Locally inertial null normal coordinates

Locally inertial coordinates are constructed by carrying Riemann normal coordinates on a codimension two spacelike surface along the geodesics normal to it. Since the normal tangents are labelled by components with respect to a null basis, these coordinates are referred to as null normal coordinates. They are convenient in the study of local causal horizons. As an application, the coordinate system is used to specify a vector field that satisfies the Killing equation approximately in a small region and the Killing identity exactly on a single null geodesic. We also construct a vector field on a surface, starting from a vector at a given point on the surface. This construction may be regarded as a generalisation of the notion of Fermi-Walker transport.