It is shown that in models including the standard electroweak theory and for some particular values of the underlying parameters, electric currents can be spontaneously generated in cosmic strings, without the need of any external field (e.g., electric or magnetic) as is required in most models. This mechanism is then shown to break spontaneously the Lorentz invariance along the initially Goto-Nambu string. The characteristic time needed for the current to build up is estimated and found to lowest order to depend only on the mass of the intermediate $W$ vector boson and the fine structure constant.

12 pages, LaTeX-RevTeX, DAMTP-93/34, 1 figure uuencoded