If two ends were a certain "length" apart were they therefore at rest (or at least rigid) to each other?

Question

Considering the definition of the SI unit of "length" [1] and [2 (" method a.")] I'm missing any requirements about the two "ends" of the required "path travelled by light" being "at rest to each other", or at least "rigid to each other".

Are such requirements perhaps presumed to be understood implicitly?

Accordingly, if some particular pair of "ends", $A$ and $B$, in some particular trial, were characterized as having been a certain "lenght of $x \, \text{m}$ apart from each other", where "$x$" is some particular positive real number and "$\text{m}$" denotes the SI base unit "metre", is it then understood:

• that ends $A$ and $B$ had observed exchanging signal pings between each other; and not only once for each trial, but for any of their signal indications throughout a sufficiently extended trial?,

• that for any two (distinct) signal indications $A_J$ and $A_K$ of end $A$ during this trial the corresponding ping durations of end $A$ were equal to each other:
  $\tau_A[ \,_J, \,_{\circledR}^{B \circledR AJ} ] = \tau_A[ \,_K, \,_{\circledR}^{B \circledR AK} ]$ ?,

• that for any two (distinct) signal indications $B_P$ and $B_Q$ of end $B$ during this trial the corresponding ping durations of end $B$ were equal to each other:
  $\tau_B[ \,_P, \,_{\circledR}^{A \circledR BP} ] = \tau_B[ \,_Q, \,_{\circledR}^{A \circledR BQ} ]$ ?, and

• that for any signal indications $A_J$ of end $A$ and any signal indication $B_P$ of end $B$ during this trial the corresponding ping durations were equal to each other:
  $\tau_A[ \,_J, \,_{\circledR}^{B \circledR AJ} ] = \tau_B[ \,_P, \,_{\circledR}^{A \circledR BP} ] = \frac{2 x}{c} \text{m} $ ?.

References:
[1] SI brochure (8th edition, 2006), Section 2.1.1.1; http://www.bipm.org/en/si/base_units/metre.html ("The metre is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second.").

[2] "the mise en pratique of the definition of the metre"; http://www.bipm.org/en/publications/mep.html

This post imported from StackExchange Physics at 2014-04-24 07:33 (UCT), posted by SE-user user12262

Comments

Is your notation (particularly the use of the circled R) explained in the links?

This post imported from StackExchange Physics at 2014-04-24 07:33 (UCT), posted by SE-user Kyle Kanos

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Most incomprehensible question ever. Also, most incomprehensible mathematical notation ever.

This post imported from StackExchange Physics at 2014-04-24 07:33 (UCT), posted by SE-user DumpsterDoofus

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@Kyle Kanos: "Is your notation [...] explained in the links?" -- Certainly not in the two linked pages themselves (I'm not sure about secondary literature). They just write "path travelled by light" without mentioning its "ends" (such as "sender" and "receiver") at all; hence there's no notation given for such "ends" (I chose "$A$", and "$B$"), nor for "duration" (a.k.a. "proper time") of any one of them (my "$\tau_A$" and "$\tau_B$"), nor for the corresponding arguments ("from some particular indication, until another"; e.g. my "from $A_J$ until $A_{\circledR}^{B \circledR AJ}$").--contd.

This post imported from StackExchange Physics at 2014-04-24 07:33 (UCT), posted by SE-user user12262

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@Kyle Kanos: "(particularly the use of the circled R)" -- It's read here "received" or "observed"; as in "$A_{\circledR}^{B \circledR AJ}$" being read as "$A$ received the signal of $B$'s indication of having received the signal of $A$ and $J$ having met and passed each other." Do you have any other suggestion and preference for denoting a "signal round trip" (a.k.a. "ping") "from $A$, to $B$, and back to $A$" ??

This post imported from StackExchange Physics at 2014-04-24 07:33 (UCT), posted by SE-user user12262

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@DumpsterDoofus: "Most incomprehensible question ever." -- Well, might it help if I give explicit names to the two "ends" of the "path travelled by light" required in the SI definition; e.g. "sender" and "receiver"? Is the/my question not comprehensible whether a particular sender and receiver, in some particular trial, had been "rigid" to each other, or even "at rest" to each other? Don't you comprehend the question concerning "mutual rigidity" as asking about ping durations being constant; and "mutual rest" as asking about ping durations being constant and (moreover) equal?

This post imported from StackExchange Physics at 2014-04-24 07:33 (UCT), posted by SE-user user12262

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I suppose my point is if you're going to introduce non-standard notation, you had better explain it. I would think that, with this question, notation is probably going to completely obscure your question and push everyone away (see DumpsterDoofus' comment). That is to say, get rid of it entirely.

This post imported from StackExchange Physics at 2014-04-24 07:33 (UCT), posted by SE-user Kyle Kanos

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@Kyle Kanos: "[...] get rid of it entirely." -- No, I certainly won't emilinate the three equations for concisely expressing my corresponding (partial) questions. But I'd be happy to adapt to whatever standard notation would be available; if it's expressive enough (1) to indicate that the quantities whose equality is in question are durations (i.e. as long as the standard symbol $\tau$ is used); and (2) to render my three distinct (partial) questions as three distinct equations. Synges notations for "trip-times" (RGT, p. 409) fails on both requirements, btw.

This post imported from StackExchange Physics at 2014-04-24 07:33 (UCT), posted by SE-user user12262

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@Kyle Kanos: Those parts of my question which can unambigously expressed using the notation of Synge mentioned above I've extracted here: physics.stackexchange.com/q/99779

This post imported from StackExchange Physics at 2014-04-24 07:33 (UCT), posted by SE-user user12262

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To close, nonsense.