Pith. sign in

REVIEW 2 major objections 3 minor 2 cited by

Revisiting the fermionic quasi-bound states around Schwarzschild black holes with improved analytic spectrum

T0 review · 2 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read New analytic spectrum sharpens fermion bound states around black holes.

desk verdict Plausible and useful unified analytic spectrum for fermionic quasibound states, but the key \tilde\ell correction is not fully controlled and the numerical comparison is visual only. read the letter →

arxiv 2501.08881 v2 pith:TGCMNKRI submitted 2025-01-15 gr-qc

Guang-Shang Chen , Cheng-Bo Yang , Shou-Shan Bao , Yong Tang , Yue-Liang Wu This is my paper · ORCID
classification gr-qc PACS 04.70.-s04.62.+v03.65.Pm
keywords fermionicquasiboundstatesSchwarzschildblackholemassiveDiracfieldmatchedasymptoticexpansionseffectiveangularquantumnumberanalyticspectrumperturbationtheoryGammafunctionregularization
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 the quasibound-state spectrum of a massive Dirac (spin-1/2) field around a Schwarzschild black hole—the discrete, slowly decaying orbital levels a fermion can occupy outside the horizon—can be written in a single compact analytic formula, Eq. (69), improving on earlier analytic results by keeping a small correction to the effective angular quantum number, $\tilde{\ell} = \sqrt{m_\ell^2 + m^2/4 - \omega^2}$. The key step is a direct two-region matching of the coupled first-order radial Dirac equations, replacing the earlier three-block matching of a second-order equation; this works because the two components of the radial spinor automatically adjust at large radius to respect the near-horizon boundary condition. The retained correction $\epsilon_\ell = \tilde{\ell} - \ell$ removes the singular behavior of the generalized factorials (Gamma functions) that appear in older formulas, and it shifts both the imaginary part, the decay rate, and, at order $(mM)^4$, the real part of the frequency. The payoff is concrete: for light fermions with $mM$ in 0.1–0.5, the improved formula tracks high-precision numerical results more closely, which matters for estimating the lifetimes of fermion clouds and dark-atom candidates around black holes.

What carries the argument

The load-bearing object is the effective angular quantum number $\tilde{\ell} = \sqrt{m_\ell^2 + m^2/4 - \omega^2} \equiv \ell+\epsilon_\ell$, a non-integer shift of the usual angular momentum label $\ell$. It sets the exponents $r^{\pm\tilde{\ell}}$ of the radial wavefunction; keeping $\epsilon_\ell$ nonzero regularizes the Gamma functions and supplies the $(mM)^4$ correction to both real and imaginary parts of the frequency. The other piece of machinery is the two-block matching scheme for the coupled first-order radial equations (30): the far-region solution (58) and the near-horizon solution (60) are matched in an overlap region that exists because light fermions with $\omega\sim m\ll 1$ have a large hierarchy of scales, and the bound-state condition (63) suppresses the $r^{-\tilde{\ell}}$ branch. The matching yields the improved spectrum Eq. (69).

What would settle it

A reader could take the full coupled radial equations (30) and integrate them numerically, or use a continued-fraction method, for $mM = 0.1, 0.3, 0.5$ with $m_\ell = +1, -1, -2$ and $n = 0, 1$, extracting the complex quasibound frequency without any large-radius approximations. If at any of those points the old formulas (51) and (52) fit the numerics better than Eq. (69), or if Eq. (69)'s real-part shift has the wrong sign, the claimed improvement is falsified.

Watch

Extended reading notes

Core claim

The paper's central result is Eq. (69), a unified analytic expression for $\omega/m$ for fermionic quasibound states with quantum numbers $n$, $\ell$, $m_\ell = \pm\ell$, written in terms of $\bar n = n+\ell+(s+1)/2$ with $s = \mathrm{sgn}(m_\ell)$. The formula keeps the improved angular parameter $\tilde{\ell}_0 = \sqrt{m_\ell^2 + m^2/4 - \omega_0^2}$, so the previously dropped regulator $\epsilon_\ell = \tilde{\ell}_0 - \ell$ is retained and all Gamma functions are well defined because $\tilde{\ell}_0$ is not an integer. The derivation solves the coupled first-order radial equations (30) directly: a Whittaker-function solution valid far from the black hole is matched, without an intermediate region, to a hypergeometric solution near the horizon, with the bound-state condition (63) that the coefficient of $r^{-\tilde{\ell}}$ be suppressed. The authors show that the far-region large component alone cannot be matched to the near-horizon solution, because the two spinor components are of equal importance near the horizon; the first-order system handles this automatically. In the limit $\tilde{\ell}\to\ell$ the formula reduces to earlier results (51) and (52), while for $mM$ in 0.1–0.5 the imaginary part is enhanced and agrees better with numerical data, especially for $m_\ell = +1$.

Load-bearing premise

The central assumption is that for light fermions ($mM\ll 1$) the approximate solution near the horizon, derived with $\tilde{\ell}$ taken close to $\ell$, and the approximate solution far away overlap in a real region where they can be matched while the rising branch is suppressed; if that matching is not consistent at order $(mM)^4$, the improved spectrum Eq. (69) does not follow.

Editorial extensions

If this is right

  • Eq. (69) gives one compact analytic expression that covers both signs of $m_\ell$, including the $n=0$ case for $m_\ell=-\ell$, through the replacement $n \to n + (1+s)/2$.
  • The retained $\epsilon_\ell$ correction removes the need for the limiting procedure that earlier work used to evaluate ill-defined Gamma functions; every quantity in Eq. (69) is finite and well defined.
  • For $m_\ell = +1$ the imaginary part is visibly enhanced toward the numerical curve for $mM$ between 0.1 and 0.5; the improvement is smaller for $m_\ell = -1$ and $-2$ and decreases as $n$ or $\ell$ grows.
  • The real part of the frequency receives a new order-$(mM)^4$ term, proportional to $(15/8 - 3\bar n/(2\ell_0)) (mM)^4/\bar n^4$, which earlier formulas did not contain.
  • The two-region matching shows that the second-order reduction used previously discards a branch of the solution that is needed for direct matching; working with the first-order system avoids constructing an intermediate solution.

Reading between the lines

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

  • Editorial extension: the same two-block matching should apply to other spherically symmetric backgrounds, such as Reissner–Nordström or Schwarzschild–de Sitter, where the first-order radial system has the same singularity structure; if so, the pattern of $\tilde{\ell}$ corrections would generalize.
  • Editorial extension: because $\epsilon_\ell$ grows with $mM$, the improved expansion may break down before the regime where the geodesic approximation starts to fail; comparing Eq. (69) against full numerics at $mM$ near 1 would map the actual radius of convergence.
  • Editorial extension: since the imaginary part controls the filling and decay of fermion levels, using the older formula would under- or over-estimate the lifetimes of fermion clouds around primordial black holes precisely in the $mM\sim 0.1$–$0.5$ window where the correction is largest.
  • Editorial extension: the observation that the second-order reduction loses a branch needed for matching suggests a diagnostic for other black-hole wave equations: when the two first-order components have comparable weight near the horizon, reduced second-order equations will typically need an intermediate region or a regulator shift.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. The paper revisits the massive Dirac quasibound-state spectrum around a Schwarzschild black hole. After reviewing the standard three-block matching applied to the second-order radial equation, it proposes a direct two-block matching of the coupled first-order radial equations. A non-integer effective angular momentum \tilde\ell = sqrt(m_\ell^2 + m^2/4 - \omega^2) is introduced, and matching in the overlap region with the bound-state condition \tilde\ell - \kappa = -n + \delta yields the compact unified spectrum Eq. (69), including an O((mM)^4) correction to the real part and enhanced imaginary parts. The improved formula is compared visually with numerical results from Ref. [34] for mM in 0.1-0.5.

Significance. If the result is correct, the paper provides a useful parameter-free analytic spectrum for fermionic quasibound states, regularizing the Gamma-function singularities of earlier treatments and improving agreement with numerical data. The derivation is detailed and self-consistent at leading order, and the use of numerical data from Ref. [34] is for validation rather than fitting, which is a strength. The central improvement claim, however, depends on a matching step whose consistency is not fully established, and the comparison is only visual. These issues need to be addressed before the result can be considered established.

major comments (2)
  1. [Sec. IV, Eqs. (59)-(60) and Eqs. (63)-(69)] The two-block matching treats \tilde\ell inconsistently between the derivation and the solution. Equation (59) is introduced under the explicit assumption \tilde\ell \simeq \ell, but the hypergeometric solution (60), its large-r expansion (61), the quantization condition (63), and the final spectrum (69) all retain the full \tilde\ell = sqrt(m_\ell^2 + m^2/4 - \omega^2). Since \tilde\ell - \ell is O((mM)^2/\ell), replacing \ell by \tilde\ell in the near-horizon solution is not a controlled higher-order correction but a resummation of O((mM)^2) effects. This same shift is precisely what generates the claimed O((mM)^4) real-part correction in Eq. (64) and the enhanced imaginary part in Eq. (69). The manuscript should show explicitly that inserting Eq. (60) into the original Eq. (30) leaves a residual of O((mM)^4) in the overlap region, or should derive Eq. (59) and its solution consistently to the required order without the \tilde\ell \simeq \ell replacement. Without such an estimate, the bound-state condition (63) and hence the improved spectrum (69) are not controlled.
  2. [Figs. 2 and 3] The central claim of improved agreement with numerical results is supported only by visual inspection. The gray curves are described as collected from Ref. [34], but no numerical residuals, error measures, or data tables are provided. This is especially important because the improvement is stated to be small for m_\ell < 0 and for larger n and \ell. Please quantify the comparison, for example by reporting the relative error of Re \omega and Im \omega for the relevant range mM \in [0.1,0.5], or by plotting the difference between the analytic and numerical curves.
minor comments (3)
  1. [Eq. (69)] The final unified formula uses both \ell_0 and \tilde\ell_0 without definitions. The text says \tilde\ell is estimated by \tilde\ell_0 \simeq \tilde\ell(\omega_0), but the first correction term contains \ell_0 while Eq. (64) contains \ell. Please define \ell_0 and clarify whether it is the integer angular momentum or the shifted \tilde\ell_0.
  2. [Eq. (73)] The notation O(ms) for the ratio of the small and large components is confusing because s = \pm 1. The expected behavior is O(m) for s = +1 and O(m^{-1}) for s = -1; please write this explicitly.
  3. [Fig. 2 and Fig. 3 captions] The vertical axes are not labeled in the extracted manuscript. Please indicate the plotted quantity, e.g., Im(\omega M) or log|Im(\omega M)|, and include units or the normalization used.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the quasibound-state spectrum is derived from the coupled radial equations via matched asymptotics, with no fitted parameter presented as a prediction.

full rationale

The central result Eq. (69) is obtained by solving the coupled first-order radial system (30). The far-region solution (58) follows from the diagonalizing transformation (54) with the algebraic parameter \tilde\ell defined in Eq. (56); the near-horizon solution (60) is a hypergeometric solution of the approximated system (59); and the quantization condition (63) is the standard requirement that the coefficient of r^{-\tilde\ell} be suppressed. Matching the asymptotic expansions (61)-(62) and solving perturbatively gives the compact spectrum (69), which reduces to the known results (51)-(52) in the limit \tilde\ell \to \ell. No parameter is fitted to the numerical data: the gray curves in Figs. 2-3 are taken from Ref. [34] only for comparison, and the previous formulas are recovered, not imposed. The nonzero angular correction \epsilon_\ell is not borrowed from self-citations: it is defined within the paper by Eq. (56) and its role as a regulator is visible from the gamma-function singularities in Eq. (50). Self-citations such as Refs. [13,49,50] motivate keeping \epsilon_\ell, but the derivation of the improved spectrum does not reduce to those citations. The skeptics' consistency concern about deriving Eq. (59) with \tilde\ell \approx \ell while using full \tilde\ell in Eq. (60) is a correctness and matched-asymptotics risk, not a circularity: Eq. (69) is not, by construction, identical to any input quantity. Overall the derivation is self-contained and benchmarked against external numerical results, so no circular step is identified.

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

The central derivation relies on standard matching asymptotics for the coupled first-order Dirac system in Schwarzschild background; the GQFT framework is reduced to GR in the relevant limit and does not enter the final spectrum. No free parameters are fit to the numerical curves.

assumptions (5)
  • domain assumption Neglect of backreaction and GQFT spin-gauge effects, reducing to GR
    Stated in Sec. II after Eq. (11); the Dirac equation (14) is then the standard equation in Schwarzschild background.
  • domain assumption Light-fermion limit ω ~ m ≪ 1 and l/m ≫ 1 ensuring overlap of near and far regions
    Invoked in Sec. IV before Eq. (59); needed for the two-block matching to be valid.
  • domain assumption Bound-state condition: coefficient of r^{-~l} in the far-region expansion is suppressed
    Eq. (63) in Sec. IV; this is the quantization condition that replaces the missing intermediate-region matching.
  • standard math Properties of Whittaker and hypergeometric functions and Gamma-function limits
    Used throughout Secs. III and IV, e.g., Eq. (50) for Gamma-function regularization.
  • domain assumption Two-component spinor matching in the overlap region can be done term by term
    Sec. IV after Eq. (60); the asymptotic expansions in (61) and (62) are matched term by term.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Revisiting the fermionic quasi-bound states around Schwarzschild black holes with improved analytic spectrum." pith.science (2026). https://pith.science/paper/TGCMNKRI

@misc{pith2026250108881,
  author       = {Pith},
  title        = {Pith review of: Revisiting the fermionic quasi-bound states around Schwarzschild black holes with improved analytic spectrum},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TGCMNKRI}},
  note         = {Machine review of arXiv:2501.08881}
}
read the original abstract

Black holes have long served as a testing ground for probing theories of gravity and quantum mechanics. Notably, fundamental fields in the neighborhood of black holes exhibit rich phenomena that could yield astrophysical observable signatures. However, exploring these structures typically requires computationally intensive numerical calculations. In this work, the dynamics of a massive Dirac field outside a Schwarzschild black hole is revisited. We propose a novel matching scheme that enables the analytical solution of the coupled first-order Dirac equation, as opposed to the conventional second-order approach. This method yields a compact and unified analytical expression for the energy spectrum, which shows improved agreement with numerical results. The improvement is due to high-order correction of angular parameter that has been ignored previously.

Figures

Figures reproduced from arXiv: 2501.08881 by the authors.

Figure 1
Figure 1. FIG. 1. Different regions used for the asymptotic solutions. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Comparison of improved result Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Comparison of improved result Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Quasi-bound states and late-time evolution of a massive fermion around a Reissner-Nordstr\"{o}m black hole

    gr-qc 2026-07 accept novelty 6.0 of 10

    Matrix matching yields improved quasi-bound spectra (fine structure + decay widths) for a massive Dirac field on RN, and branch-cut analysis plus simulations reveal an intermediate oscillatory power law followed by a ...

  2. Quasibound states of a charged Dirac field around regular black holes

    gr-qc 2026-06 unverdicted novelty 5.0 of 10

    Charged massive Dirac quasibound modes on ABG black holes stay damped; ABG and RN share hydrogenic real frequencies, but ABG’s inner barrier can make some modes much longer-lived.

Reference graph

Works this paper leans on

51 extracted references · 18 canonical work pages · cited by 2 Pith papers

  1. [34]

    Dolan and J

    S. Dolan and J. Gair, Class. Quant. Grav.26, 175020 (2009) doi:10.1088/0264-9381/26/17/175020 [arXiv:0905.2974 [gr-qc]]

    arXiv 2009

  2. [1]

    (47) into Eq

    Substituting Eq. (47) into Eq. (46), we arrive at a concrete expression for the small quantityδ, δ≈ (2p0)2˜ℓ+1 2im Γ(˜ℓ) n!Γ2(2˜ℓ+ 1) Γ( 1 2 + ˜ℓ−2iω 0) Γ( 1 2 − ˜ℓ−2iω 0) × (−)nΓ(1−2 ˜ℓ) Γ(−n−2 ˜ℓ−1) Γ(−2˜ℓ) Γ(−˜ℓ) , (49) which leads to an extremely small value forδ, i.e.,δ= O(m4˜ℓ+1). In the limit ofϵ ℓ →0, some Gamma functions in the second line of Eq....

    work page

  3. [2]

    Israel, Commun

    W. Israel, Commun. Math. Phys.8, 245-260 (1968) doi:10.1007/BF01645859

    doi:10.1007/bf01645859 1968

  4. [3]

    Carter, Phys

    B. Carter, Phys. Rev. Lett.26, 331-333 (1971) doi:10.1103/PhysRevLett.26.331

    doi:10.1103/physrevlett.26.331 1971

  5. [4]

    R. A. Konoplya and A. Zhidenko, Rev. Mod. Phys. 83, 793-836 (2011) doi:10.1103/RevModPhys.83.793 [arXiv:1102.4014 [gr-qc]]

    arXiv 2011

  6. [5]

    I. M. Ternov, V. R. Khalilov, G. A. Chizhov and A. B. Gaina, Sov. Phys. J.21, 1200-1204 (1978) doi:10.1007/BF00894575

    work page doi:10.1007/bf00894575 1978

  7. [6]

    Grain and A

    J. Grain and A. Barrau, Eur. Phys. J. C53, 641-648 (2008) doi:10.1140/epjc/s10052-007-0494-1 [arXiv:hep- th/0701265 [hep-th]]

    work page arXiv 2008

  8. [7]

    Hod, Phys

    S. Hod, Phys. Lett. B749, 167-171 (2015) doi:10.1016/j.physletb.2015.07.071 [arXiv:1510.05649 [gr-qc]]

    arXiv 2015

Show all 51 references
  1. [8]

    R. A. Konoplya and A. V. Zhidenko, Phys. Lett. B 609, 377-384 (2005) doi:10.1016/j.physletb.2005.01.078 [arXiv:gr-qc/0411059 [gr-qc]]

    2005 arXiv

  2. [9]

    Barranco, A

    J. Barranco, A. Bernal, J. C. Degollado, A. Diez- Tejedor, M. Megevand, M. Alcubierre, D. Nunez and O. Sarbach, Phys. Rev. Lett.109, 081102 (2012) doi:10.1103/PhysRevLett.109.081102 [arXiv:1207.2153 [gr-qc]]

    2012 arXiv

  3. [10]

    Arvanitaki, S

    A. Arvanitaki, S. Dimopoulos, S. Dubovsky, N. Kaloper and J. March-Russell, Phys. Rev. D81, 123530 (2010) doi:10.1103/PhysRevD.81.123530 [arXiv:0905.4720 [hep- th]]

    2010 arXiv

  4. [11]

    Brito, V

    R. Brito, V. Cardoso and P. Pani, Physics,” Lect. Notes Phys.906, pp.1-237 (2015) 2020, ISBN 978- 3-319-18999-4, 978-3-319-19000-6, 978-3-030-46621- 3, 978-3-030-46622-0 doi:10.1007/978-3-319-19000-6 [arXiv:1501.06570 [gr-qc]]

    2015 arXiv

  5. [12]

    A. A. Starobinsky, Sov. Phys. JETP37, no.1, 28-32 (1973)

    1973

  6. [13]

    S. L. Detweiler, Phys. Rev. D22, 2323-2326 (1980) 14 doi:10.1103/PhysRevD.22.2323

    1980 doi

  7. [14]

    S. Bao, Q. Xu and H. Zhang, Phys. Rev. D106, no.6, 064016 (2022) doi:10.1103/PhysRevD.106.064016 [arXiv:2201.10941 [gr-qc]]

    2022 arXiv

  8. [15]

    Unruh, Phys

    W. Unruh, Phys. Rev. Lett.31, no.20, 1265-1267 (1973) doi:10.1103/PhysRevLett.31.1265

    1973 doi

  9. [16]

    Chandrasekhar, Proc

    S. Chandrasekhar, Proc. Roy. Soc. Lond. A349, 571-575 (1976) doi:10.1098/rspa.1976.0090

    1976

  10. [17]

    W. G. Unruh, Phys. Rev. D14, 3251-3259 (1976) doi:10.1103/PhysRevD.14.3251

    1976 doi

  11. [18]

    Carter and R

    B. Carter and R. G. Mclenaghan, Phys. Rev. D19, 1093- 1097 (1979) doi:10.1103/PhysRevD.19.1093

    1979 doi

  12. [19]

    Gueven, Phys

    R. Gueven, Phys. Rev. D16, 1706-1711 (1977) doi:10.1103/PhysRevD.16.1706

    1977 doi

  13. [20]

    Martellini and A

    M. Martellini and A. Treves, Phys. Rev. D15, 3060-3061 (1977) doi:10.1103/PhysRevD.15.3060

    1977 doi

  14. [21]

    C. H. Lee, Phys. Lett. B68, 152-156 (1977) doi:10.1016/0370-2693(77)90189-7

    1977 doi

  15. [22]

    B. R. Iyer and A. Kumar, Phys. Rev. D18, 4799-4801 (1978) doi:10.1103/PhysRevD.18.4799

    1978 doi

  16. [23]

    Chandrasekhar,The mathematical theory of black holes, Vol

    S. Chandrasekhar,The mathematical theory of black holes, Vol. 69 (Oxford university press, 1998)

    1998

  17. [24]

    Villanueva-Domingo, O

    P. Villanueva-Domingo, O. Mena and S. Palomares- Ruiz, Front. Astron. Space Sci.8, 87 (2021) doi:10.3389/fspas.2021.681084 [arXiv:2103.12087 [astro- ph.CO]]

    2021

  18. [25]

    D. R. Brill and J. A. Wheeler, Rev. Mod. Phys.29, 465- 479 (1957) doi:10.1103/RevModPhys.29.465

    1957 doi

  19. [26]

    Chandrasekhar and S

    S. Chandrasekhar and S. L. Detweiler, Proc. Roy. Soc. Lond. A352, 325-338 (1977) doi:10.1098/rspa.1977.0002

    1977

  20. [27]

    H. T. Cho, Phys. Rev. D68, 024003 (2003) doi:10.1103/PhysRevD.68.024003 [arXiv:gr-qc/0303078 [gr-qc]]

    2003 arXiv

  21. [28]

    Hartman, W

    T. Hartman, W. Song and A. Strominger, [arXiv:0912.4265 [hep-th]]

    arXiv

  22. [29]

    Coutant and P

    A. Coutant and P. Millington, Class. Quant. Grav. 36, no.3, 035005 (2019) [erratum: Class. Quant. Grav. 36, no.19, 199501 (2019)] doi:10.1088/1361-6382/aaf679 [arXiv:1809.03480 [gr-qc]]

    2019 arXiv

  23. [30]

    V. I. Dokuchaev and Y. N. Eroshenko, Adv. High Energy Phys.2014, 434539 (2014) doi:10.1155/2014/434539 [arXiv:1403.1375 [astro-ph.CO]]

    2014 arXiv

  24. [31]

    I. M. Ternov, A. B. Gaina and G. A. Chizhov, Sov. Phys. J.23, 695-700 (1980) doi:10.1007/BF00891357

    1980 doi

  25. [32]

    A. B. Gaina and I. M. Ternov, Sov. Phys. J.31, 830-834 (1988) doi:10.1007/BF00920128

    1988 doi

  26. [33]

    Doran, A

    C. Doran, A. Lasenby, S. Dolan and I. Hinder, Phys. Rev. D71, 124020 (2005) doi:10.1103/PhysRevD.71.124020 [arXiv:gr-qc/0503019 [gr-qc]]

    2005 arXiv

  27. [35]

    S. R. Dolan and D. Dempsey, Class. Quant. Grav.32, no.18, 184001 (2015) doi:10.1088/0264- 9381/32/18/184001 [arXiv:1504.03190 [gr-qc]]

    2015 arXiv

  28. [36]

    Huang, D

    Y. Huang, D. J. Liu, X. h. Zhai and X. z. Li, Phys. Rev. D96, no.6, 065002 (2017) doi:10.1103/PhysRevD.96.065002 [arXiv:1708.04761 [gr-qc]]

    2017 arXiv

  29. [37]

    R. A. Konoplya and A. Zhidenko, Phys. Rev. D97, no.8, 084034 (2018) doi:10.1103/PhysRevD.97.084034 [arXiv:1712.06667 [gr-qc]]

    2018 arXiv

  30. [38]

    I. I. Cotaescu, Mod. Phys. Lett. A22, 2493- 2498 (2007) doi:10.1142/S0217732307024437 [arXiv:gr- qc/0701119 [gr-qc]]

    2007

  31. [39]

    C. A. Sporea, Mod. Phys. Lett. A30, no.28, 1550145 (2015) doi:10.1142/S021773231550145X [arXiv:1505.00470 [gr-qc]]

    2015 arXiv

  32. [40]

    J. l. Jing, Phys. Rev. D69, 084009 (2004) doi:10.1103/PhysRevD.69.084009 [arXiv:gr-qc/0312079 [gr-qc]]

    2004 arXiv

  33. [41]

    X. N. Zhou, X. L. Du, K. Yang and Y. X. Liu, Phys. Rev. D89, no.4, 043006 (2014) doi:10.1103/PhysRevD.89.043006 [arXiv:1308.2863 [gr-qc]]

    2014 arXiv

  34. [42]

    Y. L. Wu, Phys. Rev. D93, no.2, 024012 (2016) doi:10.1103/PhysRevD.93.024012 [arXiv:1506.01807 [hep-th]]

    2016 arXiv

  35. [43]

    Y. L. Wu, Sci. China Phys. Mech. Astron.66, no.6, 260411 (2023) doi:10.1007/s11433-022-2052-6 [arXiv:2208.03290 [hep-th]]

    2023 arXiv

  36. [44]

    Y. L. Wu, World Scientific, 2022, ISBN 978-981-12-5708- 7 doi:10.1142/12868

    2022 doi

  37. [45]

    Y. L. Wu, Sci. Bull.62, no.16, 1109-1113 (2017) doi:10.1016/j.scib.2017.08.005 [arXiv:1705.06365 [hep- th]]

    2017 arXiv

  38. [46]

    Y. L. Wu, Eur. Phys. J. C78, no.1, 28 (2018) doi:10.1140/epjc/s10052-017-5504-3 [arXiv:1712.04537 [hep-th]]

    2018 arXiv

  39. [47]

    Y. L. Wu, [arXiv:2411.15166 [physics.gen-ph]]

    arXiv

  40. [48]

    Y. L. Wu, doi:10.1016/j.scib.2025.03.026 [arXiv:2502.19458 [physics.gen-ph]]

    2025 arXiv

  41. [49]

    Y. K. Gao, D. Huang, Y. L. Ma, Y. Tang, Y. L. Wu and Y. F. Zhou, Phys. Rev. D109, no.6, 064072 (2024) doi:10.1103/PhysRevD.109.064072 [arXiv:2403.17619 [gr-qc]]

    2024 arXiv

  42. [50]

    S. S. Bao, Q. X. Xu and H. Zhang, Phys. Rev. D107, no.6, 064037 (2023) doi:10.1103/PhysRevD.107.064037 [arXiv:2301.05317 [gr-qc]]

    2023 arXiv

  43. [51]

    X. h. Chu, Y. q. Chu, S. s. Bao and H. Zhang, Phys. Rev. D111, no.4, 043039 (2025) doi:10.1103/PhysRevD.111.043039 [arXiv:2411.09980 [gr-qc]]

    2025 arXiv

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.