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REVIEW 4 major objections 8 minor 53 references

Superradiance of Charged Static Black Hole in Cubic Gravity

T0 review · 4 major / 8 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper builds an approximate charged static black hole metric in Einstein-cubic gravity from thermodynamics plus continued fractions, and finds charged scalar superradiance for 0<ω<qQ/r+, stronger for larger charge or coupling.

desk verdict A clever thermodynamic construction of an approximate ECG metric, but the charged-AdS extension is not validated and the ansatz is asymptotically flat, so the superradiance numbers rest on an unverified metric. read the letter →

arxiv 2505.17429 v1 pith:ICJLKXZ5 submitted 2025-05-23 gr-qc

Seyed Naseh Sajadi , Supakchai Ponglertsakul This is my paper · ORCID
classification gr-qc
keywords Einstein-cubicgravitychargedblackholessuperradiancescalarfieldcontinuedfractionexpansionholethermodynamicsSmarrformulahigher-curvature
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

Higher-curvature gravitational theories normally require solving intractable field equations to obtain black hole metrics. This paper claims a shortcut: require the first law of thermodynamics and the Smarr formula to hold, treating the cubic coupling constant as a thermodynamic variable, and use the resulting near-horizon coefficient $f_1$ to complete a continued-fraction metric for charged static black holes in Einstein-cubic-Maxwell theory. On this background the authors solve the charged Klein-Gordon equation and find that massless charged scalar waves are amplified when $0<\omega

What carries the argument

The load-bearing object is the near-horizon metric coefficient $f_1(r_+,\alpha,\Lambda,q)$ from Eq. (35), which encodes the temperature $T=f_1/4\pi$ and, through Eq. (23), determines the entropy. It is obtained from the first law and Smarr formula with $\alpha$ promoted to a thermodynamic variable and chemical potential $\psi_\alpha$ given by (25); this step stands in for the field equations. The continued-fraction ansatz (49)--(50) then extends the near-horizon data to a global approximate metric, with coefficients $a_0$--$a_4$ fixed by matching the near-horizon and asymptotic expansions. For the wave problem, the machinery is the charged Klein-Gordon equation in Schrödinger-like form with effective potential (57), the ingoing/outgoing boundary conditions (58)--(59), the Wronskian conditions (60)--(62), and the amplification factor (65).

What would settle it

Numerically solve the full Einstein-cubic-Maxwell field equations for a static, spherically symmetric metric with nonzero charge $q$ and/or cosmological constant $\Lambda$, expand the solution near the horizon, and compare the extracted $f_1$ with Eq. (35). A disagreement for generic parameters would show that the continued-fraction metric used for the superradiance integration is not a solution of the theory; a second check is to repeat the Wronskian integration on the field-equation metric and compare $Z_{\omega l}$.

Watch

Extended reading notes

Core claim

The central claim is that a consistent approximate analytic charged black hole in Einstein-cubic-Maxwell gravity can be assembled without solving the full field equations: write $f(r)=xA(x)$ with $x=1-r_+/r$ and truncate the continued fraction at fourth order, fixing the near-horizon coefficient $f_1(r_+,\alpha,\Lambda,q)$ by demanding that the first law $dM=T\,dS+\psi_\alpha\,d\alpha-\psi_\Lambda\,d\Lambda+\psi_q\,dq$ and the Smarr relation $M=2TS+4\alpha\psi_\alpha+2\psi_\Lambda\Lambda+q\psi_q$ hold coefficient by coefficient. The resulting $f_1$ has the closed form (35), the mass is (41), and the metric matches a numerical solution for the uncharged $\Lambda=0$ case. On that background, a massless scalar field with charge $Q$ obeys the radial equation (54); the effective potential (57) has asymptotic wavenumbers $k_+=\omega-qQ/r_+$ at the horizon and $k_\infty=\omega$ at infinity, and Wronskian conservation gives superradiance precisely when $0<\omega<qQ/r_+$. Numerical integration of the radial equation yields an amplification factor $Z_{\omega l}=|A_r|^2/|A_i|^2-1$ that is positive in that band, grows with $qQ$ and with $\alpha$ (for the negative $\alpha$ values considered), and vanishes at the critical frequency; including back-reaction, the black hole's mass and charge decrease in the superradiant regime, consistent with the second law.

Load-bearing premise

The result hinges on assuming that the coefficient $f_1$ determined by requiring the first law and Smarr formula to hold is the same coefficient that the field equations would produce; this identification is verified numerically only for the uncharged, $\Lambda=0$ case.

Editorial extensions

If this is right

  • Because the construction fixes $f_1$ from thermodynamics rather than from the field equations, the same pipeline yields approximate analytic black hole metrics in other one-metric-function higher-curvature theories, including Einstein-quartic gravity.
  • The superradiant threshold for charged static black holes in Einstein-cubic gravity is exactly the standard charged black hole one, $0<\omega<qQ/r_+$, so the critical frequency depends on $q$, $Q$, and $r_+$ but not on the coupling $\alpha$.
  • For the parameter ranges considered, the amplification factor $Z_{\omega l}$ grows with $qQ$ (same-sign charges) and with $\alpha$, and it vanishes at the critical frequency.
  • At second order in perturbation theory, the black hole mass and charge decrease in the superradiant band, and the loss rate tends to zero as $\omega\to\omega_c$.
  • Because the effective potential has no double-peak structure outside the horizon, the paper concludes that superradiant modes are not trapped and no superradiant instability develops in this static setting.

Reading between the lines

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

  • Editorial inference: the linchpin identification—thermodynamic $f_1$ equals the field-equation $f_1$—can be tested by numerically solving the Einstein-cubic-Maxwell equations for nonzero $q$ and/or $\Lambda$ and comparing the near-horizon expansion with Eq. (35).
  • Editorial inference: since the critical frequency $\omega_c=qQ/r_+$ is independent of $\alpha$, a measurement of the superradiant threshold would determine $qQ/r_+$, while the shape of the amplification curve would probe the higher-curvature corrections carried by the continued-fraction coefficients.
  • Editorial inference: if the thermodynamic and field-equation values of $f_1$ disagree in the charged or AdS cases, the scattering calculation is still well defined but is not necessarily a calculation in Einstein-cubic gravity; the numerical comparison above would delimit where the thermodynamic shortcut is reliable.
  • Editorial inference: the absence of a double-peak structure in the effective potential suggests that adding a confining boundary (an AdS boundary or a cavity) could trap superradiant modes and produce an instability in this theory; the paper does not follow that route.
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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

4 major / 8 minor

Summary. The paper constructs an approximate static, spherically symmetric charged black hole solution in Einstein-cubic-Maxwell gravity by combining near-horizon and asymptotic expansions with a continued-fraction metric ansatz, using thermodynamic input: the near-horizon coefficient f1 is fixed by imposing the first law and Smarr relation. Using this metric, the authors derive the standard charged-scalar superradiance condition, compute amplification factors by direct numerical integration, and argue that the black hole mass and charge decrease in the superradiant regime as a consequence of the first law and second law. The paper claims that the amplification factor increases with increasing qQ and with increasing cubic coupling α, and that the construction reproduces results that would be obtained by solving the field equations.

Significance. If the constructed metric were genuinely a solution for nonzero q and Lambda, the paper would offer a useful approximate framework for studying higher-curvature black holes and their superradiance. The Wronskian derivation of the superradiance condition is sound and metric-independent, and the known uncharged limits of the thermodynamic quantities are recovered. However, the significance is currently limited because the charged and asymptotically AdS metric is not validated against the field equations, and the continued-fraction ansatz appears incompatible with the Lambda_eff r^2 asymptotic term that the theory requires for nonzero cosmological constant. The paper would become publishable if these points were addressed by a direct validation of the metric, such as computing field-equation residuals or comparing with numerical solutions in the charged case.

major comments (4)
  1. [§2, Eqs. (27)–(35) and Fig. 1] The central near-horizon coefficient f1 is determined by the first-law/Smarr integrability conditions (31)–(34), not by solving the field equations Eab=Tab. For q≠0 and Λ≠0 there is no evidence that this thermodynamic f1 equals the coefficient of a genuine ECG-Maxwell solution; the only direct comparison with a numerical field-equation solution, Fig. 1, is for Λ=q=0. Please provide a quantitative validation for the charged case, for example by evaluating the residual Eab−Tab on the constructed metric or by comparing f1 from Eq. (35) with a numerical solution obtained directly from the field equations at nonzero q.
  2. [§2, Eqs. (17)–(19) and (49)–(51)] The large-r expansion (18) contains a Lambda_eff r^2 term, but the continued-fraction ansatz f(r)=xA(x) with x=1−r+/r satisfies A(1)=1 for any finite truncation of (50), so f(r)→1 as r→∞. The claimed charged-AdS metric therefore does not reproduce the asymptotic behavior of the theory; the continued-fraction construction as presented is only consistent with asymptotically flat boundary conditions. This requires either a corrected ansatz capable of encoding the Lambda_eff r^2 term or an explicit restriction of all AdS claims to a regime where that term is negligible.
  3. [§3, Figs. 3–6] The superradiance threshold (63) is derived correctly and is independent of the metric details, but the computed amplification factors Z_omega_l inherit the status of the metric. Because the metric is unvalidated for q≠0 and is inconsistent with Λ≠0, the quantitative claims about how q and α affect Z_omega_l are established only for the asymptotically flat, uncharged reference geometry used in Fig. 1. In particular, the statement that increasing α increases the amplification factor should be restricted to the negative values of α actually computed, and the error introduced by truncating the continued fraction at a4 should be quantified.
  4. [§4, Eqs. (66)–(70)] The thermodynamic derivation of mass loss during superradiance reuses the same first-law/Smarr input that fixed f1, so it is a consistency check rather than an independent confirmation. In addition, the step from Eq. (69) to the sufficient condition r+≥q in Eq. (70) is not derived in the text; the small-α expression for dS/dα is not monotonic in the stated way, so the inequality does not obviously follow without an explicit computation. Please show the intermediate steps or label this part as a plausibility argument.
minor comments (8)
  1. [§2, Eq. (19)] Please use consistent notation for Lambda_eff and Lambda in the cubic equation and define Lambda_eff explicitly before using it in Eq. (18).
  2. [§2, Eq. (51), Appendix B] Please clarify how the continued-fraction coefficients a1–a4 are matched to the near-horizon coefficients f1–f4; in particular, verify that a1 in Eq. (51) is consistent with the near-horizon expansion of f(r)/x.
  3. [Fig. 1] Please specify the numerical method, boundary conditions, and error tolerance used to obtain the numerical field-equation solution that is compared with the continued-fraction approximation.
  4. [Figs. 3–6] Please state how the fitted curves were obtained and what numerical accuracy is achieved in the integration of Eqs. (55)–(57).
  5. [Abstract and §2] The abstract mentions two unknown constants to be determined, but the text explicitly fixes only c1 (set to −1/2); please clarify what the second free parameter is and how it is fixed.
  6. [References] The reference entries for [21], [48], and [52] contain broken or incomplete bibliographic formatting and should be corrected.
  7. [§2.1] The statement that ECG admits h=f for static spherically symmetric solutions should be justified more explicitly, since the field equations for the two-metric ansatz (16) do not obviously enforce h=f.
  8. [§4] The derivation of Eq. (66) should state explicitly that the dot denotes an average over a wave period and that the flux formula applies at second order in the perturbation.

Circularity Check

1 steps flagged · score 3.0 of 10

Mild circularity: the final thermodynamic explanation restates the first-law input that fixed f1; the central superradiance amplification is nonetheless independently computed.

  1. other [Section 4, Eqs. (67)-(69)]
    "Even though superradiance seems to be the consequence of relativistic field equations in curved spacetime, it can also be understood by using the first and second laws of black hole thermodynamics [50]. ... dM =TdS +ψαdα +ψqdq. (67) ... dM = ωTdS/(ω −Qψq) + ωψαdα/(ω −Qψq). (69)"

    The first law (67) is the same input (27) that was imposed to fix f1 through the PDEs (31)-(34), and ψq=q/r+ was also an input. Combining it with dq/dM=Q/ω (68) gives (69), and the sign conclusion dM<0 for ω<Qψq=qQ/r+ follows algebraically from dS≥0 and ψα≤0 (25). Thus the 'thermodynamic explanation' of superradiance is a restatement of the thermodynamic frame used to construct the metric, not an independent derivation. It is not load-bearing for the central superradiance claim: the threshold (63) and amplification factor (Figs. 3-6) come from direct Wronskian/wave-integration arguments that do not invoke the first law; hence the paper is only mildly circular in this final consistency check.

full rationale

The only potentially circular element is the closing thermodynamic discussion in Section 4, where the first law that was used as input to fix f1 (Eq. 35) is reused to re-derive dM<0 in the superradiance regime. This is a secondary consistency check and does not feed back into the numerical amplification calculations, so the paper's central results remain independent. The horizon coefficient f1 is admittedly fixed by thermodynamics rather than by the ECG-Maxwell field equations ('Notice that f1 is not determined by the field equation itself'), and Fig. 1 validates the continued-fraction metric only for Λ=q=0; for q≠0 or Λ≠0 the metric's status as a solution of the theory is unverified. That is a validation/correctness gap, not a circularity: no fitted parameter is renamed as a prediction, no load-bearing self-citation is used, and the superradiance threshold (63) is the standard RN condition derived from the Wronskian, not from the thermodynamic input. The self-citations to [2,3] are background references to the same continued-fraction method and are not load-bearing. Overall score 3 reflects one minor secondary circularity in the thermodynamic explanation.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central construction relies on thermodynamic consistency rather than direct field-equation solution. The only free parameter is the integration constant c1, fixed to the Schwarzschild limit. The key axioms are the existence of h=f ECG solutions, the validity of the alpha-extended first law, the adequacy of the truncated continued fraction, and the identification of the thermodynamic f1 with the true field-equation coefficient.

free parameters (1)
  • c1 = -1/2
    Integration constant from solving the first-law PDEs for f1 (Eq. (35)); set to -1/2 so the alpha -> 0 temperature reduces to Schwarzschild, Eq. (37). It is a chosen constant, not fitted to data.
assumptions (4)
  • domain assumption Einstein-cubic-Maxwell action (14) with P as in (15) admits static spherically symmetric solutions with h=f.
    The paper restricts to h=f citing Refs. [6-11] in Section 2.1 without re-deriving this property.
  • domain assumption First law and Smarr formula with alpha as a thermodynamic variable, Eqs. (27)-(28), are valid for Einstein-cubic gravity.
    Adopted from Hajian-Tekin [20] and used to derive f1; the paper does not prove this independently.
  • ad hoc to paper The continued fraction expansion (49)-(50) truncated at a4 accurately represents the metric outside the horizon.
    The approximation is asserted; Fig. 1 checks only the uncharged case and shows disagreement at small r.
  • ad hoc to paper The thermodynamic f1 from first-law consistency equals the true Einstein-cubic-gravity field-equation coefficient for all parameters.
    This is the load-bearing assumption; validated numerically only for Lambda=q=0 in Fig. 1.

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Pith. "Pith review of Superradiance of Charged Static Black Hole in Cubic Gravity." pith.science (2026). https://pith.science/paper/ICJLKXZ5

@misc{pith2026250517429,
  author       = {Pith},
  title        = {Pith review of: Superradiance of Charged Static Black Hole in Cubic Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ICJLKXZ5}},
  note         = {Machine review of arXiv:2505.17429}
}
read the original abstract

Higher curvature gravity usually has complicated field equations, and solving them analytically is strenuous. In this work, we obtain an analytical charged black hole (BH) solution in higher curvature gravity using the thermodynamics of black holes and employing the continued fraction expansion. We investigate the thermodynamics of static black holes using the first law of thermodynamics and the Smarr formula in their proper form for Einstein-cubic gravity (ECG). Next, we obtain the thermodynamic quantities and show that our results are similar to those from solving the field equations. Then, we study the superradiance of the black hole using massless charged-scalar perturbations. We derive the superradiant conditions and compute the amplification factor through direct integration. We demonstrate how the amplification factor will change as a function of the black hole charge and frequency of the incident wave. We also show that a black hole mass and charge are decreasing in the superradiance region. Finally, we discuss superradiance as a consequence of black hole thermodynamics in ECG.

Figures

Figures reproduced from arXiv: 2505.17429 by the authors.

Figure 1
Figure 1. Comparison of numeric solution (dotted line) and continued fraction approximation (solid line) (49) for α = −0.3, r+ = 2,Λ = q = 0. In the continued fraction, we keep expansion terms up to a4. The dashed red curve is the asymptotic solution (18), including terms up to order r −100 . It should be noted that, in this figure and the following figures, we have used (41) to obtain the mass. In the following, we investiga… view at source ↗
Figure 2
Figure 2. The function Vef f (r) (57) of charged scalar field in the background of static BH, in terms of r for q = 0.2, Q = 0.4, ω = 0.2, l = 0, r+ = 0.3,Λ = 0. Now, we consider the asymptotic behavior of the effective potential and solution. For r → r+, the effective potential becomes Vef f → k 2 + = (ω − qQ/r+) 2 , and the solution for the equation (55) becomes u+(r⋆) = Ate −ik+r⋆ . (58) At the horizon, only an ingoing wav… view at source ↗
Figure 3
Figure 3. Superradiance of a complex massless scalar field wi [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Superradiance (Zωl) of a complex massless scalar field in terms of ω with l = 0, r+ = 3, α = −0.1, Q = 0.3,Λ = 0 and q = 0.46, 0.5, 0.55, 0.6, 0.65 (left) and Zωl vs q for r+ = 3,Λ = 0, α = −0.1, ω = 1.2×10−3 (right). The colored solid lines are the curves fitted with …
Figure 5
Figure 5. Figure 5: Superradiance (Zωl) of a complex massless scalar field in terms of ω with l = 0, r+ = 3, Q = 0.6, q = 0.5,Λ = 0 and α = −0.1, −0.2, −0.3 (left) and Zωl vs α for ω = 1.2×10−3 (right). By considering the back-reaction (considering the second order of perturbation) of the…
Figure 6
Figure 6. Figure 6: The rate of BH mass and charge (66) for the superradi [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]

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Works this paper leans on

53 extracted references · 14 canonical work pages

  1. [1]

    R. C. Myers, doi:10.1007/978-94-017-0934-7 8 [arXiv:gr-qc/9811042 [gr-qc]]

    work page Pith review arXiv

  2. [2]

    S. N. Sajadi, R. B. Mann, N. Riazi and S. Fakhry, doi:10.11 03/PhysRevD.102.124026 [arXiv:2010.15039 [gr-qc]]

    work page Pith review arXiv 2010

  3. [3]

    S. N. Sajadi and S. H. Hendi, Eur. Phys. J. C 82, no.8, 675 (2022) doi:10.1140/epjc/s10052- 022-10647-9 [arXiv:2207.13435 [gr-qc]]

    work page Pith review arXiv 2022

  4. [4]

    K. S. Stelle, Phys. Rev. D 16, 953 (1977). doi:10.1103/PhysRevD.16.953 17

    doi:10.1103/physrevd.16.953 1977

  5. [5]

    H. Lu, A. Perkins, C. N. Pope and K. S. Stelle, Phys. Rev. Le tt. 114, no.17, 171601 (2015) doi:10.1103/PhysRevLett.114.171601 [arXiv:1502.01028 [hep-th]]

    arXiv 2015

  6. [6]

    Bueno and P

    P. Bueno and P. A. Cano, Phys. Rev. D 94, no.10, 104005 (2016) doi:10.1103/PhysRevD.94.104005 [arXiv:1607.06463 [hep -th]]

    arXiv 2016

  7. [7]

    Bueno and P

    P. Bueno and P. A. Cano, Phys. Rev. D 94, no.12, 124051 (2016) doi:10.1103/PhysRevD.94.124051 [arXiv:1610.08019 [hep -th]]

    arXiv 2016

  8. [8]

    R. A. Hennigar, M. B. J. Poshteh and R. B. Mann, Phys. Rev. D 97, no.6, 064041 (2018) doi:10.1103/PhysRevD.97.064041 [arXiv:1801.03223 [gr- qc]]

    arXiv 2018

Show all 53 references
  1. [9]

    Cisterna, N

    A. Cisterna, N. Grandi and J. Oliva, Phys. Lett. B 805, 135435 (2020) doi:10.1016/j.physletb.2020.135435 [arXiv:1811.06523 [hep-th]]

    2020

  2. [10]

    Ahmed, R

    J. Ahmed, R. A. Hennigar, R. B. Mann and M. Mir, JHEP 05, 134 (2017) doi:10.1007/JHEP05(2017)134 [arXiv:1703.11007 [hep-th ]]

    2017 arXiv

  3. [11]

    M. B. J. Poshteh and R. B. Mann, Phys. Rev. D 99, no.2, 024035 (2019) doi:10.1103/PhysRevD.99.024035 [arXiv:1810.10657 [gr- qc]]

    2019 arXiv

  4. [12]

    A. M. Frassino and J. V. Rocha, Phys. Rev. D 102, no.2, 024035 (2020) doi:10.1103/PhysRevD.102.024035 [arXiv:2002.04071 [he p-th]]

    2020 arXiv

  5. [13]

    X. H. Feng, H. Huang, Z. F. Mai and H. Lu, Phys. Rev. D 96, no.10, 104034 (2017) doi:10.1103/PhysRevD.96.104034 [arXiv:1707.06308 [hep -th]]

    2017 arXiv

  6. [14]

    Jiang and B

    J. Jiang and B. Deng, Eur. Phys. J. C 79, no.10, 832 (2019) doi:10.1140/epjc/s10052-019- 7339-6

    2019 doi

  7. [15]

    J. D. Edelstein, N. Grandi and A. Rivadulla S´ anchez, JH EP 05, 188 (2022) doi:10.1007/JHEP05(2022)188 [arXiv:2202.05781 [hep-th ]]

    2022 arXiv

  8. [16]

    Bueno, P

    P. Bueno, P. A. Cano, R. A. Hennigar and R. B. Mann, Phys. R ev. Lett. 122, no.7, 071602 (2019) doi:10.1103/PhysRevLett.122.071602 [arXiv:1808 .02052 [hep-th]]

    2019 doi

  9. [17]

    Bueno, J

    P. Bueno, J. Camps and A. V. L´ opez, JHEP 04, 145 (2021) doi:10.1007/JHEP04(2021)145 [arXiv:2012.14033 [hep-th]]

    2021 arXiv

  10. [18]

    Bueno, P

    P. Bueno, P. A. Cano and A. Ruip´ erez, JHEP 03, 150 (2018) doi:10.1007/JHEP03(2018)150 [arXiv:1802.00018 [hep-th]]

    2018 arXiv

  11. [19]

    R. A. Hennigar and R. B. Mann, Phys. Rev. D 95, no.6, 064055 (2017) doi:10.1103/PhysRevD.95.064055 [arXiv:1610.06675 [hep -th]]

    2017 arXiv

  12. [20]

    Hajian and B

    K. Hajian and B. Tekin, Phys. Rev. Lett. 132, no.19, 191401 (2024) doi:10.1103/PhysRevLett.132.191401 [arXiv:2309.07634 [gr-qc]]

    2024 arXiv

  13. [21]

    Brito, V

    R. Brito, V. Cardoso and P. Pani, Physics,” Lect. Notes P hys. 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-q c]]

    2015 arXiv

  14. [22]

    Jung and D

    E. Jung and D. K. Park, Class. Quant. Grav. 21, 3717-3732 (2004) doi:10.1088/0264- 9381/21/15/007 [arXiv:hep-th/0403251 [hep-th]]. 18

    2004 arXiv

  15. [23]

    Doran, A

    C. Doran, A. Lasenby, S. Dolan and I. Hinder, Phys. Rev. D 71, 124020 (2005) doi:10.1103/PhysRevD.71.124020 [arXiv:gr-qc/0503019 [ gr-qc]]

    2005 arXiv

  16. [24]

    Dolan, C

    S. Dolan, C. Doran and A. Lasenby, Phys. Rev. D 74, 064005 (2006) doi:10.1103/PhysRevD.74.064005 [arXiv:gr-qc/0605031 [ gr-qc]]

    2006 arXiv

  17. [25]

    L. C. B. Crispino, S. R. Dolan and E. S. Oliveira, Phys. Re v. Lett. 102, 231103 (2009) doi:10.1103/PhysRevLett.102.231103 [arXiv:0905.3339 [ gr-qc]]

    2009 arXiv

  18. [26]

    L. C. B. Crispino, S. R. Dolan and E. S. Oliveira, Phys. Re v. D 79, 064022 (2009) doi:10.1103/PhysRevD.79.064022 [arXiv:0904.0999 [gr-q c]]

    2009 arXiv

  19. [27]

    Baake and O

    O. Baake and O. Rinne, Phys. Rev. D 94, no.12, 124016 (2016) doi:10.1103/PhysRevD.94.124016 [arXiv:1610.08352 [gr- qc]]

    2016 arXiv

  20. [28]

    Balakumar, E

    V. Balakumar, E. Winstanley, R. P. Bernar and L. C. B. Cri spino, Phys. Lett. B 811, 135904 (2020) doi:10.1016/j.physletb.2020.135904 [arXiv:2010 .01630 [gr-qc]]

    2020

  21. [29]

    C. L. Benone and L. C. B. Crispino, Phys. Rev. D 93, no.2, 024028 (2016) doi:10.1103/PhysRevD.93.024028 [arXiv:1511.02634 [gr- qc]]

    2016 arXiv

  22. [30]

    M. A. A. de Paula, L. C. S. Leite, S. R. Dolan and L. C. B. Cri spino, Phys. Rev. D 109, no.6, 064053 (2024) doi:10.1103/PhysRevD.109.064053 [ar Xiv:2401.01767 [gr-qc]]

    2024 arXiv

  23. [31]

    G. Guo, P. Wang, H. Wu and H. Yang, JHEP 07, 070 (2023) doi:10.1007/JHEP07(2023)070 [arXiv:2301.06483 [gr-qc]]

    2023 arXiv

  24. [32]

    Hod, Phys

    S. Hod, Phys. Lett. B 713, 505-508 (2012) doi:10.1016/j.physletb.2012.06.043 [arXiv:1304.6474 [gr-qc]]

    2012 arXiv

  25. [33]

    J. C. Degollado and C. A. R. Herdeiro, Phys. Rev. D 89, no.6, 063005 (2014) doi:10.1103/PhysRevD.89.063005 [arXiv:1312.4579 [gr-q c]]

    2014 arXiv

  26. [34]

    C. A. R. Herdeiro, J. C. Degollado and H. F. R´ unarsson, P hys. Rev. D 88, 063003 (2013) doi:10.1103/PhysRevD.88.063003 [arXiv:1305.5513 [gr-q c]]

    2013 arXiv

  27. [35]

    Sanchis-Gual, J

    N. Sanchis-Gual, J. C. Degollado, C. Herdeiro, J. A. Fon t and P. J. Montero, Phys. Rev. D 94 (2016) no.4, 044061 doi:10.1103/PhysRevD.94.044061 [arX iv:1607.06304 [gr-qc]]

    2016 arXiv

  28. [36]

    Sanchis-Gual, J

    N. Sanchis-Gual, J. C. Degollado, P. J. Montero, J. A. Fo nt and C. Herdeiro, Phys. Rev. Lett. 116 (2016) no.14, 141101 doi:10.1103/PhysRevLett.116.14110 1 [arXiv:1512.05358 [gr-qc]]

    2016 arXiv

  29. [37]

    Bosch, S

    P. Bosch, S. R. Green and L. Lehner, Phys. Rev. Lett. 116 (2016) no.14, 141102 doi:10.1103/PhysRevLett.116.141102 [arXiv:1601.01384 [gr-qc]]

    2016 arXiv

  30. [38]

    S. R. Dolan, S. Ponglertsakul and E. Winstanley, Phys. R ev. D 92 (2015) no.12, 124047 doi:10.1103/PhysRevD.92.124047 [arXiv:1507.02156 [gr- qc]]

    2015 arXiv

  31. [39]

    Ponglertsakul, E

    S. Ponglertsakul, E. Winstanley and S. R. Dolan, Phys. R ev. D 94 (2016) no.2, 024031 doi:10.1103/PhysRevD.94.024031 [arXiv:1604.01132 [gr- qc]]

    2016 arXiv

  32. [40]

    Ponglertsakul and E

    S. Ponglertsakul and E. Winstanley, Phys. Lett. B 764 (2017), 87-93 doi:10.1016/j.physletb.2016.10.073 [arXiv:1610.00135 [gr-qc]]. 19

    2017 arXiv

  33. [41]

    Konewko and E

    S. Konewko and E. Winstanley, Eur. Phys. J. C 84, no.6, 594 (2024) doi:10.1140/epjc/s10052-024-12910-7 [arXiv:2301.0116 9 [hep-th]]

    2024

  34. [42]

    Dappiaggi, H

    C. Dappiaggi, H. R. C. Ferreira and C. A. R. Herdeiro, Phy s. Lett. B 778, 146-154 (2018) doi:10.1016/j.physletb.2018.01.018 [arXiv:1710.08039 [gr-qc]]

    2018 arXiv

  35. [43]

    Yang and Y

    H. Yang and Y. G. Miao, Chin. Phys. C 47, no.7, 075101 (2023) doi:10.1088/1674- 1137/accdc7 [arXiv:2211.15130 [gr-qc]]

    2023 arXiv

  36. [44]

    R. M. Wald, Phys. Rev. D 48, no. 8, R3427 (1993) doi:10.1103/PhysRevD.48.R3427 [gr- qc/9307038]

    1993

  37. [45]

    Iyer and R

    V. Iyer and R. M. Wald, Phys. Rev. D 50, 846 (1994) doi:10.1103/PhysRevD.50.846 [gr- qc/9403028]

    1994

  38. [46]

    Bonanno and S

    A. Bonanno and S. Silveravalle, Phys. Rev. D 99, no.10, 101501 (2019) doi:10.1103/PhysRevD.99.101501 [arXiv:1903.08759 [gr- qc]]

    2019 arXiv

  39. [47]

    Chernyavsky and K

    D. Chernyavsky and K. Hajian, Class. Quant. Grav. 35, no.12, 125012 (2018) doi:10.1088/1361-6382/aac39a [arXiv:1710.07904 [gr-qc ]]

    2018 arXiv

  40. [48]

    Rezzolla and A

    L. Rezzolla and A. Zhidenko, Phys. Rev. D 90, no. 8, 084009 (2014) doi:10.1103/PhysRevD.90.084009 [arXiv:1407.3086 [gr-q c]]

    2014 arXiv

  41. [49]

    Wilson-Gerow and A

    J. Wilson-Gerow and A. Ritz, Phys. Rev. D 93, no.4, 044043 (2016) doi:10.1103/PhysRevD.93.044043 [arXiv:1509.06681 [hep -th]]

    2016 arXiv

  42. [50]

    J. D. Bekenstein, Phys. Rev. D 7, 949-953 (1973) doi:10.1103/PhysRevD.7.949

    1973 doi

  43. [51]

    Kokkotas, R

    K. Kokkotas, R. A. Konoplya and A. Zhidenko, Phys. Rev. D 96, no. 6, 064007 (2017) doi:10.1103/PhysRevD.96.064007 [arXiv:1705.09875 [gr- qc]]

    2017 arXiv

  44. [52]

    R. A. Konoplya and A. F. Zinhailo, Phys. Rev. D 99, no. 10, 104060 (2019) doi:10.1103/PhysRevD.99.104060 [arXiv:1904.05341 [gr- qc]]

    2019 arXiv

  45. [53]

    A. F. Zinhailo, Eur. Phys. J. C 78, no. 12, 992 (2018) [Eur. Phys. J. 78, 992 (2018)] doi:10.1140/epjc/s10052-018-6467-8 [arXiv:1809.03913 [gr-qc]]. 20

    2018 arXiv

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.