Static cylindrical symmetry and conformal flatness

TL;DR

This paper derives equations and conditions for static cylindrically symmetric matter distributions, showing conformally flat solutions are incompressible fluids and cannot be matched to Levi-Civita spacetime, with implications for spacetime symmetry at certain mass thresholds.

Contribution

It provides a complete set of equations and matching conditions for static cylindrically symmetric matter distributions and explores the limitations of conformally flat solutions in this context.

Findings

Conformally flat solutions with equal principal stresses are incompressible fluids.

Such solutions cannot be matched to Levi-Civita spacetime via Darmois conditions.

When mass per unit length reaches 1/2, spacetime exhibits plane symmetry.

Abstract

We present the whole set of equations with regularity and matching conditions required for the description of physically meaningful static cylindrically symmmetric distributions of matter, smoothly matched to Levi-Civita vacuum spacetime. It is shown that the conformally flat solution with equal principal stresses represents an incompressible fluid. It is also proved that any conformally flat cylindrically symmetric static source cannot be matched through Darmois conditions to the Levi-Civita spacetime. Further evidence is given that when the Newtonian mass per unit length reaches 1/2 the spacetime has plane symmetry.