In General Relativity, cosmic strings are well known to produce a static, locally flat spacetime with a wedge removed. If the tension exceeds a critical value, the deficit angle becomes larger than $ 2��$, leading to a compact exterior that ends in a conical singularity. In this work, we investigate dynamical solutions for cosmic strings with super-critical tensions. To this end, we model the string as a cylindrical shell of finite and stabilized transverse width and show that there is a marginally super-critical regime in which the stabilization can be achieved by physically reasonable matter. We show numerically that the static deficit angle solution is unstable for super-critical string tensions. Instead, the geometry starts expanding in axial direction at an asymptotically constant rate, and a horizon is formed in the exterior, which has the shape of a growing cigar. We are able to find the analytic form of the attractor solution describing the interior of the cosmic string. In particular, this enables us to analytically derive the relation between the string tension and the axial expansion rate. Furthermore, we show that the exterior conical singularity can be avoided for dynamical solutions. Our results might be relevant for theories with two extra dimensions, modeling our universe as a cosmic string with a three-dimensional axis. We derive the corresponding Friedmann equation, relating the on-brane Hubble parameter to the string tension or, equivalently, brane cosmological constant.

20 pages, 12 figures, 1 table; v2: matches version published in Physical Review D