Global Hyperbolicity and Self-adjointness

TL;DR

This paper demonstrates that the spatial component of the Klein-Gordon operator is essentially self-adjoint on Cauchy surfaces in certain spacetimes, linking global hyperbolicity with operator self-adjointness.

Contribution

It establishes a connection between global hyperbolicity and the self-adjointness of the Klein-Gordon operator on various spacetimes, using geodesic completeness.

Findings

Spatial Klein-Gordon operator is essentially self-adjoint on Cauchy surfaces.

Global hyperbolicity of spacetimes is proven for the studied classes.

The proof connects global hyperbolicity with geodesic completeness.

Abstract

We show that the spatial part of the Klein-Gordon operator is an essentially self-adjoint operator on the Cauchy surfaces of various classes of spacetimes. Our proof employs the intricate connection between global hyperbolicity and geodesically complete Riemannian surfaces, and concludes by proving global hyperbolicity of the spacetimes under study.