simultaneity

………. and a bag of $latex \text{ c }$s

EMROPTCE-2: Identifying the actual task at hand. (And, perhaps, a catch.)

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The title of my proposal presented in the preceding first post of this series, and consequently its acronym (initialism) as common recognizable tag, is obviously
“Expressing Mutual Rest of Participants through Coincidence Events” (“EMROPTCE”).

More specificly, the task to be undertaken had been described so:

[…] I’d like to investigate whether the following expression in terms of coincidence events is suitable for defining an explicit instance of the proposition

“being at rest to each other in a region which is flat in a suitable sense (expressed similarly)”,

regarding ten distinct, separate participants (A, B, F, G suggestively called “corners”, and J, K, P, Q, U, W as corresponding “halfway-betweens”). Require: […]

Before turning to a more detailed discussion of the exact requirements which had been suggested (which shall of course follow as this blog post series continues), surely the phrase “a region which is flat in a suitable sense (expressed similarly)” is in need of more explanation. (Recall the strict 500 words limit of the FQXi proposal which had forced me to be extremely terse on this point.)

The “scientific abstract in preparation of possible publication” (200 words maximum, which I had prepared, but which was not part of the submissions to be made for the FQXi RFP) may be instructive for being a little more elaborate in this respect:

Abstract:

Starting out from the basic notions of distinct “identifiable participants” (a.k.a. “principal identifiable points”) and of distinct “coincidence events” being described as encounters of two or more participants and/or the coincident observation, by at least one participant, of two or more signal indications stated by others, the problem stands how to express (define, construct, evaluate) further geometric or topological relations (as “spacetime propositions”) explicitly and strictly in terms of these basic notions. Concentrating on one example case which, on one hand, seems complex enough to provide some specific and distinctive proposition, and on the other hand, simple enough to be tractable, this project is concerned with defining the proposition of several participants having been “at rest to each other”, expressed as certain conditions on coincidence observations involving 10 distinct, separate participants. The suggested construction can be extended or refined, thereby establishing joint membership of numerous participants together as one “inertial frame”; in a region which is suitably “flat”. The project aim is then to prove that the suggested construction is sufficient for establishing unique, exclusive membership of (subsets of) 10 such suitable participants in the same one “inertial frame”; thus “having been at rest to each other”.

To repeat and elaborate on the two important points:

(1) The specific “construction \frak C” (or set of “requirements”) which involves exactly 10 distinct participants can readily be “extended or refined” (iterated, combined) to establish relations between many more, “tightly packed” participants, being covered in terms of overlapping (non-disjoint) subsets of 10 each. It is the ultimate goal of the proposal to prove that any one such suitable set of many participants satisfies certain geometric properties by which to establish them having been mutually “rigid” and mutually “flat” and (therefore) an “inertial frame”. Before and until having reached this goal, any such suitable set of many participants (i.e. which are related to each other in suitable subsets of 10, through repetitions "\frak C \! *" of “the construction”) is only tentatively (if at all) referred to as “inertial frame”, to avoid more cumbersome terminology.

(2) The main immediate concern is to characterize just any one suitable set \mathcal T_{\frak C} of 10 participants in relation to a, or any, “tentative inertial frame” set \mathcal F_{\frak C*} of very many participants. The concrete question to be addressed is:

If any one member of set \mathcal T_{\frak C} is explicitly a member of set \mathcal F_{\frak C*} then are all other 9 members of set \mathcal T_{\frak C} also guaranteed to be members of set \mathcal F_{\frak C*}; such that the 10 members of set \mathcal T_{\frak C} are explicitly shown having been mutually “at rest” to each other? Can this be proven from the explicit “construction” (or “requirements”) which applies to the 10 members of set \mathcal T_{\frak C} among each other, and which applies to any suitable subset of 10 members of set \mathcal F_{\frak C*}?

Trying to find such a proof (or else to establish counter-examples) is the concrete task I meant to propose to be tackled. So far the problem has been too difficult for me to reach a conclusion; but I have some hope and expectation that, with concentrated effort, within about two years, I might. (And if the desired proof were not found I’d have to feel sorry for anyone using and relying on the notions of “mutual rest” and “inertial frame” nevertheless.)

Now, in order to even attempt to find a proof of

A \in \mathcal T_{\frak C} \text{ and } A \in \mathcal F_{\frak C*} \implies \forall X \in \mathcal T_{\frak C} : X \in \mathcal F_{\frak C*},

or to look for counter-examples,
the relations between members of set \mathcal F_{\frak C*} must be characterized outright without having this desired proof available, without knowing whether such a proof could be had at all.

Either the desired proof were found then, arguably, since membership in set \mathcal F_{\frak C*} is based on suitable repetitions of the very same “construction \frak C” therefore the characterization of the relations between the members of set \mathcal F_{\frak C*} is even strengthened; they could well be considered mutually “rigid”. (That’s the “optimistic scenario” which I’ve already been investigating in some detail.)

Or counter-examples could be given, then clearly the 10 members of \mathcal T_{\frak C} could not be called “rigid” to each other; and the characterization of the relations between the members of set \mathcal F_{\frak C*} apparently remains considerably weaker. Which underscores the difficulty of characterizing set \mathcal F_{\frak C*} in the first place sufficiently to allow finding such a counter-example. Which in turn, and therein lies the catch, apparently cannot be different from the difficulty of characterizing set \mathcal F_{\frak C*} in the first place sufficiently to allow finding the desired proof. (And which therefore let’s me worry about the validity of the “optimistic scenario” described above …)

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