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We propose a new method for investigating the global properties of theretarded Green's function $G_R(x',x)$ for fields propagating on an arbitraryglobally hyperbolic spacetime. Our method combines the Hadamard form for $G_R$(this form is only valid within a normal neighbourhood of $x$) together withKirchhoff's integral representation for the field in order to calculate $G_R$outside the maximal normal neighbourhood of $x$. As an example, we apply thismethod to the case of a scalar field on a black hole toy-model spacetime, thePleba{\'n}ski-Hacyan spacetime, $\mathbb{M}_2\times\mathbb{S}^2$. The methodallows us to determine in an exact manner that the singularity structure of the`direct' term in the Hadamard form for $G_R(x',x)$ changes from a form$\delta(\sigma)$ to `$1/\pi\sigma$' after the null geodesic joining $x$ and$x'$ has crossed a caustic point, where $\sigma$ is the world function.Furthermore, there is a change of form from a $\theta(-\sigma)$ to a`$-\ln|\sigma|/\pi$' in the `tail' term, which has not been explicitly notedbefore in the literature. We complement the results from the Kirchhoff integralmethod with an analysis for large-$\ell$ of the Green function modes. Thisanalysis allows us to determine the singularity structure after null geodesicshave crossed an arbitrary number of caustics, although it raises a causalityissue which the Kirchhoff integral method resolves. Because of the similarityin the caustic structure of the spacetimes, we expect our main results for wavepropagation to also be valid on Schwarzschild spacetime.

A Kirchhoff integral approach to the calculation of Green's functions beyond the normal neighbourhood