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arXiv · 1803.05037

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

Jonathan Holland·George Sparling
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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.

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Jonathan Holland, George Sparling. 2018-03-13. The conformal, complex and non-commutative structures of the Schwarzschild solution. https://arxiv.org/abs/1803.05037

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