  We consider the Dirac operator on asymptotically static Lorentzian manifolds
with an odd-dimensional compact Cauchy surface. We prove that if
Atiyah-Patodi-Singer boundary conditions are imposed at infinite times then the
Dirac operator is Fredholm. This generalizes a theorem due to B\"ar-Strohmaier
in the case of finite times, and we also show that the corresponding index
formula extends to the infinite setting. Furthermore, we demonstrate the
existence of a Fredholm inverse which is at the same time a Feynman parametrix
in the sense of Duistermaat-H\"ormander. The proof combines methods from
time-dependent scattering theory with a variant of Egorov's theorem for
pseudo-differential hyperbolic systems.
