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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?

 14h comment I know light's speed in vacuum is constant, but what about its velocity? Michael Seifert: "the velocity of a light ray" -- Is "velocity" really attributable to an "entire light ray", or not rather (only) to a particular "piece of it" (e.g. the signal front as "tip of the ray")? "any particle that follows a geodesic in a curved spacetime is, in a very real sense, moving with "constant velocity"" -- Agreed (thus "free motion", from event to event, is made geometric-kinematically comprehensible in general); but light-like geodesics are moreover definitive of "(straight) direction between participants". 14h comment Hafele-Keating revisited with a gravity clock aepryus: "[...] The timing of such a fall would be [...]" -- Here it gets interesting: How, specificly (in terms of a thought-experiment), do you propose to "time a fall"?? Is there also a "top sensor" involved, besides a "bottom sensor"? Are they supposed to be and to remain in some particular geometric relation to each other; and how is such a relation to be measured, or "tuned" as desired? (Btw., eventually such a geometric characterization may also allow to determine, trial by trial, whether a given ball bearing "moved freely" through a "drop chamber", or in how far it did not.) 15h comment Hafele-Keating revisited with a gravity clock aepryus: "[...] the device I have imagined above is almost entirely independent of EM (and other standard model) phenomena." -- "Almost entirely"?? It's perfectly justified to worry whether any "device" (ball bearings, Cs133 atoms etc.) is being disturbed electromagnetically, or weakly, or strongly, or due to what's not even considered in the SM; throughout each trial. Your only decisive idea is about that the "tuning" (before each trial, and certainly also troughout each trial). That is: to consider and select only such devices and such trials as "valid" for which ... what exactly?? 22h comment Hafele-Keating revisited with a gravity clock aepryus: "gravity clock [...] Such an apparatus could be tuned such that each cycle took exactly one second to occur." -- We may think of all sorts of pendulum clocks, e.g. with pendulum "sizes" varying in any ways imaginable, and either being "left swinging passively" or "being jiggled actively"; and among them those being selected which maintain constant "cycle periods" (each itself) and equal "cycle periods" (pairwise between separated clocks), as measured by the Marzke-Wheeler method, throughout each and any trial. (Which pretty much determines anything else you've been asking.) 23h comment Why are neutrino and antineutrino cross sections different? Paganini: "I've added few lines of explanations that might help." -- They do; thanks, +1. "thanks for the typo" -- Well, looks like I should still lay hand on it myself. Also, there's still an entire sentence left which I find difficult to grasp as it stands: "[...] So necessarily this configuration cannot be possible explaining the null cross-section at this angle!" -- Is this perhaps supposed to mean (rather, as far as I understand the argument described): "[...] So this explains that the cross-section of this process at this angle ($\theta = \pi$) is null."? Jun28 comment Is it possible for two events happen at the exact same time? @Gennaro Tedesco: "[...] measure [...]" -- By Einstein description: The indication of "piece of embankment A" being hit by lightning, and the of "B" being hit by lightning may (or may not) be determined having been simultaneous to each other. (Where A and B shall remain separate and at rest to each other.) "times to be the same" -- No: Either way these are distinct indications; not the same. (Don't confuse "times" with "time coordinate values".) Jun27 comment Does “$\mathbf r[~t~]$” correctly denote the trajectory of a point particle wrt. a reference system w/o having to decide on a particular origin? Jun27 comment Does “$\mathbf r[~t~]$” correctly denote the trajectory of a point particle wrt. a reference system w/o having to decide on a particular origin? @CuriousOne: "You seem to have missed my answer completely: generalized coordinates." -- Perhaps you haven't quite noticed my initial reaction to your initial suggestion. There's certainly no mentioning of any coordinates in my OP. If you believe you could answer my question (read it over once again, to make sure at least for yourself) then please consider submitting a regular answer. "You are a grownup who can grab a textbook and start reading, aren't you?" -- We grownups write (textbooks, or by any means available) on subjects we don't find sufficiently covered in the TBs we read. Jun27 comment Does “$\mathbf r[~t~]$” correctly denote the trajectory of a point particle wrt. a reference system w/o having to decide on a particular origin? @CuriousOne: I strongly disagree with all parts of your assessment. (And yes: I participate especially in PSE to make my disagreement with such assessments public and archived.) However, they all seem tangential to my question here. If you have specific comments or questions to other contributions of mine, I'd appreciate it if you submit them there directly. Jun27 comment Does “$\mathbf r[~t~]$” correctly denote the trajectory of a point particle wrt. a reference system w/o having to decide on a particular origin? @CuriousOne: "Kinematics and dynamics are separated in high school physics to simplify the entry for students" -- Excellent. "but there is no obvious professional interest in separating them" -- Of course there is, and specifically in that order. Because (plainly) one shouldn't speak about "$\frac{d}{dx}|_{x_a}$" etc. without being sure how to evaluate "$x - x_a$" (in the denominator); and (fundamentally) because: "All our well-substantiated space-time propositions amount to the determination of space-time coincidences [such as] encounters between two or more [...] material points.". Jun27 comment Does “$\mathbf r[~t~]$” correctly denote the trajectory of a point particle wrt. a reference system w/o having to decide on a particular origin? CuriousOne: "[...] the chosen coordinates [...]" -- What?! Coordinates?? ... "Did you look into generalized coordinates?" -- Well, as far as I did, "generalized coordinates" seemed to presume and rely on notions of dynamics. (Which I hesitate to consider before having a grasp on geometry and kinematics. Therefore my question is specificly about geometry and kinematics.) Jun27 comment Is magnitude of velocity same as speed? @Tim Krul: "what does double modulus mean?" -- That's the notation for "norm". (Having seen this in diracpaul's answer I find it more appropriate than "modulus, or absolute value".) "what does square bracket signify?" -- In square brackets I usually enclose arguments to functions, or operators; here e.g. the argument to which the differential operator is applied. (That's "Mathematica style"; reserving parentheses for grouping only). Jun27 comment Is magnitude of velocity same as speed? Tim Krul: "what do you people mean by two moduli ?" -- The double bars which appeared in @diracpaul's answer and which I subsequently used as well (in comments and in editing my answer) is the notation for "norm". I think that's more appropriate for denoting a distance value (which in geometry/physics usually has some dimension: "length") than using only single bars which denote "modulus, or absolute value" (which is just some non-negative real number). Jun27 comment Is magnitude of velocity same as speed? Tim Krul: "Is magnitude of velocity same as speed?" -- If by "speed" you specificly mean the left term of your equation then apparently you don't mean to distinguish between "speed on the journey before reaching" e.g. some particular "origin" of the description, and "speed afterwards". However, the right term of your present equation, $$\frac{d}{dt}[\| ~\vec r \|~],$$ makes this distinction; and some answers make a point of that. So consider asking instead about $$\left\lvert \left\lvert \frac{d}{dt}[~\vec r~] \right\rvert \right\rvert =?\!\!= \frac{d}{d|t|}[~\|\vec r\|~].$$ Jun27 comment Is magnitude of velocity same as speed? yuggib: I've just suggested to the OP to ask instead about $$\left\lvert \left\lvert \frac{d}{dt}[~\vec r~] \right\rvert \right\rvert =?\!\!= \frac{d}{d|t|}[~\|\vec r\|~].$$ (I wonder if that's going to happen. Or which title I would choose if I'd try to ask this question myself, eventually ...) Jun26 comment Is magnitude of velocity same as speed? 2 (Re: eq. (01)) diracpaul: "In (01) we differentiate the first equation." -- Ah, thanks, that helps (me trying to follow your argument). So: How exactly would you argue in the particular case that $$\mathbf r \mapsto \mathbf{r_0},$$ where $$\|\mathbf{r_0}\|=0$$ ?? (Btw., this seems more or less the "subtlety" which I was concerned about in much of my own answer to the OP.) p.s. "$y=x^2$ by differentiation yields $dy=2~x~dx$" -- I find it reassuring, and even necessary to know in the first place, that $$2~x=\text{lim}_{\{\Delta x\rightarrow 0\}}[\frac{(x+\Delta x)^2-x^2}{\Delta x}].$$ Jun26 comment Is magnitude of velocity same as speed? 1 (Re: the Figure) diracpaul: "What are then $dy,dx$ ???" -- A pertinent question, deserving of a specific answer. But what I question specificly goes moreover to the consequence: Is it therefore appropriate to use "$d$" (rather than "$\Delta$") for labelling elements of your Figure, as it presently stands?!? "Sorry, but I don't know how to create editable sketches with MathJax" -- And I'm sorry, too: I don't know much more about that than to google "MathJax" and "pstricks". But consider the possibilities ... Jun26 comment Is magnitude of velocity same as speed? diracpaul: Concerning the two equations (which had been added after I made my first comment), I'm struggling already to follow the equations and implications with tag "$(01)$". Is there perhaps a "$\lt$" sign missing somewhere? And is there perhaps a version of these equations and implications in terms of "differences $\Delta$" instead of "infinitesimals $d$", to provide at least some plausibility? Jun26 comment Is magnitude of velocity same as speed? p.s. My request for making the given sketch editable is also driven by the desire to easily and unambiguously reference its salient "graphic" features (with even better resolution than to note the "color of the dots", because there seem to be several "dots" of the same color in the sketch as presently provided). Especially: How would you call the blue "dot" by which the two red arrows are connected (and two of the black arrows happen to be connected, too)? And (perhaps more importantly): why did you draw these two red arrows at all? Jun26 comment Is magnitude of velocity same as speed? diracpaul: "The two positions of a moving particle in Figure are at a differential distance apart" -- Hmm ... A "differential (infinitesimal) distance", $d$, i.e. that there is no way to measure "it"? Or instead a "differential distance", $\Delta$, which is still in the (physical) realm of "the measurable", characterizing a geometric relation between distinguishable "ends", commensurate to other (incl. "large") distance values? (Then we may consider ratios, and take limits.)
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