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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?


23h
comment Do bad clocks measure proper time?
Moonraker: "In contrast to coordinate time, proper time is independent from any geodesics geometry." -- There was no explicit mentioning of "geodesics" in the OP question statement. But, yes, the events in which the given clock took part can be identified, and interval ratios between those events may be evaluated, without referring to any coordinates. "In order to recover the proper time information, the observer must ensure synchronization of his own clock with the clock of the observed object. -- With any clock "of the observed object", or only with one of its good clocks ??
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accepted Do bad clocks measure proper time?
1d
comment Do bad clocks measure proper time?
Jonathan Gleason: "I am also un-familiar with this five-point curvature detector." -- Right ... (MTW don't seem to mention Synge's "five-point curvature detector" either; which, personally, I find unfortunate. At least, in "Box 13.1: Metric distilled from distances" they exhibit a Cayley-Menger determinant, similar to that shown and used by Synge "Relativity. The general theory", p. 409.) "[...] how to measure the metric. [...] I could only be able to give a "in principle" answer to this" -- Well, I'm certainly interested in that. No better venue than PSE to try and find out.
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comment Do bad clocks measure proper time?
Jonathan Gleason: "I had never even heard the term "bad clock" until I read this question." -- Right ... Isn't it fortunate that MTW legitimized this terminology, at least. "I think it's safe to say that, unless otherwise stated, if someone says "clock" they mean "good clock"." -- Not at all: that may be safe among (some) theoreticians. But among (decent) experimentalists, if someone says "clock" they mean "it's our damn duty to determine, trial by trial, whether it had been good, or by how much it has not been good. So how do we do that again?.".
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comment Do bad clocks measure proper time?
Jonathan Gleason: "By definition, bad clocks do not measure proper time." -- Alright, and that's my understanding, too. Now I'm curious whether those contributors whose claims I quoted in the OP question would agree and retract or correct their claims ... "to explain how one can measure the metric [...]" -- For the present question Synge's "five-point curvature detector" should suffice.
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comment Do bad clocks measure proper time?
Jonathan Gleason: I'd certainly hope for a clear-cut answer: either "Yes, all clocks (including bad clocks) measure proper time.", or "No, bad clocks don't measure proper time (only good clocks do)." "[...] arc-length of this curve (as determined by the space-time metric)." From an experimentalist's point of view I'd have to ask how to determine "space-time metric", trial by trial, in the first place ... But of course a similar objection applies to my sketch of how to define a "good clock" (PSE/a/147905).
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comment Do bad clocks measure proper time?
Kyle Kanos: "What is a "bad" clock?" -- To quote from the reference given in the OP question: "it makes the world lines of free particles through the local region of spacetime look curved (Figure 1.9)".
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asked Do bad clocks measure proper time?
Nov
23
answered Length of Day and year on massive Earth
Nov
21
comment How to theoretically define a concrete operation to perform in order to measure the length of an object?
Ben Crowell: "You might benefit from reading the linked article on operationalism." -- I agree that reading plato.stanford.edu/entries/operationalism is beneficial; and I've done so several times in the past couple of years. (Even versions from before 2009, if my memory serves.) And I'm always puzzled by the "Critiques" and by Bridgman's ultimate defeatism. Which part of "[All our well-substantiated space-time propositions amount to the determination of space-time coincidences]()" did he not understand, not know about, or not willing to suppose as universally shared ability ? ...
Nov
20
comment Is there any limit on movement of space itself?
daniel.sedlacek: "But the inflationary hypothesis says the space itself expanded [...]." -- That's far removed from Einstein's conception: "All our well-substantiated space-time propositions amount to the determination of space-time coincidences {such as} encounters between two or more recognizable material points". Hypotheses within GR must be expressed in terms of identifiable "material points", "participants", "galaxies" etc.; not "space itself" nor "its mathematical abstractions".
Nov
20
answered Is there a rigorous, explicitly geometric, general characterization for whether a given clock had been “good”, or not?
Nov
19
comment Twin paradox where the twins start at different locations
@Maxaon3000: "Shouldn't this cause A to perceive B's clock to be ticking slower?" -- As the diagram shows: during the "drift phase" A (red) perceived/received/observed the signals of two ticks stated by B within each one own tick interval; and vice versa: B (blue) also perceived/received/observed two "ticks of A" for each own one. Also noticable: from the synchronized start of the "drift phase" until the meeting there were 5 ticks by A, but only 4 ticks by B; so B "ran behind". That's how we know and agree on that their tick intervals were equal throughout.
Nov
19
comment Twin paradox where the twins start at different locations
@Maxaon3000: "Thanks!" -- You bet. And thank you in turn. "Just one more question : since B experiences length contraction" ... that's an awfully improper choice of phrase ... "he should be traversing more of A's distance per B's tick than per A's tick." -- Fair enough; and that's what the diagram suggests, too. The important point is: A was and remained a member of the inertial "race track" system whose members could determine/compare distance relations between each other in the first place (represented by the diagram grid). In contrast, B was not. [to be continued]
Nov
19
comment Are there Planck units for weak or strong “charge”, similar to the electromagnetic Planck charge $\sqrt{4~\pi~\epsilon_0~\hbar~c}~$?
Kyle Kanos: "As far as I know, neither the color charge nor the weak isospin have units" -- That's consistent with what I know; and that's certainly convenient. (Perhaps even: a convenient choice.) The only consequence is: the hypothetical "corresponding Planck units" which I'm asking about should also be plain real numbers (of some non-zero value); and the "corresponding permittivities($\epsilon_W$, $\epsilon_S$)" (as far as those are sensible quantities at all) should both have "suitable dimensions". But what would be their values, e.g. in terms of unit $\frac{1}{\text{J m}}$ ?
Nov
19
revised What do physicists mean by “information”?
corrected spelling the name of N. Wiener; cmp. http://en.wikipedia.org/wiki/Paley%E2%80%93Wiener_theorem
Nov
19
revised $L^{1}$ energy-momentum tensors in general relativity; semi-classical gravity
corrected spelling the name of N. Wiener; cmp. http://en.wikipedia.org/wiki/Wiener_process
Nov
19
suggested suggested edit on What do physicists mean by “information”?
Nov
19
suggested suggested edit on $L^{1}$ energy-momentum tensors in general relativity; semi-classical gravity
Nov
19
comment How to theoretically define a concrete operation to perform in order to measure the length of an object?
Ben Crowell: "Length transforms according to [...]" -- This seems to promote an improper notion of "length". (And I doubt that's what the OP had in mind.) "We usually have more than one way of measuring the same thing" -- Yes: different measurement operators may be applied to one-and-the-same given set of observational data. But No: genuinely different measurement operators are genuinely different. "and we need to make sure that these different methods are consistent in the regimes where they are both applicable." -- "Making sure" is a slippery slope in experimental physics.