Consider two participants (a.k.a. "material points"), P and Q, who had been coincident ("meeting, in passing") at exactly and only one event, εPQ.
Further, for all events in which P had taken part, i.e. the (ordered) set of events
{...εBP...εKP...εPQ...εPV...εPY...},
along with all events in which Q had taken part, i.e. the (ordered) set of events
{...εAQ...εJQ...εPQ...εUQ...εXQ...},
the values of Lorentzian distance ℓ between any pairs of events shall be given.
Under exactly which condition, expressed in terms of the given values ℓ, are participants P and Q said to have been "momentarily co-moving" at event εPQ ?
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Notes concerning terminology:
(1) While in 2015 the notion of Lorentzian distance ℓ 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 ℓ is also referred to as "time-separation function (or time function) τ", 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 τ remains reserved to denote durations (a.k.a. "arc lengths of timelike paths", a.k.a. "proper time"), if need be.