Shear-free ray congruences on curved space-times

A shear-free ray congruence on Minkowski space is a 3-parameter family of null geodesics along which Lie transport of a complementary 2-dimensional spacelike subspace (called the screen space) is conformal. Such congruences are defined by complex analytic surfaces in the associated twistor space ${\text{CP}}^3$ 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 3-manifold $(M^3,g_0)$ endowed with a unit vector field $U_0$ that is tangent to a conformal foliation, we require that the pair extend to a space-time $(\mathcal{M},\mathcal{G})$ endowed with a spacelike unit vector field $U_t$ in such a way that $U_t$ simultaneously generates null geodesics and is tangent to a conformal foliation on spacelike slices $t=$ const.