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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?


2h
revised Is there a name for the linear quantity corresponding to the (quadratic) “interval $\Delta s^2$”?
(v3.141592653: there had been an "end quote" mark missing in the question title.)
2d
revised Does the definition of the SI unit “second” require that possible perturbation of primary frequency standards should be measured?
(v3.141: Removed in appropriate tag "[tag:time]"; restoration of the appropriate tag "[tag:duration]" is pending.)
2d
revised Does the definition of the SI unit “second” require that possible perturbation of primary frequency standards should be measured?
(v3.14: Added two "Notes (p.s.)". Also: (trying to) restore the appropriate tag "[tag:duration]".)
May
21
revised Does the definition of the SI unit “second” require that possible perturbation of primary frequency standards should be measured?
(v3.1: Added excerpts of MTW Boxes 16.4 and 10.2; in response to comments.)
May
20
revised Lorentz Transformations and time of event
(v3.14159265: more minor copy-editing.)
May
12
revised Does the trial index of typical CHSH experiments constitute a “hidden variable”?
(v3.1415: expressions of separate constraints corrected.)
May
7
revised Time taken by a stone to reach the ground is independent of point of dropping
(v4: minor spelling correction.)
May
6
revised What's the differences between time in Physics and time in everyday use?
v3.141: spelling corrected. (This is really a minor edit.))
May
6
revised How can we define a frame of reference in general relativity?
(v3.1415926: **any** applicable timelike congruence)
Apr
23
revised What is the proper time used in relativistic non-equilibrium statistical physics?
(consistent spelling of the title)
Apr
22
revised What exactly is meant by “locating points in spacetime”, in the RT?
(v3.1: font corrected.
Apr
21
revised Naked Time ( Is there such a thing ?)
(spelling)
Apr
21
revised Evolution operator for “blending” a pair of eigenstates
(v3.14159: corrected evolution equation of bra-ket formalism; added note on evolution equation in density matrix formalism.)
Apr
18
revised Are the words “coincident” and “simultaneous” considered synonymous? Else, please explain the difference
(v3.141: dropped the incorrect characterization of grammer)
Apr
17
revised What exactly is meant by saying that two events had been “simultaneous in an inertial frame”?
(v4.0: misspelt identifier corrected)
Apr
8
revised Seeking a coordinate-free expression of the orbital period for uniform circular motion around a non-rotating point-like mass
(v3.14159: EDIT: considering the case that the problem, as it had been stated, was underdetermined. Also some copy-editing. )
Apr
7
revised Seeking a coordinate-free expression of the orbital period for uniform circular motion around a non-rotating point-like mass
rolled back to a previous revision
Apr
7
revised Seeking a coordinate-free expression of the orbital period for uniform circular motion around a non-rotating point-like mass
(v3.14159: Corrected the exponent on dimensional symbol ... again.)
Apr
7
revised Seeking a coordinate-free expression of the orbital period for uniform circular motion around a non-rotating point-like mass
(v3.1415: Corrected the exponent on dimensional symbol.)
Apr
4
revised Angular acceleration of two rods joined in the center - did I do this right?
(v2.7181: corrected expression of moment of inertia of attached mass by adding the necessary "square".)