# On intersection of null geodesics

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 $$M\setminus C^-(p)$$?

• I should clarify that of course. Here, due to splitting of spacetime, you can always associate to a null vector $v=c\partial_t+\nu$ a canonical dual null vector $\tilde{v}=-c\partial_t+\nu$. The vector $\tilde{v}$ is what I mean by the (pseudo)-normal.
– Ali
Commented Feb 18, 2021 at 0:46
• wait, I thought I understood your question, but now I am confused. Are $\gamma_{q_1}$ and $\gamma_{q_2}$ supposed to be transverse to $C^-(p)$ or not? The way the question text itself is written, I would've interpreted $\gamma_{q_1}$ as the null geodesic joining $p$ to $q_1$. But your comment suggest something else. Commented Feb 18, 2021 at 1:06
• In fact, I don't understand your comment at all! Given a $C^-(p)$ the only way I know where you can construct a "dual null vector" for $q\in C^-(p)$ is if you prescribe a priori a space-like foliation of $C^-(p)$, whereby there is a canonical null direction transverse to $C^-(p)$ that is orthogonal to the space-like leave through $q$. But this is not as simple as what you claimed in your comment. Commented Feb 18, 2021 at 1:08
• The splitting is not canonical, there are many such splittings, so it seems that the transverse geodesics that you are considering are ill defined unless you introduce a privileged timelike vector field. Commented Feb 19, 2021 at 10:51
• As Ettore Minguzzi pointed out in the previous comment, the set-up is ill-defined and the question unanswerable. Since the OP has not provided additional clarification, I am voting to close this question. Commented Jan 30 at 2:18

The geodesic from $$p$$ to $$q_i$$ is normal to $$C^-(p)$$ at $$q_i$$. So essentially you ask if two null geodesics from $$p$$ intersect in $$C^-(p)$$. Since the cut locus is empty, the answer is no.