Transport along null curves

Fermi transport is useful for describing the behaviour of spins or gyroscop es following non-geodesic, timelike worldlines. However, Fermi transport br eaks down for null worldlines. We introduce a transport law for polarizatio n vectors along non-geodesic null curves. We show how this law emerges natu rally from the geometry of null directions by comparing polarization vector s associated with two distinct null directions. We then give a spinorial tr eatment of this topic and make contact with the geometric phase of quantum mechanics. There are two significant differences between the null and timel ike cases. In the null case (a) the transport law does not approach a uniqu e smooth limit as the null curve approaches a null geodesic and (b) the tra nsport law for vectors is integrable, i.e, the result depends only on the l ocal properties of the curve and not on the entire path taken. However, the transport of spinors is not integrable: there is a global sign of topologi cal origin.