Cauchy Problem on Non-globally Hyperbolic Spacetimes

TL;DR

This paper investigates the existence and uniqueness of solutions to the wave equation's Cauchy problem on non-globally hyperbolic spacetimes with closed timelike curves, highlighting conditions for solution existence and self-consistency.

Contribution

It demonstrates the existence of discontinuous solutions and establishes conditions for their uniqueness and self-consistency in spacetimes with CTCs.

Findings

Solutions exist but are discontinuous.

Uniqueness holds under certain conditions.

Self-consistency is required if initial data intersects CTCs.

Abstract

Solutions of the Cauchy problem for the wave equation on a non-globally hyperbolic spacetime, which contains closed timelike curves (time machines) are considered. It is proved, that there exists a solution of the Cauchy problem, it is discontinuous and in some sense unique for arbitrary initial conditions, which are given on a hypersurface at time, that precedes the formation of closed timelike curves (CTC). If the hypersurface of initial conditions intersects the region containing CTC, then the solution of the Cauchy problem exists only for such initial conditions, that satisfy a certain requirement of self-consistency.