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

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.

Linear Waves in the Interior of Extremal Black Holes I

“…Price's law (1.3) gives in particular an upper bound for the decay of φ along the event horizon of subextremal Reissner-Nordström. Rigorous results pertaining to the upper bound in (1.3) have been obtained in [21,24,25,41,49]. Moreover, Luk-Oh showed in [37] that ∂ v φ cannot decay with a polynomial rate faster than v −4 along the event horizon of subextremal Reissner-Nordström, for generic, compactly supported data.…”

“…The methods in [21,24,25,37,41,49] break down in extremal Reissner-Nordström, in view of the absence of the local red-shift effect. Heuristics and numerics regarding latetime tails for extremal Reissner-Nordström in [33,44,47] suggest an extremal variant of "Price's law" that in particular predicts:…”

Asymptotic Distribution of Quasi-Normal Modes for Kerr–de Sitter Black Holes

“…We add that there have been many papers on decay of linear waves for Schwarzschild and Kerr black holes-see [2,9,18,19,22,23,28,29,46,47,49] and references given there. In that case the cosmological constant is 0 (unlike in the de Sitter case, where it is positive), and the methods of scattering theory are harder to apply because of an asymptotically Euclidean infinity.…”

Asymptotic Behavior of Massless Dirac Waves in Schwarzschild Geometry

“…This was pursued by Kronthaler in [20,21], who was able to establish a decay rate in the spherical symmetric case. Another analysis was carried out later in [13,12] for all spherical modes; they proved a t −(2l+2) local decay where l = 1, 2, · · · is the angular momentum and improved the decay rate to t −3 in the case l = 0. The decay rate for solutions with general initial data was achieved in [10,24,14]; see also [9,3,4,5,1,22,8] for related results.…”

“…The proof of Theorem 1.2 is inspired by [13,12]. The key ingredient is to justify the absence of negative eigenvalues of H +,λ and bounded zero energy solutions for H +,λ .…”