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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?


6h
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
12h
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 ??
16h
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 $K$.
16h
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.
17h
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".
19h
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.)
19h
revised Are the words “coincident” and “simultaneous” considered synonymous? Else, please explain the difference
(v3.141: dropped the incorrect characterization of grammer)
19h
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?
19h
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" ??)
20h
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).
1d
revised What exactly is meant by saying that two events had been “simultaneous in an inertial frame”?
(v4.0: misspelt identifier corrected)
1d
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.
2d
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.
2d
asked Are the words “coincident” and “simultaneous” considered synonymous? Else, please explain the difference
2d
comment What exactly is meant by saying that two events had been “simultaneous in an inertial frame”?
Emilio Pisanty: "" - Participant $M$ was the middle between $J$ and $K$, in the inertial frame of participants $A$, $B$, $M$." This is not entirely a reasonable statement, though, because [...]" -- Indeed. Consequently the statement "Entire events were simultaneous (in some inertial frame)" seems either similarly "not entirely reasonable" and/or not using the notion "simultaneous" exactly as Einstein defined it ("Relativity", Section 8, linked in the OP question).
2d
comment What exactly is meant by saying that two events had been “simultaneous in an inertial frame”?
Emilio Pisanty: "A complicated and technical way to phrase it" -- +1 for trying to answer as rigorously as I intended to ask. "$\bullet~$ There exists an event [...] which $M$ observes herself to be equidistant and collinear with $J$ and $K$" -- You're attributing "distance" values to pairs ($M$ and $J$, or $M$ and $K$) who aren't even at rest to each other ?? That's improper! You're suggesting that geometric relations (of 3 part.s) might be "observed by" (only) one of them ($M$), at (only) one event ?? $N$, $J$ and $K$ measured their relation! (And $M$ only met/passed $N$ once.)
2d
comment What exactly is meant by saying that two events had been “simultaneous in an inertial frame”?
Emilio Pisanty: "you could say that $M$ passes through the middle of $J$ and $K$" -- Right (a general consenquence in the given example, too): $M$ and $N$ met/passed each other (event $\varepsilon_{MN}$). But does this make $N$ strictly being "placed at the mid-point of the distance AB {... as observer who} perceives the two flashes of lightning at the same time", to use Einstein's exact wording? Does it make $M$ "placed at the mid-point of the distance JK" ??
Apr
15
asked What exactly is meant by saying that two events had been “simultaneous in an inertial frame”?
Apr
13
comment Lorentz contraction in continuously accelerating rod
If the rod is not to stretch not to be compressed, but if its two ends ($A$ and $B$) maintain constant ping durations between each other (which are of course not equal but $\tau A_{BA}$ and $\tau B_{AB}$, respectively) and if $A$ maintains hyperbolic motion with constant acceleration $a$ then $B$ also maintains hyperbolic motion with constant acceleration $b := a~\text{Exp}[ \frac{a}{2~c}~\tau A_{BA} ]$, where vice versa $a := b~/~\text{Exp}[ \frac{b}{2~c}~\tau B_{AB} ]$. Cmp. physics.stackexchange.com/questions/38377/…
Apr
13
answered The nature of measurement