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REVIEW 3 major objections 3 minor 93 references

Coherent graviton condensates can resolve the singularity of gravitational-decoupling hairy black holes, leaving observable strong-field fingerprints in photon rings, lensing, and ringdown.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-04 05:56 UTC pith:575IKEGZ

load-bearing objection The paper's central quantum hairy metric does not reduce to the classical GD hairy seed when the smearing scale vanishes—the signs are flipped—so the main construction is not what it claims to be, despite being explicit and fixable. the 3 major comments →

arxiv 2602.20386 v2 pith:575IKEGZ submitted 2026-02-23 hep-th gr-qc

Coherent quantum hairy black holes from gravitational decoupling: regularity, geodesics, and scalar ringdown

Henrique Navarro , Roldao da Rocha This is my paper
classification hep-th gr-qc MSC 83C5783C1083C45 PACS 04.70.-s04.60.-m
keywords coherent graviton statesblack hole singularity resolutiongravitational decoupling hairGaussian smearingquasinormal modesphoton ringMisner-Sharp massReissner-Nordström deformation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper attempts to show that gravitational-decoupling (GD) hairy black holes—spacetimes built from a Reissner–Nordström seed plus hair generated by an extra gravitational source—admit a quantum description as the mean-field limit of a finite graviton condensate. Implementing the quantum description by Gaussian smearing of the classical gravitational potential with a width Rs, the authors obtain a metric that is asymptotically flat, finite at the origin for suitable parameters, and capable of describing two-horizon, extremal, or horizonless configurations. They derive the effective stress–energy tensor and Misner–Sharp mass, and compute photon rings, critical impact parameters, light deflection, and a WKB quasinormal-mode spectrum, all of which deviate from classical and quantum-corrected Reissner–Nordström geometries. A sympathetic reader would care because the construction replaces the central singularity with a finite-size quantum core and converts that core into concrete, testable strong-field predictions.

Core claim

The central claim is that the metric function f(r)=1+2V^q_gdh(r), where V^q_gdh is the Gaussian-smeared GD hairy potential, is the coherent quantum version of the classical GD hairy black hole. The classical r^-1 and r^-2 divergences are replaced by a finite potential at r=0; curvature invariants stay finite when the coefficient f0 equals 1, and the paper derives a parameter relation involving R_Q/R_M, R_s/R_M, and the hair parameter α that selects regular geometries. On the observational side, the paper claims that the photon ring, critical impact parameter, deflection angle, and WKB quasinormal frequencies all differ from those of classical Reissner–Nordström, quantum Reissner–Nordström, a

What carries the argument

The central object is the Gaussian-regularized potential V_gdh^q(r), obtained by Fourier–Bessel transforming the classical GD hairy potential, multiplying by exp(-k^2 R_s^2/4), and transforming back; the resulting error-function and Dawson-function terms replace point-like mass and charge distributions with Gaussian profiles of width Rs. This smearing does two jobs: it makes the coherent state normalizable (the classical 1/r and 1/r^2 terms have divergent occupation numbers) and it supplies the short-distance regulator that produces the regular core. The hair parameter α and the charges ℓ and Q then determine how this regular core differs from the purely quantum-corrected Reissner–Nordström

Load-bearing premise

The load-bearing premise is that genuine quantum-gravitational corrections are equivalent to a Gaussian convolution of the classical GD hairy potential with a free width Rs; if the real quantum state does not produce this smearing, the regular core, modified horizons, and all derived photon and ringdown signatures are inputs rather than predictions.

What would settle it

Solve the exact quasinormal spectrum of the metric f(r)=1+2V_gdh^q(r) by time-domain evolution: if the frequencies coincide with the classical GD hairy or Schwarzschild spectrum, the paper's WKB difference is an artifact. Observationally, a shadow or lensing measurement that matches classical Reissner–Nordström to within the model's predicted percent-level shift at Rs/RM ≈ 0.1 would rule out the model's claimed magnitude of quantum-core effects.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If the construction is right, the classical central singularity of GD hairy black holes is an artifact of the mean-field limit; finite graviton number replaces it with a regular quantum core.
  • The photon ring radius and critical impact parameter shift with Rs and α, so black-hole-shadow observations can constrain both the quantum-core size and the hair strength.
  • The deflection angle deviates most from Reissner–Nordström in the strong-lensing regime, giving a lensing test independent of ringdown.
  • The WKB quasinormal-mode spectrum differs from both the classical GD hairy and Schwarzschild spectra, so ringdown gravitational waves could in principle distinguish the quantum-corrected geometry.
  • The asymptotic Misner–Sharp mass is M(1-α/2e^2), so the hair renormalizes the mass seen by a distant observer even though the spacetime remains asymptotically flat.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The Gaussian-convolution recipe is a one-parameter regulator: if it is accepted, all strong-field predictions are controlled by the single free width Rs, making the model a one-parameter family of deviations from classical black holes.
  • The abstract's WKB quasinormal-mode prediction is not accompanied by an explicit calculation in the body of the paper as provided; verifying it with an independent WKB or time-domain computation is the natural next step before treating the ringdown difference as a settled prediction.
  • The paper leaves Rs unfixed; a complete graviton-condensate derivation would tie Rs to the occupation number N ~ M^2/M_p^2, converting the parameter scan into a falsifiable relation between mass and quantum-core size.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper constructs a "coherent quantum" extension of gravitational-decoupling (GD) hairy black holes. The classical GD hairy metric (Eq. 27) is decomposed into an RN part and a hairy part (Eqs. 48–49); the hairy potential is then Gaussian-smeared with width R_s, interpreted as the quantum-core size of a graviton condensate, yielding the metric function f=1+2V^q_gdh (Eq. 52). The paper derives the effective stress-energy tensor, Misner–Sharp mass, horizon structure, a regularity condition (Eq. 85), and studies null geodesics (photon ring, critical impact parameter, light deflection). The abstract also claims a WKB quasinormal-mode spectrum different from GD hairy and Schwarzschild black holes.

Significance. If correct, the construction would provide an analytically tractable regular black-hole model with both GD hair and a quantum core, with potentially observable strong-field signatures. The explicit stress-tensor and mass computations, and the clean statement of the regularity condition Eq. (85), are strengths. However, the central smearing calculation contains sign and factor errors that break the claimed mean-field limit, the ringdown analysis promised in the title and abstract is absent, and the stress-tensor regularity claim is internally contradicted. These issues are load-bearing and must be addressed before the results can be considered reliable.

major comments (3)
  1. [Section III, Eqs. (48)–(53)] The "quantum hairy" potential is not the Gaussian smearing of the classical seed, and the mean-field limit fails. The classical metric (27) implies V_H = -α G_n M/(2r) e^{-r/(G_n M)} - α G_n ℓ/(2r), but Eq. (49) has a positive exponential term. Moreover, applying the smearing integral (41) to a term A e^{-μ r}/r gives (A/2r) e^{μ^2 R_s^2/4}[e^{-μ r} erfc(μ R_s/2 - r/R_s) - e^{μ r} erfc(μ R_s/2 + r/R_s)]. For A=α G_n M/2, Eq. (50)'s coefficient α G_n M/(2r) is twice the correct A/(2r); the ℓ term in Eq. (50) has the wrong sign. Consequently, as R_s→0, f_H in Eq. (52) tends to +α G_n M/r e^{-r/(G_n M)} + α G_n ℓ/r, not the negative signs of Eq. (27). All derived quantities (Eqs. (59), (61), (77), (93)) inherit this error.
  2. [Abstract and Section VII] The abstract states "we employ the WKB approximation to show that the coherent quantum GD hairy black hole has a quasinormal mode spectrum that differs from those of both the classical GD hairy and Schwarzschild black holes". The body contains no WKB or quasinormal-mode computation: no perturbed field equation, no WKB formula, no comparison with GD hairy or Schwarzschild spectra. The title's "scalar ringdown" is similarly absent. This promised result must either be added or the claim removed.
  3. [Section IV, Eqs. (61)–(73)] The text after Eq. (63) claims that "all components of the effective stress-energy tensor (60) remain finite as r→0", but Eq. (71) shows ρ(r) ~ [ ... ] R_s^2/r^2 as r→0, which diverges unless the coefficient vanishes. The text immediately after Eq. (71) acknowledges this. The same incorrect statement is repeated in the Conclusions. Regularity of the stress tensor is conditional on Eq. (85), not generic; this contradiction must be removed.
minor comments (3)
  1. [Section V, Eq. (86)] Equation (86) does not follow from the small R_s/R_M expansion of Eq. (85). For α=0, Eq. (85) gives R_s/R_M = √π R_Q^2/R_M^2, whereas Eq. (86) has an extra denominator factor 1 - 1/(2e^2) and an α in the numerator. Please re-derive or correct.
  2. [Section III, after Eq. (53)] There is a duplicated word: "two-dimensional profile of the the coherent quantum GD hairy metric function" should read "of the coherent ...".
  3. [Section IV, opening] The phrase "Conversely to the classical RN case" should be "In contrast to the classical RN case" for clarity.

Circularity Check

1 steps flagged

Partial circularity: the regular core is an input of the Gaussian regulator, but the hairy/photon-ring results are parametric consequences; the paper also has an internal mean-field inconsistency and an unsupported WKB claim.

specific steps
  1. self definitional [Section III, Eqs. (40)–(47)]
    "To cure this issue, one introduces a finite smearing scale R_s, interpreted as the characteristic size of the quantum core, and adopts a Gaussian regulator rather than a sharp UV cutoff... This smoothly suppresses high-momentum modes while preserving the large-distance behavior of the classical solution. ... Physically, this replaces point-like mass and charge distributions with Gaussian profiles, removing the UV modes responsible for the classical curvature singularity."

    The regular core is injected through the regulator: Eq. (41) defines the quantum potential as the classical potential convolved with e^{-k^2 R_s^2/4}, and Eqs. (44)/(47) then show the 1/r and 1/r^2 terms become constants. The abstract's claim that the resulting geometry is free of curvature singularities is therefore true by construction of the smearing, not derived from coherent-state dynamics; no independent equation fixes R_s from the graviton occupation number. Since R_s, alpha, ell, and Q remain free parameters, the photon-ring and lensing curves are parametric outputs, not circular fits, so this is partial self-definitionality rather than complete reduction.

full rationale

There is no load-bearing self-citation chain: the GD hairy seed (27) is from Ref. [32] and the quantum-RN core (46) from Ref. [78], neither with the present authorship; da Rocha's self-citations occur only in contextual reviews. The central observable results (photon ring, impact parameter, deflection) follow numerically from the explicit metric with free parameters and are not fitted to those observables. The only circular element is the singularity-resolution claim, which is built into the Gaussian regulator. Two non-circular but serious issues are flagged under the reviewing rule: (i) the Rs->0 limit of Eq. (50) does not recover the classical seed—the Yukawa coefficient is a factor of 2 too large and the ell term has the wrong sign relative to Eqs. (27)/(49), so the claimed mean-field limit is internally inconsistent; (ii) the abstract's WKB quasinormal-mode claim is not substantiated anywhere in the body. These affect correctness, not circularity, so they do not raise the circularity score beyond the partial self-definitionality noted above.

Axiom & Free-Parameter Ledger

4 free parameters · 4 axioms · 2 invented entities

The central construction rests on treating Gaussian smearing as quantum gravity, on the classical GD hair parameters, and on free scales; no independent derivation fixes Rs. The regularity result is a direct consequence of the smoothing choice.

free parameters (4)
  • Rs (Gaussian smearing width / quantum core size) = not fitted; scanned e.g. Rs/RM = 0.05-0.7
    Introduced in Eq. (41) as the regulator; all short-distance modifications and the regularity claim depend on it; no relation to N or Planck length is derived.
  • alpha (gravitational-decoupling hair strength) = not fitted; varied in figures
    Controls the GD deformation in the classical seed metric (Eqs. 12, 27); inherited from [32] and left free.
  • ell (GD hair charge) = set to lower bound ell = M/e^2 in most explicit formulas
    Hair charge from Eqs. (26)-(27); the DEC lower bound Eq. (28) permits ell >= M/e^2, and the paper uses the bound to close Eqs. (61)-(77).
  • Q (electric charge) = not fitted; R_Q/RM = 0.1-0.5 in figures
    Charge in the classical RN seed; free input to the quantum construction.
axioms (4)
  • domain assumption A static spherical gravitational potential can be represented as the expectation value of a free massless scalar field operator on a coherent state.
    Section III, Eqs. (31)-(37): this is the corpuscular/coherent-state modeling postulate; no derivation from full quantum gravity.
  • ad hoc to paper A Gaussian momentum-space regulator with width Rs correctly encodes the finite size of the graviton condensate.
    Section III, Eq. (41): the regulator is introduced to cure UV divergences; the identification of Rs with the quantum core size is asserted, not derived.
  • domain assumption The classical GD hairy solution saturating the DEC (Eq. 25) with hair lower bounds (Eq. 28) is a valid seed.
    Section II, Eqs. (25)-(28): taken from [32]; the quantum construction inherits these choices.
  • domain assumption The effective stress-energy tensor can be reconstructed from the quantum-corrected metric via Einstein equations and interpreted as a physical fluid.
    Section IV, Eq. (60): standard practice in effective quantum-geometry models, but an interpretive step.
invented entities (2)
  • Quantum core (Gaussian-smoothed condensate of size Rs) no independent evidence
    purpose: Replaces the point singularity and regulates curvature invariants; source of modified photon orbits.
    No independent handle outside the model: Rs is free, so the core's existence and size cannot be falsified without fixing Rs.
  • Effective anisotropic quantum fluid no independent evidence
    purpose: Stress-energy that sources the regular metric via Einstein equations (rho, p_r, p_t).
    Derived from the metric, not independently observable.

pith-pipeline@v1.3.0-alltime-deepseek · 19751 in / 17648 out tokens · 169661 ms · 2026-08-04T05:56:37.169226+00:00 · methodology

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read the original abstract

We construct a coherent-state quantum extension of gravitational-decoupling (GD) hairy black holes, in which the classical spacetime geometry emerges as the mean-field limit of a finite graviton condensate, while quantum fluctuations provide a natural short-distance regulator. The coherent quantum GD hairy black hole metric is obtained by Gaussian smearing of the gravitational potential, with an effective width encoding the size of the quantum core. The resulting geometry is free of curvature singularities over appropriate parameter ranges and exhibits a modified horizon structure. We also investigate geodesic motion in the coherent quantum GD hairy spacetime and find significant deviations from the classical Reissner-Nordstr\"om (RN) geometry. In particular, the photon ring, critical impact parameter, and light deflection are modified by the combined effects of quantum corrections and GD hair, providing potential strong-field tests of deviations from general relativity. Finally, we employ the WKB approximation to show that the coherent quantum GD hairy black hole has a quasinormal mode spectrum that differs from those of both the classical GD hairy and Schwarzschild black holes.

Figures

Figures reproduced from arXiv: 2602.20386 by Henrique Navarro, Roldao da Rocha.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: displays the normalized effective energy density ρ/ρM (left panel) and the normalized tangential pressure pt/ρM (right panel) as functions of r/RM for different values of the hairy parameter α. As α increases, both quantities are suppressed in the inner region, while their radial profiles remain monotonic and rapidly decay toward zero at large radii. This behavior indicates that increasing α weakens the ef… view at source ↗
Figure 5
Figure 5. Figure 5: shows the radial behavior of the normalized mass function m(r)/M for different values of the hairy parameter α, with fixed Rs/RM = 0.1 and two representative values of RQ/RM. For RQ/RM = 0.1 (left panel), the normalized mass function m(r)/M increases monotonically from the origin and rapidly approaches its asymptotic value, with larger α delaying the saturation. On the other hand, for RQ/RM = 0.3 (right pa… view at source ↗
Figure 6
Figure 6. Figure 6: illustrates the relation between the normalized parameters RQ/RM and Rs/RM for different values of the hairy parameter α. In all cases, RQ/RM increases monotonically with Rs/RM, defining a well-behaved parameter space for the solutions. Increasing α shifts the curves downward, indicating that stronger GD hairy effects reduce the allowed charge scale RQ/RM for a fixed stellar radius Rs/RM. On the other hand… view at source ↗
Figure 7
Figure 7. Figure 7: shows the photon radius Rγ (in units of RM) as a function of Rs/RM for different values of the hairy parameter α, with RQ/RM = 0.1 (left panel) and RQ/RM = 0.3 (right panel). For each fixed value of α, the photon ring becomes slightly smaller as the ratio RQ/RM increases, while for fixed RQ/RM the photon radius increases with α, indicating the photon ring increases due to GD hair. At small and intermediate… view at source ↗
Figure 8
Figure 8. Figure 8: shows the critical impact parameter bc (in units of RM) as a function of Rs/RM for different values of the hairy parameter α, with RQ/RM = 0.1 (left panel) and RQ/RM = 0.3 (right panel). For fixed RQ/RM, increasing α enhances the critical impact parameter and extends the admissible range of Rs/RM before the loss of unstable photon orbits. In the weak￾hair regime, all curves approach the classical value, co… view at source ↗
Figure 9
Figure 9. Figure 9: shows the light deflection angle as a function of the normalized impact parameter b/RM for the coherent quantum GD hairy black hole, compared with the classical RN case. For fixed Rs/RM = 0.1, increasing the hairy parameter α slightly reduces the deflection angle at small impact parameters, while all curves sharply converge at large b/RM. The left and right [PITH_FULL_IMAGE:figures/full_fig_p026_9.png] view at source ↗

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Reference graph

Works this paper leans on

93 extracted references · 63 linked inside Pith

  1. [1]

    B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. D100, 104036 (2019), 1903.04467

    Pith/arXiv arXiv 2019

  2. [2]

    Ovalle, Phys

    J. Ovalle, Phys. Rev. D95, 104019 (2017), 1704.05899

    Pith/arXiv arXiv 2017

  3. [3]

    Ovalle, Phys

    J. Ovalle, Phys. Lett. B788, 213 (2019), 1812.03000

    Pith/arXiv arXiv 2019

  4. [4]

    Yousaf, K

    Z. Yousaf, K. Bamba, B. Almutairi, S. Khan, and M. Z. Bhatti, Class. Quant. Grav.41, 175001 (2024), 2407.10451

    Pith/arXiv arXiv 2024

  5. [5]

    Contreras, J

    E. Contreras, J. Ovalle, and R. Casadio, Phys. Rev. D103, 044020 (2021), 2101.08569

    Pith/arXiv arXiv 2021

  6. [6]

    Leon and C

    P. Leon and C. Las Heras, Eur. Phys. J. C83, 260 (2023)

    2023

  7. [7]

    Ramos, C

    A. Ramos, C. Arias, E. Fuenmayor, and E. Contreras, Eur. Phys. J. C81, 203 (2021), 2103.05039

    Pith/arXiv arXiv 2021

  8. [8]

    Sharif and T

    M. Sharif and T. Naseer, Phys. Dark Univ.42, 101324 (2023), 2310.00872

    Pith/arXiv arXiv 2023

  9. [9]

    Morales and F

    E. Morales and F. Tello-Ortiz, Eur. Phys. J.C78, 841 (2018), 1808.01699

    Pith/arXiv arXiv 2018

  10. [10]

    Panotopoulos and A

    G. Panotopoulos and A. Rinc´ on, Eur. Phys. J.C78, 851 (2018), 1810.08830

    Pith/arXiv arXiv 2018

  11. [11]

    K. N. Singh, S. K. Maurya, M. K. Jasim, and F. Rahaman, Eur. Phys. J. C79, 851 (2019). 29

    2019

  12. [12]

    Gavassino and J

    L. Gavassino and J. Noronha, Phys. Rev. D109, 096040 (2024), 2305.04119

    Pith/arXiv arXiv 2024

  13. [13]

    Berkimbayev, Eur

    D. Berkimbayev, Eur. Phys. J. C86, 55 (2026), 2511.18485

    arXiv 2026

  14. [14]

    C. L. Heras and P. Leon, Fortsch. Phys.66, 1800036 (2018), 1804.06874

    Pith/arXiv arXiv 2018

  15. [15]

    V. A. Torres-S´ anchez and E. Contreras, Eur. Phys. J.C79, 829 (2019), 1908.08194

    Pith/arXiv arXiv 2019

  16. [16]

    Hensh and Z

    S. Hensh and Z. Stuchl ´ ık, Eur. Phys. J. C79, 834 (2019), 1906.08368

    Pith/arXiv arXiv 2019

  17. [17]

    S. K. Maurya, G. Mustafa, S. Ray, B. Dayanandan, A. Aziz, and A. Errehymy, Phys. Dark Univ.42, 101284 (2023)

    2023

  18. [18]

    Iqbal, M

    N. Iqbal, M. Amir, M. Alshammari, W. W. Mohammed, and M. Ilyas, Eur. Phys. J. C85, 428 (2025)

    2025

  19. [19]

    Tello-Ortiz, P

    F. Tello-Ortiz, P. Bargue˜ no, A. Alvarez, and E. Contreras, Fortsch. Phys.71, 2200170 (2023)

    2023

  20. [20]

    Khatoon, I

    M. Khatoon, I. Mahmood, H. Sohail, A. Ditta, H. O. Elansary, X.-Y. Liu, A. Ashraf, and S. Mannanova, Eur. Phys. J. C85, 1102 (2025)

    2025

  21. [21]

    Zubair, H

    M. Zubair, H. Sohail, S. Waheed, A. Ilyas, and I. Mahmood, Chin. Phys. C50(2026)

    2026

  22. [22]

    da Rocha, Phys

    R. da Rocha, Phys. Rev. D95, 124017 (2017), 1701.00761

    Pith/arXiv arXiv 2017

  23. [23]

    Meert and R

    P. Meert and R. da Rocha, Nucl. Phys. B967, 115420 (2021), 2006.02564

    Pith/arXiv arXiv 2021

  24. [24]

    Casadio, P

    R. Casadio, P. Nicolini, and R. da Rocha, Class. Quant. Grav.35, 185001 (2018), 1709.09704

    Pith/arXiv arXiv 2018

  25. [25]

    da Rocha and J

    R. da Rocha and J. M. Hoff da Silva, Phys. Rev. D85, 046009 (2012), 1202.1256

    Pith/arXiv arXiv 2012

  26. [26]

    da Rocha and A

    R. da Rocha and A. A. Tomaz, Eur. Phys. J. C80, 857 (2020), 2005.02980

    Pith/arXiv arXiv 2020

  27. [27]

    R. T. Cavalcanti, A. G. da Silva, and R. da Rocha, Class. Quant. Grav.33, 215007 (2016), 1605.01271

    Pith/arXiv arXiv 2016

  28. [28]

    X.-J. Wang, Y. Meng, X.-M. Kuang, and K. Liao, Phys. Rev. D112, 124016 (2025), 2508.02355

    arXiv 2025

  29. [29]

    Liang, X

    Y. Liang, X. Lyu, and J. Tao, Commun. Theor. Phys.76, 085402 (2024)

    2024

  30. [30]

    Gabbanelli, A

    L. Gabbanelli, A. Rinc´ on, and C. Rubio, Eur. Phys. J. C78, 370 (2018), 1802.08000

    Pith/arXiv arXiv 2018

  31. [31]

    Li, Phys

    Z. Li, Phys. Lett. B841, 137902 (2023), 2212.08112

    Pith/arXiv arXiv 2023

  32. [32]

    Ovalle, R

    J. Ovalle, R. Casadio, E. Contreras, and A. Sotomayor, Phys. Dark Univ.31, 100744 (2021), 2006.06735

    Pith/arXiv arXiv 2021

  33. [33]

    Rinc´ on, L

    A. Rinc´ on, L. Gabbanelli, E. Contreras, and F. Tello-Ortiz, Eur. Phys. J. C79, 873 (2019), 1909.00500

    Pith/arXiv arXiv 2019

  34. [34]

    Avalos, P

    R. Avalos, P. Bargue˜ no, and E. Contreras, Fortsch. Phys.2023, 2200171 (2023), 2303.04119

    Pith/arXiv arXiv 2023

  35. [35]

    A. M. Albalahi, Z. Yousaf, A. Ali, and S. Khan, Eur. Phys. J. C84, 9 (2024)

    2024

  36. [36]

    Sharif, T

    M. Sharif, T. Naseer, and H. Shadab, Chin. J. Phys.97, 1386 (2025)

    2025

  37. [37]

    Andrade, D

    J. Andrade, D. Santana, T. Naseer, E. P. Valdiviezod, and D. T. C. Ortiz, Eur. Phys. J. C85, 1174 (2025)

    2025

  38. [38]

    H. L. Prihadi, D. Dwiputra, F. Khairunnisa, and F. P. Zen, Eur. Phys. J. C85, 946 (2025), 2501.01680

    Pith/arXiv arXiv 2025

  39. [39]

    S. K. Maurya, M. K. Jasim, A. Errehymy, K. Boshkayev, G. Mustafa, and B. Dayanandan, Phys. Dark Univ.46, 101665 (2024)

    2024

  40. [40]

    O. A. Almatroud, M. Rizwan, M. Alshammari, M. Z. Bhatti, S. Alshammari, and Z. Yousaf, Eur. Phys. J. C85, 1285 (2025)

    2025

  41. [41]

    S. K. Maurya, F. Al Khayari, A. Ashraf, M. K. Jasim, S. T. T., and P. Channuie, Chin. J. Phys.96, 621 (2025)

    2025

  42. [42]

    Zhang, M

    C.-M. Zhang, M. Zhang, and D.-C. Zou, Chin. Phys. C47, 015106 (2023), 2208.06830

    Pith/arXiv arXiv 2023

  43. [43]

    J. Lin, M. Bravo-Gaete, and X. Zhang, Phys. Rev. D109, 104039 (2024), 2401.02045

    Pith/arXiv arXiv 2024

  44. [44]

    V. F. Guimar˜ aes, R. T. Cavalcanti, and R. da Rocha, Class. Quant. Grav.42, 175011 (2025), 2506.20044

    Pith/arXiv arXiv 2025

  45. [45]

    R. T. Cavalcanti, R. C. de Paiva, and R. da Rocha, Eur. Phys. J. Plus137, 1185 (2022), 2203.08740

    Pith/arXiv arXiv 2022

  46. [46]

    G. P. Ribeiro, R. B. Magalh˜ aes, and L. C. B. Crispino, Phys. Rev. D112, 124082 (2025), 2512.04377

    arXiv 2025

  47. [47]

    Y. Yang, D. Liu, A. ¨Ovg¨ un, Z.-W. Long, and Z. Xu, Phys. Rev. D107, 064042 (2023), 2203.11551

    Pith/arXiv arXiv 2023

  48. [48]

    Avalos and E

    R. Avalos and E. Contreras, Eur. Phys. J. C83, 155 (2023), 2302.09148. 30

    Pith/arXiv arXiv 2023

  49. [49]

    Al-Badawi, S

    A. Al-Badawi, S. K. Jha, and A. Rahaman, Eur. Phys. J. C84, 145 (2024)

    2024

  50. [50]

    Tello-Ortiz, R

    F. Tello-Ortiz, R. Avalos, Y. G´ omez-Leyton, and E. Contreras, Phys. Dark Univ.46, 101547 (2024)

    2024

  51. [51]

    Ditta, F

    A. Ditta, F. Javed, S. K. Maurya, G. Mustafa, and F. Atamurotov, Phys. Dark Univ.42, 101345 (2023)

    2023

  52. [52]

    Mansour, T

    N. Mansour, T. Toghrai, A. El Boukili, A. Benami, A. K. Daoudia, and M. B. Sedra, Int. J. Mod. Phys. A 39, 2450151 (2024)

    2024

  53. [53]

    Mahapatra and I

    S. Mahapatra and I. Banerjee, Phys. Dark Univ.39, 101172 (2023), 2208.05796

    Pith/arXiv arXiv 2023

  54. [54]

    Misyura, A

    M. Misyura, A. Rincon, and V. Vertogradov, Phys. Dark Univ.46, 101717 (2024), 2405.05370

    Pith/arXiv arXiv 2024

  55. [55]

    Astefanesei, R

    D. Astefanesei, R. Ballesteros, P. Cabrera, G. Casanova, and R. Rojas, Phys. Rev. D110, 024045 (2024), 2404.15566

    Pith/arXiv arXiv 2024

  56. [56]

    Rehman and G

    H. Rehman and G. Abbas, Chin. Phys. C47, 125106 (2023)

    2023

  57. [57]

    Casadio, A

    R. Casadio, A. Giusti, I. Kuntz, and G. Neri, Phys. Rev. D103, 064001 (2021), 2101.12471

    Pith/arXiv arXiv 2021

  58. [58]

    Casadio, A

    R. Casadio, A. Giugno, A. Giusti, and M. Lenzi, Phys. Rev. D96, 044010 (2017), 1702.05918

    Pith/arXiv arXiv 2017

  59. [59]

    Casadio, Int

    R. Casadio, Int. J. Mod. Phys. D31, 2250128 (2022), 2103.00183

    Pith/arXiv arXiv 2022

  60. [60]

    M¨ uck, Eur

    W. M¨ uck, Eur. Phys. J. C76, 374 (2016), 1606.01790

    Pith/arXiv arXiv 2016

  61. [61]

    Casadio, A

    R. Casadio, A. Giugno, and A. Orlandi, Phys. Rev. D91, 124069 (2015), 1504.05356

    Pith/arXiv arXiv 2015

  62. [62]

    Giusti, Int

    A. Giusti, Int. J. Geom. Meth. Mod. Phys.16, 1930001 (2019)

    2019

  63. [63]

    Casadio, A

    R. Casadio, A. Giugno, and A. Giusti, Phys. Lett. B763, 337 (2016), 1606.04744

    Pith/arXiv arXiv 2016

  64. [64]

    Dvali and C

    G. Dvali and C. Gomez, Fortsch. Phys.61, 742 (2013), 1112.3359

    Pith/arXiv arXiv 2013

  65. [65]

    Dvali and C

    G. Dvali and C. Gomez, Phys. Lett. B719, 419 (2013), 1203.6575

    Pith/arXiv arXiv 2013

  66. [66]

    Flassig, A

    D. Flassig, A. Pritzel, and N. Wintergerst, Phys. Rev. D87, 084007 (2013), 1212.3344

    Pith/arXiv arXiv 2013

  67. [67]

    Hofmann and T

    S. Hofmann and T. Rug, Nucl. Phys. B902, 302 (2016), 1403.3224

    Pith/arXiv arXiv 2016

  68. [68]

    Casadio and A

    R. Casadio and A. Orlandi, JHEP08, 025 (2013), 1302.7138

    Pith/arXiv arXiv 2013

  69. [69]

    Casadio, R

    R. Casadio, R. T. Cavalcanti, A. Giugno, and J. Mureika, Phys. Lett. B760, 36 (2016), 1509.09317

    Pith/arXiv arXiv 2016

  70. [70]

    Casadio, A

    R. Casadio, A. Giusti, and J. Ovalle, Phys. Rev. D105, 124026 (2022), 2203.03252

    Pith/arXiv arXiv 2022

  71. [71]

    Cadoni, R

    M. Cadoni, R. Casadio, A. Giusti, and M. Tuveri, Phys. Rev. D97, 044047 (2018), 1801.10374

    Pith/arXiv arXiv 2018

  72. [72]

    Casadio, Ukr

    R. Casadio, Ukr. J. Phys.69, 466 (2024)

    2024

  73. [73]

    W. Feng, R. da Rocha, and R. Casadio, Eur. Phys. J. C84, 586 (2024), 2401.14540

    Pith/arXiv arXiv 2024

  74. [74]

    Calmet, R

    X. Calmet, R. Casadio, S. D. H. Hsu, and F. Kuipers, Phys. Rev. D108, 086012 (2023), 2305.09466

    Pith/arXiv arXiv 2023

  75. [75]

    Casadio, R

    R. Casadio, R. da Rocha, A. Giusti, and P. Meert, Phys. Rev. D110, 104067 (2024), 2407.04146

    Pith/arXiv arXiv 2024

  76. [76]

    Casadio, R

    R. Casadio, R. da Rocha, A. Giusti, and P. Meert, Phys. Lett. B849, 138466 (2024), 2310.07505

    Pith/arXiv arXiv 2024

  77. [77]

    W. Feng, A. Giusti, and R. Casadio, Eur. Phys. J. Plus140, 145 (2025), 2408.17091

    Pith/arXiv arXiv 2025

  78. [78]

    This geometry is a particular case of coherent quantum GD hairy black holes forα= 0 andℓ= 0

    investigated how a purely coherent quantum RN black hole geometry affects observational properties of photons. This geometry is a particular case of coherent quantum GD hairy black holes forα= 0 andℓ= 0. Ref. [80] considered a similar approach for a quantum Schwarzchild- like geometry and analyzed the dynamics of both massive and massless particles. The s...

    2025

  79. [79]

    Antonelli, M

    T. Antonelli, M. Sebastianutti, and A. Giusti, Eur. Phys. J. C85, 1219 (2025), 2506.02231

    arXiv 2025

  80. [80]

    Meert, A

    P. Meert, A. Giusti, and R. Casadio, Phys. Lett. B867, 139613 (2025), 2504.02786

    Pith/arXiv arXiv 2025

Showing first 80 references.