In general relativistic spacetimes, convex neighbourhoods are guaranteed to exist. As a reminder: a convex neighbourhood ##U## is a neighbourhood ##U## such that for any two points ##p## and ##q## in U there exists a unique geodesic connecting ##p## and ##q## staying within ##U##.(adsbygoogle = window.adsbygoogle || []).push({});

With that established, does it somehow follow that within a small enough neighbourhood there exist a unique null-geodesic connecting them?

In this paper the author seem to deduce this in his Proposition 1 -- something that he uses to establish the existence of neighbourhoods for which one can assign coordinates to events by sending and receiving light signals.

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# Convex (null) neighbourhoods

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