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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?


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Apr
8
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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?
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Apr
3
revised Kramers-Kronig relations for the electron Self-Energy Σ
corrected spelling the name of H. A. Kramers (in the question title; consistent with the spelling already present in the article); cmp. http://en.wikipedia.org/wiki/Kramers%E2%80%93Kronig_relations
Mar
30
revised Why are the proper time and the proper length not defined in the same frame of reference?
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Mar
11
revised What spacelike, timelike and lightlike really mean?
(v3.14159: grammar corrected. (Really only a minor edit.))
Mar
3
revised Questions about “geodesic path” and “arc length of a geodesic path” in the context of GTR and “Lorentzian manifolds”
(v3.141592653: Terminology gathered from (the accepted answer to) PSE/q/167291: "Given the Wikipedia notion of “arc length”, how is its manifestly real “signed variant” to be called and denoted?".)
Feb
26
revised Given the Wikipedia notion of “arc length”, how is its manifestly real “signed variant” to be called and denoted?
(v3.1415926535: Added some documentation of prior research.)
Feb
23
revised Given the Wikipedia notion of “arc length”, how is its manifestly real “signed variant” to be called and denoted?
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Feb
23
revised Deriving (dimensionless) physical constants from theory
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Feb
22
revised Given the Wikipedia notion of “arc length”, how is its manifestly real “signed variant” to be called and denoted?
(v3.14159265: Note on notation)
Feb
20
revised Questions about “geodesic path” and “arc length of a geodesic path” in the context of GTR and “Lorentzian manifolds”
(v3.14159265: tl;dr)