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Given the description of a clock $\mathcal A$, as

  • (1) a set $A$ of all (more than 2) distinct indications of this clock, in no particular order
    (where the individual indications contained in set $A$ are denoted below as "$A_J$", "$A_Q$", "$A_V$" etc.), together with

  • (2) a function $t_{\mathcal A} : A \rightarrow \mathbb R$
    (such that we may speak of set $A$ and function $t_{\mathcal A}$ together as "a clock, $\mathcal A$" at all),

and, in line with Einstein's assertion that:
"All our well-substantiated space-time propositions amount to the determination of space-time coincidences {such as} encounters between two or more {... participants}"
,
given

  • (3) a sufficient suitable account, in no particular order, of participants (such as $A$, $J$, or $Q$) and their encounters (where $A_J$ is $A$'s indication at the encounter of $A$ and $J$, $A_Q$ is $A$'s indication at the encounter of $A$ and $Q$, and so on) --

How can the order of indications of clock $\mathcal A$ be derived?

Or to consider the simplest case:
How can be determined whether
$A_J$ was between $A_Q$ and $A_V$, or
$A_Q$ was between $A_V$ and $A_J$, or
$A_V$ was between $A_J$ and $A_Q$
?

Note:
It should not be assumed that clock $\mathcal A$ was "monotonous";
in other words: it is not an acceptable answer to argue
"if $(t_{\mathcal A}[ A_V ] - t_{\mathcal A}[ A_Q ]) \times (t_{\mathcal A}[ A_Q ] - t_{\mathcal A}[ A_J ]) > 0$ then $A_Q$ was between $A_V$ and $A_J$".

Another note in response to a comment by Jim:
The encounters (coincidence events) in which the participant took part whose set of indications at those encounters is called $A$ (and who is therefore him/her/itself conveniently called $A$ as well) may be denoted as
"$\mathscr E_{AJ}$", "$\mathscr E_{AQ}$", "$\mathscr E_{AV}$",
among others;
corresponding to the indications "$A_J$", "$A_Q$", "$A_V$" of $A$ (and, together with function $t_{\mathcal A}$, therefore indications of clock $\mathcal A$) which had been named explicitly above.

As far as a Lorentzian manifold may be assigned to those events in which $A$ took part (together with additional events, as may be required) and as far as the corresponding Causal structure would be determined, those events in which $A$ took part were all elements of one chronological (or timelike) Curve.

My question is consequently, in other words, whether and how elements of such a chronological (or timelike) curve may be ordered, given their parametrization only by some arbitrary function $t_{\mathcal A} : A \rightarrow \mathbb R$,
together, of course, with a sufficient suitable account (3, above) of all participants involved, and their encounters, if any, in no particular order.

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  • $\begingroup$ I must ask, are you referring to the indications of only one clock for each event, or encounter? Or do you consider all participants to have their own "clock"; i.e. are the observations of indications and encounters occurring in the same locations or is there a common observer recording indications for distantly observed encounters? $\endgroup$ – Jim Feb 4 '14 at 22:49
  • $\begingroup$ Jim: "[...] referring to the indications of only one clock for each event, or encounter?" -- At the encounter (coincidence event) of (physical participants) $A$ and $J$ their corresponding indications are distinguishable and denoted $A_J$ vs. $J_A$. "Or do you consider all participants to have their own "clock"" -- Rather, the same participant with the same set of given indications may be part of many distinct clocks; for different functions "$t_{index}$" that are assigned. (Some of those clocks would be thereby "monotonous", others would not; some would be "good" in the sense of MTW ...) $\endgroup$ – user12262 Feb 5 '14 at 6:26
  • $\begingroup$ Then from what I can see, this seems to be a verbose way of asking about the relativity of simultaneity for encounters that may or may not be space-like separated. If so, why obfuscate the problem? Surely experience has taught you that questions worded thusly are doomed to be unanswered and inevitably closed as unclear? $\endgroup$ – Jim Feb 5 '14 at 13:58
  • $\begingroup$ Jim: "[...] seems to be a verbose way of asking about the relativity of simultaneity for encounters that may or may not be space-like separated." -- I'm flabbergasted by your conclusion. AFAIK, the indications of any one clock (about which I asked my question) are parts of events which are strictly pairwise time-like to each other; thus my question is not about simultaneity: neither explicitly, nor implied. "Surely experience has taught you [...]" -- Surely experience has taught that there's no further assistance to be expected. However, I can squeeze an answer in 600 chars. as follows: $\endgroup$ – user12262 Feb 5 '14 at 18:06
  • 1
    $\begingroup$ Jim: "I see your point" -- Thanks. Please also note that meanwhile I had tried accordlingly to add some appendage to my question; FWIW. $\endgroup$ – user12262 Feb 5 '14 at 19:41
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Before addressing the question directly, it should be helpful to sketch some relevant relations involving the notions timelike curve and light cone of an event (and here at first also distinguishing its "past" or "future" parts):

Considering the light cone of one particular event, and an (open) timelike curve containing another identified event inside the "future part" (or otherwise: the "past part") of this light cone then

  • the part of this timelike curve which is "on the future side" (or otherwise: "on the past side") of the identified event is entirely inside this light cone, too; and

  • any elements of this timelike curve which are not inside this light cone are necessarily "on the past side" of the identified event.

In additon, consider a suitable second light cone such that there remains some finite, bounded section of the timelike curve outside either of the two light cones. Then each of those elements of the timelike curve which are outside (or at least: not inside) these two light cones is thereby necessarily "between" the one identified event and any element of this timelike curve which is inside the "future part" (or otherwise: the "past part") of at least one of those two light cones.

Now, relevant aspects of these relations may be expressed by using only the terminology of the question (and as indicated by Einstein), in terms of (distinguishable) "coincidences" of corresponding subsets of (distinguishable) "participants", as provided by point (3) of the question:

My direct answer:

Along with participants $A$, $J$, $Q$, and $V$ (who took part in the correspondingly denoted coincidence events "$\mathscr E_{AJ}$", "$\mathscr E_{AV}$", and "$\mathscr E_{AQ}$")
there exist participants $K$ and $L$ who took part in coincidence event "$\mathscr E_{KL}$" where

  • event $\mathscr E_{KL}$ is distinct from event $\mathscr E_{AJ}$,

  • participant $K$ took part in coincidence event $\mathscr E_{AJ}$, too (which is therefore more completely denoted as "$\mathscr E_{AJK}$"),

  • there was noone having taken part both in events $\mathscr E_{KL}$ and $\mathscr E_{AV}$, and

  • there was noone having taken part both in events $\mathscr E_{KL}$ and $\mathscr E_{AQ}$,

and in addition there exist participants $N$ and $Q$ who took part in coincidence event "$\mathscr E_{NP}$" where

  • event $\mathscr E_{NP}$ is distinct from event $\mathscr E_{AQ}$,

  • participant $P$ took part in coincidence event $\mathscr E_{AQ}$, too (which is therefore more completely denoted as "$\mathscr E_{AQP}$"),

  • there was noone having taken part both in events $\mathscr E_{NP}$ and $\mathscr E_{AV}$, and

  • there was noone having taken part both in events $\mathscr E_{NP}$ and $\mathscr E_{AJ}$.

If these conditions are satisfied then I'd suggest to call indication $A_V$ "between" the indications $A_J$ and $A_Q$" (and just as well $A_V$ "between" the indications $A_Q$ and $A_J$).

Caveats concerning transitivity and "consistency":

The definition suggested above, of how to determine "between-ness" of indications of a given clock, apparently does not guarantee outright transitivity, i.e. such that

  • if $A_V$ was "between" the indications $A_J$ and $A_Q$ and

  • if (for yet another indictation "$A_F$" of participant $A$, and of clock $\mathcal A$) $A_J$ was "between" the indications $A_F$ and $A_V$

then necessarily

  • $A_J$ was "between" the indications $A_F$ and $A_Q$ and

  • $A_V$ was "between" the indications $A_F$ and $A_Q$, too.

Nor does the suggested definition guarantee outright "consistency" in the sense that

  • given events $\mathscr E_{AQ}$, $\mathscr E_{AV}$ and $\mathscr E_{KL}$ (as required for determining that $A_V$ had been "between" the indications $A_J$ and $A_Q$) such that there was noone who participated in both $\mathscr E_{KL}$ and $\mathscr E_{AQ}$, or in both $\mathscr E_{KL}$ and $\mathscr E_{AV}$, and

  • given (for yet another indictation "$A_U$" of participant $A$, and of clock $\mathcal A$) $A_U$ was "between" the indications $A_Q$ and $A_V$

there shouldn't anyone having participated both in events $\mathscr E_{KL}$ and $\mathscr E_{AU}$, either.

I can think of two general approaches for trying to address there caveats (separately, or jointly), namely:
either to suitably modify the definition of "between-ness" (which I certainly cannot attempt at the moment),
or to simply assume that the "account of participants and their encounters" (which is required as point (3) of the question) is "complete and correct" such that transitivity and "consistency" are thereby guaranteed as described (which may however not be easily satisfied by actually collected experimental data).

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