# simultaneity

## EMROPTCE-3: Let’s calculate! — “Constraints” vs. “Degrees of freedom”

Keeping in mind the caveats sketched in the preceding part 2 of this blog post series let’s now focus on the specific basic “construction $\frak C$” being proposed for expressing mutual rest of participants through coincidence events; namely the ping relations between four “corners” (A, B, F, G), and six “half-way-betweens” (J, K, P, Q, U, W):

Require:

– for any signal event in which A had taken part (and had thus stated a signal indication), A observed in coincidence that

* B and F and G had observed this signal event, and

* likewise for K and P;

– for any signal event in which B had taken part, B observed in coincidence that

* A and F and G had observed this signal event, and

* likewise for Q and U;

– etc. for F and for G;

– for any signal event in which J had taken part, J observed in coincidence that

* A and B and K and P and Q and U had observed this signal event;

– etc. for K, P, Q, U and for W.

(At this point some may ask for some additional guidance being provided through a sketch. But I’m hesitant, for fear of a sketch being even more “leading” than referring to A, B, F, G as “corners”, and to J, K, P, Q, U, W as “half-way-betweens”. Let’s say I’ll add a sketch here if and when WordPress supports(1) drawing with PSTricks commands.)

Perhaps the best and decisive illustration is by noticing and comprehending that the set of these requirements provides constraints on the geometric relations between these 10 participants (A through W, a.k.a. “set $\mathcal T_{\frak C}$“), and by spelling out concretely how many actual contraints are thereby obtained (which can be expressed as actual algebraic “constraint equations”, as they will appear later in this series of blog posts):

for any one signal event in which participant A had taken part (and had thus stated a signal indication) A is supposed to determine one coincidence event in which A’s observations of ping echos regarding six distinct other participants (namely concerning B, F, G, J, K, and P) were made strictly together (and not in any way “sequentially”). This gives $5$ “actual constraints” provided by participant A, for any one signal event in which participant A had taken part. The exact same count applies to the constraints provided by B, F, and G, respectively, since the “requirements on” B, F, and G are exactly similar to those on A.

The “requirements on” J, K, P, Q, U, and W are likewise exactly similar to each other (but somewhat different from those on A, B, F, and G). However, incidentally, J (for instance) is also supposed to determine one coincidence event in which J’s observations of ping echos (from any one signal event in which J had taken part) regarding six distinct other participants (namely concerning A, B, K, P, Q, and U) were made strictly together (and not in any way “sequentially”). Therefore J provides $5$ “actual constraints” for any one signal event, too; and K, P, Q, U, and W likewise.

Now (“Let’s calculate!”) try to compare the number of these constraints to the number of “degrees of freedom” of these 10 participants (set $\mathcal T_{\frak C}$) under consideration (which (optimistically!) are provided and indeed limited by their characterizations with respect to any suitable “tentative inertial frame $\mathcal F_{\frak C*}$“):

Considering 10 signal events (which must be distinct events, because –oh!, seems I forgot to mention — the 10 participants of set $\mathcal T_{\frak C}$ ar of course supposed to never meet each other) that’s (optimistically!?) $\approx 40$ initial degrees of freedom. (Strictly, it might actually some fewer; “effectively” perhaps rather $\approx 40 - 4 - 3 = \approx 33$; or even only $\approx 33 - 6 = \approx 27$, considering that the initial signal events in which J, K, P, Q, U, and W took part, respectively, are not independent of the initial signal events pertaining to A, B, F, or G).

Since any subsequent events are not completely independent of the initial signal events under consideration, but related by (mutual) observation and (even) pings (a.k.a. “signal round-trips”), the number of “degrees of freedom” increases by $3$ (rather than by $4$) for each participant at each subsequent events.

For the explicit (basic) set of “requirements” as quoted above, the count of “degrees of freedom vs. constraints” therefore stands (roughly, but optimistically) at

$27 + 2*3*4 + 2*3*6 - 50 = 27 + 24 + 36 - 50 = 37,$

where the number $2*3*4 = 24$ refers to the “degrees of freedom” of A, B, F, and G in two events each (following the initial signal events), and the number $2*3*6 = 36$ refers to the “degrees of freedom” of J, K, P, Q, U, and W, in two subsequent events each.
Clearly, that’s “too few constraints” to draw some distinctive conclusion such as the proposed “mutual rest”.

However: as noted earlier already, these “requirements” (i.e. the “contruction $\frak C$“) can be “extended or refined (iterated, combined)“; it can be repeated over and over because any event in which one of the ten participants took part may in turn serve as initial signal event for the next/appended iteration of the “construction”. The important point is: the number of “constraints” obtained by appending successive “construction” steps grows more rapidly than the corresponding number of “degrees of freedom”; specificly as

$27 + 2*3*4 + 3*3*6 - 50 - 5*6 = 37 + 18 - 30 = 25,$

$27 + 3*3*4 + 3*3*6 - 50 - 5*6 - 5*4 = 25 + 12 - 20 = 17,$

$27 + 3*3*4 + 4*3*6 - 50 - 2*5*6 - 5*4 = 17 + 18 - 30 = 5,$

$27 + 4*3*4 + 4*3*6 - 50 - 2*5*6 - 2*5*4 = 5 + 12 - 20 = -3$

There are already (optimistically) more “constraints” than “degrees of freedom”; meaning more practically “more equations than unknowns”, which indicates that there is some “solution” to be evaluated, as I propose and hope.

A “practical difficulty” is of course immediately obvious, too: any desired concrete “solution” would seemingly have to be obtained from a system of about one hundred or more “constraint equations”. (That’s certainly a bit too much of a problem for an old laptop running an almost as old version of Mathematica (TM) …)

[To be continued.]

(1: Instant PSTricks test: $\begin{pspicture}(5,5) %% Triangle in red: \psline[linecolor=red](1,1)(5,1)(1,4)(1,1) %% Bezier curve in green: \pscurve[linecolor=green,linewidth=2pt,% showpoints=true](5,5)(3,2)(4,4)(2,3) %% Circle in blue with radius 1: \pscircle[linecolor=blue,linestyle=dashed](3,2.5){1} \end{pspicture}$
)

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

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”).

[…] 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 …)

## We’re sorry to let you know that … (EMROPTCE-1)

Sometime in December of last year I read about the latest FQXi request for proposals, for the “Physics of What Happens Grant Program”, which I found quite interesting and motivating, to say the least. A few weeks later, still days ahead of the deadline, I registered with the FQXi (for the first time) and I submitted the proposal which I had prepared in the meantime; in the required form, see appendix below.

The receipt of my proposal entry was duely confirmed. Yesterday I received the notification of the decision of the initial review panel (see appendix below). Of course, I had not quite expected to be dismissed already at the initial round of selection, without receiving any individual feedback to my entry, or even some hint of substantive recognition … Let me not dwell on that.

Foremost, I like to thank the organizers and sponsors of FQXi, and of this particular RFP especially, for having dangled enough of a carrot for me to develop my proposal as far as it presently stands. And good luck to those entries which are still in competition.

Further, I’ve got this blog set up so far already … Let me not dwell on its shortcomings (for now); time to put it to use!:

Let’s just see anyways whether two years are enough (as prescribed by the grant proposal; or, given my circumstances, rather until the end of 2017) for me to achieve (and to document in this blog) what I undertook by submitting my proposal: “Expressing Mutual Rest of Participants through Coincidence Events“.

Appendices

(1): My proposal, as submitted to the FQXi RFP in Feb. 2015

(Note: The FQXi RFP submission procedures were really quite strict in allowing precisely only 500 words in the proposal text, i.e. excluding the title. So I wasn’t able to submit my fairly carefully laid-out one page PDF document of about 530 words, but I reverted my LaTeX file to plain text, shortening it just enough by skipping the references.)

PROPOSAL INFORMATION

Project Title: Expressing Mutual Rest of Participants through Coincidence Events

Project Summary: The notion “event”, or suitably synonymous: “coincidence”, has basic self-evident importance in contemporary conceptions of geometric relations; implicitly for instance in Einstein’s definition of “simultaneity” [1]:

“If the observer perceives the two flashes of lightning at the same time, then …”,

and especially prominent in Einstein’s foundational considerations [2a]:

“We assume the possibility for stating … the immediate space-time adjacency (coincidence) [for observational contents of] events … without giving a definition for this fundamental expression.”

along with [2b]:

“All our well-substantiated space-time propositions amount to the determination of space-time coincidences. If, for example, the [course of events] consisted in the motion of material points, then, for this last case, nothing else are really observable except for encounters between two or more of these material points.”

Such commendation suggests a research program to explore how “space-time propositions” may be explicitly expressed, and thus be firmly comprehended, in terms of descriptions of coincidence events; i.e. by detailling all their observational contents

– who (among all distinct identifiable participants) had “encountered (and passed)” whom (and, case by case, whom not); and/or

– who had observed whose signal indications “at the same time” (or else: in sequence, or not at all).

Applicable propositions would concern causal relations (such as, for any three distinct coincidence events with at least one common participant to determine which one had been “between” the other two) as well as suitably generalized metric relations (foremost to evaluate the ratio of any two non-zero “space-time intervals”, as a real number).

Arguably, such a research program has barely been recognized (e.g. within the broader context by which Einstein’s “point-coincidence argument” has received continued attention [3]); only few though well-known cases (e.g. [1]) can serve at least as partial examples.

As a short-term contribution 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:

– for any signal event in which A had taken part (and had thus stated a signal indication), A observed in coincidence that

* B and F and G had observed this signal event, and

* likewise for K and P;

– for any signal event in which B had taken part, B observed in coincidence that

* A and F and G had observed this signal event, and

* likewise for Q and U;

– etc. for F and for G;

– for any signal event in which J had taken part, J observed in coincidence that

* A and B and K and P and Q and U had observed this signal event;

– etc. for K, P, Q, U and for W.

References:

[1]…

(2): E_mail from fqxi.org received on March 17th, 2015:

FQXi grant proposal decision [Physics of What Happens]

Dear […],

Thank you for your submission to FQXi’s Physics of What Happens Grant Program. Our initial review panel has now concluded. We’re sorry to let you know that they did not select your proposal as one of our finalists.

Due to our limited budget and a strong pool of entries, competition was tough. We had to decline many promising entries. This two-stage review process avoids inflicting the community with a very small acceptance rate for long proposals that take great effort to prepare.

We can’t provide individual feedback on the entries, but in general, the panel looked closely at a proposal’s relevance to the FQXi mission and the program topic, and to the estimated scientific impact-per-dollar. This year’s panel was particularly careful on the question of relevance, as they interpreted it.

Thank you again for the work and time spent on preparing your proposal. Please look out for more FQXi programs in the future, including further Grant competitions.

Sincerely,

[two personalized signatures]
FQXi

## Can you derive the Lorentz factor?

Anyone who claims to comprehend RT should certainly be able to make sense of the famous Lorentz factor $\gamma := 1 / \sqrt{1 - \beta^2}$, and to provide a derivation based (ideally strictly and exclusively) on foundational principles and definitions of RT relating to geometric or kinematic notions, as laid out by A.Einstein; instead, in particular, sticking to the derivation by H.A.Lorentz (or G.FitzGerald) with its appeal to dynamics such as to “classical electro-magnetic theory”, prior to Einstein’s (or H.Poincaré’s) contributions.

The actual derivations presented by Einstein, or by Poincaré, or any other, are themselves not beyond critique nor necessarily already ideal in the above sense. Rather, the goal of presenting a compreshensibe and convincing derivation of $" \sqrt{1 - \beta^2} "$ or various related expressions seems to invite some extend of “didactic-scientific creativity”. Different disseminators of RT, who well may each consider themselves careful educators, do take noticable liberties in “their style”.

Here, then, is my favorite derivation (for now …):

Consider the (hopefully familiar) cast of characters and the following (hopefully familiar) setup requirements (which I hope to illustrate eventually, too):

three distinct participants $A$, $B$, and $F$ who were

(1)

• at rest to each other, and
• straight to each other, i.e. such that the distance ratios between them satisfied $\frac{AB}{AF} + \frac{BF}{AF} = 1$, and

another three distinct participants $J$, $K$, and $Q$ who were likewise

(2)

• at rest to each other, and
• straight to each other, i.e. such that the distance ratios between them satisfied $\frac{JK}{JQ} + \frac{KQ}{JQ} = 1$, and such that further

(3)

• $J$, $K$, and/or $Q$ were not at rest to $A$, $B$, and/or $F$, but they were mutually moving uniformly “straight along, against” each other and satisfied the following further setup conditions:

(4)

• $F$ and $Q$ met each other (in passing) in coincidence with (or “just as”) both $F$ and $Q$ observed that $A$ and $K$ had met each other (in passing), i.e. the two corresponding events are (called) “light-like to each other”, i.e. $\mathcal{E}_{A \circ K} \nearrow \mathcal{E}_{F \circ Q}$ and

(5)

• $A$ and $K$ met each other (in passing), such that $A$‘s indication of having met (and being passed by) $K$ is simultaneous to $B$‘s indication of having met (and being passed by) $Q$, i.e. $A_{\circ K} \equiv A_{\circledS}^{B \oslash Q}$, and similarly
• $A$ and $J$ met each other (in passing), such that $J$‘s indication of having met (and being passed by) $A$ is simultaneous to $Q$‘s indication of having met (and being passed by) $F$, i.e. $J_{\circ A} \equiv J_{\circledS}^{Q \oslash F}$, and finally:

(6)

• that the two distance ratios $\frac{AB}{AF}$ and $\frac{JK}{KQ}$ are equal to each other (and, of course, non-zero) where their corresponding real number value is called $" \beta "$, $\frac{AB}{AF} = \frac{JK}{KQ} := \beta$.

(If it might be found that these requirements had not been satisfied for some particular set of participants $\{ A, B, F, J, K, Q \}$, in some particular trial, then this particular trial should be considered “invalid” and its data, if any, “to be discarded”. The following derivation of $" \sqrt{1 - \beta^2} "$ is meant only a concern of “valid” trials, in whith the stated requirements were found satisfied.

A point not to be missed is the question whether each and all of the stated requirements are even necessary and effective in distinguishing “valid” from “invalid” trials, or whether these “valid setup” requirements constitute an overconstraint. To address this question, however, the definitions and interdependencies must of course be considered of what is even meant at all by the notions “mutual rest”, “distance ratio”, and not least “simultaneity” at all.)

Now, given these requirements, can any conclusions be drawn (agreements reached, measurements obtained) regarding distance ratios $\frac{JQ}{AF}$ or $\frac{BF}{KQ}$, and so on? How can the given “two systems of participants” compare distances between each other at all, given that they are supposed to “move against each other” throughout the trial?

The two parts of requirement (5) suggest a particular symmetric relation, between the pairs $B$, $F$ and $K$, $Q$ on one hand, and in turn the pairs $J$, $Q$ and $A$, $F$ on the other. (It is possible to contruct other even more explicitly symmetric setup requirements, which however involve two systems of five participants each and are accordingly more lengthy to express.)

Therefore (and barring possible other arguments or finds related to “dynamics”) the two systems may agree, as an obvious convention, to compare distances between each other based on setting $\frac{JQ}{AF} = \frac{BF}{KQ}$.
Suitably combining the equations of requirements (1), (2) and (6):

$1 - \beta = 1 - \frac{AB}{AF} = \frac{BF}{AF}$

and

$1 + \beta = 1 + \frac{JK}{KQ} = \frac{JQ}{KQ}$

then allows to calculate

$\frac{JQ}{AF} = \frac{BF}{KQ} = \frac{BF}{AF} \frac{AF}{JQ} \frac{JQ}{KQ} = (1 - \beta) (1 + \beta) \frac{AF}{JQ} = (1 - \beta^2) \frac{AF}{JQ} = \sqrt{ (1 - \beta^2) }$.

There it is already — the inverse of the Lorentz factor $\gamma$ having been derived in the course of identifying a mutually agreeable convention for how to compare distances, i.e. to measure distance ratios, between systems of participants moving “against each other on straight lines”.

As a sequel, it follows how participants who are not at rest to each other can agree on comparisons of durations between each other; where the Lorentz factor will make another appearance.

The challenge remains of course to define the notions employed above (“mutual rest”, “distance ratio”, “simultaneity”) and corresponding measurement operations in the first place, based on foundational principles which will have to be introduced more broadly to begin with.

Supplement (June 23, 2015):

Only recently I noticed that the setup conditions required above also imply that

• $A$ and $J$ met each other (in passing) in coincidence with (or “just as”) both $A$ and $J$ observed that $B$ and $Q$ had met each other (in passing), i.e. that the two corresponding events had been “light-like to each other”; $\mathcal{E}_{B \circ Q} \nearrow \mathcal{E}_{A \circ J}$.

In consequénce, it can be shown that the four events $\mathcal{E}_{A \circ K}$, $\mathcal{E}_{A \circ J}$, $\mathcal{E}_{B \circ Q}$, and $\mathcal{E}_{F \circ Q}$ (which contain the entire observational content of the above description) are plane with respect to each other; i.e. their Cayley-Menger determinant (in terms of their (pairwise, symmetric) spacetime intervals) vanishes.

We thus require a definition of simultaneity such that this definition supplies us with the method by means of which, in the present case, he can decide by experiment whether or not both the lightning strokes occurred simultaneously.

As long as this requirement is not satisfied, I allow myself to be deceived as a physicist (and of course the same applies if I am not a physicist), when I imagine that I am able to attach a meaning to the statement of simultaneity.

(I would ask the reader not to proceed farther until he is fully convinced on this point.)

— A. Einstein