On Pointwise Decay of Linear Waves on a Schwarzschild Black Hole Background

Abstract

We prove sharp pointwise t −3 decay for scalar linear perturbations of a Schwarzschild black hole without symmetry assumptions on the data. We also consider electromagnetic and gravitational perturbations for which we obtain decay rates t −4 , and t −6 , respectively. We proceed by decomposition into angular momentum ℓ and summation of the decay estimates on the Regge-Wheeler equation for fixed ℓ. We encounter a dichotomy: the decay law in time is entirely determined by the asymptotic behavior of the Regge-Wheeler potential in the far field, whereas the growth of the constants in ℓ is dictated by the behavior of the Regge-Wheeler potential in a small neighborhood around its maximum. In other words, the tails are controlled by small energies, whereas the number of angular derivatives needed on the data is determined by energies close to the top of the Regge-Wheeler potential. This dichotomy corresponds to the well-known principle that for initial times the decay reflects the presence of complex resonances generated by the potential maximum, whereas for later times the tails are determined by the far field. However, we do not invoke complex resonances at all, but rely instead on semiclassical Sigal-Soffer type propagation estimates based on a Mourre bound near the top energy.1 1 The notation a± stands for a± ε where ε > 0 is arbitrary (the choice determines the constants involved). Also, instead of (/ ∇ 10 , / ∇ 9 ) in (1.3) one needs less, namely (/ ∇ σ+1 , / ∇ σ ) where σ > 8 is arbitrary, see the proof in Section 5 for details.2 From the point of view of the decay estimates in [17], these values need to be excluded as they are precisely the ones that give rise to a zero energy resonance.

Uniform energy bound and asymptotics for the Maxwell field on a slowly rotating Kerr black hole exterior

“…Decay in L ∞ loc has been proved without a rate [17]. For solutions of the ReggeWheeler equation, pointwise decay has been proved [15].…”

“…The location of the orbiting null geodesics are determined by the double roots of the potential appearing in an ODE for the radial components. The potential is R(r; M, a; e, ℓ z , Q), where e, ℓ z , and Q are constant of the geodesic motion, but R remains defined by the function given in equation (15). With respect to a convenient, non-affine parameterisation, λ, of the null geodesics, the ODE is (dr/dλ) 2 = R. Standard ODE analysis, as in a Newtonian potential problem, dictates that there are orbiting null geodesics when R = 0 and ∂ r R = 0.…”

Resolvent Estimates for Normally Hyperbolic Trapped Sets

“…Our paper is concerned only with the analysis near the trapped set. Unlike in most other mathematical works on Schwarzschild and Kerr black holes -see for instance [8], [3], [17,18], [22,23], [34], [43]-this analysis of the trapped set does not use separation of variables and properties of the Regge-Wheeler potential. It is carried out in a way applicable to the perturbations of the metric.…”

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.…”