# Can a value of "length, in meters" be attributed to a pair of ends which are rigid (but not at rest) to each other?

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The definition of the SI base unit "metre" [1] doesn't seem to rule out explicitly that a certain value of "length, in meters" could be attributed to a pair of ends which are rigid to each other, but not at rest to each other.

Consider, therefore, two such ends, $A$ and $B$, which both find constant but unequal ping durations between each other, i.e. in the notation of [2]$\! {\,}^{(\ast)}$:

$[ \, A \, B \, A \, ] \ne [ \, B \, A \, B \, ]$.

Is there a value of "length, in meters" attributable to this pair of ends, $A$ and $B$ ?

If so, what is that value?,
i.e. if the SI definition allowed to express the value of "the lenght $AB$" as "$x \, \text{m}$", for some positive real number $x$, then how should $x$ be expressed in terms of the two (given) unequal ping duration values $[ \, A \, B \, A \, ]$ and $[ \, B \, A \, B \, ]$, and the SI base unit "second" ("$\text{s}$")?
(Is perhaps: "$x := \left( \frac{[ \, A \, B \, A \, ]}{2 \, \text{s}} + \frac{[ \, B \, A \, B \, ]}{2 \, \text{s}} \right) \times \frac{299 \, 792 \, 458}{2}$"?
Or perhaps: "$x := \sqrt{ \frac{[ \, A \, B \, A \, ]}{2 \, \text{s}} \times \frac{[ \, B \, A \, B \, ]}{2 \, \text{s}} } \times 299 \, 792 \, 458$"? ...)

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."). Together with "the mise en pratique of the definition of the metre"; http://www.bipm.org/en/publications/mep.html

[2] J.L.Synge, "Relativity. The general Theory", North-Holland, 1960; p.409:
" [...] light signals passing between a source $0$ and mirrors $1$, $2$, [...]
Trip-times such as $[ \, 0 \, 1 \, 0 \, ]$ [...] are measureable [...]
"

$(\ast$: Suggestions for more standard and/or expressive notation for ping durations are welcome.$)$

This post imported from StackExchange Physics at 2014-04-24 07:32 (UCT), posted by SE-user user12262

retagged Apr 24, 2014
I don't understand your question very well. Why can't you measure the distance between $A$ and $B$?

This post imported from StackExchange Physics at 2014-04-24 07:32 (UCT), posted by SE-user jinawee
jinawee: "Why can't you measure the distance between $A$ and $B$?" -- Well, how to use the SI prescription "the path travelled by light in vacuum during" in the setup scenario with unequal ping durations $[ \, A \, B \, A \, ] \ne [ \, B \, A \, B \, ]$? How to get one result ("the length separating $A$ and $B$ from each other; in meters") at all? (Instead e.g. describing these two ends by two values of quasi-distances, such as the two separate values "$c/2 [ \, A \, B \, A \, ]$" and "$c/2 [ \, B \, A \, B \, ]$"?) Why is the SI definition not rigorous and unambiguous on this point??

This post imported from StackExchange Physics at 2014-04-24 07:32 (UCT), posted by SE-user user12262
What do you mean ping duration? Is there a physical example of your problem? Could you do a sketch?

This post imported from StackExchange Physics at 2014-04-24 07:32 (UCT), posted by SE-user jinawee
jinawee: "What do you mean [by] ping duration?" -- The duration of some particular participant (such as $A$) from having stated some particular signal indication until observing the (echo-)indication of some other participant (e.g. $B$) of having observed the signal indication. In other words: just what Synge called "trip-time". (That's why I used Synge's notation in my question here ...) "Is there a physical example of your problem?" -- Sure. Can $A$ and $B$ of that example be attributed a value of "length, in meters" by the SI prescription??

This post imported from StackExchange Physics at 2014-04-24 07:32 (UCT), posted by SE-user user12262

Just use the instantaneous position of either end. The length is ambiguous otherwise, because you need to specify the frame.

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