A question on null geodesics in Lorentzian geometry Suppose that $(M,g)$ is a smooth Lorentzian manifold, that $I\subset \mathbb R$ is a closed finite interval and that $\gamma: I \to M$ is a smooth null geodesic on $M$, that is to say, it satisfies
$$ \nabla_{\dot{\gamma}(s)}\dot\gamma(s)=0 \quad \text{and} \quad g(\dot{\gamma}(s),\dot{\gamma}(s))=0\quad \text{for all $s \in I$}.$$
Assume also that $\beta: I\to M$ is a smooth curve such that
$$g(\dot{\beta}(s),\dot{\beta}(s)) =0 \quad \text{for all $s \in I$},$$
and additionally that $\beta$ is not completely overlapping with $\gamma$ at any point (this just means that there is no non empty open interval $J \subset I$ such that $\beta|_{J}$ is a reparametrization of $\gamma$ on some interval of $I$).
Does it follow that $\gamma$ and $\beta$ can only intersect a finite number of times? If the answer is no in general, can one impose some soft assumptions to make the intersection points finite?
Thanks,
 A: We argue by contradiction. Suppose there are infinitely intersections. By reversing time orientation if necessary, there exists a monotonically increasing sequence of times $s_n \in I$ such that at each $\gamma(s_n)$ the curves $\gamma$ and $\beta$ intersect. Since $I$ is compact $s_n$ has a limit $s'\in I$. For convenience label the points $p_n = \gamma(s_n)$, and $p' = \gamma(s')$.
Since $I$ is compact, we have that the image $\beta$ is compact, and hence closed, in $M$. Since $p_n \to p'$ and $p_n \in \beta$, we have that $p'\in \beta$ also. By taking a subsequence if necessary, we can find a increasing sequence of times $t_n$ such that $\beta(t_n) = p_n$ and $t_n$ converges to $t'$, for which $\beta(t') = p'$.
Case I: suppose there exists some $n_0$ such that for all $n > n_0$, the causal segment $\beta|_{[t_n, t']}$ is geodesic (up to reparametrization).

*

*$\dot{\beta}(t')$ cannot be equal to $\dot{\gamma}(s')$: other wise by the uniqueness theorem for geodesics $\beta$ and $\gamma$ must overlap on some interval.

*Since the tangent vectors are not equal, this means that on every open neighborhood of the origin in $T_{p'}M$, the exponential map will have to be non-injective, which contradicts the theorem on existence of normal neighborhoods.

So case I is ruled out.
Case II: the alternative is that $\beta|_{[t_n,t']}$ is not geodesic for any $n$. This implies that $p_n$ is to the chronological past of $p'$ (See Theorem 8.1.2 and Corollary in Wald's General Relativity.)
Theorem 2.14 from https://arxiv.org/abs/gr-qc/0609119 states that every point in a Lorentzian manifold has a causally convex globally hyperbolic neighborhood. Fix such a neighborhood $V'$ of $p'$. Since $\beta$ is compact, there is some $n_0$ such that $\beta|_{[t_{n_0}, t']}$ and $\gamma|_{[s_{n_0}, s']}$ remain both in $V'$. For each $n > n_0$, that $p_n$ is to the chronological past of $p'$ implies there exists a sequence of timelike tangent vectors $v_n\in T_{p'}M$ with $\exp_{p'}(v_n) = p_n$.
$v_n$ must converge to zero, as the exponential map is a diffeomorphism on a sufficiently small neighborhood of zero, and $p_n\to p'$. But this shows that for any neighborhood of zero there exists simultaneously a timelike and a null vector within that neighborhood both of which exponentiates to $p_n$, contradicting the existence of normal neighborhoods.
This rules out case II.
