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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?


Jul
25
revised metric-space wiki description
(still proof-reading ...)
Jul
25
revised metric-space wiki excerpt
added 242 characters in body
Jul
25
revised metric-space wiki description
added 1903 characters in body
Jul
25
revised Is there some other name used for “ping rigidity”?
(v3.141: à propos "rigid".)
Jul
25
revised Is there some other name used for “ping rigidity”?
(v3.14: still fixing ...)
Jul
23
revised Are there more distinctive names of “null curves” with certain additional properties (absence of “chord curves”)?
(v3.1415: defined the "cord curves" explicitly as "segments". And specified that the question about "geodesics" is in this case a question about "null geodesics".)
Jul
22
revised Are there more distinctive names of “null curves” with certain additional properties (absence of “chord curves”)?
(v3.141: interchanged the index counts of the coordinate range, to better match the pattern "{ t, x, y }". Also corrected some other inconsequential oversight.)
Jul
22
revised Are there more distinctive names of “null curves” with certain additional properties (absence of “chord curves”)?
(v3.141: put it the missing "t" component ot the example "chord curve".)
Jul
21
revised The Euler equations as a RNG fixed point
corrected spelling the name of L. Prandtl; cmp. http://en.wikipedia.org/wiki/Prandtl_number
Jul
21
revised Saturation of the Cauchy-Schwarz Inequality
corrected spelling the name of H. A. Schwarz, cmp. http://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality; also some copy-editing.
Jul
21
revised Coupling constant in electroweak theory
corrected spelling the name of S. Weinberg; cmp. http://en.wikipedia.org/wiki/Weinberg_angle
Jul
21
revised Calculus used by Gauss to find Ceres' position?
corrected spelling the name of C. F. Gauss, the great mathematician ( http://en.wikipedia.org/wiki/Gauss ), and some copy-editing.
Jul
21
revised Why supra-conductivity became super-conductivity?
corrected spelling the name of H. Kamerlingh Onnes; cmp. http://en.wikipedia.org/wiki/Heike_Kamerlingh_Onnes
Jul
21
revised Double slit experiment and single particles. Is the wave function just a mathematical model?
spelling: Niels Bohr, etc.
Jul
21
revised Energy-momentum tensor of Bosonic Ghost Action in String Theory
corrected spelling the name of H. Grassmann, cmp. http://en.wikipedia.org/wiki/Grassmann_variable; as well as the name of L. D. Faddeev
Jul
21
revised Stabilizer formalism for symmetric spin-states?
corrected spelling the name of W. Heisenberg; cmp. http://en.wikipedia.org/wiki/Heisenberg_picture
Jul
21
revised Labelling representations using isospin and hypercharge
corrected spelling the name of P. Gordan; cmp. http://en.wikipedia.org/wiki/Clebsch%E2%80%93Gordan_coefficients
Jul
21
revised Triangle inequality Clebsch-Gordan coeffcients
corrected spelling the name of P. Gordan; cmp. http://en.wikipedia.org/wiki/Clebsch%E2%80%93Gordan_coefficients
Jul
21
revised Shouldn't the addition of angular momentum be commutative?
corrected spelling the name of P. Gordan; cmp. http://en.wikipedia.org/wiki/Clebsch%E2%80%93Gordan_coefficients
Jul
21
revised Clebsch-Gordan Coefficients for two spin-1 particles - Why is there a ∣0⟩∣0⟩ ket?
corrected spelling the name of P. Gordan; cmp. http://en.wikipedia.org/wiki/Clebsch%E2%80%93Gordan_coefficients