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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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  • $\begingroup$ 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. $\endgroup$
    – Ali
    Commented Feb 18, 2021 at 0:46
  • $\begingroup$ 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. $\endgroup$
    – Willie Wong
    Commented Feb 18, 2021 at 1:06
  • $\begingroup$ 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. $\endgroup$
    – Willie Wong
    Commented Feb 18, 2021 at 1:08
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    $\begingroup$ 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. $\endgroup$
    – Ettore Minguzzi
    Commented Feb 19, 2021 at 10:51
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    $\begingroup$ 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. $\endgroup$
    – Willie Wong
    Commented Jan 30 at 2:18

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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.

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    $\begingroup$ Any comment about negative vote? $\endgroup$
    – Anton Petrunin
    Commented Jan 29 at 12:23
  • $\begingroup$ Your answer is posted two years after the OP's clarifying comment, in which it was clearly stated that the intention is for geodesics transverse to the cone, not the geodesics tangent to the cone. So it doesn't answer the question at all. $\endgroup$
    – Willie Wong
    Commented Jan 30 at 2:11
  • $\begingroup$ As posed by the OP (including even the clarifying comment) the question is unanswerable, since the initial setup is ill-defined. $\endgroup$
    – Willie Wong
    Commented Jan 30 at 2:16
  • $\begingroup$ @WillieWong The null vector that is normal to the null cone is tangent to it (the metric is not Riemannian). $\endgroup$
    – Anton Petrunin
    Commented Jan 30 at 4:32
  • $\begingroup$ Yes, I am aware that's how it actually works. But the OP is evidently confused, if you look at the comment thread below the question. $\endgroup$
    – Willie Wong
    Commented Jan 30 at 12:12

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