Morawetz Estimates Without Relative Degeneration and Exponential Decay on Schwarzschild–de Sitter Spacetimes

Abstract

We use a novel physical space method to prove relatively non-degenerate integrated energy estimates for the wave equation on subextremal Schwarzschild–de Sitter spacetimes with parameters (M,Λ)$$(M,\Lambda )$$. These are integrated decay statements whose bulk energy density, though degenerate at highest order, is everywhere comparable to the energy density of the boundary fluxes. As a corollary, we prove that solutions of the wave equation decay exponentially on the exterior region. The main ingredients are a previous Morawetz estimate of Dafermos–Rodnianski and an additional argument based on commutation with a vector field which can be expressed in the form r1-2Mr-Λ3r2∂∂r,$$\begin{aligned} r\sqrt{1-\frac{2M}{r}-\frac{\Lambda }{3}r^2}\frac{\partial }{\partial r}, \end{aligned}$$where ∂r$$\partial _r$$ here denotes the coordinate vector field corresponding to a well-chosen system of hyperboloidal coordinates. Our argument gives exponential decay also for small first-order perturbations of the wave operator. In the limit Λ=0$$\Lambda =0$$, our commutation corresponds to the one introduced by Holzegel–Kauffman (A note on the wave equation on black hole spacetimes with small non-decaying first-order terms, 2020. arXiv:2005.13644).