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REVIEW 3 major objections 3 minor 34 references

For a special parameter choice in mass-varying massive gravity, the tensor sector of Schwarzschild-(A)dS black holes reduces exactly to general relativity.

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2026-08-01 19:12 UTC pith:ZESDPELF

load-bearing objection Useful eigendecomposition method with explicit δX formulas, but the β=α² GR-reduction claim in the abstract only holds on the general branch and a measure-zero intersection of the special branch — Eq. (46) gives δX ∝ κ12 ≠ 0 generically. the 3 major comments →

arxiv 2607.17046 v2 pith:ZESDPELF submitted 2026-07-19 gr-qc

Perturbations of black holes in mass-varying massive gravity

De-Jun Wu This is my paper
classification gr-qc
keywords massive gravitymass-varying massive gravityblack hole perturbationsmatrix perturbation theoryeigendecompositionSchwarzschild-(A)dSlinear stabilityscalar-tensor theory
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.

The paper develops a new way to compute gravitational perturbations in mass-varying massive gravity, using eigendecomposition and matrix perturbation theory instead of differentiating the matrix square root. It applies this method to all constant-scalar Schwarzschild-(A)dS black hole solutions and claims that when the two parameters satisfy β=α², the tensor perturbation equations become identical to those of general relativity while the scalar sector reduces to a free massive scalar field. With suitable scalar potentials, these black holes are linearly stable. The paper also reveals that the solution manifolds of the generic and special parameter branches meet at a locus where the perturbation vanishes, making transitions between them smooth.

Core claim

The central claim is that under the condition β=α², the first-order metric perturbation equation for every constant-scalar Schwarzschild-(A)dS black hole in mass-varying massive gravity is identical to the linearized Einstein equation, because the graviton's effective energy-momentum perturbation δX vanishes identically. The same condition turns the scalar perturbation equation into a free, minimally coupled massive scalar equation whose effective mass squared is −V₂U, so stability depends only on the sign of that term and the Breitenlohner–Freedman bound for AdS. The argument proceeds through a new eigendecomposition method: one diagonalizes the matrix K, perturbs its eigenvalues using matr

What carries the argument

The central object is the effective graviton energy-momentum tensor X built from the matrix square root K = δ − √(g⁻¹η). The paper's key identity is δX = P′δX̄P′⁻¹ + [δP′P′⁻¹, X], where P′ is the eigenvector matrix adapted to degenerate eigenvalues and X̄ is the diagonal matrix of the eigenvalues of K. On Schwarzschild-(A)dS backgrounds X is proportional to the identity, so the commutator drops out, and the remaining task is computing the eigenvalue perturbations δkᵢ. Degenerate eigenvalues are treated by projecting the metric perturbation onto the degenerate eigenspace and diagonalizing the projected matrix Ω̃; the final expression for δX is proportional to κ₁₃, which vanishes when β=α² bec

Load-bearing premise

The method requires the matrix K to be diagonalizable, and for the fourfold-degenerate solution the paper has not shown that the eigenvector perturbation δP′ is small; it assumes smoothness and evaluates perturbations by taking an x→0 limit, so the reduction to general relativity for those backgrounds rests on an unproven regularity assumption.

What would settle it

Find a metric perturbation mode—odd or even parity, any multipole—on a β=α² Schwarzschild-(A)dS background for which δX is nonzero, which would break the claimed reduction to general relativity. Alternatively, for the c₀=1/(k₃−1)⁴ solution, solve the eigenvalue perturbation problem of the defective matrix exactly: if δP′ is of the same order as the metric perturbation rather than parametrically smaller, the x→0 limiting procedure is not a valid perturbation expansion.

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

If this is right

  • If the claim is correct, the linear gravitational-wave ringdown and quasinormal modes of these massive-gravity black holes (with β=α²) are indistinguishable from those of Schwarzschild-(A)dS in general relativity.
  • The scalar sector behaves as a free massive scalar, so tachyonic instability is decided solely by the sign of −V₂U (and the BF bound in AdS), giving a clean parametric handle on scalar stability.
  • The eigendecomposition method provides a template for perturbation analyses in other modified-gravity theories with a matrix-valued graviton potential, avoiding the complicated Fréchet derivative of the matrix square root.
  • The intersection of the generic and special solution manifolds at δX=0 shows that the constant-graviton-mass limit of hairy solutions is smooth at first order, which is relevant for spontaneous scalarization scenarios.
  • Even the fourfold-degenerate, non-diagonalizable solution can be handled by a Schur-decomposition limit, extending perturbation theory to seemingly singular backgrounds.

Where Pith is reading between the lines

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

  • The reduction to general relativity in the tensor sector suggests that precision ringdown tests of massive gravity may be insensitive to the graviton mass for these special backgrounds; observable deviations would have to come from the scalar channel or from nonlinearities.
  • The method, if extended to non-constant scalar profiles, could provide a practical route to study spontaneous scalarization and superradiance in mass-varying massive gravity; the paper sketches but does not complete this extension.
  • A natural test of the method's robustness would be to apply it to a simplified toy model with a known exact solution and compare against numerical perturbation results, which the paper does not do.
  • The fact that δX vanishes exactly at β=α² may point to a hidden symmetry or duality between the special and general branches that the paper does not explore.

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 manuscript develops an alternative method for computing first-order perturbations of the matrix square root K in massive gravity, based on eigendecomposition and degenerate matrix perturbation theory. It applies the method to all constant-scalar Schwarzschild-(A)dS black hole solutions in mass-varying massive gravity, deriving explicit expressions for δX on the special branch (β=α², k3=-1/α with k1,k2 distinct) and on the general branch (k2=k3=k4). The central claim is that, for β=α², δX=0 so that tensor perturbations reduce to those of general relativity, while scalar perturbations obey a free massive scalar equation with effective mass squared −V2U; this is used to conclude linear stability under appropriate parameter choices. The paper also discusses the smoothness of the constant graviton mass limit and treats the 4-fold degenerate non-diagonalizable solution via Schur decomposition and a limiting procedure.

Significance. The proposed method is a useful technical contribution: on backgrounds with X∝I it eliminates the commutator term in δX, and the general-branch calculation (Eq. (50)) cleanly exhibits the reduction δX∝κ13, with κ13=0 for β=α² and k3=-1/α. The treatment of the λ2−λ3 degeneracy limit via cancellation of O(δ^{-1}) terms is a genuine technical achievement. The paper is also commendably candid about the limitations of the eigendecomposition approach. If the central claim held for all claimed solutions, the identification of a locus where massive-gravity tensor perturbations coincide with GR would be physically significant. However, as detailed below, the claim is not established for the generic special branch, and the singular-regime applicability is explicitly not proven for the 4-fold degenerate non-diagonalizable solution.

major comments (3)
  1. [Sec. IV, Eq. (46)] The headline claim that β=α² gives δX=0 for all constant-scalar solutions is proven only on the general branch (k2=k3=k4): Eq. (50) has δX∝κ13, and β=α² with k3=−1/α forces κ13=0. On the special branch, Eq. (46) gives δX=κ12/[2f(k3−1)](h_ab−γ^cd h_cd γ_ab), where κ12=1/2(α(k1+k2)+βk1k2+1) is not forced to vanish by β=α². Example: EF frame, e=0, b=√c0, c0=1, α=1, k3=−1, at a radius where the M/r term is negligible, k1≈0.482, k2≈−0.932, so κ12≈0.050≠0. Thus Eq. (21) is not the GR equation on this branch. The Sec. IV escape clause (choosing e(r) to approach the general branch) selects only a measure-zero locus k2→k3 (or k1→k3); it does not cover the generic special branch. Consequently the abstract's unqualified statement that the tensor sector 'of these black holes' reduces to GR, and the resulting stability conclusion, are unsupported for generic special-branch solutions.
  2. [Sec. III and Appendix 3] The abstract claims the method is valid 'even in seemingly singular regimes'. For the 4-fold degenerate non-diagonalizable solution c0=1/(k3−1)^4, Appendix 3 states 'we have not yet found a way to ensure [δP'] is small', and perturbations are evaluated by taking x→0 'assuming smoothness at this boundary'. This is an explicit admission that first-order perturbation theory has not been justified at a defective-matrix point. The claim that the O(1/x) terms in Eq. (59) combine with κ13∝δ (the unnumbered κ13 equation in Appendix 3) to give a finite result is a formal cancellation, not a proof that the perturbation expansion is controlled. Since this solution is included in 'all constant-scalar solutions', the proof of the paper's central claim is incomplete at this boundary. Moreover, the statement in Sec. V that verification extends straightforwardly to fourfold degeneracy appears to contrad
  3. [Sec. IV, scalar stability] The stability analysis for the scalar sector is presented only for dS backgrounds; for AdS, the manuscript says 'It can be satisfied by multiple parameter regimes, which are lengthy and omitted here.' This is a placeholder rather than a verification. If the AdS stability conclusion is part of the claimed results, the BF-bound check should be stated explicitly or deferred to future work with the claim appropriately qualified. As written, the conclusion that 'these black holes are stable' is too broad.
minor comments (3)
  1. [Eqs. (48) and (53)] The denominators in δλ1 and δλ2 appear as x(x+z)^2 and x(x−z)^2 in the text. Please verify whether the square is intended; the standard non-degenerate first-order formula would suggest x(x+z) and x(x−z). The current typesetting is ambiguous.
  2. [Appendix 3, Eq. (83)] The symbol δ is overloaded: it denotes both a perturbation (e.g., δP′, δX) and the small parameter used in the x→0 limit. Using ε for the latter would improve readability.
  3. [Sec. IV, paragraph after Eq. (50)] The phrase 'as the corresponding constraint vanishes' is vague; specify whether this refers to Eq. (35), the condition κ12=0, or the k2→k3 limit. Clarifying this would help the reader follow the claimed intersection of the two branches.

Circularity Check

0 steps flagged

No significant circularity: the β=α² tensor-GR reduction is independent algebra from explicit eigendecomposition; self-citations are auxiliary and non-load-bearing, while flagged gaps (non-diagonalizable Ω̃, Appendix 3 δP′ smallness) and the abstract's overbroad claim are correctness/rigor concerns, not circular steps.

full rationale

The derivation chain is self-contained and does not reduce to its inputs. δX for the special branch (Eq. 46) and general branch (Eqs. 27–28, 50, 59) is obtained by exact eigendecomposition plus first-order matrix perturbation theory, with no fitted parameters and no data. The key reduction β=α² ⇒ δX=0 is an algebraic identity, not a construction: with the branch relation k3=−1/α, κ13=½(α(k1+k3)+βk1k3+1) vanishes, and at the intersection k2→k3 or k1→k3, κ12=½(α(k1+k2)+βk1k2+1)=0 follows from the same definition; neither κij is defined in terms of δX. Refs. [25] (Tolley–Wu–Zhou, including the present author) and [26] are cited only for the existence of hairy backgrounds and the auxiliary identity c′=½rcφ′²; the no-hair-for-β=α² conclusion is re-derived in Appendix 1 from the paper's own Eq. (39), so the self-citation is not load-bearing. Per the reviewing rule, flagged limitations are real but are rigor/validity gaps rather than circularity: Sec. III assumes results 'remain generally applicable even when Ω̃ is non-diagonalizable' without proof; Appendix 3 concedes 'we have not yet found a way to ensure this condition' (δP′ small) and evaluates the c0=1/(k3−1)^4 solution by 'assuming smoothness at this boundary.' The most serious concern is not circularity but an overbroad claim: the abstract states 'under the condition β=α², the tensor sector of these black holes reduces to that of general relativity,' yet for generic special-branch solutions the paper's own Eq. (46) gives δX ∝ κ12 with κ12 generically nonzero, and Sec. IV only forces κ12→0 at the measure-zero intersection with the general branch. This unsupported generalization should be weighed as correctness risk, not as a circular step.

Axiom & Free-Parameter Ledger

4 free parameters · 6 axioms · 0 invented entities

No data fitting appears. The ledger lists the hand-chosen parameter conditions and the unproved smoothness/diagonalizability assumptions of the method. The special beta=alpha^2 branch is essential to the main claim, and the degenerate-limit treatment is the least supported element.

free parameters (4)
  • beta (or alpha3, alpha4) = beta = alpha^2
    Hand-chosen parameter relation that forces k3=-1/alpha and kappa_13=0; without it the tensor sector does not reduce to GR. This is a theory condition, not a data fit, but it is the crux of the stability claim.
  • V2 (scalar potential curvature) = V2 > 0 (S-dS stability condition)
    Chooses the potential V=m^2+1/2 V2 phi^2; S-dS scalar stability requires -V2 U >= 0, i.e. V2>0 under the stated ranges. Parameter choice, not fitted.
  • c0 (integration constant) = c0 > (alpha+1)^2/alpha^2 for S-dS
    Integration constant of the background solution; the scalar stability window and eigenvalue degeneracies depend on it. Not fitted to data.
  • alpha (or alpha3) = alpha > 0 for the stated S-dS stability
    Theory parameter appearing in Eq. (30); the non-tachyonic condition for the S-dS scalar uses alpha>0.
axioms (6)
  • domain assumption K^mu_nu is diagonalizable for the solutions treated by the eigendecomposition method (except the 4-fold degenerate case handled separately).
    Stated in Sec. III: 'A limitation of this method is that it requires K^mu_nu to be diagonalizable.' All perturbation formulas (14)-(20) are built on this.
  • ad hoc to paper Non-diagonalizable Omega-tilde is harmless because only its trace and the matrix itself enter the final expressions.
    Sec. III: 'Strictly speaking, this method requires Omega-tilde to be diagonalizable. In practice... it is reasonable to assume the results... remain generally applicable even when Omega-tilde is non-diagonalizable.'
  • domain assumption phi=0 is a local minimum of V and W, so the trivial background solves the scalar equation and scalar perturbation decouples.
    Sec. II: 'setting phi(r)=0 yields trivial solutions, provided that this configuration corresponds to a local minimum of both V and W.'
  • domain assumption The reference metric eta_mu_nu is the flat Minkowski metric in spherical coordinates.
    Eq. (7) and the definition K^mu_nu = delta^mu_nu - sqrt(g^{mu rho} eta_{rho nu}); inherent to the dRGT/MVMG framework.
  • domain assumption Schwarzschild-(A)dS in GR and a minimally coupled massless scalar on it are linearly stable (via refs [29-31]).
    Used in Sec. IV to transfer GR stability to the beta=alpha^2 branch.
  • ad hoc to paper The 2->3 and 3->4 degeneracy limits are smooth, and perturbations at the 4-fold degenerate boundary can be obtained by x->0.
    Appendix 3: 'assuming smoothness at this boundary, we could evaluate perturbations by taking an appropriate limit'; Appendix 2c uses commutator scaling O(delta^-1)*O(delta) ~ O(1).

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read the original abstract

Based on eigendecomposition and matrix perturbation theory, we propose an alternative method for analyzing perturbations in massive gravity that is applicable to both the base theory and its various modifications. We demonstrate the validity of this approach even in seemingly singular regimes. Applying this framework to mass-varying massive gravity, we compute perturbations for all constant-scalar Schwarzschild-(A)dS black hole solutions. We show that, under the condition $\beta = \alpha^2$, the tensor sector of these black holes reduces to that of general relativity, while the scalar sector remains free from tachyonic instabilities given appropriate parameter choices. A deeper relationship is also revealed between the solutions with generic parameters and those with $\beta = \alpha^2$: their solution manifolds intersect at a specific locus where the perturbation vanishes.

discussion (0)

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Reference graph

Works this paper leans on

34 extracted references · 29 linked inside Pith

  1. [1]

    The conclusion holds regardless of the configuration of the scalar field given it is static

    Background solutions The t r component of the field equations is b βk3 2 + 2αk3 + 1 = 0.(31) 10 It implies thatk 3 = ± √ α2−β−α β in non-diagonal solutions since the possibility ofb= 0 has been excluded, and the solutions split into two branches. The conclusion holds regardless of the configuration of the scalar field given it is static. It can be solved ...

  2. [2]

    It is convenient to lower the contravariant index ofδX µ ν and express it in terms of spherical harmonics once the calculation is completed

    Perturbations In this appendix, we present a detailed calculation ofδX µ ν for different backgrounds. It is convenient to lower the contravariant index ofδX µ ν and express it in terms of spherical harmonics once the calculation is completed. BecauseδX µν =δg µλX λ ν +g µλδX λ ν andX λ ν is proportional to the identity matrix,g τ µδX µ ν is symmetric. The...

  3. [3]

    We try to circumvent the problem by using Schur decomposition

    Schur decomposition In this appendix we discuss a peculiar solution withc 0 = 1 (k3−1)4 , whose existence cannot be confirmed by utilizing eigendecomposition. We try to circumvent the problem by using Schur decomposition. 19 c0 = 1 (k3−1)4 is equivalent tob=± a−e 2 , which yields two identical eigenvalues ofK, given by k1 =k 2 = 1− r 2 z .(78) However, th...

  4. [4]

    B. P. Abbottet al.[LIGO Scientific and Virgo], Phys. Rev. Lett.116, no.6, 061102 (2016) doi:10.1103/PhysRevLett.116.061102 [arXiv:1602.03837 [gr-qc]]

    Pith/arXiv arXiv 2016

  5. [5]

    Cardoso, S

    V. Cardoso, S. Biswas and S. Sarkar, [arXiv:2511.14841 [gr-qc]]

    arXiv

  6. [6]

    B. P. Abbottet al.[LIGO Scientific and Virgo], Phys. Rev. Lett.116, no.22, 221101 (2016) [erratum: Phys. Rev. Lett.121, no.12, 129902 (2018)] doi:10.1103/PhysRevLett.116.221101 [arXiv:1602.03841 [gr-qc]]

    Pith/arXiv arXiv 2016

  7. [7]

    Abbottet al.[LIGO Scientific, VIRGO and KAGRA], Phys

    R. Abbottet al.[LIGO Scientific, VIRGO and KAGRA], Phys. Rev. D112, no.8, 084080 (2025) doi:10.1103/PhysRevD.112.084080 [arXiv:2112.06861 [gr-qc]]

    Pith/arXiv arXiv 2025

  8. [8]

    Yunes, X

    N. Yunes, X. Siemens and K. Yagi, [arXiv:2408.05240 [gr-qc]]

    Pith/arXiv arXiv

  9. [9]

    de Rham, G

    C. de Rham, G. Gabadadze, Phys. Rev.D82, 044020 (2010), [arXiv:1007.0443]

    Pith/arXiv arXiv 2010

  10. [10]

    de Rham, G

    C. de Rham, G. Gabadadze and A. J. Tolley, Phys. Rev. Lett.106, 231101 (2011), [arXiv:1011.1232]

    Pith/arXiv arXiv 2011

  11. [11]

    S. F. Hassan and R. A. Rosen, Phys. Rev. Lett.108, 041101 (2012) [arXiv:1106.3344 [hep-th]]

    Pith/arXiv arXiv 2012

  12. [12]

    S. F. Hassan and R. A. Rosen, JHEP1204(2012) 123 [arXiv:1111.2070 [hep-th]]

    Pith/arXiv arXiv 2012

  13. [13]

    Hinterbichler, Rev

    K. Hinterbichler, Rev. Mod. Phys.84, 671-710 (2012) doi:10.1103/RevModPhys.84.671 [arXiv:1105.3735 [hep-th]]

    Pith/arXiv arXiv 2012

  14. [14]

    de Rham, Living Rev

    C. de Rham, Living Rev. Rel.17, 7 (2014) doi:10.12942/lrr-2014-7 [arXiv:1401.4173 [hep-th]]

    Pith/arXiv arXiv 2014

  15. [15]

    Babichev and R

    E. Babichev and R. Brito, Class. Quant. Grav.32, 154001 (2015) doi:10.1088/0264- 9381/32/15/154001 [arXiv:1503.07529 [gr-qc]]

    Pith/arXiv arXiv 2015

  16. [16]

    K. Wood, P. M. Saffin and A. Avgoustidis, Phys. Rev. D109, no.12, 124006 (2024) doi:10.1103/PhysRevD.109.124006 [arXiv:2402.17835 [gr-qc]]. 21

    Pith/arXiv arXiv 2024

  17. [17]

    Guarato and R

    P. Guarato and R. Durrer, Phys. Rev. D89, no.8, 084016 (2014) doi:10.1103/PhysRevD.89.084016 [arXiv:1309.2245 [gr-qc]]

    Pith/arXiv arXiv 2014

  18. [18]

    Kodama and I

    H. Kodama and I. Arraut, PTEP2014, 023E02 (2014) doi:10.1093/ptep/ptu016 [arXiv:1312.0370 [hep-th]]

    Pith/arXiv arXiv 2014

  19. [19]

    Bernard, C

    L. Bernard, C. Deffayet and M. von Strauss, Phys. Rev. D91, no.10, 104013 (2015) doi:10.1103/PhysRevD.91.104013 [arXiv:1410.8302 [hep-th]]

    Pith/arXiv arXiv 2015

  20. [20]

    Bernard, C

    L. Bernard, C. Deffayet and M. von Strauss, JCAP06, 038 (2015) doi:10.1088/1475- 7516/2015/06/038 [arXiv:1504.04382 [hep-th]]

    Pith/arXiv arXiv 2015

  21. [21]

    Kobayashi, M

    T. Kobayashi, M. Siino, M. Yamaguchi and D. Yoshida, PTEP2016, no.10, 103E02 (2016) doi:10.1093/ptep/ptw145 [arXiv:1509.02096 [gr-qc]]

    Pith/arXiv arXiv 2016

  22. [22]

    Cusin, R

    G. Cusin, R. Durrer, P. Guarato and M. Motta, JCAP04, 051 (2016) doi:10.1088/1475- 7516/2016/04/051 [arXiv:1512.02131 [astro-ph.CO]]

    Pith/arXiv arXiv 2016

  23. [23]

    Bernard, C

    L. Bernard, C. Deffayet, A. Schmidt-May and M. von Strauss, Phys. Rev. D93, no.8, 084020 (2016) doi:10.1103/PhysRevD.93.084020 [arXiv:1512.03620 [hep-th]]

    Pith/arXiv arXiv 2016

  24. [24]

    Mazuet and M

    C. Mazuet and M. S. Volkov, Phys. Rev. D96, no.12, 124023 (2017) doi:10.1103/PhysRevD.96.124023 [arXiv:1708.03554 [hep-th]]

    Pith/arXiv arXiv 2017

  25. [25]

    Mazuet and M

    C. Mazuet and M. S. Volkov, JCAP07, 012 (2018) doi:10.1088/1475-7516/2018/07/012 [arXiv:1804.01970 [hep-th]]

    Pith/arXiv arXiv 2018

  26. [26]

    Wood, Phys

    K. Wood, Phys. Rev. D113, no.8, 084054 (2026) doi:10.1103/vspq-pkfy [arXiv:2509.15055 [hep-th]]

    arXiv 2026

  27. [27]

    Q. G. Huang, Y. S. Piao and S. Y. Zhou, Phys. Rev. D86, 124014 (2012) doi:10.1103/PhysRevD.86.124014 [arXiv:1206.5678 [hep-th]]

    Pith/arXiv arXiv 2012

  28. [28]

    A. J. Tolley, D. J. Wu and S. Y. Zhou, Phys. Rev. D92, no.12, 124063 (2015) doi:10.1103/PhysRevD.92.124063 [arXiv:1510.05208 [hep-th]]

    Pith/arXiv arXiv 2015

  29. [29]

    Zhang and S

    J. Zhang and S. Y. Zhou, Phys. Rev. D97, no.8, 081501 (2018) doi:10.1103/PhysRevD.97.081501 [arXiv:1709.07503 [gr-qc]]

    Pith/arXiv arXiv 2018

  30. [30]

    Koyama, G

    K. Koyama, G. Niz and G. Tasinato, Phys. Rev. D84, 064033 (2011) doi:10.1103/PhysRevD.84.064033 [arXiv:1104.2143 [hep-th]]

    Pith/arXiv arXiv 2011

  31. [31]

    Berezhiani, G

    L. Berezhiani, G. Chkareuli, C. de Rham, G. Gabadadze and A. J. Tolley, Phys. Rev. D85, 044024 (2012) doi:10.1103/PhysRevD.85.044024 [arXiv:1111.3613 [hep-th]]

    Pith/arXiv arXiv 2012

  32. [32]

    Kodama and A

    H. Kodama and A. Ishibashi, Prog. Theor. Phys.110, 701-722 (2003) doi:10.1143/PTP.110.701 [arXiv:hep-th/0305147 [hep-th]]

    Pith/arXiv arXiv 2003

  33. [33]

    Dafermos and I

    M. Dafermos and I. Rodnianski, [arXiv:0709.2766 [gr-qc]]

    Pith/arXiv arXiv

  34. [34]

    Holzegel and J

    G. Holzegel and J. Smulevici, Commun. Math. Phys.317, 205-251 (2013) doi:10.1007/s00220- 012-1572-2 [arXiv:1103.3672 [gr-qc]]. 22

    Pith/arXiv arXiv 2013