The wave equation on Schwarzschild-de Sitter spacetimes

TL;DR

This paper proves that solutions to the wave equation on Schwarzschild-de Sitter spacetimes decay to a constant at a polynomial rate, with uniform bounds on energy decay, regardless of initial data symmetry.

Contribution

It establishes the first rigorous decay results for solutions with arbitrary smooth initial data on Schwarzschild-de Sitter backgrounds, including decay along horizons.

Findings

Solutions decay to a constant faster than any polynomial rate.

Uniform decay bounds for energy measured by Killing fields.

Decay rates established along horizons.

Abstract

We consider solutions to the linear wave equation $\Box_g\phi=0$ on a non-extremal maximally extended Schwarzschild-de Sitter spacetime arising from arbitrary smooth initial data prescribed on an arbitrary Cauchy hypersurface. (In particular, no symmetry is assumed on initial data, and the support of the solutions may contain the sphere of bifurcation of the black/white hole horizons and the cosmological horizons.) We prove that in the region bounded by a set of black/white hole horizons and cosmological horizons, solutions $\phi$ converge pointwise to a constant faster than any given polynomial rate, where the decay is measured with respect to natural future-directed advanced and retarded time coordinates. We also give such uniform decay bounds for the energy associated to the Killing field as well as for the energy measured by local observers crossing the event horizon. The results in particular include decay rates along the horizons themselves. Finally, we discuss the relation of these results to previous heuristic analysis of Price and Brady et al.