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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?


2d
revised What exactly is meant by “locating points in spacetime”, in the RT?
(v3.1: font corrected.
2d
asked What exactly is meant by “locating points in spacetime”, in the RT?
2d
comment Is Einstein's characterization of “time” as “the position of the little hand of my watch” definitive and binding in the RT?
Jimmy360: "Yes, I'm referring to coordinate time." -- Then you've missed the intended gist of my question. (But I'm still looking whether "Is there a difference between time and coordinate time, within RT? (If so: What is it?)" has been asked at PSE already.) "The clock you have will obey your perception of time, for your reference frame." -- It ? Will ?? ... (Not each real-valued parametrization of an ordered set (e.g. of your own indications, or of my own indications) is monotonous to that order; and any two such monotonous parametrizations are not necessarily affine to each other.)
2d
comment Is Einstein's characterization of “time” as “the position of the little hand of my watch” definitive and binding in the RT?
@Earwicker: "I prefer the time term to be [...]" -- At least here you wrote "time term" (corresponding to "sq. space terms"), and not plainly "time". Note that the (temporal) correspondent of "distance" (or generally: of "spatial separation") is correctly/distinctly called "duration". "proper time is basically [the] "interval" between events)" -- By convention the "interval" $s^2[A,B]$ between two time-like related events is the maximum square of the duration of any participant at both events, from his/her/its indication at one event until his/her/its indication at the other.
2d
awarded  Popular Question
Apr
22
comment Is Einstein's characterization of “time” as “the position of the little hand of my watch” definitive and binding in the RT?
John Rennie: "[Please] bear with me ..."-- Even a charitable reading seems to show your attempt, on a physics site, to address a question which is not explicitly about coordinates by conjuring up coordinates. "choose a ruler for measuring distance" -- "Distance" between what, or whom? Even more relevant: "a clock for measuring time" -- "Measuring time" (durations, or at least ratios) between what (of whom)?? (Can you conceive/admit that any "ruler" has identifiable "ends"?; that each "end", at any applicable "point in spacetime" has an identifiable [... cmp. OP title].)
Apr
22
comment Is Einstein's characterization of “time” as “the position of the little hand of my watch” definitive and binding in the RT?
Jimmy360: "time is relative [...] time is different at different points." -- Indeed; distinct "material points", or "principal identifiable points " (MTW, Box 13.1), or plainly "participants" all have distinct individual "positions of little hands" (generally: "indications"). "Therefore, your time is the time on your clock [...]" -- If you're referring to "coordinate time" (i.e. assigning real number values "$t$" to given distinct indications) then: which of my (possible) many different clocks??
Apr
22
comment Is Einstein's characterization of “time” as “the position of the little hand of my watch” definitive and binding in the RT?
p.s. ... v. Laue! Herglotz! ...
Apr
22
comment Is Einstein's characterization of “time” as “the position of the little hand of my watch” definitive and binding in the RT?
@dmckee: "What would "binding" even mean beyond uncritical hero worship?" -- This is meant as Einstein's "time" definition being strictly bound to what's called "Theory of Relativity"; in possible distinction to its various interpreations (by Minkowski, Robb, Pauli, Weyl, Born ...). "perfectly compatible with the notion of proper time as that is what is measured by the observer [...]" -- Fine (consider expanding this into an answer). But keep in mind that any one "position of the little hand" is just one indication; not a duration of that observer/watch between a pair of indications.
Apr
22
asked Is Einstein's characterization of “time” as “the position of the little hand of my watch” definitive and binding in the RT?
Apr
22
comment How does color of galaxies explain their distance?
Kyle: "Galaxies come in a range of colours" -- Presumably this means a range of proper colours, i.e. as determined, case by case, by members of any galaxy under consideration themselves. "but more distant galaxies appear redder than nearby galaxies." -- Therefore more correctly: more distant galaxies appear redder than nearby galaxies of equal proper colour.
Apr
21
comment Schwarzschild metric circular orbits and kepler's 3rd law
+1; my interest in this question is especially to effect another appraisal of answers such as this: PSE/a/174711.
Apr
21
revised Naked Time ( Is there such a thing ?)
(spelling)
Apr
21
answered Naked Time ( Is there such a thing ?)
Apr
21
suggested approved edit on Naked Time ( Is there such a thing ?)
Apr
21
comment Are the words “coincident” and “simultaneous” considered synonymous? Else, please explain the difference
Ernie: ""coincident" may be considered an absolute way of describing an event [...]" -- I appreciate that your edit took accout of my previous comment. "[...] an event that is not simultaneous." -- This characterization I find utterly puzzling ... (I just submitted an answer myself; as I had intended from the start. Perhaps it can further clarify my objection to your use and formulation concerning the word "simultaneous", in Relativity.)
Apr
21
answered Are the words “coincident” and “simultaneous” considered synonymous? Else, please explain the difference
Apr
21
revised Evolution operator for “blending” a pair of eigenstates
(v3.14159: corrected evolution equation of bra-ket formalism; added note on evolution equation in density matrix formalism.)
Apr
21
comment Evolution operator for “blending” a pair of eigenstates
@Mark Mitchison: (I saw your recent comment only after I had started to prepare my reply to By Symmetry; thanks anyways ...) So, for the given problem, you're suggesting not (just) $$\hat H := \frac{-i~\hbar~\hat\sigma_y}{t_{\text{blend}}},$$ but more generally: $$\hat H := \frac{-i~\hbar~\mathbf x\cdot\hat{\mathbf \sigma}}{t_{\text{blend}}}.$$ This does indeed seem to allow more "freedom" when trying to derive an expression for $\hat H[~t~]$ for the corresponding general problem (PSE/q/177107) (where there may even be an explicit $t$ dependence).
Apr
21
comment Evolution operator for “blending” a pair of eigenstates
By Symmetry: "[...] straightforwardly [... with]" $\hat\sigma_y^2=\hat I$ -- Now I realize: $$\text{Exp}[-i~\frac{\pi}{4}~\hat\sigma_y~]=\text{Cos}[~\frac{\pi}{4}~\hat \sigma_y~]-i~\text{Sin}[~\frac{\pi}{4}~\hat\sigma_y~]=\text{Cos}[~ \frac{\pi}{4} ~]~\hat I-i~\text{Sin}[~\frac{\pi}{4}~]~\hat\sigma_y,$$ etc. This makes for the given problem: $$\hat H := \frac{-i~\hbar~\hat \sigma_y}{t_{\text{blend}}}.$$ So, thanks again! Now, having solved this homework, how about trying to tackle the corresponding general problem (PSE/q/177107) ... ?