Static solutions of Einstein's equations with cylindrical symmetry

TL;DR

This paper classifies all static, cylindrically symmetric vacuum solutions of Einstein's equations, including known and new solutions, and explores their matching with interior solutions with matter.

Contribution

It provides a comprehensive derivation and classification of all such solutions, including new numerical solutions connecting to interior matter distributions.

Findings

Identified all vacuum solutions with cylindrical symmetry.

Discovered new solutions that connect to interior matter configurations.

Analyzed the singularity structure on the axis.

Abstract

In analogy with the standard derivation of the Schwarzschild solution, we find all static, cylindrically symmetric solutions of the Einstein field equations for vacuum. These include not only the well known cone solution, which is locally flat, but others in which the metric coefficients are powers of the radial coordinate and the space-time is curved. These solutions appear in the literature, but in different forms, corresponding to different definitions of the radial coordinate. Because all the vacuum solutions are singular on the axis, we attempt to match them to "interior" solutions with nonvanishing energy density and pressure. In addition to the well known "cosmic string" solution joining on to the cone, we find some numerical solutions that join on to the other exterior solutions.