Adiabatic Deformations of Black Hole Moduli: I. Canonical Slices and Fredholm Solvability

Anna Chrysostomou; Karim Benakli

REVIEW 3 major objections 4 minor 21 references

Adiabatic Deformations of Black Hole Moduli: I. Canonical Slices and Fredholm Solvability

T0 review · 3 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read One scalar check decides black hole moduli response

desk verdict A carefully executed conditional theorem: the operator-level Fredholm criterion is solid, but the physical reduction to the quasistatic radial problem is an admitted assumption, not a proof. read the letter →

arxiv 2608.05275 v1 pith:DRM7KW4K submitted 2026-08-05 hep-th gr-qc

Karim Benakli , Anna Chrysostomou This is my paper · ORCID

classification hep-thgr-qc MSC 83C5783C0547A5334B05

keywords adiabaticblackholedeformationEinstein-Maxwell-dilatontheoryFredholmsolvabilityfinite-rankperturbationcanonicaltangentdirectionlocalsliceslowlyvaryingmodulusmoduli

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

This paper establishes a general criterion for when a static black hole, forced by a slowly varying exterior scalar modulus, possesses a unique leading-order adiabatic deformation. It shows that the problem splits into a geometric part and a functional-analytic part, and that each part is decided by explicit scalar data. Geometrically, varying the modulus at fixed static horizon mass defines the unique tangent direction on the static solution manifold whose residual mass-integration constant vanishes. Functionally, after a local slice removes the remaining tangential freedom, existence and uniqueness of the completed radial boundary-value problem hold exactly when the outer matching functional does not annihilate the one-dimensional kernel generator of the sliced radial operator. If that scalar condition fails, some exterior data admit no unique leading adiabatic response.

What carries the argument

The machinery is a one-dimensional radial reduction. Differentiating the static family defines the covector $C_\mu$ on the tangent space of the static solution manifold, and its kernel selects the canonical tangent direction. Promoting the static family to an instantaneous representative leaves a lag field whose scalar component obeys a single reduced radial equation with a residual integration constant $C_\mu^{\rm lag}$. A local slice $\mu(v,r_m)=0$ fixes the tangential ambiguity and adds a rank-one term, producing the sliced operator $L_{sl}=L+f_\phi(r_m)K(r)\,{\rm ev}_{r_m}$. The outer matching condition is a bounded linear functional $B_m$, and the completed operator $A=(L_{sl}, B_m)$ is Fredholm of index zero. The decisive object is $\psi_\star$, the generator of the one-dimensional kernel of $L_{sl}$, because $B_m\psi_\star \neq 0$ is exactly the condition for invertibility.

What would settle it

Solve the first-order evolution equation (5.50) with fixed exterior data and initial data containing a nonzero homogeneous mode whose time derivative is of first adiabatic order; if the late-time radial profile differs from the solution of the quasistatic equation $L_{sl}\eta_{ad}=S_{ext}$ by $O(\epsilon)$, the causal-selection assumption fails. For a concrete family such as magnetic GHS, compute $\psi_\star$ and the matching functional; $B_m\psi_\star=0$ would make the completed operator non-invertible, giving either no solution or a one-parameter family for generic data.

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Extended reading notes

Core claim

The central claim is Theorem 2: after fixing a representation by the local slice $\mu(v,r_m)=0$, the completed operator $A=(L_{sl}, B_m): X \to Y\times\mathbb{R}$ is Fredholm of index zero and is bijective if and only if $B_m\psi_\star \neq 0$, where $\psi_\star$ generates the one-dimensional kernel of the sliced radial operator $L_{sl}$. Equivalently, existence and uniqueness of the leading-order adiabatic boundary-value problem reduce to a single scalar matching condition on that kernel generator. The companion geometric result, the Canonical Tangent Theorem, states that in coordinates $(\Phi, M_{H,0})$ the kernel of the intrinsic covector $C_\mu$ is spanned by $\partial_\Phi|_{M_{H,0}}$, so the direction selected by a slowly varying exterior modulus is exactly the tangent direction whose residual mass-integration constant vanishes. When the slice is transverse and $K\not\equiv 0$, the canonical tangent mode is displaced from the kernel of $L_{sl}$, but the Fredholm index forces a generally different one-dimensional kernel to remain; only the outer matching condition can fix its amplitude.

Load-bearing premise

The argument assumes that on the slowly driven branch the lag field's time derivatives are of second adiabatic order, so the transport terms drop out and the quasistatic radial equation holds; this causal selection is not derived from the full evolution problem.

Editorial extensions

If this is right

The same solvability criterion applies to any smooth non-extremal static solution manifold with a Fredholm reduced radial operator, not only to the Einstein–Maxwell–dilaton family used in the derivation.
The horizon-mass parameter $M_{H,0}$ is the natural second collective coordinate because the exact dynamical horizon mass evolves only at second adiabatic order, making the canonical tangent direction $\partial_\Phi|_{M_{H,0}}$ the adiabatic trajectory selected by a slowly varying modulus.
A transverse local slice changes the reduced operator only by a finite-rank term, preserving the Fredholm index; the original tangent zero mode is displaced from the kernel exactly when slice transversality and $K\not\equiv 0$ both hold.
When $B_m\psi_\star \neq 0$, the completed boundary-value operator is bijective, so for every exterior source and matching datum the leading adiabatic deformation exists and is unique.
The companion specialization to the magnetic GHS family can evaluate $\varphi_\Phi$, the slice-transversality factor $m_\Phi(r_m)$, the kernel generator $\psi_\star$, and the matching condition explicitly.

Reading between the lines

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

If $B_m\psi_\star = 0$ for some exterior model, the leading adiabatic response is either absent or non-unique, which could signal that slow homogeneous hair modes at first adiabatic order must be included in the physical solution.
Because the slice is local and does not require a metric on the solution space, the construction suggests a route to adiabatic tracking in moduli spaces where $L^2$ orthogonality projections are unavailable or ill-defined.
A numerical solution of the full first-order evolution equation with the same exterior data and explicit initial data could reveal whether the $O(\epsilon^2)$ transport-term estimate holds on the retarded branch; any $O(\epsilon)$ contamination from excited transients would falsify the quasistatic reduction.
The framework could be extended toward extremal horizons by replacing the non-extremality-based implicit-function step with a limiting argument, although the rank-one conclusion for $C_\mu$ and the Fredholm setup would need separate treatment.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

This paper is worth engaging with seriously. The central abstract result, Theorem 2, is a real formal contribution: under the stated Fredholm and transversality hypotheses, existence and uniqueness of the leading-order adiabatic deformation is equivalent to a single scalar condition B_m psi_* != 0. I traced the key identities — (3.48), (3.51), (3.63), (5.29), (5.44) — and they are consistent. The canonical tangent theorem (ker C_mu = span{partial_Phi | M_H,0}) is new and elegant, and the reduction of the coupled linearized system to a single scalar operator is genuinely non-trivial. Appendix A's Volterra argument is a coherent way to pin down the Fredholm properties, and the paper is unusually honest about its own limits: it explicitly says the causal selection of the adiabatic branch is not derived and requires the full IBVP. That is rare and should be credited.

The main soft spot is exactly the one flagged by the stress-test note. The passage from the exact evolution equation (5.50) to the quasistatic radial problem (5.59) rests on the amplitude bound (5.54)-(5.55). The Fredholm hypotheses do not imply that bound. Generic retarded data can include homogeneous modes whose time-derivative terms stay at first order, and a near-extremal background can make the forced response scale as epsilon/(kappa_H r_H) rather than epsilon. So the scalar criterion B_m psi_* != 0 governs the physical deformation only if the slowly forced branch really is the causally selected branch. The authors acknowledge this, so the paper should be read as a conditional theorem, not a complete physical result. The second caveat is that the hypotheses themselves — one-dimensional kernel, transversality, nontriviality of the finite-rank update — are not verified for any concrete family here; that is deferred to the companion paper. Neither caveat is a flaw in the internal logic, but both are load-bearing for the advertised physical conclusion.

This paper is for researchers working on slowly driven black holes in scalar-tensor theories, especially the moduli-space approximation. They will get a careful formal toolkit: the canonical covector, the slice construction, and the Fredholm criterion. The reader gets value from it primarily as a rigorous statement of what would need to be true for the adiabatic reduction to work. I would send this to peer review. A serious referee should engage with the operator-level theorem and the appendix; the Section 5.2 gap should be made prominent, but it is not a desk-reject. If I were editing, I would ask the authors to either strengthen the adiabatic branch argument or explicitly state the additional spectral-gap assumption needed for the quasistatic reduction.

Referee Report

3 major / 4 minor

Summary. The paper develops a general framework for the leading adiabatic response of a static, non-extremal black hole to a slowly varying asymptotic modulus in Einstein–Maxwell–dilaton theory. After an exact near-zone reduction, the linearized system is reduced to a single radial scalar operator L. A canonical tangent direction on the static solution manifold is identified through the kernel of an intrinsic covector C_mu, and a local slice mu(r_m)=0 converts the remaining tangential ambiguity into a finite-rank perturbation L_sl = L + f_phi(r_m) K tensor ev_{r_m}. The central result, Theorem 2, states that the completed operator A=(L_sl, B_m): X -> Y x R is Fredholm of index zero and is bijective if and only if B_m psi_star != 0, where psi_star generates the one-dimensional kernel of L_sl. The derivation is explicit: the reduction of the metric perturbations, the mass-balance cancellation, the Volterra-based Fredholm analysis in Appendix A, and the finite-rank index argument are all carried out in detail.

Significance. The paper's formal core is careful and checkable. It provides an exact reduction to a single scalar operator, a clean geometric identification of a canonical tangent direction, and a transparent finite-rank Fredholm argument that reduces solvability to a single scalar matching condition. It also avoids presupposing an inner product on the solution space, which distinguishes it from standard moduli-space projection approaches. The general framework is potentially useful as a foundation for explicit studies of adiabatic black-hole deformations, and the companion GHS specialization should make the abstract criteria testable. If the physical reduction in Section 5.2 is supplied, this would be a substantive methodological contribution. The main weakness is that the advertised physical conclusion is conditional on an unproven causal-selection step, which the authors explicitly acknowledge.

major comments (3)

[Sec. 5.2, Eqs. (5.50)-(5.59)] The central reduction from the sliced evolution equation (5.50) to the quasistatic radial problem (5.59) relies on the amplitude estimates (5.54)-(5.55): on the slowly forced branch, eta_ad/M_Pl = O_ad(epsilon) and (r_H,0/M_Pl) partial_v eta_ad = O_ad(epsilon^2). These estimates are not implied by the Fredholm hypotheses of Appendix A, and the paper itself states that establishing the causal selection requires analysis of the full initial-boundary-value problem (5.50) and is not attempted there. Indeed, Eq. (5.61) explicitly allows homogeneous modes with (r_H,0/M_Pl) partial_v eta_hom = O_ad(epsilon), and such modes can be excited by retarded forcing without being freely prescribed. Unless a spectral-gap or mode-stability argument rules them out, the physical leading-order deformation is not governed by (5.59), and the criterion B_m psi_star != 0 is not established for the actual retarded solution. Since Theorem 2 is internally consistent but applies only to a branch that has not been shown to contain the physical solution, this is a load-bearing gap in the paper's main claim.
[Sec. 5.2 and Abstract] The abstract's claim that existence and uniqueness of the leading-order adiabatic boundary-value problem 'reduce to a single scalar matching condition' is stronger than what is proved. What is proved is a conditional statement about the frozen radial operator A once the adiabatic branch is selected. The matching condition B_m psi_star != 0 is a theorem about A under the hypotheses of Lemma 1, but it does not by itself select the causal branch, nor does it control transients or the time-dependent evolution problem. The paper should either reframe the conclusion as explicitly conditional on a causal-selection hypothesis or supply the missing analysis of (5.50). This distinction is essential because the actual physical solution is a solution of the evolution problem, not of the quasistatic radial problem.
[Sec. 5.3, Theorem 2 and Remark 7] The statement of Theorem 2 depends on the identification ker L = span{phi_Phi}, which is assembled from the Fredholm results of Appendix A and the additional admissibility assumption (A.17)-(A.18). This is stated in Remark 6, but it is worth emphasizing that the theorem does not prove that phi_Phi is the only kernel generator; it assumes it. The companion paper and any application must verify this one-dimensional kernel identification explicitly. This is not a flaw in the theorem, but it is a nontrivial input that should appear in the theorem statement or in an explicit hypothesis list.

minor comments (4)

[Sec. 5.2, Eq. (5.64)] The notation B_m is used both for the matching functional and for the coefficient of the derivative evaluation in B_m = A_m ev_{r_m} + B_m partial_r ev_{r_m}; this is confusing and should be disambiguated.
[Sec. 4.1, Eq. (4.7)] The choice dot M_H^inst = O_ad(epsilon^2) is imposed as a condition on the instantaneous representative, but the text should more explicitly separate this convention from the exact quadratic horizon-flux result for M_H^exact, since the two notions are carefully distinguished throughout.
[Sec. 5.1.3, Eq. (5.35)] The rank-one operator notation |K><ev_{r_m}| is clear, but it may be helpful to state explicitly that the operator acts as eta -> f_phi(r_m) K(r) eta(r_m), since the bra-ket notation is otherwise uncommon in this context.
[References] The companion paper [18] is cited as arXiv:2607.xxx; providing the full identifier or stating that it is forthcoming would improve reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Fredholm solvability criterion is derived from the stated hypotheses by finite-rank perturbation theory and linear algebra; the self-citations are programmatic and not load-bearing.

full rationale

The paper's central claim, that under the stated transversality and Fredholm hypotheses the completed operator A = (L_sl, B_m) is bijective iff B_m psi_star != 0, follows directly from the one-dimensionality of ker L_sl and the definition of A (Eqs. 5.107-5.119). This is a mathematical consequence, not a fitted input or a prediction equivalent to its own assumption: psi_star is constructed from the sliced operator, and B_m is supplied by the outer matching functional. Theorem 1 identifies the canonical tangent direction using the identity M_H,0 = (M_Pl^2/2) r_H,0 and non-extremality; Appendix A establishes the Fredholm properties of L from a regular Volterra problem with an explicit right inverse. The finite-rank perturbation argument is standard and is cited to an external textbook [3]. The only self-references [12] and [18] concern companion applications and are not used to justify the abstract theorem. The paper explicitly flags the one step it does not derive: the causal selection of the adiabatic branch, stating in Section 5.2 that 'Establishing this causal selection requires analysis of the complete initial-boundary-value problem Eq. (5.50)' and 'is not attempted here.' That is an unproved validity assumption about the existence of a slowly forced branch, not a circular reduction: the quasistatic radial problem (5.59) does not presuppose the scalar matching criterion it is used to derive. No circular step can be quoted, so the score is 0.

Assumptions & free parameters 1 free parameters · 8 assumptions · 0 invented entities

No new physical entities are introduced; the covector C_mu, the lag field, and the sliced operator L_sl are mathematical constructs internal to the framework. The only hand-chosen scale is the matching radius r_m.

free parameters (1)

near-zone matching radius r_m = not fitted; chosen in (r_H, L_cos)
Defines the finite near-zone and the slice evaluation point; the solvability condition and operators depend on it. The framework requires only r_H << r_m << L_cos, so it is a hand-chosen scale rather than a fitted number.

assumptions (8)

domain assumption V, B in C^2 on the attained scalar range and B(phi) > 0
Section 2.1; positivity ensures standard Maxwell kinetic term, smoothness is needed for linearization.
domain assumption S is a smooth two-dimensional manifold of non-extremal static EMD solutions, regular local coordinates (Phi, M_H,0), with common radial neighborhood and phi_Phi in X nonzero
Section 3.2 and Appendix A; this is the core geometric hypothesis; companion paper II is to verify it for GHS.
domain assumption Uniform non-degeneracy kappa_H r_H >> epsilon (ultra-near-extremal excluded)
Section 2.3 Eqs (2.28),(2.38); needed for the adiabatic hierarchy and horizon regularity.
ad hoc to paper Slowly forced adiabatic branch: eta_ad = O(epsilon), (r_H,0/M_Pl) partial_v eta_ad = O(epsilon^2), so transport terms are subleading
Section 5.2 Eqs (5.54)-(5.59); assumed for the leading response, explicitly not derived from the full IBVP.
domain assumption Fredholm properties of L: surjective, index one, one-dimensional kernel span{phi_Phi}, with bounded right inverse
Appendix A proves these for the reduced operator under regularity conditions; Lemma 1 takes them as hypotheses.
domain assumption Slice transversality m_Phi(r_m) != 0 (equivalently f_phi(r_m) phi_Phi(r_m) != 0)
Section 5.1.2 Eq (5.15); required for the local slice mu(v,r_m)=0 to select a unique representative.
domain assumption Outer matching functional B_m: X -> R is a bounded linear functional
Section 5.3 Eqs (5.63)-(5.64); represents the exterior problem; boundedness is used in the Fredholm argument.
standard math Volterra integral equation well-posedness and standard Fredholm theory for compact perturbations
Appendix A and Section 5.3, refs [3],[21].

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Cite this review

Pith. "Pith review of Adiabatic Deformations of Black Hole Moduli: I. Canonical Slices and Fredholm Solvability." pith.science (2026). https://pith.science/paper/DRM7KW4K

@misc{pith2026260805275,
  author       = {Pith},
  title        = {Pith review of: Adiabatic Deformations of Black Hole Moduli: I. Canonical Slices and Fredholm Solvability},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DRM7KW4K}},
  note         = {Machine review of arXiv:2608.05275}
}

read the original abstract

The leading adiabatic response of a static black hole to a slowly evolving asymptotic modulus separates into a geometric problem and a functional-analytic one: the former identifies the tangent direction on the manifold of static solutions actually followed by the black hole, and the latter determines whether the resulting deformation admits a unique solution. We show that both can be characterized by explicit scalar criteria. Differentiating a smooth family of static solutions defines an intrinsic covector on its tangent space. When the family is parameterized by the asymptotic modulus and the static horizon-mass parameter, the kernel of this covector identifies a canonical tangent direction that coincides with the one selected by a slowly varying exterior modulus. Promoting the static family to a slowly evolving instantaneous representative does not solve the exact field equations; the resulting discrepancy defines a lag field governed, after elimination of the metric perturbations, by a single reduced radial operator. A representation convention separates the exact dynamical horizon from that of the instantaneous representative. A local slice then removes the remaining tangential ambiguity and induces a finite-rank perturbation of the reduced operator. The Fredholm index is preserved, and, whenever the finite-rank update is nontrivial, the canonical tangent zero mode is displaced from the kernel of the reduced operator. Under the transversality and Fredholm hypotheses, existence and uniqueness of the completed leading-order adiabatic boundary-value problem reduce to a single scalar matching condition on the resulting kernel generator. The construction depends only on a smooth non-extremal static solution manifold and the Fredholm properties of the reduced operator. A companion paper specializes the framework to the magnetic GHS family.

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Works this paper leans on

21 extracted references · 5 canonical work pages

[1]

Gibbons and K.-i

G.W. Gibbons and K.-i. Maeda,Black Holes and Membranes in Higher Dimensional Theories with Dilaton Fields,Nucl. Phys. B298(1988) 741

1988
[2]

Garfinkle, G.T

D. Garfinkle, G.T. Horowitz and A. Strominger,Charged black holes in string theory,Phys. Rev. D 43(1991) 3140

1991
[3]

Kato,Perturbation Theory for Linear Operators, Classics in Mathematics, Springer, Berlin (1995)

T. Kato,Perturbation Theory for Linear Operators, Classics in Mathematics, Springer, Berlin (1995)

work page 1995
[4]

Jacobson,Primordial black hole evolution in tensor-scalar cosmology,Phys

T. Jacobson,Primordial black hole evolution in tensor-scalar cosmology,Phys. Rev. Lett.83(1999) 2699 [astro-ph/9905303]

arXiv 1999
[5]

Horbatsch and C.P

M.W. Horbatsch and C.P. Burgess,Cosmic Black-Hole Hair Growth and Quasar OJ287,JCAP05 (2012) 010 [1111.4009]

arXiv 2012
[6]

Chadburn and R

S. Chadburn and R. Gregory,Time dependent black holes and scalar hair,Class. Quant. Grav.31 (2014) 195006 [1304.6287]

arXiv 2014
[7]

No-Scalar-Hair

J.D. Bekenstein,Novel “No-Scalar-Hair” Theorem for Black Holes,Phys. Rev. D51(1995) R6608

1995
[8]

Hui and A

L. Hui and A. Nicolis,No-Hair Theorem for the Galileon,Phys. Rev. Lett.110(2013) 241104 [1202.1296]

arXiv 2013

Show all 21 references

[9]

Sotiriou and S.-Y

T.P. Sotiriou and S.-Y. Zhou,Black Hole Hair in Generalized Scalar-Tensor Gravity,Phys. Rev. Lett.112(2014) 251102 [1312.3622]

2014 arXiv
[10]

Babichev and C

E. Babichev and C. Charmousis,Dressing a black hole with a time-dependent Galileon,JHEP08 (2014) 106 [1312.3204]

2014 arXiv
[11]

M.S. Volkov,Hairy black holes in the XX-th and XXI-st centuries, in14th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Astrophysics, and Relativistic Field Theories, vol. 2, pp. 1779–1798, 2017, DOI [1601.08230]

2017 arXiv
[12]

Benakli and A

K. Benakli and A. Chrysostomou,The Fate of Black Hole-Induced Moduli Excursions in the Presence of Scalar Potentials,2607.05488

arXiv
[13]

Delgado, M

M. Delgado, M. Montero and C. Vafa,Black Holes as Probes of Moduli Space Geometry,JHEP04 (2023) 045 [2212.08676]. – 51 –

2023 arXiv
[14]

Ferrara, G.W

S. Ferrara, G.W. Gibbons and R. Kallosh,Black Holes and Critical Points in Moduli Space,Nucl. Phys. B500(1997) 75 [hep-th/9702103]

1997 arXiv
[15]

Manton,A Remark on the Scattering of BPS Monopoles,Phys

N.S. Manton,A Remark on the Scattering of BPS Monopoles,Phys. Lett. B110(1982) 54

1982
[16]

Ferrell and D.M

R.C. Ferrell and D.M. Eardley,Slow-Motion Scattering and Coalescence of Maximally Charged Black Holes,Phys. Rev. Lett.59(1987) 1617

1987
[17]

Gibbons and P.J

G.W. Gibbons and P.J. Ruback,The Motion of Extreme Reissner–Nordstrom Black Holes in the Low Velocity Limit,Phys. Rev. Lett.57(1986) 1492

1986
[18]

Benakli and A

K. Benakli and A. Chrysostomou,Adiabatic Deformations of Black-Hole Moduli: II. The Magnetic GHS Family,2607.xxx
[19]

Misner and D.H

C.W. Misner and D.H. Sharp,Relativistic equations for adiabatic, spherically symmetric gravitational collapse,Phys. Rev.136(1964) B571

1964
[20]

Arnowitt, S

R.L. Arnowitt, S. Deser and C.W. Misner,Canonical variables for general relativity,Phys. Rev.117 (1960) 1595

1960
[21]

Coddington and N

E.A. Coddington and N. Levinson,Theory of Ordinary Differential Equations, McGraw-Hill, New York (1955). – 52 –

1955

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [1]

Gibbons and K.-i

G.W. Gibbons and K.-i. Maeda,Black Holes and Membranes in Higher Dimensional Theories with Dilaton Fields,Nucl. Phys. B298(1988) 741

1988

[2] [2]

Garfinkle, G.T

D. Garfinkle, G.T. Horowitz and A. Strominger,Charged black holes in string theory,Phys. Rev. D 43(1991) 3140

1991

[3] [3]

Kato,Perturbation Theory for Linear Operators, Classics in Mathematics, Springer, Berlin (1995)

T. Kato,Perturbation Theory for Linear Operators, Classics in Mathematics, Springer, Berlin (1995)

work page 1995

[4] [4]

Jacobson,Primordial black hole evolution in tensor-scalar cosmology,Phys

T. Jacobson,Primordial black hole evolution in tensor-scalar cosmology,Phys. Rev. Lett.83(1999) 2699 [astro-ph/9905303]

arXiv 1999

[5] [5]

Horbatsch and C.P

M.W. Horbatsch and C.P. Burgess,Cosmic Black-Hole Hair Growth and Quasar OJ287,JCAP05 (2012) 010 [1111.4009]

arXiv 2012

[6] [6]

Chadburn and R

S. Chadburn and R. Gregory,Time dependent black holes and scalar hair,Class. Quant. Grav.31 (2014) 195006 [1304.6287]

arXiv 2014

[7] [7]

No-Scalar-Hair

J.D. Bekenstein,Novel “No-Scalar-Hair” Theorem for Black Holes,Phys. Rev. D51(1995) R6608

1995

[8] [8]

Hui and A

L. Hui and A. Nicolis,No-Hair Theorem for the Galileon,Phys. Rev. Lett.110(2013) 241104 [1202.1296]

arXiv 2013