On well-posedness of the Cauchy problem for the wave equation in static spherically symmetric spacetimes

TL;DR

This paper establishes simple, general conditions for the well-posedness of the Cauchy problem for scalar fields in static, spherically symmetric spacetimes, focusing on properties of the spatial wave operator.

Contribution

It introduces the concept of quasi essentially self-adjointness and characterizes conditions ensuring well-posedness beyond standard self-adjointness assumptions.

Findings

Conditions for well-posedness related to operator properties

Introduction of quasi essentially self-adjointness as a key concept

Application to various spacetime examples

Abstract

We give simple conditions implying the well-posedness of the Cauchy problem for the propagation of classical scalar fields in general (n+2)-dimensional static and spherically symmetric spacetimes. They are related to properties of the underlying spatial part of the wave operator, one of which being the standard essentially self-adjointness. However, in many examples the spatial part of the wave operator turns out to be not essentially selfadjoint, but it does satisfy a weaker property that we call here quasi essentially self-adjointness, which is enough to ensure the desired well-posedness. This is why we also characterize this second property. We state abstract results, then general results for a class of operators encompassing many examples in the literature, and we finish with the explicit analysis of some of them.