# How to determine whether two participants in the same event had been momentarily co-moving at this event?

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Consider two participants (a.k.a. "material points"), $P$ and $Q$, who had been coincident ("meeting, in passing") at exactly and only one event, $\varepsilon_{P Q}$.

Further, for all events in which $P$ had taken part, i.e. the (ordered) set of events
$\{ ... \, \varepsilon_{B P} \, ... \, \varepsilon_{K P} \, ... \, \varepsilon_{P Q} \, ... \, \varepsilon_{P V} \, ... \, \varepsilon_{P Y} \, ... \}$,
along with all events in which $Q$ had taken part, i.e. the (ordered) set of events
$\{ ... \, \varepsilon_{A Q} \, ... \, \varepsilon_{J Q} \, ... \, \varepsilon_{P Q} \, ... \, \varepsilon_{U Q} \, ... \, \varepsilon_{X Q} \, ... \}$,
the values of Lorentzian distance $\ell$ between any pairs of events shall be given.

Under exactly which condition, expressed in terms of the given values $\ell$, are participants $P$ and $Q$ said to have been "momentarily co-moving" at event $\varepsilon_{P Q}$ ?

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Notes concerning terminology:

(1) While in 2015 the notion of Lorentzian distance $\ell$ was denounced as "a mathematical function that isn't used in physics", its use in physics appears suitably reputable since 2016.

(2) Apparently, Lorentzian distance $\ell$ is also referred to as "time-separation function (or time function) $\tau$", e.g. here and elsewhere. For the purposes of my question above, I'd like to ask that the terms "time-separation function" (or "time-function") are not used where "Lorentzian distance" could be used equivalently instead, and that the symbol $\tau$ remains reserved to denote durations (a.k.a. "arc lengths of timelike paths", a.k.a. "proper time"), if need be.

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