Non stationary Einstein-Maxwell elds interacting with a superconducting cosmic string

Non stationary cylindrically symmetric exact solutions of the Einstein-Maxwell equations are derived as single soliton perturbations of a Levi{Civita metric, by an application of Alekseev inverse scattering method. We show that the metric derived by L. Witten, interpreted as describing the electrogravitational eld of a straight, stationary, conducting wire may be recovered in the limit of a `wide' soliton. This leads to the possibility of interpreting the solitonic solutions as representing a non stationary electrogravitational eld exterior to, and interacting with, a thin, straight, superconducting cosmic string. We give a detailed discussion of the restrictions that arise when appropriate energy and regularity conditions are imposed on the matter and elds comprising the string, considered as `source', the most important being that this `source' must necessarily have a non-vanishing minimum radius. We show that as a consequence, it is not possible, except in the stationary case, to assign uniquely a current to the source from a knowledge of the electrogravitational elds outside the source. A discussion of the asymptotic properties of the metrics, the physical meaning of their curvature singularities, as well as that of some of the metric parameters, is also included.