Geometric properties of a 2-D space-time arising in 4-D black hole physics

The Schwarzschild exterior space-time is conformally related to a direct product space-time, ${\mathcal{M}}_2\times S_2$, where ${\mathcal{M}}_2$ is a two-dimensional space-time. This direct product structure arises naturally when considering the wave equation on the Schwarzschild background. Motivated by this, we establish some geometrical results relating to ${\mathcal{M}}_2$ that are useful for black hole physics. We prove that ${\mathcal{M}}_2$ has the rare property of being a causal domain. Consequently, Synge’s world function and the Hadamard form for the Green function on this space-time are well-defined globally. We calculate the world function and the van Vleck determinant on ${\mathcal{M}}_2$ numerically and point out how these results will be used to establish global properties of Green functions on the Schwarzschild black hole space-time.