2 Shear - Free Ray Congruences

A shear-free ray congruence on Minkowski space is a 3-parameter family of null geodesics along which Lie transport of a complementary 2dimensional spacelike subspace (called the screen space) is conformal. Such congruences are defined by complex analytic surfaces in the associated twistor space CP 3 and are the basis of the construction of massless fields. On a more general space-time, it is unclear how to couple the massless field with the gravitational field. In this article we do this by considering the following Cauchy-type problem: given a Riemannian 3manifold (M, g0) endowed with a unit vector field U0 that is tangent to a conformal foliation, we require that the pair extend to a space-time (M, G) endowed with a spacelike unit vector field Ut in such a way that Ut simultaneously generates null geodesics and is tangent to a conformal foliation on spacelike slices t = const. 2 SHEAR-FREE RAY CONGRUENCES