Baer, Christian Strohmaier, Alexander An index theorem for Lorentzian manifolds with compact spacelike Cauchy boundary We show that the Dirac operator on a compact globally hyperbolic Lorentzian spacetime with spacelike Cauchy boundary is a Fredholm operator if appropriate boundary conditions are imposed. We prove that the index of this operator is given by the same expression as in the index formula of Atiyah-Patodi-Singer for Riemannian manifolds with boundary. The index is also shown to equal that of a certain operator constructed from the evolution operator and a spectral projection on the boundary. In case the metric is of product type near the boundary a Feynman parametrix is constructed. Lorentzian manifolds;Cauchy boundary;Mathematical Sciences not elsewhere classified 2019-09-20
    https://repository.lboro.ac.uk/articles/journal_contribution/An_index_theorem_for_Lorentzian_manifolds_with_compact_spacelike_Cauchy_boundary/9386471