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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?


1d
comment Why do some of the dimensionful physical constants take on the values they do?
user12262 wrote: "Related (though not exhaustively covered): "Understanding the “$\pi$” of a rotating disk", PSE/q/121889" -- Oops!, sorry, I had misread the title of this question as asking about "dimension- less constants", of course. Now addressing your actual question instead: I doubt that "value of a dimensionful constant" is a meaningful notion at all, since it depends on arbitrary choices of "units". Physics is done entirely in terms of dimensionless real number ratios.
1d
comment In general relativity, are light-like curves light-like geodesics?
joshphysics: "consider the following curve $$\begin{align} t(\lambda)=\lambda,\qquad x(\lambda)=R\cos(\alpha\lambda),\qquad y(\lambda)=R\sin(\alpha\lambda). \end{align}$$ Let's refer to it as$$\gamma : \mathbb R\rightarrow\cal M.$$ Now, for any two different real numbers $a$ and $b$ is$$ s[ \gamma[ a ], \gamma[ b ] ] = 0~?,$$ where $s$ is (a suitable generalization of) a metric, as induced by the signals exchanged between events which are elements of the region $\cal M$ under consideration. If not, should $\gamma$ really be called "null curve"?
2d
comment In terms of physics, does the phrase “time slows down” mean the same thing as “things happen more slowly?”
user3169700: Adding to the above: in typical "twin problem" setups both protagonists are presumed to conform to some particular equal proper rate; e.g. their individual proper rates of developing liver spots having been and remaining equal throughout their lifes; $\nu_A=\nu_B$. That's why we consider twins, after all. Comparing the numbers of spots over unequal durations, say say $\Delta \tau_A$ (the duration of one twin from separation to reunion) vs. $\Delta \tau_B$ (the corresponding duration of the other twin) then the corresponding increases $\Delta t_A$ vs. $\Delta t_B$ are unequal, too.
2d
comment In general relativity, are light-like curves light-like geodesics?
joshphysics: "consider the following curve $$\begin{align} t(\lambda)=\lambda, \qquad x(\lambda)=R\cos(\alpha\lambda), \qquad y(\lambda)=R\sin(\alpha\lambda). \end{align}$$ -- Along with that let's also consider curve $$\begin{align} t(\lambda)=\lambda, \qquad x(\lambda)=R\cos(-\alpha\lambda), \qquad y(\lambda) = R\sin(-\alpha\lambda). \end{align}$$ Does either or both of these curves correspond to "the propagation trajectory" of a light signal (e.g. a photon)? If so, should the given region, with its parametrization "$\mathbb R^{2,1}$" be called "three-dimensional flat space with [...]"?
2d
comment Why do some of the dimensionful physical constants take on the values they do?
Related (though not exhaustively covered): "Understanding the “$\pi$” of a rotating disk", PSE/q/121889
2d
comment In terms of physics, does the phrase “time slows down” mean the same thing as “things happen more slowly?”
user3169700: "Ok [...]" -- Glad to hear. "but what do you mean by "under the condition of equal (proper) rate?"" -- Given the ordered set $\cal A$ of indications of one participant, and considering one specific ("way of") parametrizing these indications: $$t := \cal A \rightarrow \mathbb R,$$ then the corresponding proper rate (or frequency) is $$\nu := \frac{d}{d\tau}[~t~]$$ (if the derivative exists; otherwise consider "average proper rate"). The parametrization $t$ may for instance be given as "number of ticks", or "number of liver spots", etc.
2d
comment Why Light and Observers have different laws of physics
@ACuriousMind: "I'm voting to close [...]" -- I agree with your vote (FWIW while I'm not (yet) permitted to vote on such matters) because the OP has greatly "moved the target" of the question, and not dealt with the substance of answers provided so far. However: "because it is about nothing well-defined. You keep throwing words around like "the laws of physics", [...]" -- Well, we can hardly blame the OP when he (Derek Roberts) and the larger public is confronted with ill-defined assertions such as "A law of physics is just some set of equations that we use to predict what happens."
2d
answered In terms of physics, does the phrase “time slows down” mean the same thing as “things happen more slowly?”
2d
revised The Euler equations as a RNG fixed point
corrected spelling the name of L. Prandtl; cmp. http://en.wikipedia.org/wiki/Prandtl_number
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suggested suggested edit on The Euler equations as a RNG fixed point
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revised Saturation of the Cauchy-Schwarz Inequality
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2d
suggested suggested edit on Saturation of the Cauchy-Schwarz Inequality
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revised Coupling constant in electroweak theory
corrected spelling the name of S. Weinberg; cmp. http://en.wikipedia.org/wiki/Weinberg_angle
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suggested suggested edit on Coupling constant in electroweak theory
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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.
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suggested suggested edit on Understanding Cherns-Simons-Witten Theory
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suggested suggested edit on Calculus used by Gauss to find Ceres' position?
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revised Why supra-conductivity became super-conductivity?
corrected spelling the name of H. Kamerlingh Onnes; cmp. http://en.wikipedia.org/wiki/Heike_Kamerlingh_Onnes
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suggested suggested edit on Why supra-conductivity became super-conductivity?
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revised Double slit experiment and single particles. Is the wave function just a mathematical model?
spelling: Niels Bohr, etc.