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Let $M$ be a globally hyperbolic manifold with complete spacelike Cauchyhypersurface $\Sigma \subset M$. Building on past and recent works of B\"ar andStrohmaier, we extend their Fredholm result of the Atiyah-Singer Dirac operatoron compact Lorentzian spaces to the case, where $M$ is diffeomorphic to aproduct of $\Sigma$ with a compact time intervall and the hypersurface is aGalois covering with respect to a group $\Gamma$. We follow the first approachof both authors in this extended setting, where a well-posedness result of theCauchy problem for the Dirac operator on non-compact manifolds is needed inpreparation. After employing von Neumann algebras and further ingredients forGalois coverings, the well-posedness result is specified for the setting ofinterest, which leads to $\Gamma$-Fredholmness of the Dirac operator under APSboundary conditions.

Atiyah-Singer Dirac Operator on spacetimes with non-compact Cauchy hypersurface