TY - JOUR

T1 - A proof of Price's Law on Schwarzschild black hole manifolds for all angular momenta

AU - Donninger, Roland

AU - Schlag, Wilhelm

AU - Soffer, Avy

N1 - Funding Information:
* Corresponding author. E-mail addresses: [email protected] (R. Donninger), [email protected] (W. Schlag), [email protected] (A. Soffer). 1 The author is an Erwin Schrödinger Fellow of the FWF (Austrian Science Fund) Project No. J2843 and he wants to thank Peter C. Aichelburg for his support. Furthermore, all three authors would like to thank Piotr Bizoń for a number of helpful remarks on a first version of this paper. 2 The author was partly supported by the National Science Foundation DMS-0617854. 3 The author wants to thank A. Ori and T. Damour for helpful discussions, the IHES France for the invitation and the NSF DMS-0903651 for partial support.

PY - 2011/1/15

Y1 - 2011/1/15

N2 - 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 L1 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.

AB - 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 L1 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.

KW - Dispersive estimates for wave equation

KW - Schwarzschild black hole

KW - Spectral and scattering theory

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U2 - 10.1016/j.aim.2010.06.026

DO - 10.1016/j.aim.2010.06.026

M3 - Article

AN - SCOPUS:77958476020

SN - 0001-8708

VL - 226

SP - 484

EP - 540

JO - Advances in Mathematics

JF - Advances in Mathematics

IS - 1

ER -