Type: Article
Publication Date: 2018-01-01
Citations: 74
DOI: https://doi.org/10.24033/asens.2358
We develop a definitive physical-space scattering theory for the scalar wave equation ◻gψ = 0 on Kerr exterior backgrounds in the general subextremal case a < M .In particular, we prove results corresponding to "existence and uniqueness of scattering states" and "asymptotic completeness" and we show moreover that the resulting "scattering matrix" mapping radiation fields on the past horizon H - and past null infinity I -to radiation fields on H + and I + is a bounded operator.The latter allows us to give a time-domain theory of superradiant reflection.The boundedness of the scattering matrix shows in particular that the maximal amplification of solutions associated to ingoing finite-energy wave packets on past null infinity I -is bounded.On the frequency side, this corresponds to the novel statement that the suitably normalised reflection and transmission coefficients are uniformly bounded independently of the frequency parameters.We further complement this with a demonstration that superradiant reflection indeed amplifies the energy radiated to future null infinity I + of suitable wave-packets as above.The results make essential use of a refinement of our recent proof [M.Dafermos, I. Rodnianski and Y. Shlapentokh-Rothman, Decay for solutions of the wave equation on Kerr exterior spacetimes III: the full subextremal case a < M , arXiv:1402.6034] of boundedness and decay for solutions of the Cauchy problem so as to apply in the class of solutions where only a degenerate energy is assumed finite.We show in contrast that the analogous scattering maps cannot be defined for the class of finite non-degenerate energy solutions.This is due to the fact that the celebrated horizon red-shift effect acts as a blue-shift instability when solving the wave equation backwards.Contents * t(t, r) ≐ tt(r), * φ(t, r) ≐ φφ(r) mod 2π, * θ ≐ θ,