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Which geometric relations obtain between two distinct rest systems?

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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 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?

This question was deleted from Physics Stack Exchange and restored from an archive.  

asked Apr 24, 2014 in Experimental Physics by Frank Wappler (0 points) [ revision history ]
edited Apr 24, 2014 by dimension10

The answer to this question can be worked out in a few minutes by writing down the trajectories of all the observers (I didn't do it, but you should do it), voting to close.

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