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Static Cylindrical Symmetric Solutions in the Einstein-Aether Theory

In this work we present all the possible solutions for a static cylindrical symmetric spacetime in the Einstein-Aether (EA) theory. As far as we know, this is the first work in the literature that considers cylindrically symmetric solutions in the theory of EA. One of these solutions is the generalization in EA theory of the Levi-Civita (LC) spacetime in General Relativity (GR) theory. We have shown that this generalized LC solution has unusual geodesic properties, depending on the parameter $c_{14}$ of the aether field. The circular geodesics are the same of the GR theory, no matter the values of $c_{14}$. However, the radial and $z$ direction geodesics are allowed only for certain values of $σ$ and $c_{14}$. The $z$ direction geodesics are restricted to an interval of $σ$ different from those predicted by the GR and the radial geodesics show that the motion is confined between the origin and a maximum radius. The latter is not affected by the aether field but the velocity and acceleration of the test particles are Besides, for $0\leqσ<1/2$, when the cylindrical symmetry is preserved, this spacetime is singular at the axis $r=0$, although for $σ>1/2$ exists interval of $c_{14}$ where the spacetime is not singular, which is completely different from that one obtained with the GR theory, where the axis $r=0$ is always singular.