Cosmic strings with cosmological constant.

Abstract

All static and cylindrically symmetric vacuum solutions of Einstein's field equation with cosmological constant \ensuremath{\Lambda} are found. We use these solutions to represent the exterior metric of a cosmic string. If \ensuremath{\Lambda} is negative, the exterior approaches the anti--de Sitter metric away from the string. If \ensuremath{\Lambda} is positive, the exterior metric is either an scrR $^{3}\mathrm{scrS}$ $^{1}$ universe with a curvature singularity, an scrR $^{2}\mathrm{scrS}$ $^{2}$ universe with a conical singularity, or an scrR $^{2}\mathrm{scrS}$ $^{2}$ universe with no singularity.