Let us consider a globally hyperbolic Lorentzian manifold $(M,g)$ with empty cut locus. Suppose that $p$ is a point in $M$ and consider $C^-(p)$ to be the past null cone in $M$ emanating from the point $p$. Next, suppose that $q_1,q_2 \in C^-(p)$ and denote by $\gamma_{q_1}$ and $\gamma_{q_2}$ the null geodesics that emanate from $q_1$ and $q_2$ respectively in the direction of the null vector that is normal to the null cone there.

Is it true that $\gamma_{q_1}$ and $\gamma_{q_2}$ can never intersect in $M\setminus C^-(p)$?

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