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

TL;DR

This paper explores the geometric structure of a two-dimensional space-time derived from the Schwarzschild black hole exterior, establishing its properties and calculating key functions to aid in understanding wave propagation in black hole physics.

Contribution

It demonstrates that the 2D space-time is a causal domain and computes the world function and Van Vleck determinant, facilitating the analysis of Green functions in black hole backgrounds.

Findings

$ ext{M}_2$ is a causal domain.

World function and Van Vleck determinant are computed.

Results aid in understanding Green functions on Schwarzschild space-time.

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.