The Levi-Civita spacetime

TL;DR

This paper analyzes two exact solutions of Einstein's equations involving cylindrical dust distributions, exploring their junction conditions with Levi-Civita spacetime, and reveals new limits on energy density and symmetry properties.

Contribution

It provides new upper limits for energy density per unit length in cylindrical dust solutions and shows additional symmetries at specific density values.

Findings

Upper limit for energy density: 1/2 for homogeneous dust

Extended the range of energy density beyond previous limits

Identified extra symmetries in Levi-Civita spacetime at specific densities

Abstract

We consider two exact solutions of Einstein's field equations corresponding to a cylinder of dust with net zero angular momentum. In one of the cases, the dust distribution is homogeneous, whereas in the other, the angular velocity of dust particles is constant [1]. For both solutions we studied the junction conditions to the exterior static vacuum Levi-Civita spacetime. From this study we find an upper limit for the energy density per unit length $\sigma$ of the source equal ${1\over 2}$ for the first case and ${1\over 4}$ for the second one. Thus the homogeneous cluster provides another example [2] where the range of $\sigma$ is extended beyond the limit value ${1\over 4}$ previously found in the literature [3,4]. Using the Cartan Scalars technics we show that the Levi-Civita spacetime gets an extra symmetry for $\sigma={1\over 2}$ or ${1\over 4}$. We also find that the cluster of homogeneous dust has a superior limit for its radius, depending on the constant volumetric energy density $\rho_0$.