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Complex null geodesics in the extended Schwarzschild universe

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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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Authors and Affiliations

  1. Rochester Institute of Technology, Rochester, NY, USA

    Jonathan Holland

  2. University of Pittsburgh, Pittsburgh, PA, USA

    George Sparling

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  1. Jonathan Holland
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  2. George Sparling
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Correspondence to Jonathan Holland.

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Dedicated to Professor James G. Holland, 1927–2018.

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Holland, J., Sparling, G. Complex null geodesics in the extended Schwarzschild universe. Gen Relativ Gravit 50, 86 (2018). https://doi.org/10.1007/s10714-018-2407-z

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