Antipodal identification in the Schwarschild spacetime

TL;DR

This paper investigates the effects of antipodal identification on Schwarzschild spacetime, revealing changes in topology, light cone structure, and singularities through a Möbius transformation analysis.

Contribution

It introduces a novel analysis of antipodal identification in Schwarzschild spacetime, including topology and geometric properties, and addresses the suppression of conical singularities.

Findings

Non simply-connected topology: R^{2*} x S^2

Bending of light cones observed

Transformation affects horizons and singularities

Abstract

Through a M\"obius transformation, we study aspects like topology, ligth cones, horizons, curvature singularity, lines of constant Schwarzschild coordinates $r$ and $t$, null geodesics, and transformed metric, of the spacetime $(SKS/2)^\prime$ that results from: i) the antipode identification in the Schwarzschild-Kruskal-Szekeres ($SKS$) spacetime, and ii) the suppression of the consequent conical singularity. In particular, one obtains a non simply-connected topology: $(SKS/2)^\prime\cong \mathbb{R}^{2*}\times S^2$ and, as expected, bending light cones.