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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?


Nov
14
comment How to identify a “measuring rod”, and how to compare separated “measuring rods” with each other?
... my question may invite descrptions of different notions of "measuring rod"; and I wouldn't necessarily downvote a given answer for providing just one such description while failing to recognize (all) possible others. A goal would rather be to collect and to name such different notions explicitly. (Perhaps I'll add the tag terminology to my question, accordingly.) Still, I'm critical of your answer, not only because you use (heaven forbid! &) coordinates, but especially: you seem to conflate "pairs of ends" with "pairs of events" (in which one or the other end had taken part).
Nov
14
comment How to identify a “measuring rod”, and how to compare separated “measuring rods” with each other?
John Rennie: "And the downvote is because?" -- First, to fend off possible misunderstanding: I did not cast that irresponsive downvote (which is still recognizable by the negative total score of your answer). And I consider a downvote to a given answer a lot more reputable if it's accompanied by a comment, or an own answer, or at least an upvote to another given answer (if there is one present; which here, at the moment, is not). Secondly, I recognize that even in the given context of relativity, my question may invite descrptions of different notions [to be continued]
Nov
14
revised How to identify a “measuring rod”, and how to compare separated “measuring rods” with each other?
(v3.141592: wording corrected)
Nov
14
comment Circumference of a circle in a co-rotating frame of reference
Rok: "I've seen this one [PSE/q/121889] but unfortunately it didn't dispel my confusion" -- Then, surely, there exists a question with a less confusing answer which you have not yet considered. Find it! (My suggestion: How to identify a "measuring rod", and how to compare separated "measuring rods" with each other? (PSE/q/146693).)
Nov
14
asked How to identify a “measuring rod”, and how to compare separated “measuring rods” with each other?
Nov
13
comment Circumference of a circle in a co-rotating frame of reference
Related, and in some sense even a better question: "Understanding the "$\pi$" of a rotating disk" (PSE/q/121889).
Nov
13
comment How to describe arbitrary accelerations in special relativity
I've just retracted a comment I had made earlier, because it had been partially mistaken; sorry for any inconvenience. I'll only note that I'm more familiar with the derivation $$\Delta u \sim \frac{u + a~\Delta\tau}{1 + (u~a~\Delta\tau) / c^2} - u = \frac{(1 - u^2/c^2)~a~\Delta\tau}{1 + (u~a~\Delta\tau) / c^2} \approx (1 - u^2/c^2)~a~\Delta\tau,$$ from which follow the familiar equations of hyperbolic motion in case of constant $a$.
Nov
12
answered How to describe arbitrary accelerations in special relativity
Nov
11
asked Can “uniform motion” (or “mutual rest”) be determined intrinsically, by members of Synge's “five-point curvature detector”?
Nov
7
comment Is $\overline{B}^0$ really the antiparticle of $B^0$?
@ACuriousMind: "@user12262: That's an answer, I think" -- No, I've been merely addressing a wrinkle in the question statement. A suggestion of an answer was instead apparently given by Melquíades above; and I am not at all competent to determine its possible credibility, or even submit one myself.
Nov
7
revised Born approximation to Lippmann-Schwinger integral equation
corrected spelling the name of B. A. Lippmann; cmp. http://en.wikipedia.org/wiki/Lippmann%E2%80%93Schwinger_equation . Also some copy-editing.
Nov
7
revised Lippmann-Schwinger Equation with Outgoing Solutions
corrected spelling the name of B. A. Lippmann; cmp. http://en.wikipedia.org/wiki/Lippmann%E2%80%93Schwinger_equation . Also some copy-editing.
Nov
7
suggested approved edit on Born approximation to Lippmann-Schwinger integral equation
Nov
7
revised Total angular momentum in multielectron atoms
corrected spelling the name of A. Clebsch; cmp. http://en.wikipedia.org/wiki/Clebsch%E2%80%93Gordan_coefficients
Nov
7
suggested approved edit on Lippmann-Schwinger Equation with Outgoing Solutions
Nov
7
suggested approved edit on Total angular momentum in multielectron atoms
Nov
7
comment Is $\overline{B}^0$ really the antiparticle of $B^0$?
Harold: "$\overline B^0$ and $B^0$ have a different mass" -- No: the mass eigenstates of the $\overline B^0$-$B^0$-system which do indeed have different masses are called the light eigenstate $B_L$ and the heavy eigenstate $B_H$; and they are (expressed as) linear combinations of $\overline B^0$ and $B^0$ states. Cmp. the PDG review.
Nov
6
revised Is it theoretically possible to have a universe where sound travels faster than light $c$?
(v3.1415926: remarks on the recent OP "Update", and on possible "improvements". Also: somehow I caught on again on the correct "[tag:<insert_tagname_here>]" syntax.)
Nov
6
comment How to determine “timelike”-ness without using a coordinate system?
Incnis Mrsi: "But in Minkowski’s space partial order relations perform well." -- You may find my related, somewhat generalized question interesting: "Which causal structures are absent from any nice patch of Minkowski space?" (PSE/q/66636) ...
Nov
6
comment How to determine “timelike”-ness without using a coordinate system?
Incnis Mrsi: "relational formalism [...]" -- I didn't know this technical term for what I'm trying to get at (and what I believe Einstein/Robb/EPS/Schelb/Schröter/... were trying to get at); so: thanks for pointing that out. "has some bugs there" -- Quite possibly. Can they be addressed/fixed systematically? "and can’t substitute for differential geometry." -- I'm not asking for substitution; but any comprehensible physical foundation in the first place ...