Cosmic strings with cosmological constant

All static and cylindrically symmetric vacuum solutions of Einstein’s field equation with cosmological constant Λ are found. We use these solutions to represent the exterior metric of a cosmic string. If Λ is negative, the exterior approaches the anti–de Sitter metric away from the string. If Λ 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.