Hidden symmetries and decay for the wave equation on the Kerr spacetime

Andersson, Lars; Blue, Pieter

doi:10.48550/arxiv.0908.2265

Cited by 62 publications

(167 citation statements)

References 0 publications

Supporting

Mentioning

162

Contrasting

Order By: Relevance

“…In [42], [41], [55] and [39] it is shown how to apply such estimates to obtain Strichartz type decay estimates. We also mention the recent work [3]. After submission of the present paper, Tataru announced a proof of the sharp pointwise t −3 decay for general data without symmetry assumptions, see [54].…”

Section: Introduction and Main Resultsmentioning

confidence: 77%

A proof of Price's Law on Schwarzschild black hole manifolds for all angular momenta

Donninger

Schlag

Soffer

2011

Advances in Mathematics

122

View full text Add to dashboard Cite

Price's Law states that linear perturbations of a Schwarzschild black hole fall off as t −2ℓ−3 for t → ∞ provided the initial data decay sufficiently fast at spatial infinity. Moreover, if the perturbations are initially static (i.e., their time derivative is zero), then the decay is predicted to be t −2ℓ−4 . We give a proof of t −2ℓ−2 decay for general data in the form of weighted L 1 to L ∞ bounds for solutions of the Regge-Wheeler equation. For initially static perturbations we obtain t −2ℓ−3 . The proof is based on an integral representation of the solution which follows from self-adjoint spectral theory. We apply two different perturbative arguments in order to construct the corresponding spectral measure and the decay bounds are obtained by appropriate oscillatory integral estimates.

show abstract

Section: Introduction and Main Resultsmentioning

confidence: 77%

A proof of Price's Law on Schwarzschild black hole manifolds for all angular momenta

Donninger

Schlag

Soffer

2011

Advances in Mathematics

122

View full text Add to dashboard Cite

show abstract

“…As a consequence of Theorem 3 in [3], it follows that D is contained in the domain of the Friedrichs extension A F of the restriction of A 0 to C ∞ 0 (Ω s , C) and that A F f = A 0 f for every f ∈ D. In this connection, note that the addition of a multiple of the identity operator 'does not affect' the Friedrichs extension of an operator. 1 Since D is a core for Ā0 , from this follows that A F ⊃ Ā0 and hence, since A F is in particular symmetric and Ā0 is self-adjoint, that A F = Ā0 . Lemma 3.7.…”

Section: Basic Properties Of Operators In the Equationmentioning

confidence: 94%

“…Definition 4.2. We denote by W 1 A the complex Hilbert space 1 given by D(A 1/2 ) equipped with the scalar product | 1 , defined by…”

Section: Formulation Of An Initial Value Problemmentioning

confidence: 99%

“…This absence complicates the application of methods from operator theory and of so called "energy methods" that are both employed in estimating the decay of solutions of hyperbolic partial differential equations. 1 On the other hand, two facts are worth noting. For this, note that in the following any reference to coordinates implicitly assumes use of Boyer-Lindquist coordinates [8].…”

Section: Introductionmentioning

confidence: 99%

“…Quite differently from the case of a Schwarzschild background, the results for these test cases suggest an asymmetry between the cases µ = 0 and µ = 0. In the case of the wave equation, i.e., µ = 0, results point to the stability of the solutions [34,16,12,1,23,24], whereas for µ = 0, there are a number of results pointing in the direction of instability of the solutions under certain conditions [13,14,36,17,22,9,19].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the stability of the massive scalar field in Kerr space-time

Beyer¹

2011

Journal of Mathematical Physics

View full text Add to dashboard Cite

The current early stage in the investigation of the stability of the Kerr metric is characterized by the study of appropriate model problems. Particularly interesting is the problem of the stability of the solutions of the Klein-Gordon equation, describing the propagation of a scalar field in the background of a rotating (Kerr-) black hole. Results suggest that the stability of the field depends crucially on its mass µ. Among others, the paper provides an improved bound for µ above which the solutions of the reduced, by separation in the azimuth angle in Boyer-Lindquist coordinates, KleinGordon equation are stable. Finally, it gives new formulations of the reduced equation, in particular, in form of a time-dependent wave equation that is governed by a family of unitarily equivalent positive self-adjoint operators. The latter formulation might turn out useful for further investigation. On the other hand, it is proved that from the abstract properties of this family alone it cannot be concluded that the corresponding solutions are stable.

show abstract