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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?


Sep
11
revised Naturalness and experiments
corrected spelling; such as the name of M. Planck; cmp. http://en.wikipedia.org/wiki/Planck_scale
Sep
11
revised ${1 \over T} e^{-i/T}$ for Boltzmann-Gibbs distribution
corrected spelling the names of L. Boltzmann (cmp. http://en.wikipedia.org/wiki/Boltzmann_distribution) and of M. Planck (cmp. http://en.wikipedia.org/wiki/Black-body_radiation#Planck.27s_law_of_black-body_radiation)
Sep
10
suggested suggested edit on Greenhouse gases
Sep
10
suggested suggested edit on Naturalness and experiments
Sep
10
suggested suggested edit on ${1 \over T} e^{-i/T}$ for Boltzmann-Gibbs distribution
Sep
10
revised How do the energy eigenvalues of rotational degrees of freedom in statistical mechanics come about?
corrected spelling; such as the name of M. Planck; cmp. http://en.wikipedia.org/wiki/Planck_constant
Sep
10
suggested suggested edit on How do the energy eigenvalues of rotational degrees of freedom in statistical mechanics come about?
Sep
10
revised What is information?
corrected spelling, such as the name of M. Planck; cmp. http://en.wikipedia.org/wiki/Planck_area
Sep
10
revised Why is the absolute zero a rational number in Celcius?
corrected spelling the name of M. Planck; cmp. http://en.wikipedia.org/wiki/Planck_constant
Sep
10
suggested suggested edit on What is information?
Sep
10
revised Could strings be geons?
corrected spelling the name of M. Planck; cmp. http://en.wikipedia.org/wiki/Planck_length
Sep
10
suggested suggested edit on Why is the absolute zero a rational number in Celcius?
Sep
10
revised Is string length in string theory quantized?
corrected spelling the name of M. Planck; cmp. http://en.wikipedia.org/wiki/Planck_length
Sep
10
suggested suggested edit on Could strings be geons?
Sep
10
suggested suggested edit on Is string length in string theory quantized?
Sep
10
comment How to define a convex surface in case of refraction?
akrasia: "A convex surface is one that bulges out towards the person who is talking about it." -- You've got a (virtual! ;) point there: a cave is usually understood from the perspective of "being (wrapped) inside"; hence it's said to have a "concave surface". Then as the opposite (watching a cave from the outside): a "convex surface". But: Does a surface whose cross section is shaped like the capital letter "B" "bulge out towards" a person (looking at it from the right-hand side), for instance? "we do not live inside glass" -- But, typically: air. Some, part-time, even: water.
Sep
10
revised How to define a convex surface in case of refraction?
(v3.141: Added a note explaining the relation to standard/Wikipedia terminology.)
Sep
10
answered Radioactive decay - What mechanism decides when an unstable nucleus decays?
Sep
10
comment How to define a convex surface in case of refraction?
@akrasia: "talk about using a sledgehammer!" -- Talk about giving a forthright, self-respecting, defensible answer.
Sep
10
answered How to define a convex surface in case of refraction?