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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?


Jun
8
comment Taking selfies while falling, would you be able to notice a horizon before hitting a singularity?
Rob Jeffries: "imagine that the worldlines of face and camera are much closer together. In that case it is feasible to send a signal from your face to the camera and for light from the camera to make the journey back to your face before your face reaches the event horizon." -- Well, I have to take your word on that ... So: are you quite sure about your conclusion? "However, there will come a point where the light cannot make the round trip before your face hits the singularity." -- Of course I'd appreciate a more quantitative description.
Jun
8
comment Taking selfies while falling, would you be able to notice a horizon before hitting a singularity?
@Kyle Kanos: "possible duplicate of How can anything ever fall into a black hole as seen from an outside observer? {PSE/q/21319}" -- My question doesn't specify any of the named participants as "outside"; instead, the person (of whom the selfies are taken and who reviews them as they are displayed) and the camera/smartphone (who takes and displays the pictures of the person), or even several of those, are (merely) supposed "to fall" and to be and remain separate from each other (at least until both have "hit a singularity"). Thus: no duplcate
Jun
8
comment Taking selfies while falling, would you be able to notice a horizon before hitting a singularity?
@CuriousOne: "The only correct answer is: we don't know. [...]" -- ?? Well, I still like to encourage you to expand this into an answer; but I'm certainly strongly looking forward to any answers delving extensively into "light cone structure" and "members of timelike congruences (exchanging pings between each other)"; not least in order to define those otherwise very very mysterious notions "to fall", "horizon" and "to hit a singularity" in the first place.
Jun
8
comment Taking selfies while falling, would you be able to notice a horizon before hitting a singularity?
@Rob Jeffries: "Possible duplicate of [PSE/q/187917]" -- There is no explicit mentioning of "pings" or "signal round trips" between distinct participants in PSE/q/187917; and, perhaps more importantly, not in any answers submitted there either. (Not even the apparent mere absence of pings in the sketch provided with this answer there (PSE/q/187925) has been noted there explicitly ...) Consequently, my question isn't a duplicate of PSE/q/187917; certainly not in terms of which answers are acceptable. (I've added a note on this to my question.)
Jun
8
revised Taking selfies while falling, would you be able to notice a horizon before hitting a singularity?
(v3.141592: Added a note to empasize the notions of "pings"/"signal round trips" in the statement of my question; in notable distinction to the question statement of PSE/q/187917 and any answers there.)
Jun
8
asked Taking selfies while falling, would you be able to notice a horizon before hitting a singularity?
May
31
comment Is there a name for the linear quantity corresponding to the (quadratic) “interval $\Delta s^2$”?
@yuggib: "that my interpretation is far from being established" -- IMHO it's almost past due to establish it (or sth. sufficiently similar). "[...] know if two events are actually happening to one observer one after another" -- The question how to establish "between-ness" is of course interesting. (I had assigned it a while ago as homework.)
May
31
answered Is there a name for the linear quantity corresponding to the (quadratic) “interval $\Delta s^2$”?
May
31
comment Do clocks measure conformal time (new argument)?
John Eastmond: "a model of a clock that consists of a rigid ruler of fixed proper length" -- Does this mean specifically two ends which were and remained bilaterally chronometrically rigid to each other?, i.e. such that the ping durations of either end wrt. the other remained (separately) constant (but not necessarily equal to each other). "[...] co-moving spatial Cartesian co-ordinates." -- Physics is co-ordinate independent.
May
30
comment Is there a name for the linear quantity corresponding to the (quadratic) “interval $\Delta s^2$”?
@yuggib: "[...] the total (relativistic) length." -- Well: if I squint really hard (and you squint as well) and we read your suggestion for the name I asked about rather as "the (linear, relativistic) separation", and if you expand your comment (interpreted this way) into an answer, then I'd be happy to accept it. (And even Wikipedia may eventually articulate it, too.) "[...] the version that takes into account the path taken" -- But even the "interval" $s^2[~\varepsilon_A, \varepsilon_B~]$ is unique, for given events $\varepsilon_A, \varepsilon_B$; so why not $\overline s$ ??
May
30
comment Is there a name for the linear quantity corresponding to the (quadratic) “interval $\Delta s^2$”?
@yuggib: "The curve is the path in ST taken between the two events." -- But which path ?? Or do you mean any path; incl. even paths containing elements (i.e. events) such that some pairs of them are timelike related, and other pairs spacelike? "If you look at the arclength formula, [...]" -- Well, I generally think of "the arclength formula" in this form ... "[...] discrete difference between coordinates is not so useful, since [...]" -- The OP question doesn't mention coordinates. (And yes, coordinates aren't by themselves useful.)
May
30
comment Is there a name for the linear quantity corresponding to the (quadratic) “interval $\Delta s^2$”?
@yuggib: "[...] a "signed" version [...] of the arclength of a curve in Minkowski ST" -- What "curve" ?? The OP question only mentioned discrete events; and only few of those. "a pseudo-Riemannian manifold" -- The section you linked (actually called "Arclength and the line element") mentions "the first fundamental form $ds^2$" and the corresponding "(signed) line element $ds$", btw. However, these don't seem to take two explicit arguments; and therefore have little to do with the OP question.
May
30
revised Is there a name for the linear quantity corresponding to the (quadratic) “interval $\Delta s^2$”?
(v3.141592653: there had been an "end quote" mark missing in the question title.)
May
30
asked Is there a name for the linear quantity corresponding to the (quadratic) “interval $\Delta s^2$”?
May
29
comment Is it possible to speak about changes in a physical constant which is not dimensionless?
"The ruler is essentially a scaled-up H atom" -- If you agree calling (any) one H atom "a ruler" outright (with its unique "Max" and any "Max/e^2" part as two relevant ends/marks) then ... what/how should be "monitored" at all?? "stipulating that the relative size of [distinct] atoms could measurably change" -- In reference to M-W measurement this can at least be contemplated. But less so by "atomic artifact rulers". "I'll be happy for the protocol to fail" -- Which "protocol"? M-W, or "your a/m-ically minded" one? And importantly: What is "failure of a protocol", in your view?
May
29
comment Is it possible to speak about changes in a physical constant which is not dimensionless?
Emilio Pisanty: "[...] skim read" -- Apologies; relevant is: each of the H atoms of your "atomistically/metreologically minded ruler" shall have distinguishable parts "Max" and "Max/e^2". And AFAIU you suggest monitoring whether "Max/e^2 parts of neighboring Hs" were coincident. (In principle, "before each trial"; why not during, too. And what do you mean by a "linear arrangement" of several such parts, btw.?) Now: Do "pieces of equipment" which are thus monitored/seleted therefore remain "proportional" in reference to M-W comparison? (Not even to mention anything "atomic" in general.)
May
28
comment Is it possible to speak about changes in a physical constant which is not dimensionless?
Emilio Pisanty: "follows from the special considerations I took regarding my ruler." -- Such as? Considerations ("diligence") applicable in each individual trial?, or (only) expectations extrapolating previous trials? "not really achievable by a mere change in $\alpha$." -- That's considering just one "known unknown" (dismissing its "being responsible by itself"). But what about any/all "unknown unknowns" (wisely mentioned elsewhere), and even any/all other "known unknowns"?? "(The alternative is that not all hydrogen atoms are alike.)" -- Some being "ends"; some "bulk" etc.
May
28
comment Is it possible to speak about changes in a physical constant which is not dimensionless?
Emilio Pisanty: "if my ruler has "shrunk" then so have I (as I'm made of atoms) and so has every piece of equipment in my lab and elsewhere on Earth." -- How so?? Different artefacts (you, "your ruler", other "pieces of equipment") which have no relation with each other except for all being in turn made/constituated "of atoms" don't necessarily have any particular geometric relations with each other at all. There's surely no expectation that they should remain "proportional" to each other, trial by trial. (But in the exceptional case that they all did you may call them "undisturbed".)
May
28
answered Is it possible to speak about changes in a physical constant which is not dimensionless?
May
27
comment Is it possible to speak about changes in a physical constant which is not dimensionless?
Emilio Pisanty: "I have little to add beyond recommending a thorough reading of Duff and Duff, Okun and Veneziano." -- Having done so (already almost when those arXiv "preprints" had appeared), I'm still and again struck that they don't even seem to consider the solution provided by Einstein (Kretschmann?, Comstock?, Poincaré? ...) in the guise of "point coincidence"; much less endorse and promote it (as I try to do). Now: How to turn this observation into a PSE question? ...