On Solutions to the Wave Equation on Non-globally Hyperbolic Manifold

TL;DR

This paper investigates the wave equation on a non-globally hyperbolic manifold with closed timelike curves, establishing conditions for the existence and uniqueness of classical solutions to the Cauchy problem.

Contribution

It provides the first analysis of the wave equation on a Minkowski plane with a handle, identifying necessary conditions for well-posedness in non-globally hyperbolic spacetimes.

Findings

Classical solutions exist and are unique under specific initial data conditions.

The study characterizes the additional constraints required for well-posedness.

It extends understanding of wave behavior in spacetimes with closed timelike curves.

Abstract

We consider the Cauchy problem for the wave equation on a non-globally hyperbolic manifold of the special form (Minkowski plane with a handle) containing closed timelike curves (time machines). We prove that the classical solution of the Cauchy problem exists and is unique if and only if the initial data satisfy to some set of additional conditions.