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The following questions (in no particular order) which I had submitted have been "removed from PSE for reasons of moderation":

  1. Which geometric relations obtain between two distinct rest systems?

Consider, as a thought experiment, a set of participants who measure throughout the experiment having been at rest to each other; among them explicitly participants ${\mathbf A}$, ${\mathbf B}$ and ${\mathbf F}$ who determine the ratios of their (chronogeometric) distances between each other as real number values $\frac{{\mathbf A}{\mathbf B}}{{\mathbf A}{\mathbf F}}$, $\frac{{\mathbf B}{\mathbf F}}{{\mathbf A}{\mathbf F}}$, and $\frac{{\mathbf A}{\mathbf B}}{{\mathbf B}{\mathbf F}} = \frac{{\mathbf A}{\mathbf B}}{{\mathbf A}{\mathbf F}} / \frac{{\mathbf B}{\mathbf F}}{{\mathbf A}{\mathbf F}}$.

Further let there be another set of participants (of which neither ${\mathbf A}$, nor ${\mathbf B}$, nor ${\mathbf F}$ are a member) who measure throughout the experiment having been at rest to each other as well; among them ${\mathbf J}$, ${\mathbf K}$ and ${\mathbf Q}$, who determine the ratios of their (chronogeometric) distances between each other as real number values $\frac{{\mathbf J}{\mathbf K}}{{\mathbf J}{\mathbf Q}}$, $\frac{{\mathbf K}{\mathbf Q}}{{\mathbf J}{\mathbf Q}}$, and $\frac{{\mathbf J}{\mathbf K}}{{\mathbf K}{\mathbf Q}} = \frac{{\mathbf J}{\mathbf K}}{{\mathbf J}{\mathbf Q}} / \frac{{\mathbf K}{\mathbf Q}}{{\mathbf J}{\mathbf Q}}$,

such that

  • ${\mathbf J}$ passed ${\mathbf A}$, then passed ${\mathbf B}$,

  • ${\mathbf A}$ passed ${\mathbf J}$, then passed ${\mathbf K}$,

  • ${\mathbf Q}$ passed ${\mathbf F}$, in coincidence with ${\mathbf Q}$ and ${\mathbf F}$ observing ${\mathbf J}$ and ${\mathbf A}$ having passed each other,

  • ${\mathbf B}$ and ${\mathbf F}$ determined that ${\mathbf B}$'s indication of the passage of ${\mathbf J}$ was simultaneous to ${\mathbf F}$'s indication of the passage of ${\mathbf Q}$, and

  • ${\mathbf K}$ and ${\mathbf Q}$ determined that ${\mathbf K}$'s indication of the passage of ${\mathbf A}$ was simultaneous to ${\mathbf Q}$'s indication of the passage of ${\mathbf F}$.

Question:
Is thereby guaranteed that for these distance ratios obtains

(1)
$\frac{{\mathbf A}{\mathbf B}}{{\mathbf A}{\mathbf F}} = \frac{{\mathbf J}{\mathbf K}}{{\mathbf J}{\mathbf Q}}$ ?,

and (moreover)

(2)
$\left( \left(\frac{{\mathbf B}{\mathbf F}}{{\mathbf A}{\mathbf F}}\right)^2 + 1 - \left(\frac{{\mathbf A}{\mathbf B}}{{\mathbf A}{\mathbf F}}\right)^2 \right) \left( \left(\frac{{\mathbf K}{\mathbf Q}}{{\mathbf J}{\mathbf Q}}\right)^2 + 1 - \left(\frac{{\mathbf J}{\mathbf K}}{{\mathbf J}{\mathbf Q}}\right)^2 \right) = 4 \left( 1 - \left( \frac{{\mathbf A}{\mathbf B}}{{\mathbf A}{\mathbf F}} \right) \left( \frac{{\mathbf J}{\mathbf K}}{{\mathbf J}{\mathbf Q}} \right) \right)$ ?

Or otherwise:
What could be concluded if (1) and/or (2) were not found satisfied?


3h
comment Relativity of simultaneity example in Resnick
Nevertheless: +1 for picking Resnick's pics which explicitly identify participant A' in distinction to participant A, and B' in distinction to B. Because: this allows (also in accordance with Einstein's definition) for instance to conclude and say that (A)'s indication of "being passed by (A')" was simultaneous to (B)'s indication of "being passed by (B')" as well as (A')'s indication of "being passed by (A)" was not simultaneous to (B')'s indication of "being passed by (B)". (I plan to elaborate this and submit as an answer, but I can get to that only in more than a week.)
3h
comment Relativity of simultaneity example in Resnick
KartMan: "When the event happens at A, A' coincides with A, and when the event happens at B, B' coincides with B." -- Rather, the one event is the coincidence event of A and A' "meeting in passing" (consisting explicitly of of (A)'s indication of "being passed by (A')", and of (A')'s indication of "being passed by (A)"); and the other event is the coincidence event of B and B' "meeting in passing" (consisting explicitly of of (B)'s indication of "being passed by (B')", and of (B')'s indication of "being passed by (B)"). You notice that "primes" as part of names are inconvenient, btw.
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revised metric-space wiki description
(still proof-reading ...)
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suggested suggested edit on metric-space tag wiki
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awarded  Tag Editor
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revised metric-space wiki excerpt
added 242 characters in body
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revised metric-space wiki description
added 1903 characters in body
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wiki created metric-space description
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suggested suggested edit on metric-space tag wiki
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revised Is there some other name used for “ping rigidity”?
(v3.141: à propos "rigid".)
19h
revised Is there some other name used for “ping rigidity”?
(v3.14: still fixing ...)
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asked Is there some other name used for “ping rigidity”?
23h
comment why are the anthropometric units (which are about as big as we are) as large as they are relative to their corresponding Planck units?
robert bristow-johnson: "the $10^{44}$ question [...] to form a thought" -- That's hard; but it may be "factorized" into a question about "ingredients of thought", i.e. the $$\frac{G~m_e^2}{\hbar~c} \approx ?$$ question above (which I'm sure can be addressed at PSE, if it wasn't already); and another about "emergence/reduction of thought itself", i.e. a $$\text{Bohr duration / in anthropometric units} \approx \frac{\hbar}{m_e~\alpha} / \text s~ \approx 10^{-19}$$ question. The latter may receive (for now) more interest at other SE venues, but note physics.stackexchange.com/q/14959
1d
comment $\Delta^+$ decay in GZK process
@rob ("user 44126") -- I just noticed (with much surprise) that I have been awared the 300 as reputation which you had invested as bounty on Jul 16, 2014. I like to point out that I had submitted my answer as (my very first) community-wiki under the assumption that "Community wiki posts ... do not confer any reputation.". (Reflecting on how I had earned my reputation points until then I considered "+300" pretty much an utterly disproportionate amount.)
1d
comment why are the anthropometric units (which are about as big as we are) as large as they are relative to their corresponding Planck units?
robert bristow-johnson: "@user12262, i tried to address your suggestion [...]" -- (I first checked whether the question title had changed; when I noticed that it had not, I figured that at least the "EDIT" should be interesting ...) "why $c \approx 10^9$ in anthropometric units." -- Well, that's a question about "Why humans work and appear the way they do?". From a physics perspective this may relate to phrases (which I think I've read somewhere) such as "(scale of) emergence", or "(at which ratios) difference in quantity turns to difference in quality". (I know little about either.)
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awarded  Mortarboard
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comment Are there more distinctive names of “null curves” with certain additional properties (absence of “chord curves”)?
ACuriousMind: "You would have to define "null interval" different from "There is a null geodesic joining the two points" to not make that a tautology." -- Certainly. How do you like "Given two events such that either one of them is 'on the light cone' of the other, do they constitute a 'null interval'?", PSE/q/127859 ? "In Minkowski space, it is certainly true" -- I bet. But where's the rub e.g. with "closed time-like curves" or similar ... errm ... aspects of "the field of "Lorentzian geometry" as we know it today"? ...
2d
comment Given two events such that either one of them is 'on the light cone' of the other, do they constitute a 'null interval'?
Danu: Thanks (+1) for the emphatically unambiguous answer. (To me, that's some relief after having had a brush with some intricacies of "the field of "Lorentzian geometry" as we know it today", PSE/a/127734 ...) Now: what might be the next more complicated but still answerable question? ...))