The conformal, complex and non-commutative structures of the Schwarzschild solution

TL;DR

This paper explores the complex, conformal, and non-commutative geometric structures of the Schwarzschild solution, revealing a rich mathematical framework involving elliptic curves, branched covers, and hyperelliptic surfaces that enhance understanding of black hole singularities.

Contribution

It introduces a novel complexification of Schwarzschild null geodesics as hyperelliptic curves, ensuring geodesic completeness and conservation laws through branched covers and involutions.

Findings

Null geodesics form elliptic and hyperelliptic curves.

A branched cover resolves cusp singularities in the complex structure.

The resulting space-time has a null geodesically complete Hamiltonian.

Abstract

The generic null geodesic of the Schwarzschild--Kruskal--Szekeres geometry has a natural complexification, an elliptic curve with a cusp at the singularity. To realize that complexification as a Riemann surface without a cusp, and also to ensure conservation of energy at the singularity, requires a branched cover of the space-time over the singularity, with the geodesic being doubled as well to obtain a genus two hyperelliptic curve with an extra involution. Furthermore, the resulting space-time obtained from this branch cover has a Hamiltonian that is null geodesically complete. The full complex null geodesic can be realized in a natural complexification of the Kruskal--Szekeres metric.