Perfect-fluid cylinders and walls - sources for the Levi-Civita space-time

TL;DR

This paper derives solutions for perfect-fluid cylinders and walls as sources for Levi-Civita space-time, analyzing their matching conditions and the range of mass per unit length for different configurations.

Contribution

It introduces a method to generate and match perfect-fluid cylinder and wall solutions to Levi-Civita space-time using a generating function approach.

Findings

Cylinder sources match for $-rac{1}{2}<\sigma<rac{1}{2}$

Wall sources are possible for $|\sigma|>rac{1}{2}$ without a Newtonian limit

Previous work matched sources only for $0\leq\sigma<rac{1}{2}$

Abstract

The diagonal metric tensor whose components are functions of one spatial coordinate is considered. Einstein's field equations for a perfect-fluid source are reduced to quadratures once a generating function, equal to the product of two of the metric components, is chosen. The solutions are either static fluid cylinders or walls depending on whether or not one of the spatial coordinates is periodic. Cylinder and wall sources are generated and matched to the vacuum (Levi--Civita) space--time. A match to a cylinder source is achieved for $-\frac{1}{2}<\si<\frac{1}{2}$, where $\si$ is the mass per unit length in the Newtonian limit $\si\to 0$, and a match to a wall source is possible for $|\si|>\frac{1}{2}$, this case being without a Newtonian limit; the positive (negative) values of $\si$ correspond to a positive (negative) fluid density. The range of $\si$ for which a source has previously been matched to the Levi--Civita metric is $0\leq\si<\frac{1}{2}$ for a cylinder source.