Cauchy Problem for the Dirac operator on spatially non-compact spacetimes

TL;DR

This paper establishes the well-posedness of the Cauchy problem for the Dirac operator on globally hyperbolic, non-compact spacetimes with complete Cauchy hypersurfaces, laying groundwork for future Fredholmness results.

Contribution

It proves well-posedness of the Dirac Cauchy problem on non-compact, globally hyperbolic manifolds with complete hypersurfaces, extending previous simpler setting results.

Findings

Proved well-posedness of the Dirac Cauchy problem in the specified setting.

Prepared groundwork for Fredholmness results on non-compact hypersurfaces.

Focused on a modified setting compared to earlier work.

Abstract

Let $M$ be a globally hyperbolic manifold with complete spacelike Cauchy hypersurface $\Sigma$. We prove well-posedness of the Cauchy problem for the Dirac operator on globally hyperbolic manifolds with complete Cauchy hypersurfaces. This result is needed as preparation in showing a Fredholmness result in the manner, provided by B\"ar and Strohmaier, for certain non-compact Cauchy hypersurfaces in future work. The results already have been published for a simpler setting in arxiv:2107.08532. This version is only focused on the Cauchy problem for a slightly modified setting.