I'm looking for a reference for the following theorem:
Theorem
Let $\mathcal M$ be a smooth manifold with smooth Lorentzian metric $g$. Let $p$ and $q$ be two points on the same (causal) geodesic, with the tangent vector at $p$ pointing towards $q$ ($p$,$q$ separated by a finite affine parameter). Let $\mathcal U (q) $ be an open neighbourhood of $q$ in $\mathcal{M}$ then there exists an open neighbourhood of $(p, \dot \gamma|_p)$ in $T\mathcal M$ such that any geodesic in this neighbourhood intersects $\mathcal{U}(q)$
Now let $e_0$, $e_1$, $e_2$, $e_3$ be an orthonormal basis at $p$ then the tangent vector to any null geodesic can be written as $$\dot \gamma|_p=\alpha(e_0+k^1 e_1+ k^2 e_2+ k^3 e_3) $$ with $|k|=1$ and hence $k\in S^2$ which is usually referred to as the celestial sphere of a time like observer $e_0$ at $p$. With these notions the following corollary should be clear from my point of view
Corollary
Let $\mathcal{M}$ be a smooth manifold with smooth Lorentzian metric $g$. Let $p$ and $q$ be two points on the same null geodesic with the tangent vector $\dot \gamma(k|_p)$ at $p$ pointing towards $q$ ($p$,$q$ separated by a finite affine parameter). Let $\mathcal U (q) $ be an open neighbourhood of $q$ in $\mathcal{M}$ then there exists an open neighbourhood $B_\epsilon(k)$ of $k$ on $S^2$ such that any null geodesic $\gamma(\tilde k|_p)$ with $\tilde k \in B_\epsilon(k)$ intersects $\mathcal{U}(q)$.
In principle the theorem should follow from the continuous dependence on initial data for ODEs. Together with a partitioning of the reference geodesics into normal neighborhoods. As it is a rather fundamental question i would expect this to have been worked out in detail somewhere. However so far i was unable to find a sufficiently precise source.
