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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?


Aug
31
revised Why is the cross product between two vectors calculated by the determinant of a matrix
(v3.1415926535: Correction of mistake pointed out in comment by firtree.)
Aug
31
comment Why is the cross product between two vectors calculated by the determinant of a matrix
firtree: "The last term in the last formula, [...] seems to be a mistake." -- That's right, thanks! Explicitly: $$ \Big(|\vec{a}\times\vec{b}|\Big)^2:=$$ $$ \left((a_j b_k-a_k b_j)^2+(a_k b_i-a_i b_k)^2+(a_i b_j-a_j b_i)^2\right)~(|\vec{i}|)^4=$$ $$\left((a_i^2+a_j^2+a_k^2)~(b_i^2+b_j^2+b_k^2)\right)~(|\vec{i}|)^4-\left((a_i b_i+a_j b_j+a_k b_k)~(|\vec{i}|)^2\right)^2:=$$ $$\Big(|\vec{a}|\Big)^2~\Big(|\vec{b}|\Big)^2-|\vec{a}|~| \vec{b}|~a_b~b_a;$$ where I've tried hard, of course, not to introduce any notions (such as "angle") which don't appear in the question itself.
Aug
31
comment As the universe expands, the wavelengths of photons are stretched, and energy is lost. What about electrons?
@Jim: "Something seems off about this answer. I reread it and I can't put my finger on it [...]" -- I believe that I can put my finger on something specificly "off" with Rob Jeffries' answer: <br> Rob Jeffries: "Neutrinos are an example of a particle with a non-zero mass. They decouple from the rest of the Universe at about 1 second and freely propagate. The expansion then reduces their momenta [...]" -- Why should a neutrino whose momentum (magnitude) and/or speed decreases in the course of a trial be called "freely propagating"??
Aug
31
revised Why is the cross product between two vectors calculated by the determinant of a matrix
(v3.1415926535: ... small correction ...)
Aug
31
revised Why is the cross product between two vectors calculated by the determinant of a matrix
(v3.14.159265: ... still correcting ...)
Aug
31
answered Why is the cross product between two vectors calculated by the determinant of a matrix
Aug
30
answered “Delayed choice” quantum imaging experiment - why wouldn't it work?
Aug
30
comment Can Allan variance be generalized such that the “Oscillator model” is not presumed?
+1, I certainly recognize some step towards answering my question. "the same reason that the conventional variance is not a good measure for processes that do not, at least approximately, follow a normal distribution." -- Now it'd be helpful to quantify your notions "good" and "approximately". (Unfortunately, I know next to nothing about statistics; yet.) "applications that I had to deal with" -- You might also be qualified and interested in addressing my question "How to express Allan variance without neglecting clock drift" (PSE/q/132970).
Aug
30
comment “Delayed choice” quantum imaging experiment - why wouldn't it work?
Nathaniel: "thanks!" -- You bet! "Without a subscription I can see the figure, but I can only read the first line or so of its caption." -- Right, me neither. And the figure is shown so damn small that I can't quite make out the labelling either. So, we got to be thankful for what Nature put on their webpage; allowing us to figure out the essentials of the experiment. But perhaps someone could kindly point us to a public/arxiv version of the write-up ... "So I guess that pretty much answers my question." -- As I expected. (Seems always the same trivial question in various guises ...)
Aug
29
comment “Delayed choice” quantum imaging experiment - why wouldn't it work?
It should be instructive to look at the "Schematic of the experiment.", i.e. Figure 1 of the corresponding article "Quantum imaging with undetected photons", G. B. Lemos et al., Nature 512, 409-412 (28 August 2014), which is available online at nature.com/nature/journal/v512/n7515/full/nature13586.html Apparently for any one event $A$ in which the object (cardboard cat) is illuminated, and for the corresponding event $B$ in which some screen element is illuminated, event $B$ is either within, or certainly at least on, the forward lightcone of $A$.
Aug
29
comment Difference between Urca process and $\beta$-decay in neutron stars cooling
Py-ser: "Now, the general expression for the Urca process is: $B_1 \rightarrow B_2 + l + \overline\nu$" -- No; but the general expression for the Urca process is (as far as I understand) instead rather: $$\left\{\sum_{k=1}^N B_k\right\}_{\text{energetic}} \rightarrow \left\{\left(\sum_{k=1}^{N-1} B_k\right) + B_{\text{trans}}\right\} + \ell + \overline\nu \rightarrow \left\{\sum_{k = 1}^N B_k\right\}_{\text{relaxed}} + \overline\nu + \nu,$$ or the conjugate, where in the transition step there appears a $\overline\ell$ together with a $\nu$. (The Wikipedia could be improved correspondingly.)
Aug
29
asked Can Allan variance be generalized such that the “Oscillator model” is not presumed?
Aug
29
comment Two clocks along different worldlines
@Jim: "you were continuing apace at your initial level of speech after several subtextual requests to "tone it down"." -- "Subtext" seems kinda hard to quantify; but I certainly wouldn't disagree with your assessment. Now: What about my untiring subtextual requests to "tone it <expletive_conciliatorily_suppressed> up" ?!? ... "[...] a warning" -- Yeah, thanks. (I had looked up "things Socratic", and not for the first time, after user Void's latest comment. &)
Aug
29
comment Two clocks along different worldlines
@Jim: "think you misinterpreted me. [...] you were being too "formal"" -- Not to worry: I think this was exactly the interpretation based on which I wrote my preceding comment. (The word "too" being perfectly unambiguous; and I'm not quite sure about what you meant by "formal". But I like to believe it refers especially to the catch phrase in our other correspondence above following the OP question.) p.s. Apologies for "cluttering Void's and/or user41616's comment space": If only there were some other more appropriate public and archived venue on PSE ... [to be continued]
Aug
28
comment Two clocks along different worldlines
@Jim: "[...] reluctance to appeal succinctly to your co-conversant's expected level of formality" -- I strive to be as "formal" as I can reasonably muster, because public archived correspondence is not just addressing the incidental co-conversant(s) but posterity; and less "formal" terminology may be established, if desirable. "[...] becomes offensive." -- I consider it my courtesy (even obligation, as a physicst) that I do ask for "formal" underpinnings of incidental terminology. That seems less offensive than presuming outright that co-conversants were "not even wrong".
Aug
28
comment Does the definition of the SI unit “second” require that possible perturbation of primary frequency standards should be measured?
p.p.s. I appreciate that you let your answer stand despite my criticism; thus allowing my comments to stand.
Aug
28
revised How to express Allan variance without neglecting clock drift
(v3.14: trying to optimize the appearance on the http://physics.stackexchange.com/questions page ...)
Aug
28
asked How to express Allan variance without neglecting clock drift
Aug
27
comment Two clocks along different worldlines
@Jim: "I agree." -- Splendid. ... Now, considering again the recent answer below and subsequent follow-up questions and responses would it be too much asked for user Void to kindly weigh in? ...
Aug
27
comment Two clocks along different worldlines
@Jim: "Einstein probably recognized his own insight, but he wrote with such effective simplicity in his works that most of almost everyone could understand exactly what he was saying." -- This seems rather questionable; hence the OP's question and related activities, such as our's. But the relevant, grokable subject matter of "(being able to judge) coincidence (vs. sequence)" had surely been communicated effectively as: If the observer perceives the two flashes of lightning at the same time, then ...