Closed curve through an event in spacetime

Physics Asked on January 2, 2022

I’m studying the book Techniques of Differential Topology in Relativity by Roger Penrose and I’m stuck in an exercise he left to the reader. We say that the spacetime $M$ is strongly causal in $p$ if and only if there exists an event $q$ such that $q prec p$ and for all events $x,y$ with $x ll p$ and $q ll y$ we have that $x ll y$. I have to show that in such a situation if $q ll p$, then there exists a closed trip (a timelike, future-oriented curve) through $p$. If anyone could help me with an answer or my thoughts below, I would be very grateful.

Here, $a ll b$ means that there exists a piecewise future-oriented timelike geodesic from $a$ to $b$, while $a prec b$ means that there exists a piecewise future-oriented causal geodesic, where the tangent vector is always null or timelike.

I tried many times but still can’t figure out how. One thing that came to my mind is that we could extend the trip from $q$ to $p$ to a point $p’$ and then apply the condition of strong causality with the same $q$ but I’m not quite sure that it is OK to do that. If you want deeper contextualization, you can look at lemma $4.16$ and remark $4.17$ of the book, on pages $31$ and $32$.

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