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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?


Apr
5
revised Query into the cumulative velocity of mounted platforms
(v3.141: cobbling to stick to my lasts.)
Apr
4
revised The scissor paradox: can we pass the information faster than light?
(v3.14159265: better description of motion in response to signal; which is not perfectly rigid ...)
Apr
4
revised The scissor paradox: can we pass the information faster than light?
(v3.1415926: having proof-read to sloppily ...)
Apr
1
revised Extended Rigid Bodies in Special Relativity
(v3.141592: ...)
Apr
1
revised Extended Rigid Bodies in Special Relativity
(formalities ...)
Mar
29
revised Physical meaning of non-trivial solutions of vacuum Einstein's field equations
(some spelling corrected)
Mar
27
revised Experimentally Verifying a Clock's Accuracy
(v3.1415: Ockham's razor swing-back.)
Mar
4
revised Did a racer on a “restless” course race at some particular average velocity? (1: Starting blocks, finish line moving uniformly wrt. each o.)
(v3.14159: changed "starting block" to the correct "starting blocks" in the title. Thereby, subsequently, being forced to limit the title to 150 characters.))
Mar
2
revised What are the generators of spherical symmetry?
Identified the required at least six characters to correct; arguably legitimately. (No claim of factual accuracy or pertinency implied.)
Mar
2
revised About the complex nature of the wave function?
Identified the required at least six characters to correct legitimately. (No claim of factual accuracy or pertinency.)
Mar
2
revised Relativistic Doppler effect on gamma rays
Identified the required at least six characters to correct formally. Refer to submitted answer for additional factual corrections.
Feb
22
revised Rigorous definition of frame of reference
(v3.14159: some more "mathematical" terminology: in the special case of (Lorentzian) manifolds, a reference frame is also called "(timelike) congruence".)
Feb
18
revised If two ends were a certain “length” apart were they therefore at rest (or at least rigid) to each other?
(v3.141592: Removed the tag `special-relativity`(which I had not applied initially, either). There is no restriction to flat regions intended. Also note section 1.5 of the "SI brochure"; titled "SI units in the framework of general relativity".)
Feb
14
revised If two ends were a certain “length” apart were they therefore at rest (or at least rigid) to each other?
(v:3.14159: seems noone had thought or asked about "length" as a notion meriting definition, and thus tagging, 'round here before.)
Feb
14
revised If two ends were a certain “length” apart were they therefore at rest (or at least rigid) to each other?
(v:3.1415: using the variable "x")
Feb
14
revised Relativity on a moving train
(v3.1415926: Oopsie corrected. (Well -- at least without invoking any coordinates without any foundation for how to determine whether any such coordinates had been "good", or not.))
Feb
12
revised How to determine the order of indications of a clock?
(v3.1415926: one more typo corrected)
Feb
5
revised How to determine the order of indications of a clock?
(v3.141592: another note, in response to a comment by Jim.)
Jan
14
revised Synchronisation of clocks
(v:3.1415926: spelling.)
Jan
14
revised Synchronisation of clocks
(v:3.141592: fixed links.)