Static Cylindrical Symmetric Solutions in the Einstein-Aether Theory

TL;DR

This paper derives all static cylindrical symmetric solutions in Einstein-Aether theory, including a generalization of Levi-Civita spacetime, revealing unique geodesic behaviors and singularity properties influenced by the aether field parameter.

Contribution

It presents the first known cylindrical symmetric solutions in Einstein-Aether theory, extending Levi-Civita spacetime and analyzing their geodesic and singularity features.

Findings

Radial and z-direction geodesics depend on parameters, with confinement between origin and maximum radius.

For certain parameters, the spacetime is non-singular at the axis, unlike in General Relativity.

Geodesic properties are significantly affected by the aether field parameter c_{14}.

Abstract

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 $\sigma$ and $c_{14}$. The $z$ direction geodesics are restricted to an interval of $\sigma$ 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 $0\leq\sigma<1/2$, when the cylindrical symmetry is preserved, this spacetime is singular at the axis $r=0$, although for $\sigma>1/2$ 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.