Decay of Solutions of the Wave Equation in the Kerr Geometry

In this paper we present a spectral decomposition of solutions to relativistic wave equations described on horizon-penetrating hyperboloidal slices within a given Schwarzschild-black-hole background. The wave equation in question is Laplace-transformed which leads to a spatial differential equation with a complex parameter. For initial data which are analytic with respect to a compactified spatial coordinate, this equation is treated with the help of the Mathematica-package in terms of a sophisticated Taylor series analysis. Thereby, all ingredients of the desired spectral decomposition arise explicitly to arbitrarily prescribed accuracy, including quasinormal modes, quasinormal mode amplitudes as well as the jump of the Laplace-transform along the branch cut. Finally, all contributions are put together to obtain via the inverse Laplace transformation the spectral decomposition in question. The paper explains extensively this procedure and includes detailed discussions of relevant aspects, such as the definition of quasinormal modes and the question regarding the contribution of infinity frequency modes to the early time response of the black hole.

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“…2), we gather 1. QNM contributions: Writing according to (60), (96), (102) and (28) the Laplace transformV in the vicinity of the QNM s n aŝ…”

Section: Spectral Decompositionmentioning

confidence: 99%

“…Rigorous mathematical results including integral representations have been obtained in the case of Cauchy problems for the massive Dirac equation as well as for the Teukolskyequation in the nonextreme Kerr-Newman geometry outside the event horizon, see e.g. [25][26][27][28][29].…”

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confidence: 99%

Spectral decomposition of black-hole perturbations on hyperboloidal slices

2016

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“…The linearization of the Einstein equations around the Kerr solution has been shown to have no unstable modes [42]. For each component in such a decomposition, the L ∞ loc norm of solutions to the wave equation have been shown to decay [24] and solutions to the Dirac equation decay at a rate of t −5/6 in L ∞ loc [23]. The problem of fields coupled to the Einstein equation is very challenging.…”

Section: Introductionmentioning

confidence: 97%

Phase space analysis on some black hole manifolds

Blue

Soffer

2009

Journal of Functional Analysis

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The Schwarzschild and Reissner-Nordstrøm solutions to Einstein's equations describe space-times which contain spherically symmetric black holes. We consider solutions to the linear wave equation in the exterior of a fixed black hole space-time of this type. We show that for solutions with initial data which decay at infinity and at the bifurcation sphere, a weighted L 6 norm in space decays like t −1/3 . This weight vanishes at the event horizon, but not at infinity. To obtain this control, we require only an loss of angular derivatives.

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“…For instance, a definite sign of K(u) in the case of a vanishing mass of the field would open up the road to a simpler and independent proof of the stability of the solutions of the wave equation in a Kerr background [26]. In addition, it could lead to a sharper estimate on a possible onset of instability in the case of a non-vanishing mass of the field [1,2,4,12,20,22,28,38,39,56].…”

Section: Discussion and Open Problemsmentioning

confidence: 99%

On a new symmetry of the solutions of the wave equation in the background of a Kerr black hole

Beyer¹,

Holmes²

2008

Class. Quantum Grav.

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This paper derives a new symmetry operator S that commutes with a normalized form of the wave operator in a Kerr background. Differently to previously known symmetry operators for the wave operator, S contains only a partial time derivative of the first order, but not of higher order. The key for the proof of commutation is an abstract lemma that might also be applicable to other separable wave equations. As a consequence, in formulations of the initial value problem for the wave equation in terms of first-order systems of PDE and related formulations, S leads on a operator that formally commutes with the infinitesimal generator of time evolution and therefore is a candidate for the generator of a strongly continuous semigroup or group of symmetries. Finally, a new conservation law is derived for the scalar field in the gravitational field of a Kerr black hole that is associated with a Killing tensor of that spacetime.

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Decay of Solutions of the Wave Equation in the Kerr Geometry

Cited by 116 publications

References 12 publications

Spectral decomposition of black-hole perturbations on hyperboloidal slices

Spectral decomposition of black-hole perturbations on hyperboloidal slices

Phase space analysis on some black hole manifolds

On a new symmetry of the solutions of the wave equation in the background of a Kerr black hole

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