Static solutions of Einstein's equations with cylindrical symmetry

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.