  It is shown that the space of null geodesics of a star-shaped causally simple
subset of Minkowski space is contactomorphic to the canonical contact structure
in the spherical cotangent bundle of $\mathbb{R}^n$. In the $3$-dimensional
case we prove a similar result for a large class of causally simple
contractible subsets of an arbitrary globally hyperbolic spacetime applying
methods from the theory of contact-convex surfaces. Moreover we prove that
under certain assumptions the space of null geodesics of a causally simple
spacetime embeds with smooth boundary into the space of null geodesics of a
globally hyperbolic spacetime. The characteristic foliation of this boundary
provides an invariant of the conformal class of the causally simple spacetime.
