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arXiv · 2002.11432

Gödel spacetime, planar geodesics and the Möbius map

Donato Bini·Andrea Geralico·Robert T. Jantzen·Wolfango Plastino
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Abstract

Timelike geodesics on a hyperplane orthogonal to the symmetry axis of the Gödel spacetime appear to be elliptic-like if standard coordinates naturally adapted to the cylindrical symmetry are used. The orbit can then be suitably described through an eccentricity-semi-latus rectum parametrization, familiar from the Newtonian dynamics of a two-body system. However, changing coordinates such planar geodesics all become explicitly circular, as exhibited by Kundt's form of the Gödel metric. We derive here a one-to-one correspondence between the constants of the motion along these geodesics as well as between the parameter spaces of elliptic-like versus circular geodesics. We also show how to connect the two equivalent descriptions of particle motion by introducing a pair of complex coordinates in the 2-planes orthogonal to the symmetry axis, which brings the metric into a form which is invariant under Möbius transformations preserving the symmetries of the orbit, i.e., taking circles to circles.

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Donato Bini, Andrea Geralico, Robert T. Jantzen, Wolfango Plastino. 2020-02-26. Gödel spacetime, planar geodesics and the Möbius map. https://arxiv.org/abs/2002.11432

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