Morawetz estimates without relative degeneration and exponential decay on Schwarzschild-de Sitter spacetimes

TL;DR

This paper introduces a novel physical space method to establish non-degenerate energy estimates and exponential decay for wave equations on Schwarzschild-de Sitter spacetimes, improving understanding of wave behavior in these geometries.

Contribution

It presents a new approach to prove integrated energy estimates and exponential decay without degeneracy at the horizon, extending previous methods to include small perturbations.

Findings

Proved exponential decay of wave solutions on Schwarzschild-de Sitter backgrounds.

Established energy estimates that are non-degenerate at the horizon.

Extended decay results to small perturbations of the wave operator.

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 $(M,\Lambda)$. 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 $r\sqrt{1-\frac{2M}{r}-\frac{\Lambda}{3}r^2}\frac{\partial}{\partial r}$, where $\partial_r$ 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 $\Lambda=0$, our commutation corresponds to the one introduced by Holzegel-Kauffman.