Null congruences and spin-coefficients

Congruences of null curves in space–time – referred to here as null congruences – and especially of rays (geodetic null curves), play an important part in electromagnetic and gravitational radiation theory and in the construction of exact solutions of Einstein's equations. We recall that a congruence is a family of curves, surfaces, etc., with the property that precisely one member of the family passes through each point of a given domain of the space under consideration. (The tangent-space elements to a congruence constitute what is known as a foliation cf. Hermann 1968.) In fact, all calculations in this chapter are local in space–time, so it will not matter if certain congruences globally violate this one-point one-member condition. The null congruences one encounters are frequently many-sheeted globally, in the sense that as one moves continuously from a point of the space–time and returns to that point, one may find that the associated line of the congruence has shifted; but such behaviour will not affect our local considerations. Moreover, there are likely to be specific points (such as branch loci of the congruence, or ‘source’ world-lines from each of whose points many rays diverge) which have to be regarded as singularities of the congruence and must lie outside the domain of interest.