The Levi-Civita spacetime

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 $\frac{1}{2}$ for the first case and $\frac{1}{4}$ for the second one. Thus the homogeneous cluster provides another example [2] where the range of $\sigma$ is extended beyond the limit value $\frac{1}{4}$ previously found in the literature [3,4]. Using the Cartan Scalars technics we show that the Levi-Civita spacetime gets an extra symmetry for $\sigma =\frac{1}{2}$ or $\frac{1}{4}$. We also find that the cluster of homogeneous dust has a superior limit for its radius, depending on the constant volumetric energy density ${\rho }_0$.

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