  We discuss the chiral anomaly for a Weyl field in a curved background and
show that a novel index theorem for the Lorentzian Dirac operator can be
applied to describe the gravitational chiral anomaly. A formula for the total
charge generated by the gravitational and gauge field background is derived in
a mathematically rigorous manner. It contains a term identical to the integrand
in the Atiyah-Singer index theorem and another term involving the
$\eta$-invariant of the Cauchy hypersurfaces.
