Antipodal identification in the Schwarschild spacetime

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.