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Riemannian manifolds dual to static spacetimes

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Abstract

We establish a one-to-one correspondence between static spacetimes and Riemannian manifolds that maps causal geodesics to geodesics, as suggested by L. C. Epstein. We explore constant curvature spacetimes—such as the de Sitter and the anti-de Sitter spacetimes—and find that they map to constant curvature Riemannian manifolds, namely the Euclidean space, the sphere and the hyperbolic space. By imposing the conditions required to map to the sphere, we obtain the metrics for which there is radial oscillatory motion with a period independent of the amplitude. We then consider the case of a perfect fluid and an Einstein cluster and determine the conditions required to find this type of motion. Finally, we give examples of surfaces corresponding to certain types of motion for metrics that do not exhibit constant curvature, such as the Schwarzschild, Schwarzschild de Sitter and Schwarzschild anti-de Sitter solutions, and even for a simplified model of a wormhole.

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Notes

  1. This idea is similar to the Eisenhart lift [9], and may be considered as an application of the Kaluza–Klein trick without the electromagnetic field; it has been used before in the slightly more general context of stationary spacetimes and Randers metrics in [4,5,6,7,8].

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Acknowledgements

CF gratefully acknowledges the Calouste Gulbenkian Foundation for the scholarship program Novos Talentos em Matemática. JN was partially supported by FCT/Portugal through Projects UIDB/MAT/04459/2020 and UIDP/MAT/04459/2020 and Grant (GPSEinstein) PTDC/MAT-ANA/1275/2014.

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  1. CAMGSD, Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal

    Carolina Figueiredo & José Natário

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  1. Carolina Figueiredo
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  2. José Natário
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Correspondence to José Natário.

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Figueiredo, C., Natário, J. Riemannian manifolds dual to static spacetimes. Gen Relativ Gravit 52, 84 (2020). https://doi.org/10.1007/s10714-020-02736-5

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