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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