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REVIEW 2 major objections 5 minor 83 references

Quasinormal modes and grey-body factors of Morris-Thorne wormholes

T0 review · 2 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper derives analytic sixth-order WKB series for quasinormal frequencies and grey-body factors of three Morris-Thorne wormhole families and claims they satisfy the eikonal shadow-radius correspondence.

desk verdict Useful sixth-order WKB formulas for wormhole QNMs, but the shadow-correspondence check and grey-body eikonal limit are internally inconsistent as written. read the letter →

arxiv 2412.13385 v1 pith:DRB6SA2J submitted 2024-12-17 gr-qc

Zainab Malik This is my paper · ORCID
classification gr-qc PACS 04.30.Nk04.50.+h
keywords quasinormalmodesgrey-bodyfactorsMorris-ThornewormholesWKBapproximationeikonalwormholeshadoweffectivepotentialtraversable
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

The paper claims that for a broad class of Morris-Thorne wormholes, the perturbation effective potential has its maximum exactly at the throat, and that this single fact makes the sixth-order WKB method fully analytic. Working from that premise, it derives closed-form expansions, in powers of the inverse multipole number $\kappa=\ell+1/2$, for electromagnetic and scalar quasinormal frequencies, together with matching grey-body factors $\Gamma_\ell(\Omega)=1/(1+e^{2\pi i K})$, for three wormhole families: the Ellis-Bronnikov wormhole, a tideless model with shape function $b(r)=\sqrt{b_0 r}$, and a family $b(r)=b_0(b_0/r)^q$ with optional redshift function $1/r^p$. In the eikonal limit the frequencies reduce to compact forms whose real parts are proportional to $\kappa$, and the paper concludes that they satisfy the correspondence $\omega_R=(\ell+1/2)/R_s$ with the wormhole shadow radius. If this is right, it provides an analytic bridge between wormhole ringdown, grey-body transmission spectra, and shadow measurements, which could help distinguish wormholes from black holes observationally.

What carries the argument

The central object is the effective potential $V(r)$ of the scalar and electromagnetic perturbations. For the Morris-Thorne class considered, its maximum lies exactly at the throat radius $r=b_0$, and this peak location is known exactly rather than found by an expansion. The machinery is the sixth-order WKB formula for quasinormal modes, whose correction terms $\Lambda_i$ depend on derivatives of $V$ at the peak; because the peak is at $b_0$, all derivatives can be evaluated in closed form, giving the presented series in $\kappa=\ell+1/2$. For grey-body factors the same peak data enter the phase $K$ in $\Gamma_\ell(\Omega)=1/(1+e^{2\pi i K})$, kept unexpanded in inverse multipole number, and the eikonal interpretation is carried by the shadow correspondence $\omega_R=(\ell+1/2)/R_s$.

What would settle it

Integrate the null geodesic equation in the Ellis-Bronnikov metric ($\Phi=0$, $b(r)=b_0^2/r$) to find the critical impact parameter; if the resulting shadow radius is not $b_0$, then the paper's own eikonal frequency $\omega_R=(\ell+1/2)/b_0$ cannot satisfy the shadow correspondence $\omega_R=(\ell+1/2)/R_s$, and the claimed extension fails.

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Extended reading notes

Core claim

The paper claims that the throat-peak property of Morris-Thorne wormholes turns the usually numerical WKB calculation into an analytic one. It presents explicit sixth-order expansions for the quasinormal frequencies of scalar and electromagnetic perturbations of the Ellis-Bronnikov wormhole, of the tideless wormhole with $b(r)=\sqrt{b_0 r}$, and of the family $b(r)=b_0(b_0/r)^q$ with $\Phi(r)=1/r^p$ (both tideless and redshifted variants). It gives the corresponding grey-body factors through the compact formula $\Gamma_\ell(\Omega)=1/(1+e^{2\pi i K})$, where $K$ is obtained without an additional inverse-multipole expansion. In the eikonal limit $\ell\to\infty$ the frequencies reduce to simple forms whose real parts are $\kappa$ divided by a radius, and the paper concludes that these satisfy $\omega_R=(\ell+1/2)/R_s$ with $R_s$ the wormhole shadow radius. The intended upshot is that wormhole ringdown and shadow observations probe the same geometric quantity.

Load-bearing premise

The load-bearing premise is that the shadow-radius formula used to check the eikonal correspondence is the right one; for the tideless wormholes it gives no finite value, so the claimed agreement depends on correcting or replacing it.

Editorial extensions

If this is right

  • The sixth-order expansions give immediate closed-form approximations for scalar and electromagnetic ringdown of the three wormhole families, removing the need for numerical integration at high multipoles and low overtones.
  • The grey-body factors $\Gamma_\ell(\Omega)=1/(1+e^{2\pi i K})$ become explicit functions of frequency and wormhole parameters, so transmission and reflection spectra can be compared directly with black-hole templates.
  • The eikonal relation $\omega_R=(\ell+1/2)/R_s$ turns a measured high-frequency ringdown into a direct estimate of the wormhole shadow radius, and vice versa, for these spacetimes.
  • The same throat-peak argument can generate analytic quasinormal modes and grey-body factors for any Morris-Thorne throat where the effective potential peaks at $r=b_0$, making the method a template for other shape and redshift functions.
  • For the redshifted family the eikonal frequency is multiplied by the throat redshift factor, so the ringdown carries direct information about the lapse function at the throat.

Reading between the lines

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

  • Beyond the paper: the method's WKB grey-body-factor ansatz excludes superradiant amplification, so extending the calculation to rotating wormholes will need a different approach when the reflection coefficient exceeds unity.
  • Beyond the paper: the non-tideless eikonal frequency is rescaled by the throat redshift factor; if the shadow correspondence is exact, the effective shadow radius should be the throat radius divided by that factor, a prediction ray tracing could test.
  • Beyond the paper: the closed-form grey-body factors could be integrated against a thermal spectrum to predict the wormhole's emission signature, which the paper does not do; that would turn the analytic transmission coefficients into an observational template.
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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

2 major / 5 minor

Summary. The paper derives sixth-order WKB analytic expressions for the quasinormal modes and grey-body factors of three Morris-Thorne wormhole models (Ellis-Bronnikov, a tideless model with b(r)=sqrt(b0 r), and a model with non-zero tidal force). The derivation exploits the claim that the effective potential maximum lies exactly at the throat, which simplifies the WKB expansion. The paper further claims that in the eikonal limit the quasinormal frequencies satisfy the shadow-radius correspondence omega_R = (l+1/2)/R_s. The central claims are the accuracy of the analytic expressions and the eikonal correspondence.

Significance. The idea of using the exact location of the potential maximum at the throat to obtain compact analytic WKB formulas is attractive and could be useful for phenomenological studies of wormhole ringdown and absorption. If correct, the eikonal expressions would provide a simple link between wormhole geometry and observable gravitational-wave signatures. However, the manuscript contains load-bearing inconsistencies in the grey-body factor formulas and in the shadow-radius verification, so the claimed results are not supported as written.

major comments (2)
  1. [V.A, Eq. (19)] The full WKB relation for K that determines the grey-body factor is inconsistent with the eikonal limit reported in Sec. V.D. Setting Omega=0 in Eq. (19) and keeping the leading term in kappa gives i K approximately -0.943/kappa, i.e. K approximately +i 0.943/kappa, so the transmission probability Gamma_l(0) is approximately 1/2 for large l. In contrast, the eikonal expression K = i(b0^2 Omega^2 - kappa^2)/(2 kappa) gives K = -i kappa/2 at Omega=0, which yields Gamma_l(0) tending to 0. The contradiction shows that the grey-body factor formulas in Eqs. (19) and (21) do not reduce to the claimed eikonal limit and are not reliable for the stated purpose.
  2. [V.D, Eq. (22)] The shadow radius formula R_s = b0/Phi(b0) is incorrect for the models studied. For the tideless models (Phi=0) the denominator vanishes, while the eikonal QNMs in Sec. V.D imply R_s = b0. For model 3, with b0=1 and Phi(r)=1/r^p, the formula gives R_s = 1, whereas the eikonal frequency omega_R = e kappa/b0 requires R_s = b0/e = e^{-1}. The correct impact parameter for null geodesics in this metric is R_s = b0 e^{-Phi(b0)}. Thus the claimed verification of the eikonal QNM-shadow correspondence is unsupported.
minor comments (5)
  1. [Sec. II] The assertion that the effective potential has its maximum at the throat is used throughout but is not proved for the potentials in Eqs. (9)-(10); a short argument or a citation would make the paper more self-contained.
  2. [Sec. V.D] The symbol e appears without definition in the eikonal formulas for model 3; it should be introduced explicitly, for example e = exp(Phi(b0)), to avoid confusion with the base of the natural logarithm.
  3. [Sec. V.D] The equation numbers for the definitions of alpha and beta (printed as Eqs. (18) and (19)) collide with the equation numbers used for the grey-body relations in Sec. V.A; the numbering should be corrected.
  4. [Throughout] There are several typos, including 'disccussed' (Sec. I), 'other other method' (Sec. III), 'eseentially' (Sec. IV), 'becuase' (Sec. IV), and 'does require' in Sec. IV which should read 'does not require'.
  5. [Abstract and Sec. V] The abstract claims 'accurate' analytic expressions, but the manuscript provides no numerical comparison with exact or independent results; in light of the inconsistency in Eq. (19), the accuracy claim needs either a correction of the formulas or a benchmark validation.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the WKB derivation and shadow-correspondence check use external standard results; the paper's self-citations are not load-bearing.

full rationale

The derivation chain is self-contained. The quasinormal-mode condition (Eq. 12) is the standard higher-order WKB formula taken from Schutz-Will, Iyer-Will, Konoplya, and Matyjasek-Opala (refs. 30-33), and the grey-body ansatz (Eq. 15) is the standard transmission formula; neither is defined in terms of the wormhole results being derived. The location of the maximum of the effective potential at the throat is read off directly from the potentials in Eqs. (9)-(10), and the analytic expansions in Sec. V are applications of the WKB formula to the specific shape and lapse functions. The eikonal forms in Sec. V.D are Taylor limits of those expansions, not fits to the shadow data. The shadow-correspondence check is an external benchmark: Eq. (29) is the known relation and the shadow radius is not used as an input in computing the QNMs. The self-citations (refs. 48-49, 59-60) are cited only as methodological analogies for expanding in inverse multipole number, and the WKB input itself is standard external machinery. A separate correctness issue exists in Sec. V.D: the quoted shadow radius R_s = b0/Phi(b0) diverges for the tideless models and does not match the eikonal QNMs for model 3, so the claimed correspondence check is not demonstrated. That inconsistency is a mathematical error rather than a circular reduction, because the shadow formula is not an input to the QNM derivation. No step in the paper reduces a prediction to an input by construction.

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

The paper introduces no free parameters or new entities. It relies on standard WKB results and the geometric properties of the chosen wormhole metrics.

assumptions (4)
  • domain assumption Morris-Thorne metric ansatz (Eq. 1) with shape function b(r) and redshift function Φ(r) satisfying conditions (3)-(5)
    The entire analysis is restricted to this class of wormholes from Ref. [1].
  • domain assumption The effective potential maximum is located at the throat r=b0 for the considered wormholes and fields
    Used in Secs. III and IV to evaluate WKB derivatives at the throat. This is verified for the three explicit models but asserted for a 'broad class'.
  • standard math Higher-order WKB quantization condition Eq. (12) and grey-body factor ansatz Eq. (15) from Refs. [30-33] and [28,53-57]
    The paper relies on these standard formulas without re-deriving them.
  • domain assumption Eikonal shadow correspondence ω_R = (ℓ+1/2)/R_s from Ref. [14]
    Used in Sec. V.D as a consistency check. The paper's own shadow radius formula is likely a typo.

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Pith. "Pith review of Quasinormal modes and grey-body factors of Morris-Thorne wormholes." pith.science (2026). https://pith.science/paper/DRB6SA2J

@misc{pith2026241213385,
  author       = {Pith},
  title        = {Pith review of: Quasinormal modes and grey-body factors of Morris-Thorne wormholes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DRB6SA2J}},
  note         = {Machine review of arXiv:2412.13385}
}
read the original abstract

Using the fact that, for a broad class of Morris-Thorne wormholes, the maximum of the effective potential is located at the throat, we derive accurate analytic WKB expressions for the quasinormal modes and grey-body factors of various traversable wormholes. In the eikonal limit, these analytic expressions acquire a compact form and satisfy the correspondence between the quasinormal modes and the radii of the wormhole shadows.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

83 extracted references · 3 canonical work pages

  1. [1]

    M. S. Morris and K. S. Thorne, American Journal of Physics 56, 395 (1988)

    1988

  2. [2]

    Dutta Roy, S

    P. Dutta Roy, S. Aneesh, and S. Kar, Eur. Phys. J. C 80, 850 (2020), 1910.08746

    arXiv 2020

  3. [3]

    B. Azad, J. L. Bl´ azquez-Salcedo, X. Y. Chew, J. Kunz, and D.-h. Yeom, Phys. Rev. D 107, 084024 (2023), 2212.12601

    arXiv 2023

  4. [4]

    J. L. Bl´ azquez-Salcedo, X. Y. Chew, and J. Kunz, Phys. Rev. D 98, 044035 (2018), 1806.03282

    arXiv 2018

  5. [5]

    K. A. Bronnikov, R. A. Konoplya, and T. D. Pappas, Phys. Rev. D 103, 124062 (2021), 2102.10679

    arXiv 2021

  6. [6]

    M. S. Churilova, R. A. Konoplya, Z. Stuchlik, and A. Zhi- denko, JCAP 10, 010 (2021), 2107.05977

    arXiv 2021

  7. [7]

    Oliveira, D

    R. Oliveira, D. M. Dantas, V. Santos, and C. A. S. Almeida, Class. Quant. Grav. 36, 105013 (2019), 1812.01798

    arXiv 2019

  8. [8]

    A Unified Spectral Approach for Quasinormal Modes of Morris-Thorne Wormholes

    D. Batic and D. Dutykh, Class. Quant. Grav. 41, 215003 (2024), 2410.05979

    work page Pith review arXiv 2024

Show all 83 references
  1. [9]

    Ou, M.-Y

    M.-Y. Ou, M.-Y. Lai, and H. Huang, Eur. Phys. J. C 82, 452 (2022), 2111.13890

    2022 arXiv

  2. [10]

    M. S. Churilova, R. A. Konoplya, and A. Zhidenko, Phys. Lett. B 802, 135207 (2020), 1911.05246

    2020 arXiv

  3. [11]

    Zhang, A

    C. Zhang, A. Wang, and T. Zhu, JCAP 05, 059 (2023), 2303.08399

    2023 arXiv

  4. [12]

    Azad, Lect

    B. Azad, Lect. Notes Phys. 1022, 3 (2023)

    2023

  5. [13]

    P. D. Roy, Eur. Phys. J. C 82, 673 (2022), 2110.05019

    2022 arXiv

  6. [14]

    Jusufi, Gen

    K. Jusufi, Gen. Rel. Grav. 53, 87 (2021), 2007.16019

    2021 arXiv

  7. [15]

    Mitra and S

    A. Mitra and S. Ghosh, Phys. Rev. D 109, 064005 (2024), 2310.17479

    2024 arXiv

  8. [16]

    D. J. Gogoi and U. D. Goswami, JCAP 02, 027 (2023), 2208.07055

    2023 arXiv

  9. [17]

    Alfaro, P

    S. Alfaro, P. A. Gonz´ alez, D. Olmos, E. Papantonopou- los, and Y. V´ asquez, Phys. Rev. D 109, 104009 (2024), 2402.15575

    2024 arXiv

  10. [18]

    K. A. Bronnikov and R. A. Konoplya, Phys. Rev. D 101, 064004 (2020), 1912.05315

    2020 arXiv

  11. [19]

    B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. Lett. 116, 061102 (2016), 1602.03837

    2016 arXiv

  12. [20]

    B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. Lett. 119, 161101 (2017), 1710.05832

    2017 arXiv

  13. [21]

    Abbott et al

    R. Abbott et al. (LIGO Scientific, Virgo), Astrophys. J. Lett. 896, L44 (2020), 2006.12611

    2020 arXiv

  14. [22]

    Oshita, Phys

    N. Oshita, Phys. Rev. D 109, 104028 (2024), 2309.05725

    2024 arXiv

  15. [23]

    R. A. Konoplya and A. Zhidenko, Phys. Rev. D 81, 124036 (2010), 1004.1284

    2010 arXiv

  16. [24]

    R. F. Rosato, K. Destounis, and P. Pani (2024), 2406.01692

    2024 arXiv

  17. [25]

    R. A. Konoplya and A. Zhidenko, JCAP 09, 068 (2024), 2406.11694

    2024 arXiv

  18. [26]

    Oshita, K

    N. Oshita, K. Takahashi, and S. Mukohyama, Phys. Rev. D 110, 084070 (2024), 2406.04525

    2024 arXiv

  19. [27]

    Dubinsky (2024), 2412.00625

    A. Dubinsky (2024), 2412.00625

    2024

  20. [28]

    Skvortsova (2024), 2411.06007

    M. Skvortsova (2024), 2411.06007

    2024 arXiv

  21. [29]

    S. V. Bolokhov and M. Skvortsova (2024), 2412.11166

    2024 arXiv

  22. [30]

    B. F. Schutz and C. M. Will, Astrophysical Journal 291, L33 (1985)

    1985

  23. [31]

    Iyer and C

    S. Iyer and C. M. Will, Phys. Rev. D 35, 3621 (1987)

    1987

  24. [32]

    R. A. Konoplya, Phys. Rev. D 68, 024018 (2003), gr- qc/0303052

    2003

  25. [33]

    Matyjasek and M

    J. Matyjasek and M. Opala, Phys. Rev. D 96, 024011 (2017), 1704.00361

    2017 arXiv

  26. [34]

    R. A. Konoplya and C. Molina, Phys. Rev. D 71, 124009 (2005), gr-qc/0504139

    2005 arXiv

  27. [35]

    A. F. Zinhailo, Eur. Phys. J. C 79, 912 (2019), 1909.12664

    2019 arXiv

  28. [36]

    Dubinsky (2024), 2405.13552

    A. Dubinsky (2024), 2405.13552

    2024 arXiv

  29. [37]

    R. A. Konoplya and E. Abdalla, Phys. Rev. D 71, 084015 (2005), hep-th/0503029

    2005 arXiv

  30. [38]

    Skvortsova, Grav

    M. Skvortsova, Grav. Cosmol. 30, 279 (2024), 2405.15807

    2024 arXiv

  31. [39]

    G. Guo, P. Wang, H. Wu, and H. Yang, JHEP 07, 070 (2023), 2301.06483

    2023 arXiv

  32. [40]

    S. V. Bolokhov, Phys. Lett. B 856, 138879 (2024), 2310.12326

    2024 arXiv

  33. [41]

    Paul, Eur

    P. Paul, Eur. Phys. J. C 84, 218 (2024), 2312.16479

    2024 arXiv

  34. [42]

    M. A. Cuyubamba, R. A. Konoplya, and A. Zhidenko, Phys. Rev. D 93, 104053 (2016), 1604.03604

    2016 arXiv

  35. [43]

    K. D. Kokkotas, R. A. Konoplya, and A. Zhidenko, Phys. Rev. D 83, 024031 (2011), 1011.1843

    2011 arXiv

  36. [44]

    H. Gong, S. Li, D. Zhang, G. Fu, and J.-P. Wu (2023), 2312.17639

    2023 arXiv

  37. [45]

    Zhidenko, Phys

    A. Zhidenko, Phys. Rev. D 78, 024007 (2008), 0802.2262

    2008 arXiv

  38. [46]

    A. F. Zinhailo, Eur. Phys. J. C 78, 992 (2018), 1809.03913

    2018 arXiv

  39. [47]

    Skvortsova, Fortsch

    M. Skvortsova, Fortsch. Phys. 72, 2400132 (2024), 2405.06390

    2024 arXiv

  40. [48]

    Malik, EPL 147, 69001 (2024), 2410.04306

    Z. Malik, EPL 147, 69001 (2024), 2410.04306

    2024 arXiv

  41. [49]

    Malik, Z

    Z. Malik, Z. Naturforsch. A 79, 1063 (2024)

    2024

  42. [50]

    S. V. Bolokhov, Phys. Rev. D 110, 024010 (2024), 2311.05503

    2024 arXiv

  43. [51]

    Dubinsky and A

    A. Dubinsky and A. Zinhailo, Eur. Phys. J. C 84, 847 (2024), 2404.01834. 8

    2024 arXiv

  44. [52]

    R. A. Konoplya, Gen. Rel. Grav. 34, 329 (2002), gr- qc/0109096

    2002

  45. [53]

    R. A. Konoplya, D. Ovchinnikov, and B. Ahmedov, Phys. Rev. D 108, 104054 (2023), 2307.10801

    2023 arXiv

  46. [54]

    Dubinsky and A

    A. Dubinsky and A. F. Zinhailo (2024), 2410.15232

    2024 arXiv

  47. [55]

    Toshmatov, A

    B. Toshmatov, A. Abdujabbarov, Z. Stuchl ´ ık, and B. Ahmedov, Phys. Rev. D 91, 083008 (2015), 1503.05737

    2015 arXiv

  48. [56]

    S. V. Bolokhov and R. A. Konoplya (2024), 2410.10419

    2024 arXiv

  49. [57]

    R. A. Konoplya, A. F. Zinhailo, and Z. Stuchlik, Phys. Rev. D 102, 044023 (2020), 2006.10462

    2020 arXiv

  50. [58]

    R. A. Konoplya and A. Zhidenko, Class. Quant. Grav. 40, 245005 (2023), 2309.02560

    2023 arXiv

  51. [59]

    Malik, Int

    Z. Malik, Int. J. Mod. Phys. A 39, 2450024 (2024)

    2024

  52. [60]

    Malik, Int

    Z. Malik, Int. J. Theor. Phys. 63, 128 (2024), 2409.09872

    2024 arXiv

  53. [61]

    Takahashi and J

    T. Takahashi and J. Soda, Class. Quant. Grav. 29, 035008 (2012), 1108.5041

    2012 arXiv

  54. [62]

    Takahashi and J

    T. Takahashi and J. Soda, Prog. Theor. Phys. 124, 711 (2010), 1008.1618

    2010 arXiv

  55. [63]

    R. J. Gleiser and G. Dotti, Phys. Rev. D 72, 124002 (2005), gr-qc/0510069

    2005 arXiv

  56. [64]

    Dotti and R

    G. Dotti and R. J. Gleiser, Phys. Rev. D 72, 044018 (2005), gr-qc/0503117

    2005 arXiv

  57. [65]

    R. A. Konoplya and A. Zhidenko, Phys. Rev. D 95, 104005 (2017), 1701.01652

    2017 arXiv

  58. [66]

    Takahashi, Prog

    T. Takahashi, Prog. Theor. Phys. 125, 1289 (2011), 1102.1785

    2011 arXiv

  59. [67]

    R. A. Konoplya and A. Zhidenko, JCAP 05, 050 (2017), 1705.01656

    2017 arXiv

  60. [68]

    R. A. Konoplya and A. F. Zinhailo, Eur. Phys. J. C 80, 1049 (2020), 2003.01188

    2020 arXiv

  61. [69]

    Khanna and R

    G. Khanna and R. H. Price, Phys. Rev. D 95, 081501 (2017), 1609.00083

    2017 arXiv

  62. [70]

    R. A. Konoplya, A. F. Zinhailo, and Z. Stuchl ´ ık, Phys. Rev. D 99, 124042 (2019), 1903.03483

    2019 arXiv

  63. [71]

    R. A. Konoplya, Phys. Lett. B 784, 43 (2018), 1805.04718

    2018 arXiv

  64. [72]

    K. A. Bronnikov, Acta Phys. Polon. B 4, 251 (1973)

    1973

  65. [73]

    H. G. Ellis, J. Math. Phys. 14, 104 (1973)

    1973

  66. [74]

    Y. B. Zeldovich, Soviet Journal of Experimental and The - oretical Physics Letters 14, 180 (1971)

    1971

  67. [75]

    A. A. Starobinskil and S. M. Churilov, Sov. Phys. JETP 65, 1 (1974)

    1974

  68. [76]

    A. A. Starobinsky, Sov. Phys. JETP 37, 28 (1973)

    1973

  69. [77]

    J. D. Bekenstein and M. Schiffer, Phys. Rev. D 58, 064014 (1998), gr-qc/9803033

    1998 arXiv

  70. [78]

    W. E. East and F. Pretorius, Phys. Rev. Lett. 119, 041101 (2017), 1704.04791

    2017 arXiv

  71. [79]

    Hod, Phys

    S. Hod, Phys. Lett. B 708, 320 (2012), 1205.1872

    2012 arXiv

  72. [80]

    R. A. Konoplya, Phys. Lett. B 666, 283 (2008), 0801.0846

    2008 arXiv

  73. [81]

    R. A. Konoplya and A. Zhidenko, Phys. Rev. D 90, 064048 (2014), 1406.0019

    2014 arXiv

  74. [82]

    M. G. Richarte, E. L. Martins, and J. C. Fabris, Phys. Rev. D 105, 064043 (2022), 2111.01595

    2022 arXiv

  75. [83]

    Zhu, S.-J

    Z. Zhu, S.-J. Zhang, C. E. Pellicer, B. Wang, and E. Ab- dalla, Phys. Rev. D 90, 044042 (2014), [Addendum: Phys.Rev.D 90, 049904 (2014)], 1405.4931

    2014 arXiv

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