Klein-Gordon Solutions on Non-Globally Hyperbolic Standard Static Spacetimes

TL;DR

This paper develops a method to construct and analyze solutions to the Klein-Gordon equation on non-globally hyperbolic static spacetimes, extending previous work and exploring various properties of these solutions.

Contribution

It introduces a general construction of Klein-Gordon solutions based on self-adjoint extensions of the Laplace-Beltrami operator on static spacetimes, extending Wald's work.

Findings

Existence and uniqueness of solutions established.

Support and energy properties analyzed.

Examples including non-bounded below extensions provided.

Abstract

We construct a class of solutions to the Cauchy problem of the Klein-Gordon equation on any standard static spacetime. Specifically, we have constructed solutions to the Cauchy problem based on any self-adjoint extension (satisfying a technical condition: "acceptability") of (some variant of) the Laplace-Beltrami operator defined on test functions in an $L^2$-space of the static hypersurface. The proof of the existence of this construction completes and extends work originally done by Wald. Further results include the uniqueness of these solutions, their support properties, the construction of the space of solutions and the energy and symplectic form on this space, an analysis of certain symmetries on the space of solutions and of various examples of this method, including the construction of a non-bounded below acceptable self-adjoint extension generating the dynamics.