Discontinuity cylinder model of gravitating U(1) cosmic strings

Abstract

We introduce a model for an infinite-length, straight U(1) cosmic string as a cylindrical, singular shell enclosing a region of false vacuum. The properties of the geometry for the region exterior to the string are fully determined under the assumption that changes in the scalar and gauge field variables occur only at the cylindrical shell. This is consistent with a limiting form of the scalar potential V(\ensuremath{\varphi}) where a minimum at \ensuremath{\Vert}\ensuremath{\varphi}\ensuremath{\Vert}=0 is separated by a large barrier from a global minimum at \ensuremath{\Vert}\ensuremath{\varphi}\ensuremath{\Vert}=\ensuremath{\eta}\ensuremath{\ne}0. The introduction of an approximately singular ``surface'' for the string allows the definition of a \ensuremath{\delta}-function stress-energy density that characterizes discontinuities in the fields. We show consistency of the model with the full coupled equations for the metric, and the scalar and gauge fields in curved space-time. It is found that for this model, in the absence of an ``external'' cosmological constant, the exterior geometry of the string approaches Minkowski space-time with a deficit angle, and it is shown that in the limit when the string becomes a line source, i.e., its radius vanishes, the deficit angle reduces to the well-known expression \ensuremath{\Delta}\ensuremath{\theta}=8\ensuremath{\pi}\ensuremath{\mu}, with \ensuremath{\mu} the proper mass per unit length of the string.