We prove an index theorem for the Dirac operator on compact
Lorentzian manifolds with spacelike boundary. Unlike in the
Riemannian situation, the Dirac operator is not elliptic. But
it turns out that under Atiyah-Patodi-Singer boundary conditions,
the kernel is finite dimensional and consists of smooth sections.
The corresponding index can be expressed by a curvature integral,
a boundary transgression integral and the eta-invariant of the
boundary operator just as in the Riemannian case. There is a natural
physical interpretation in terms of particle-antiparticle creation.
This is joint work with Alexander Strohmaier.