Stationary Spacetimes and Self-Adjointness in Klein-Gordon Theory

TL;DR

This paper investigates the conditions under which the spatial part of the Klein-Gordon operator is essentially self-adjoint in stationary spacetimes, extending previous results from static to more general cases, including some non-globally hyperbolic spacetimes.

Contribution

It generalizes essential self-adjointness results for the Klein-Gordon operator from static to stationary and certain non-globally hyperbolic spacetimes, under smoothness and semi-boundedness assumptions.

Findings

Essential self-adjointness holds in globally hyperbolic stationary spacetimes with smooth metrics.

Extension of self-adjointness results to some non-globally hyperbolic spacetimes.

The spatial Klein-Gordon operator is a Laplace-Beltrami operator plus a potential.

Abstract

We consider the problem of essential self-adjointness of the spatial part of the Klein-Gordon operator in stationary spacetimes. This operator is shown to be a Laplace-Beltrami type operator plus a potential. In globally hyperbolic spacetimes, essential selfadjointness is proven assuming smoothness of the metric components and semi-boundedness of the potential. This extends a recent result for static spacetimes to the stationary case. Furthermore, we generalize the results to certain non-globally hyperbolic spacetimes.