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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 $\varepsilon_{P Q}$ ?


A note concerning terminology:

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.

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    $\begingroup$ Could you expand on this a bit more... Lorentzian distance is just meant to be a generalization of the geodesic length, is it not? That is, for small distances, $\ell$ is just the distance in Minkowski spacetime? So you are asking, given that two bodies are at the same spacetime point $\epsilon_{PQ}$, under what conditions are they comoving? If the events $\epsilon$ are 'dense' in time (i.e. you can assume the velocity between any consecutive events is constant), then can't one just check that the distance between the next events after $\epsilon_{PQ}$ for both $P$ and $Q$ are equal? $\endgroup$ – Bobak Hashemi Jun 8 '17 at 5:28
  • $\begingroup$ Sorry, the last line should have read: that the next and prior points in the sequence are the same for both $P$ and $Q$ $\endgroup$ – Bobak Hashemi Jun 8 '17 at 5:43
  • $\begingroup$ @Bobak Hashemi: "Lorentzian distance is just meant to be a generalization of the geodesic length, is it not?" -- Arguably a generalization of arc length of timelike geodesics, if at all. (This would presume, however, that values of "arc length" could be determined without requiring values of Lorentzian distance in the first place.) [contd.] $\endgroup$ – user12262 Jun 8 '17 at 21:02
  • $\begingroup$ "That is, for small distances, $\ell$ is just the distance in Minkowski spacetime?" -- In flat spacetime interval values $s^2$ are defined and for (any) two timelike related events $s^2$ is simply equal to the square of the Lorentzian distance, $\ell^2$, for these two events (in suitable order, modulo a sign according to the interval sign convention of choice). But in general curved spacetime interval values are apparently not defined at all (regardless how small), while values Lorentzian distance still are defined. $\endgroup$ – user12262 Jun 8 '17 at 21:03
  • $\begingroup$ @Bobak Hashemi: "So you're asking, given that two bodies are at the same spacetime point [...] under what conditions are they comoving?" -- Correct. "If the events ϵ are 'dense' [...]" -- Well, I wouldn't object to assuming this; e.g. specificly that for any two distinct events $\varepsilon_{AQ}$ and $\varepsilon_{JQ}$ in which $Q$ had taken part $Q$ also took part in event $\varepsilon_{FQ}$ for which $$0 \lt\text{max}[ \,\{\ell[\,\varepsilon_{AQ},\varepsilon_{FQ}\, ], \ell[\,\varepsilon_{FQ},\varepsilon_{JQ}\, ]\}\, ]\le\sqrt{\frac{1}{2}}\,\ell[\,\varepsilon_{AQ}, \varepsilon_{JQ}\, ].$$ $\endgroup$ – user12262 Jun 8 '17 at 21:06

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