values of ℓ when keeping l = 1 (the the top panel) and for different values of l for a fixed value of ℓ = 0.1. For both cases, the Schwarzschild case is compared. For electromagnetic perturbations analyzed thr…
Figure 6
Figure 6. Figure 6: The greybody factors T v bmetric is displayed for different values of ℓ when keeping l = 1 (the the top panel) and for different values of l for a fixed value of ℓ = 0.1. For both cases, the Schwarzschild case is compared. can be recast as: p A(r)B(r) −1  r p A(r) F …
Figure 7
Figure 7. Figure 7: The effective potential V t metric is shown for different values of ℓ for l = 1. Also, the Schwarzschild case is compared in this analysis. or, more explicitly V t metric(r) =  1 − 2M r  l(l + 1) r 2 − 2(ℓr + 3M) (ℓ + 1)r 3  . (85) It is worth noting that in the li…
Figure 8
Figure 8. Figure 8: The greybody factors T t bmetric is displayed for different values of ℓ when keeping l = 1 (the the top panel) and for different values of l for a fixed value of ℓ = 0.1. For both cases, the Schwarzschild case is compared. corresponding Dirac equations, which govern th…
Figure 9
Figure 9. Figure 9: The effective potential V + metric is shown for different values of ℓ. Also, the Schwarzschild case is compared in this analysis. 0 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0 0 5 10 15 20 0.0 0.2 0.4 0.6 0.8 1.0 [PITH_FULL_IMAGE:figures/full_fig_p029_9.png]
Figure 10
Figure 10. Figure 10: The greybody factors Tbmetric is displayed for different values of ℓ when keeping l = 1 (the the top panel) and for different values of l for a fixed value of ℓ = 0.1. For both cases, the Schwarzschild case is compared. with positive energy can escape the gravitationa…
Figure 11
Figure 11. Figure 11: The emission rate for different values of [PITH_FULL_IMAGE:figures/full_fig_p030_11.png]
Figure 12
Figure 12. Figure 12: The evaporation time tmetric is shown for different values of ℓ. A comparison with the Schwarzschild and Kalb–Ramond cases is shown. as the photon capture cross–section, given by σ = π(3√ 3M) 2 , leading to ˆ tmetric 0 ξdτ = − ˆ Mf Mi  27ξ 4096π 3 (1 + ℓ) 2M2 −1 dM.…
Figure 13
Figure 13. Figure 13: The Hawking temperature Tmetric–affine is exhibited for different values of X. The Schwarzschild case is compared. so that Pmet–aff(ω, ℓ) = dω 2π 1 e  8πM(−(X−4)3) 3/4√ 3X+4 (4−X) 11/4  ω − 1 . (114) It is worth mentioning that, when compared to Planck distribution,…
Figure 14
Figure 14. Figure 14: The particle density nmet–aff is shown for different values of X. The Schwarzschild and Kalb–Ramond cases are also compared. It, shaped by its additional dependence on ω, diverges from the conventional blackbody form, a difference that becomes apparent upon closer ana…
Figure 15
Figure 15. Figure 15: Comparison of n for Schwarzschild case, Kalb–Ramond, bumblebee in the metric formalism and bumblebee with metric–affine approach. Here, it is considered X = 0.1 = ℓ = 0.1 and M = 1. B. Fermionic modes With the definitions established so far, the particle density for f…
Figure 16
Figure 16. Figure 16: The particle density nψmet–aff is shown for different values of X. The Schwarzschild and Kalb–Ramond cases are also compared. 0.2 0.4 0.6 0.8 1.0 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.1250 0.1252 0.1254 0.1256 0.1258 0.1260 0.036 0.038 0.040 0.042 0.044 0.046 0.…
Figure 17
Figure 17. Figure 17: Comparison of nψ for the bumblebee black hole in the metric and the metric–affine formulations (for X = ℓ = 0.1. In this comparison, the Schwarzschild and the Kalb–Ramond cases are also compared. C. Greybody factors for bosons 1. Scalar perturbations Using the same me…
Figure 18
Figure 18. Figure 18: Comparison of n, and nψ. The configuration employed is the following: nSchw (Schwarzschild for bosons): M = 1; nψSchw (Schwarzschild for fermions): M = 1; nKR (Kalb–Ramond for bosons): M = 1, and ℓ = 0.2; nψKR (Kalb–Ramond for fermions): M = 1 and ℓ = 0.2; nmetric (Bu…
Figure 19
Figure 19. Figure 19: The effective potential V s met–aff is shown for different values of X. Also, the Schwarzschild case is compared in this analysis. 0 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0 0 5 10 15 20 0.0 0.2 0.4 0.6 0.8 1.0 [PITH_FULL_IMAGE:figures/full_fig_p039_19.png]
Figure 20
Figure 20. Figure 20: The greybody factors T s bmet–aff is displayed for different values of X when keeping l = 1 (the the top panel) and for different values of l for a fixed value of X = 0.1. For both cases, the Schwarzschild case is compared. 2. Vector perturbations Following the analog…
Figure 21
Figure 21. Figure 21: The effective potential V v met–aff is shown for different values of X. Also, the Schwarzschild case is compared in this analysis. Therefore, the greybody factors can be presented below T v bmet–aff ≥ sech2 ˆ +∞ −∞ V v met–aff 2ω dr ∗  = sech2 "ˆ +∞ rh V v met–aff 2…
Figure 22
Figure 22. Figure 22: The greybody factors T v bmet–aff is displayed for different values of X when keeping l = 1 (the the top panel) and for different values of l for a fixed value of X = 0.1. For both cases, the Schwarzschild case is compared. Naturally, if X → 0, we recover the effectiv…
Figure 23
Figure 23. Figure 23: The effective potential V t met–aff is shown for different values of X for l = 1. Also, the Schwarzschild case is compared in this analysis. 0 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0 0 5 10 15 20 0.0 0.2 0.4 0.6 0.8 1.0 [PITH_FULL_IMAGE:figures/full_fig_p042_23.png]
Figure 24
Figure 24. Figure 24: The greybody factors T t bmet–aff is displayed for different values of X when keeping l = 1 (the the top panel) and for different values of l for a fixed value of X = 0.1. For both cases, the Schwarzschild case is compared. D. Greybody factors for fermions Following t…
Figure 25
Figure 25. Figure 25: The comparison of the greybody factors for the bosonic case when [PITH_FULL_IMAGE:figures/full_fig_p043_25.png]
Figure 26
Figure 26. Figure 26: The effective potential V + met–aff is shown for different values of X. Also, the Schwarzschild case is compared in this analysis. 0 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0 0 5 10 15 20 0.0 0.2 0.4 0.6 0.8 1.0 [PITH_FULL_IMAGE:figures/full_fig_p044_26.png]
Figure 27
Figure 27. Figure 27: The greybody factors Tbmet–aff is displayed for different values of X when keeping l = 1 (the the top panel) and for different values of l for a fixed value of X = 0.1. For both cases, the Schwarzschild case is compared. verify that the non–metricity is responsible fo…
Figure 28
Figure 28. Figure 28: The comparison of Tb for the metric and metric–affine formalisms for fixed values of ℓ and X, i.e., X = ℓ = 0.1. 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.05 0.10 0.15 [PITH_FULL_IMAGE:figures/full_fig_p045_28.png]
Figure 29
Figure 29. Figure 29: The emission rate for different values of [PITH_FULL_IMAGE:figures/full_fig_p045_29.png]
Figure 30
Figure 30. Figure 30: The comparison of the emission rates for the bumblebee in the [PITH_FULL_IMAGE:figures/full_fig_p046_30.png]
Figure 31
Figure 31. Figure 31: The evaporation time tmet–aff is shown for different values of X. The Schwarzschild case is also compared. 0 2 ×109 4 ×109 6 ×109 8 ×109 1 ×1010 0 5.0 ×1032 1.0 ×1033 1.5 ×1033 2.0 ×1033 [PITH_FULL_IMAGE:figures/full_fig_p048_31.png]
Figure 32
Figure 32. Figure 32: The blackhole lifetime comparison is shown for bumblebee ( [PITH_FULL_IMAGE:figures/full_fig_p048_32.png]
Figure 33
Figure 33. Figure 33: The greybody factors ΓS metric(ω) for scalar perturbations, computed from the quasinormal modes, are presented as functions of the frequency ω for l = 1 and M = 1. and ∆f = − (ω 2 − ω 2 0R) 3 32ω 5 0Rω0I ( 1 + ω0R(ω0R − ω1R) 4ω0I 2 + ω 2 0R " (ω0R − ω1R) 2 16ω 4 0I − …
Figure 34
Figure 34. Figure 34: The greybody factors Γt metric(ω) for tensor perturbations, obtained from quasinormal modes, are plotted as functions of the frequency ω for l = 1 and M = 0.5. behavior observed earlier. This effect might be associated with the decrease in the real part of the quasino…
Figure 35
Figure 35. Figure 35: The greybody factors ΓS met–aff(ω) for scalar perturbations are obtained from quasinormal modes and plotted as functions of the frequency ω, considering l = 1 and M = 1. B. The metric–affine case 1. Scalar perturbations We now explore how greybody factors relate to qu…
Figure 36
Figure 36. Figure 36: The greybody factors ΓV met–aff(ω) for vector perturbations are computed from quasinormal modes and displayed as functions of the frequency ω, considering l = 1 and M = 1. 2. Vector perturbations This subsection examines the connection between greybody factors and qua…
Figure 37
Figure 37. Figure 37: The greybody factors Γt met–aff(ω) for tensor perturbations are computed from quasinormal modes and shown as functions of the frequency ω, with parameters l = 1 and M = 0.5 [PITH_FULL_IMAGE:figures/full_fig_p054_37.png]

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Forward citations

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Works this paper leans on

181 extracted references · 70 canonical work pages · cited by 3 Pith papers

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    Specifi- cally, analogous to Eq

    The Hawking radiation Following a similar approach to the one employed in the previous section for the bumble- bee model in the metric formalism, we now focus on the metric–affine framework. Specifi- cally, analogous to Eq. (18), we write rmet–aff = 2M − 1 4 Eλ 4√ 4 − X 4 p −(X − 4)3, (107) where the negative solution of the square root in Eq. (14) is als...

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    The tunneling process Following the methodology employed in the previous section for the bumblebee black hole, we now focus on: ∆(r)met–aff = 1 4 r 2 p −(X − 4)3(M − ω′) r √ 3X + 4 − p −(X − 4)3 √ 3X + 4 + 4 ! (117) in a such way that the integral present in Eq. (118) is cast below Im Smet–aff = Im ˆ ω 0 −dω′ ˆ rf ri dr 4 r √ −(X−4)3 (4−X)7/2 1 − q ∆(r)me...

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    Scalar perturbations Using the same methodology applied in the metric approach, we now derive the effective potential for the metric–affine bumblebee black hole V s met–aff = A(r) " l(l + 1) r2 + 1 r p A(r)B(r)−1 d dr p A(r)B(r) # = 4l2(r − 2M ) + 4l(r − 2M ) − 2M (X − 4) q 4−X 3X+4 q (r−2M )2 r2 r3 p −((X − 4)(3X + 4)) , (124) 37 0.0 0.2 0.4 0.6 0.8 1.0 ...

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    Vector perturbations Following the analogy with the scalar perturbation in the metric–affine framework, the effective potential is expressed as V v met–aff = − 4l(l + 1)(2M − r) r3 p −((X − 4)(3X + 4)) . (126) 39 2 4 6 8 100.00 0.02 0.04 0.06 0.08 0.10 Figure 21: The effective potential V v met–aff is shown for different values of X. Also, the Schwarzschi...

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    Tensor perturbations Following the analogy with the scalar and vector perturbations in themetric–affine frame- work, the effective potential is expressed as V t met–aff = 1 − 2M r l(l + 1) r2 + 6M (X − 4)2 − 2r(X − 4)2 + 8r p −((X − 4)(3X + 4)) r3(X − 4)(3X + 4) ! . (128) 40 0 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0 0 5 10 15 20 0.0 0.2 0.4 0.6 0.8 1.0 Figure 2...

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    Scalar perturbations In this subsection, the correspondence between the greybody factors and the quasinormal modes is derived specifically for scalar perturbations in the metric case. To this end, we employ Eq. (58) to carry out the calculations. Fig. 33 presents the greybody factors Γ S metric(ω) as functions of the frequency ω for l = 1 and M = 1, inclu...

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    Vector perturbations As discussed in the previous section and consistent with Eq. (70), the Lorentz–violating contribution to the effective potential for vector perturbations vanishes, implying that only the trivial contribution remains

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