On several static cylindrically symmetric solutions of the Einstein equations

Abstract

We study the Einstein equations in the static cylindrically symmetric case with the stress–energy tensor of the form [Formula: see text], where [Formula: see text] is an unknown function and [Formula: see text], [Formula: see text], [Formula: see text] are arbitrary real constants ([Formula: see text] is assumed to be nonzero). The stress–energy tensor of this form includes as special cases several well-known solutions, such as the perfect fluid solution with the barotropic equation of state, the solution with the static electric field and the solution with the massless scalar field. We solve the Einstein equations with this stress–energy tensor and study some properties of the obtained metric.