On the Global Hadamard form for the Green Function in Schwarzschild space-time

The retarded Green function of a wave equation on a 4-dimensional curved background space-time is a (generalized) function of two space-time points and diverges when these are connected by a null geodesic. The Hadamard form makes explicit the form of this divergence but only when one of the points is in a normal neighbourhood of the other point. In this paper we derive a global in time representation for the retarded Green function for a scalar field in Schwarzschild space-time which makes explicit its complete singularity structure. We interpret this representation as a sum of Hadamard forms, the summation being taken over the number of times the null wavefront has passed through a caustic point. Our representation is valid everywhere except in a neighbourhood of caustic points. We deal with these points by providing a separate representation for the Green function which makes explicit its (different) singularity structure specifically at caustics.Comment: 29 pages, 3 figures. In version 3 we include numerical evidence for our analytical results as well as small corrections with respect to the previous versio