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In analogy with the standard derivation of the Schwarzschild solution, wefind all static, cylindrically symmetric solutions of the Einstein fieldequations for vacuum. These include not only the well known cone solution,which is locally flat, but others in which the metric coefficients are powersof the radial coordinate and the space-time is curved. These solutions appearin the literature, but in different forms, corresponding to differentdefinitions of the radial coordinate. Because all the vacuum solutions aresingular on the axis, we attempt to match them to "interior" solutions withnonvanishing energy density and pressure. In addition to the well known "cosmicstring" solution joining on to the cone, we find some numerical solutions thatjoin on to the other exterior solutions.

Static solutions of Einstein's equations with cylindrical symmetry