  We consider how causality and micro-causality are realised in QED in curved
spacetime. The photon propagator is found to exhibit novel non-analytic
behaviour due to vacuum polarization, which invalidates the Kramers-Kronig
dispersion relation and calls into question the validity of micro-causality in
curved spacetime. This non-analyticity is ultimately related to the generic
focusing nature of congruences of geodesics in curved spacetime, as implied by
the null energy condition, and the existence of conjugate points. These results
arise from a calculation of the complete non-perturbative frequency dependence
of the vacuum polarization tensor in QED, using novel world-line path integral
methods together with the Penrose plane-wave limit of spacetime in the
neighbourhood of a null geodesic. The refractive index of curved spacetime is
shown to exhibit superluminal phase velocities, dispersion, absorption (due to
\gamma \to e^+e^-) and bi-refringence, but we demonstrate that the wavefront
velocity (the high-frequency limit of the phase velocity) is indeed c, thereby
guaranteeing that causality itself is respected.
