Complete Model of a Self-gravitating Cosmic String I. A New Class of Exact Solutions and Gravitational Lensing

Abstract

We find solutions of Einstein's field equation for topologically stable strings associated with the breaking of a U(1) symmetry. Strings form in many GUTs and are expected whenever the homotopy group $\Pi_1(M_0)$ is non-trivial. The behavior of the fields making up the string is described by the Euler-Lagrange equations. These fields appear in the energy-momentum tensor so we must solve simultaneously for the coupled Einstein-scalar-gauge field equations. Numerical results are obtained using a Taylor-series method. We obtain a 5-parameter family of solutions and discuss their physical characteristics. Significant gravitational lensing can occur due to strings based on this model and are shown for different solutions. Finally, we prove that the assumption of regularity at the string axis is not necessary by looking at the physical properties of the string solutions.