arXiv:0908.4292v2 [gr-qc] 12 Jul 2010 A PROOF OF PRICE’S LAW ON SCHWARZSCHILD BLACK HOLE MANIFOLDS FOR ALL ANGULAR MOMENTA ROLAND DONNINGER, WILHELM SCHLAG, AND AVY SOFFER Abstract. 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. 1. Introduction and main result In General Relativity, the dynamics of spacetime is governed by Einstein’s equation which, in the absence of matter, takes the form R µν (g)=0 where R µν (g) is the Ricci tensor of the Lorentz metric g. Exact solutions (i.e., solutions which are known in closed form) include the free flat Minkowski spacetime as well as the Schwarzschild metric and, more generally, the Kerr solution. The Schwarzschild solution is spherically symmetric and corresponds to a nonrotating black hole whereas rotating black holes are described by the axially symmetric Kerr spacetime. A fundamental mathematical problem in General Relativity is the understanding of the stability of these solutions. The stability of the flat Minkowski spacetime under small perturbations was shown in the seminal work of Christodoulou and Klainerman [15] in the late 1980’s. A simpler proof was later developed by Lindblad and Rodnianski [37]. However, we are very far from understanding the dynamics near a black hole. Yet, latest experimental setups are crucially dependent on such an analysis, in order to observe gravitational waves (see for example [21], [23], [24], [22] and cited ref.). Most efforts are now focused on understanding the linear dynamics and stability of such solutions, see e.g. [32], [7] and cited ref., as well as [43]. The mathematical aspects of the problem will be discussed below in more detail. We also refer the reader to the survey [20] which gives an excellent overview of recent developments in the field from the mathematical perspective. 1.1. Wave evolution on the Schwarzschild manifold. As a first approximation to the lin- ear stability problem of a nonrotating black hole one may consider the wave equation on a fixed Schwarzschild background. One is then typically interested in decay estimates for the evolution. To simplify things even more, one restricts the analysis to the exterior region of the black hole which, however, is physically reasonable: such a model describes a black hole subject to a small external The first author is an Erwin Schr¨ odinger 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 Bizo´ n for a number of helpful remarks on a first version of this paper. The second author was partly supported by the National Science Foundation DMS-0617854. The third 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. 1