Gravitational time delay due to a spinning string

We show that gravity around a straight cosmic string that carries angular momentum per unit length (J) induces a delay between the arrival time of two particles that describe an arbitrary closed trajectory around the string with the same speed at each point but in opposite directions. The time delay is 16GJ, where G is Newton's constant. The particle delayed is the one that circles the closed path in the sense contrary to the angular momentum. As a consequence, double images of a star produced by the string acting as a gravitational lens will acquire a relative time delay, thus providing a mechanism, at least in principle, to measure the string angular momentum. We argue, however, that in any realistic model the time delay effect is negligible. We also discuss other possible implications of this effect in 2+1 dimensions.