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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?


Jul
18
comment Does time move slower at the equator?
E. P.: "Each of them gives the proper time on their worldline" -- What do mean by "give(s)"?? Do you perhaps claim that all (atomic) clocks are exactly accurate? Do you understand/agree with how to evaluate [Accuracy and Error of Atomic Clocks (p.se/q/98398) ](physics.stackexchange.com/questions/98398/…) ? "but of course they measure time at different rates - that's the whole point of relativity." -- No: That's awfully improper. Better ask (yourself) properly: "How to compare rates (i.e. "proper" rates) between separated clocks?".
Jul
18
comment Does time move slower at the equator?
Emilio Pisanti: "essentially the same problem as the two identical clocks that differ in altitude by a foot." -- Sure, but: Since "identical clocks" is not the same as "the (very) same clock, in the (very) same trial", do you mean two clocks that are/were equal by some measure, and if so by which measure, exactly? (Related: What's meant by "a foot of altitude", rather than plainly a foot?). [end part 1/3; to be continued]
Jul
18
comment Does time move slower at the equator?
Emilio Pisanty: "I haven't read the proposal paper [ arxiv.org/pdf/1209.2889v1.pdf ] in detail, but it presumably contains an in-depth discussion" -- Well, they (Bondarescu, et al.) seem to uncritically accept claims about "accuracy level in clock rate" from secondary literature. Now, I haven't seen the possibility of "changes of the geoid" and their impact on the determination of "accuracy" (or "systematic error") addressed anywhere else... (But at least MTW § 16.4.)
Jul
18
comment Does time move slower at the equator?
Emilio Pisanti: "Each atomic clock gives (a reasonable enough implementation of) the SI second - wrt. the local time." -- Well, thanks for providing a concrete justification to submit a question about that notion which I find so utterly mysterious: "local". (I plan it along the lines of: "Is a Cs133 atom defined as local?; Are changes of the geoid on scales typical for a Cs133 atom ruled out by definition?"). "comparing two clocks at different locations, whose local time itself runs at different rates" -- That seems a bit improper. (Does "1 Hz" not mean "equal rate, proper"?)
Jul
18
awarded  Quorum
Jul
17
comment Does time move slower at the equator?
Emilio Pisanty: "the geoid can indeed change" -- Should the SI "second" definition consequently refer to a ceasium atom unperturbed by changes of the geoid ? (Cmp. "Does the definition of the SI unit “second” require that possible perturbation of primary frequency standards should be measured?", http://p.se.com/q/123563). "see your atomic clocks as a way of measuring the geoid," -- How should be determined whether (or to which accuracy) a given atomic clock had been "good" before and without having "measured the geoid" already?
Jul
17
answered is there a metatime required for space-time to change
Jul
17
comment Does time move slower at the equator?
Related: Understanding the "$\pi$" of a rotating disk. Also, the present title of this question, "Does time move slower at the equator?" appears utterly improper. Consider instead asking properly "How to compare the (proper) rates of a clock placed at the North Pole to a clock at the Equator (which rotates around the North Pole once a day)?"
Jul
17
comment $\Delta^+$ decay in GZK process
@Chris White: "I also threw in some formatting. [...]" -- Exemplary, IMHO; I like it a lot, FWIW. (Indeed: honestly nothing but formatting thrown in, too...) I recall my frustration when logging in here for the very first time (or was it even right before that) that I was not (yet) permitted to read the source "code" of answers (nor comments); so I felt unable to match some essential ways of formatting. (I'll check if at least community wiki source content is readable without login. (How to search for community wiki, btw. ?))
Jul
17
comment $\Delta^+$ decay in GZK process
@Qmechanic: "That specific branching ratio is also discussed in Chapter 8, eq. (8.11) of these 't Hooft's lecture notes. [ www.phys.uu.nl/~thooft/lectures/lieg07.pdf (p. 46) ]" -- With the same result (eq. 8.11:$$ \Gamma[\Delta^+\rightarrow n~\pi^+] : \Gamma[\Delta^+\rightarrow p~\pi^0] = 1 : 2$$), thank goodness. A bit above (on that same page) 't Hooft refers to two different irreducible representations of the isospin group in default of which the present version of this answer had coined the phrase "state representations to evaluate transition probabilities". Close enough, I suppose ...
Jul
16
revised $\Delta^+$ decay in GZK process
(formatting; hope this helps.)
Jul
16
awarded  Revival
Jul
16
comment $\Delta^+$ decay in GZK process
The initial version of this answer has been formulated based on the comments of user12262 to the OP question above; and in consideration of remarks stated there (meta).
Jul
16
answered $\Delta^+$ decay in GZK process
Jul
16
comment $\Delta^+$ decay in GZK process
p.s. Above I wrote mistakenly "CL coefficients", while I meant "Clebsch-Gordan coefficients; a.k.a. CG coefficients". Note how to spell the name of Paul Gordan, btw.
Jul
16
comment $\Delta^+$ decay in GZK process
jazzwhiz: "I am assuming that you are using the fact that the Delta is a $|3/2,1/2 \rangle$ particle" -- The $\Delta^+$, right. (Also, I didn't point that out explicitly in my above comment since I had been using up pretty much all of the permitted 600 characters already &). "It is a long time since I have done Clebsch-Gordon" -- Same here. ($\approx$ 20 years since I first and last saw the similar problem of calculating $\frac{ \sigma(p \pi^+ \rightarrow p \pi^+)}{\sigma(p \pi^- \rightarrow p \pi^-)}$ etc.). So I'm a bit hesitant to "argue away" the remaining $|1/2,1/2 \rangle$ parts ...
Jul
16
revised $\Delta^+$ decay in GZK process
(added tag [particle-physics].)
Jul
16
suggested suggested edit on $\Delta^+$ decay in GZK process
Jul
16
comment $\Delta^+$ decay in GZK process
Some relevant calculations as comment; I'm not quite sure about the appropriate wording to formulate an entire answer: With suitable isospin CL coefficients$$p+\pi^0\equiv |1/2,1/2\rangle\,|1,0\rangle=\sqrt{2/3}~|3/2,1/2\rangle-\sqrt{1/3}~|1/2,1/2 \rangle,$$and$$n+\pi^+ \equiv|1/2,-1/2\rangle\,|1,1\rangle=\sqrt{1/3}~|3/2,1/2\rangle+\sqrt{2/3}~|1/2,1‌​/2\rangle.$$The branching ratio is thereby $$\frac{\Delta^+\rightarrow p+\pi^0}{\Delta^+\rightarrow n+\pi^+}\approx (\sqrt{2/3}~/\sqrt{1/3})^2=2.$$
Jul
16
comment Why do we say that light travels at a speed?
Derek Roberts: "So what do you want me to say, at the speed of light, time "nullifies"" -- No: duration "nullifies", if you like. "Explain how your equation proves that you've observed a photon" -- I've been referring to a sender and a receiver exchanging a signal between each other. (If any arguments are to be made then about which notions should be considered and admitted for anybody as self-evident.)