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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?


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) ... ?
Apr
20
comment Evolution operator for “blending” a pair of eigenstates
@Mark Mitchison: "Just replace $\pi/4$ in the expression for $\hat U$ by anything of the form $\Omega~t$ [with] $\Omega~t_{\text{blend}} = \pi/4$. Can you please prove that: $$\hat U[~t_{\text{blend}}~]=\frac{1}{\sqrt 2}\left(\array{1&-1\cr 1&1}\right) =?= \text{Exp}[~i~\frac{\pi}{4}~\hat \sigma_y~] = \text{Exp}[~i~\frac{\pi}{4}~{\left(\array{0&-i \cr i&0}\right)}~]=\text{Exp}[~\frac{\pi}{4}~\left(\array{ 0 & 1 \cr -1 & 0}\right)~]$$ ? (Meanwhile I ponder this ...)
Apr
20
comment Evolution operator for “blending” a pair of eigenstates
By Symmetry: "OK firstly operators act on kets as vectors [...] $|\psi\rangle \rightarrow \hat U |\psi\rangle$" -- Aw; thanks for clarifying (+1). So, what I wrote at least doesn't work with "bra-ket" notation ... (I may need a while for deciding whether and how to correct/edit my OP regarding this "formality".) "[...] clearest to use matrix notation." -- Great. However: I'm missing some explicit $t$ dependence in your expressions for operator $\hat U[~t~]$. (This still seems a substantive part of my question which you answer didn't address.)
Apr
19
asked Evolution operator for “blending” a pair of eigenstates
Apr
19
comment What exactly is meant by saying that two events had been “simultaneous in an inertial frame”?
@Angelika: "what problem will it bring" -- "Problems" are not mentioned in my OP question; your remark is rather concerned with the motivation for asking. Yes, I consider it a problem if differently defined notions are given the same "overloaded" name (such as speaking of "simultaneous events", event though $M$ is not "the middle between events $\varepsilon_{AJ}$ and $\varepsilon_{BK}$"). I consider a problem what detracts from the details of how to determine "mutual rest" etc..
Apr
18
comment Does the speed of light have a range of speeds due to medium-dependency?
John Duffield: "[...] because of the "spacetime tilt", not the spacetime curvature. That's to do with the tidal force [...]" -- Well, it really doesn't matter how you name the geometric relations to be measured; if only they allow you to conclude the (most probable) distributions of "mass-energy-momentum-stress-charges-fields", so you can optimally derive your expectations on which geometric relations might be found in the next trial; so you minimize your risk of being surprised by some pencils falling (or perhaps even more so: not falling). p.s. Coordinates are as superfluous as cat antlers
Apr
18
comment What exactly is meant by saying that two events had been “simultaneous in an inertial frame”?
@Angelika: "Why not think like this, when $A$ met $J$ they touched each other [...] the particular indications of meetings are same for $A$ and $J$" -- Well, as long as you admit that all participants are and remain distinct, even at meetings/passings (and of course everyone is always meeting/passing some others) we can still speak of "$A$, at the meeting/passing of $J$" vs. "$J$, at the meeting/passing of $A$"; resolving event $\varepsilon_{AJ}$ into indications $A_J$ vs. $J_A$. But if we're not even supposed to distinguish "who honked" vs. "who hollered" at some meeting/passing ... then ??
Apr
18
comment What exactly is meant by saying that two events had been “simultaneous in an inertial frame”?
@Angelika: "In the statements 5 and 6 frames are not mentioned." -- "Frames" are mentioned in already in (1) and (2); e.g. "(1) participants $A$, $B$ [...] were jointly members of an inertial frame"; and $A$ and $B$ are of course explicitly mentioned in (5). (Likewise $J$, $K$ in statements (2) and (6).) "I think its assumed that $M$ can receive indications of $A$ and $B$ only and not of $J$ and $K$" -- Wrong. $M$ observed all that, in coincidence. "or $M$ doesn't "know" about the frame of $J, K, N$" -- $M$ and $N$ even met/passed each other. $M$ is just not "middle between" $J$ and $K$.
Apr
18
comment Does the speed of light have a range of speeds due to medium-dependency?
John Duffield: "Einstein [wrote]: A curvature of rays of light can only take place when the velocity [speed] of propagation of light varies" -- Which illustrates that it's a lot more sensible to consider and evaluate curvature of "spacetime" regions. "current teaching wherein the speed of light is usually taken to mean the locally measured speed of light." -- Current teaching is to express measurements (of "spatial" relations) in terms of the non-zero symbol "$c_0$". "called the coordinate speed of light, which is IMHO most unfortunate." -- Appeal to coordinates is unfortunate, indeed.
Apr
18
comment Does the speed of light have a range of speeds due to medium-dependency?
nick lee: "Does the speed of light have a range of [values ...] ?" -- Do you mean "phase speed", or "group speed"? These can have a wide range of values (i.e. in comparison to "$c_0$") characteristic of "medium properties"; even negative values. Or do you mean "signal front speed"? -- That's just a symbolic non-zero constant, $c_0$, which appears in the (chrono-geometric) definition of (how to evaluate) "distance".
Apr
18
comment Are the words “coincident” and “simultaneous” considered synonymous? Else, please explain the difference
@Duck Joe: "BTW, "coincident" and "simultaneous" are adjectives, not adverbs." -- Thanks!, I already made corrections. (I first went to check that "coincident" is not possibly a noun, either, as I had mistakenly assumed while writing the first version of this question.)
Apr
18
revised Are the words “coincident” and “simultaneous” considered synonymous? Else, please explain the difference
(v3.141: dropped the incorrect characterization of grammer)
Apr
18
comment What exactly is meant by saying that two events had been “simultaneous in an inertial frame”?
@Angelika: "[...] right that $M$ doesn't determine simultaneity in $N$'s frame and vice-versa." -- O.k. So: $~~~~$ $M$ doesn't determine/assert/judge simultaneity of entire events, and neither does $N$, and consequently saying that "entire events were simultaneous, or entire events were dis-simultaneous (in one frame, or the other)" is meaningless or at best an imprecise rendition of the conclusions (5) and (6) stated in the OP. Correct?
Apr
18
comment What exactly is meant by saying that two events had been “simultaneous in an inertial frame”?
@Angelika: "I [am] really not getting your concept of $M$'s or $N$'s "business or privilege to judge"" -- Then how do you interpret the phrasing of Einstein's defintion: "If the observer perceives ... then ..." ?? (Then whose privilege and business is the determination of simultaneity, or of dis-simultaneity, instead, if not that of the mentioned "observer placed in/at the middle" ??)
Apr
18
comment What exactly is meant by saying that two events had been “simultaneous in an inertial frame”?
@Angelika: "could you elaborate the difference between $A$'s indication and $J$'s? [...]" -- Besides calling them distinctive "sparkle" vs. "flicker", and distinguishing them in notation: $A_J$ vs. $J_A$ ? Well, I can make up another, perhaps more impressive example derived from Einstein's original: "The locomotive and I met/passed each other, as the locomotive honked and I got startled.". (It wasn't me who honked; and the locomotive didn't get and look startled).
Apr
17
revised What exactly is meant by saying that two events had been “simultaneous in an inertial frame”?
(v4.0: misspelt identifier corrected)
Apr
17
comment Are the words “coincident” and “simultaneous” considered synonymous? Else, please explain the difference
Ernie: "[...] in the context of relativity, "coincident" may be considered an absolute way of describing two events [...]" -- Well (+1 for effort), but I find this/your formulation incorrect and unacceptable. No: in relativity, "coincident" means pertaining to exactly only one event (several participants "meeting/passing" each other; several observations being made "together, at once"). "reconciles events occurring in more than one inertial frame of reference." -- Even one event bundles/contains/reconciles "occurences" of several participants who aren't at rest to each other.
Apr
16
comment Are the words “coincident” and “simultaneous” considered synonymous? Else, please explain the difference
@santiago: "This probably should be on English.SE" -- Arguably it should. There's a difficulty, though: These two words might be considered and used as good as synonymous when discussing Relativity, while perhaps referring to distinct notions only in some other context(s). I wouldn't quite trust the good contributors at English.SE to understand and appreciate the difference ... (In fact, I wouldn't care much about any context other than Relativity.) p.s. The first example quote actually gives an instance of using "coincidence", rather than "coincident". I'll try to find a more fitting quote.
Apr
16
asked Are the words “coincident” and “simultaneous” considered synonymous? Else, please explain the difference