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

Abstract

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.

19 pages, 6 figures

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• Self-Force Calculations with Matched Expansions and Quasinormal Mode Sums
• Self-Force and Green Function in Schwarzschild spacetime via Quasinormal Modes and Branch Cut
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Cited by in corpus (4)

• A recipe for echoes from exotic compact objects
• Communication through quantum fields near a black hole
• Regularized calculation of the retarded Green function in Schwarzschild spacetime
• Global Hadamard form for the Green Function in Schwarzschild space-time