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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?


Dec
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Nov
19
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corrected spelling the name of N. Wiener; cmp. http://en.wikipedia.org/wiki/Paley%E2%80%93Wiener_theorem
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user12262: a correction in (4), and minor copy-editing
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revised Twin paradox where the twins start at different locations
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Nov
17
revised How, in practice, could instantaneous signalling violate causality?
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Nov
14
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Nov
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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
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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.
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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
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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
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Sep
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revised When has the speed of light been measured, recently?
(v3.1415926: tiny typos corrected.)
Sep
15
revised Can isotropy (or anisotropy) be expressed in terms of intervals ($s^2$) between pairs of events?
(v3.1: corrected question title.)
Sep
15
revised Can isotropy (or anisotropy) be expressed in terms of intervals ($s^2$) between pairs of events?
(v3: named the quantity "s^2"; and some formatting.)
Sep
15
revised Can isotropy (or anisotropy) be expressed in terms of intervals ($s^2$) between pairs of events?
rolled back to a previous revision
Sep
15
revised Can isotropy (or anisotropy) be expressed in terms of intervals ($s^2$) between pairs of events?
(v3: naming the quantity
Sep
12
revised How to define a convex surface in case of refraction?
(v3.1415: corrected my off-the-seat-of-my-pants spelling of "off the seat of my pants".)