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 A, B and F who determine the ratios of their (chronogeometric) distances between each other as real number values ABAF, BFAFand ABBF=ABAF/BFAF.
Further let there be another set of participants (of which neither A, nor B, nor F are a member) who measure throughout the experiment having been at rest to each other as well; among them J, K and Q, who determine the ratios of their (chronogeometric) distances between each other as real number values JKJQ, KQJQ, and JKKQ=JKJQ/KQJQ,
such that
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J passed A, then passed B,
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A passed J, then passed K,
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Q passed F, in coincidence with Q and F observing J and A having passed each other,
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B and F determined that B's indication of the passage of J was simultaneous to F's indication of the passage of Q, and
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K and Q determined that K's indication of the passage of A was simultaneous to Q's indication of the passage of F.
Question:
Is thereby guaranteed that for these distance ratios obtains
(1)
ABAF=JKJQ?,
and (moreover)
(2)
((BFAF)2+1−(ABAF)2)((KQJQ)2+1−(JKJQ)2)=4(1−(ABAF)(JKJQ))?
Or otherwise:
What could be concluded if (1) and/or (2) were not found satisfied?
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