An index theorem for Lorentzian manifolds

Christian Bär

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.

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