Riemannian manifolds dual to static spacetimes
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Riemannian manifolds dual to static spacetimes Carolina Figueiredo1 · José Natário1 Received: 28 April 2020 / Accepted: 27 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
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. Keywords Static spacetimes · Geodesics · Epstein metric · Fermat metric · Constant curvature spaces · Isochronous motions · Einstein cluster · Surface embeddings
Contents Introduction . . . . . . . . . . . . . . . . 1 Geodesic correspondence . . . . . . . 2 Constant curvature spacetimes . . . . . 2.1 Minkowski spacetime . . . . . . . 2.2 Rindler spacetime . . . . . . . . . 2.3 de Sitter spacetime . . . . . . . . 2.4 Flat anti-de Sitter spacetime . . . . 2.5 Hyperbolic anti-de Sitter spacetime 2.6 Spherical anti-de Sitter spacetime . 3 Radial isochronous oscillatory motion . 3.1 Perfect fluid . . . . . . . . . . . .
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José Natário [email protected] CAMGSD, Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal 0123456789().: V,-vol
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3.2 Einstein cluster . . . . . . . . . . . . 4 Non-constant curvature spacetimes . . . . 4.1 Schwarzschild spacetime . . . . . . . Null geodesics . . . . . . . . . . . . . . . Radial motion . . . . . . . . . . . . . . . 4.2 Interior solution . . . . . . . . . . . . Null geodesics . . . . . . . . . . . . . .
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