The massive wave equation in asymptotically AdS spacetimes

TL;DR

This paper studies the massive wave equation in asymptotically AdS spacetimes, establishing well-posed boundary conditions and regularity results without assuming stationarity or separability.

Contribution

It demonstrates well-posedness and regularity for the massive wave equation with various boundary conditions in asymptotically AdS spaces, extending previous results.

Findings

Well-posedness of initial-boundary value problems at H^1 regularity

Existence of higher regularity and asymptotic expansions near scri

Applicable to a range of negative mass parameters including conformally coupled case

Abstract

We consider the massive wave equation on asymptotically AdS spaces. We show that the timelike scri behaves like a finite timelike boundary, on which one may impose the equivalent of Dirichlet, Neumann or Robin conditions for a range of (negative) mass parameter which includes the conformally coupled case. We demonstrate well posedness for the associated initial-boundary value problems at the $H^1$ level of regularity. We also prove that higher regularity may be obtained, together with an asymptotic expansion for the field near scri. The proofs rely on energy methods, tailored to the modified energy introduced by Breitenlohner and Freedman. We do not assume the spacetime is stationary, nor that the wave equation separates.