Linear Waves in the Interior of Extremal Black Holes II

Gajic, Dejan

doi:10.1007/s00023-017-0614-x

Cited by 30 publications

(30 citation statements)

References 38 publications

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“…The results of the follow-up paper [28] in the extremal Kerr interior allow us to make the following conjecture for axisymmetric perturbations of extremal Kerr characteristic initial data for (1.8). We do not even venture a conjecture in the case of non-axisymmetric initial data for (1.8), because there are as of yet no boundedness and decay estimates for nonaxisymmetric solutions φ to (1.1) in the exterior, nor in the interior of extremal Kerr.…”

Section: Conjectures For the Vacuum Einstein Equationsmentioning

confidence: 85%

Linear Waves in the Interior of Extremal Black Holes I

Gajic

2016

Commun. Math. Phys.

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Abstract:We consider solutions to the linear wave equation in the interior region of extremal Reissner-Nordström black holes. We show that, under suitable assumptions on the initial data, the solutions can be extended continuously beyond the Cauchy horizon and, moreover, that their local energy is finite. This result is in contrast with previously established results for subextremal Reissner-Nordström black holes, where the local energy was shown to generically blow up at the Cauchy horizon.

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Section: Conjectures For the Vacuum Einstein Equationsmentioning

confidence: 85%

Linear Waves in the Interior of Extremal Black Holes I

Gajic

2016

Commun. Math. Phys.

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“…These kind of r-weighted multipliers are adapted to the singular geometry of the Schwarzschild black hole at r = 0 and are thus very different from the multipliers used in the interior of the Kerr and Reissner-Nordström black holes, cf. [21], [23], [24], [32], [33].…”

Section: Sobolev Embedding On the Spheres Givesmentioning

confidence: 99%

Generic Blow-Up Results for the Wave Equation in the Interior of a Schwarzschild Black Hole

Fournodavlos

Sbierski

2019

Arch Rational Mech Anal

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We study the behaviour of smooth solutions to the wave equation, g ψ = 0, in the interior of a fixed Schwarzschild black hole. In particular, we obtain a full asymptotic expansion for all solutions towards r = 0 and show that it is characterised by its first two leading terms, the principal logarithmic term and a bounded second order term. Moreover, we characterise an open set of initial data for which the corresponding solutions blow up logarithmically on the entirety of the singular hypersurface {r = 0}. Our method is based on deriving weighted energy estimates in physical space and requires no symmetries of solutions. However, a key ingredient in our argument uses a precise analysis of the spherically symmetric part of the solution and a monotonicity property of spherically symmetric solutions in the interior.

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“…In this section, we review the results established in [10,11] concerning the behaviour of solutions to the linear wave equation g φ = 0 in the interior of extremal black holes. The results concern the following cases:…”

Section: Previous Results On the Linear Wave Equationmentioning

confidence: 99%

“…Mathematically, the only partial progress was made for a related linear problem, namely the study of the linear scalar wave equation on extremal black hole backgrounds. For the linear scalar wave equation, the first author established [10,11] that in the extremal case, the Cauchy horizon is more stable than its subextremal counterpart. In particular, the solutions to linear wave equations are not only bounded, as in the subextremal case, but they in fact obey higher regularity bounds which fail in the subextremal case (see Section 1.1 for a more detailed discussion).…”

Section: Introductionmentioning

confidence: 99%

The interior of dynamical extremal black holes in spherical symmetry

Gajic

Luk

2019

Pure Appl. Analysis

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We study the nonlinear stability of the Cauchy horizon in the interior of extremal Reissner-Nordström black holes under spherical symmetry. We consider the Einstein-Maxwell-Klein-Gordon system such that the charge of the scalar field is appropriately small in terms of the mass of the background extremal Reissner-Nordström black hole. Given spherically symmetric characteristic initial data which approach the event horizon of extremal Reissner-Nordström sufficiently fast, we prove that the solution extends beyond the Cauchy horizon in C 0, 1 2 ∩ W 1,2 loc , in contrast to the subextremal case (where generically the solution is C 0 \ (C 0, 1 2 ∩ W 1,2 loc )). In particular, there exist non-unique spherically symmetric extensions which are moreover solutions to the Einstein-Maxwell-Klein-Gordon system. Finally, in the case that the scalar field is chargeless and massless, we additionally show that the extension can be chosen so that the scalar field remains Lipschitz. * and the solution cannot be extended as a spherically symmetric solution to the Einstein-Maxwell-scalar field system 2 .Remark 1.3 (Regularity of the metric and extensions as solutions to (1.1)). The fact that we can extend the solutions beyond the Cauchy horizon is intimately connected to the regularity of the solutions up to the Cauchy horizon. In particular this relies on the fact the metric, the scalar field and the electromagnetic potential remain in (spacetime) C 0 ∩ W 1,2 loc . In fact, the solutions are at a level of regularity for which the Einstein equations are still locally well-posed 3 . One can therefore construct extensions beyond the Cauchy horizon by solving appropriate characteristic initial value problems; see Section 10.In this connection, note that we emphasised in the statement of the theorem that the solution can be extended beyond the Cauchy horizon as a spherically symmetric solution to (1.1). The emphasis on the spherical symmetry of the extension is made mostly to contrast with the situation in the subextremal case (cf. Remark 1.2). This should not be understood as implying that the extensions necessarily are spherically symmetric: In fact, with the bounds that we establish in this paper, one can in principle construct using the techniques in [23] extensions (still as solutions to (1.1)) without any symmetry assumptions (cf. Footnote 1).Remark 1.4 (Assumptions on the event horizon). The assumptions we impose on the event horizon are consistent with the expected late-time behavior of the solutions in the exterior region of the black hole, at least in the e = m = 0 case if one extrapolates from numerical results [25]. In particular, the transversal derivative of the scalar field is not required to decay along the event horizon, and is therefore consistent with the Aretakis instability [2]. Of course, in order to completely understand the structure of the interior, one needs to prove that the decay estimates along the event horizon indeed hold for general dynamical solutions approaching these extremal black holes. This remains a...

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Linear Waves in the Interior of Extremal Black Holes II

Cited by 30 publications

References 38 publications

Linear Waves in the Interior of Extremal Black Holes I

Linear Waves in the Interior of Extremal Black Holes I

Generic Blow-Up Results for the Wave Equation in the Interior of a Schwarzschild Black Hole

The interior of dynamical extremal black holes in spherical symmetry

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