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

Aug, 2009
41 pages
Published in:
  • Adv.Math. 226 (2011) 484-540
e-Print:

Citations per year

2009201320172021202502468
Abstract: (arXiv)
Price's Law states that linear perturbations of a Schwarzschild black hole fall off as t23t^{-2\ell-3} for tt \to \infty 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 t24t^{-2\ell-4}. We give a proof of t22t^{-2\ell-2} decay for general data in the form of weighted L1L^1 to LL^\infty bounds for solutions of the Regge--Wheeler equation. For initially static perturbations we obtain t23t^{-2\ell-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.
  • black hole: Schwarzschild
  • perturbation: linear
  • measure: spectral
  • space-time: stability
  • potential: decay
  • resonance: energy
  • asymptotic behavior
  • black hole: angular momentum