  In this paper we consider an Abelian Gauge Theory in R^4 equipped with the
Minkowski metric. This theory leads to a system of equations, the
Klein-Gordon-Maxwell equations, which provide models for the interaction
between the electromagnetic field and matter. A three dimensional vortex is a
finite energy solution of these equations in which the magnetic field looks
like the field created by a finite solenoid. Under suitable assumptions, we
prove the existence of vortex-solutions.
