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

TL;DR

This paper introduces a novel Kirchhoff integral method to analyze the global properties of Green's functions in curved spacetimes, revealing how singularity structures change after crossing caustics, with applications to black hole models.

Contribution

The paper develops a new approach combining Hadamard form and Kirchhoff integrals to compute Green's functions beyond normal neighborhoods in arbitrary spacetimes, providing exact singularity structure analysis.

Findings

Singularity structure of the direct term changes from δ(σ) to 1/πσ after crossing a caustic.

Tail term transitions from θ(-σ) to -ln|σ|/π, a new observation in the literature.

Method applies to black hole toy-model spacetime and predicts similar behavior in Schwarzschild spacetime.

Abstract

We propose a new method for investigating the global properties of the retarded Green's function $G_R(x',x)$ for fields propagating on an arbitrary globally hyperbolic spacetime. Our method combines the Hadamard form for $G_R$ (this form is only valid within a normal neighbourhood of $x$) together with Kirchhoff's integral representation for the field in order to calculate $G_R$ outside the maximal normal neighbourhood of $x$. As an example, we apply this method to the case of a scalar field on a black hole toy-model spacetime, the Pleba{\'n}ski-Hacyan spacetime, $\mathbb{M}_2\times\mathbb{S}^2$. The method allows 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 noted before in the literature. We complement the results from the Kirchhoff integral method with an analysis for large-$\ell$ of the Green function modes. This analysis allows us to determine the singularity structure after null geodesics have crossed an arbitrary number of caustics, although it raises a causality issue which the Kirchhoff integral method resolves. Because of the similarity in the caustic structure of the spacetimes, we expect our main results for wave propagation to also be valid on Schwarzschild spacetime.