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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
1
revised Extended Rigid Bodies in Special Relativity
(v3.141592: ...)
Apr
1
revised Extended Rigid Bodies in Special Relativity
(formalities ...)
Apr
1
suggested suggested edit on Extended Rigid Bodies in Special Relativity
Apr
1
answered Extended Rigid Bodies in Special Relativity
Mar
30
comment Are signal fronts in a beam not at rest to each other?
auxsvr: "[...] because that's how we do measurements" -- Explicate how "we (time-like guys) do measurements" (hint: google.com/#q=%22chronogeometry%22 ) and prove that therefore "null-like objects" are incapable of "doing likewise" (and consequently they're indeed not called "observers"). "the light-cone frame [...] has nothing to do with an inertial frame of the photon" -- Correct (given the justification I keep asking about). Though only looking at coordinates seems inconclusive: e.g. the set "$\{ \forall \theta \in \mathbb R: (\theta, 0, 0, 0) \}$".
Mar
30
comment Are signal fronts in a beam not at rest to each other?
auxsvr: "An inertial frame is the coordinate system of [...]" -- The argument should be made without referring to any coordinates; because otherwise it would just beg the question how such coordinates might have been assigned to the participants (and their individual indications, or even their coincidence events) in the first place. "Wave fronts [...] are the parts of a wave with equal phase" -- Not really; cmp. en.wikipedia.org/wiki/Front_velocity "here exists no inertial frame in which a photon can be at rest" -- Why not e.g. a google.com/search?q="light+cone+frame";?
Mar
30
comment Are signal fronts in a beam not at rest to each other?
auxsvr: "in order for something to be at rest with respect to something else, both of them must be [...] represented by time-like four-vectors." -- However: being (separately) "represented by time-like four-vectors" is apparently not sufficient for two "something"s to be at rest wrt. each other. A satisfactory answer should point out which condition is sufficient for determining "mutual rest", and derive explicitly that such a condition cannot be satisfied by "somethings which are (separately) represented by null-like four-vectors". (contd.)
Mar
29
answered Alice and Bob moving in a circular ring of radius $R$
Mar
29
comment How can photons interact with anything?
anna v: "Here is a diagram that describes what happens when a photon hits your eye." -- As far as that's related to vision, the individual electron in the indicated diagram ought to be replaced by a "retinal molecule", as shown for instance here: google.com/…
Mar
29
awarded  Promoter
Mar
29
comment Are signal fronts in a beam not at rest to each other?
This question has been asked in reaction to the statement "there is no reference frame moving with speed *c* relative to another reference frame" in this answer and, as it turned out, also in anticipation of statements such as "An object moving at $c$ does not have a rest frame" in this more recent answer.
Mar
29
comment Physical meaning of non-trivial solutions of vacuum Einstein's field equations
Robin Ekman: You answer starts out with: "The Newtonian vacuum field equation $\nabla^2 \phi = \rho$ where $\phi$ is the gravitational potential and $\rho$ is proportional to mass density [...]" -- Is this equation as such not more correctly called "the Newtonian field equation"? (p.s. I noticed a typo in the first paragraph, and in the penultimate.)
Mar
29
revised Physical meaning of non-trivial solutions of vacuum Einstein's field equations
(some spelling corrected)
Mar
29
suggested suggested edit on Physical meaning of non-trivial solutions of vacuum Einstein's field equations
Mar
29
comment Experimentally Verifying a Clock's Accuracy
Chris Mueller: "["An Optical Lattice Clock with Accuracy and Stability at the 10^{-18} Level", B.J. Bloom et al., arxiv.org/ftp/arxiv/papers/1309/1309.1137.pdf ...] How did they measure it? [...]" -- They wrote: [N]eutral atom clocks with many ultracold atoms confined in magic-wavelength optical lattices have the potential for much greater precision than ion clocks. This potential has been realized only very recently [...] We have used this measurement precision to systematically evaluate important effects How to relate such precision (good for them!) to accuracy (MTW §16.4) ??
Mar
28
comment Experimentally Verifying a Clock's Accuracy
Chris Mueller: "Identical clocks [...]" -- Clocks which are distinguishable but equal by some (or several) particular measure(s). But by which measure(s), in particular?? "how would we ever establish the accuracy of any clock?" -- How about MTW Box 16.4 ("Ideal Clocks Built from Geodesics")? But surely there's a bit more to it (which makes the OP's question so interesting), such as "Characterizability of Free Motion ...", U. Schelb, FP 30(6):867 (2000) which (unfortunately) still needs/presumes "clock readings $t_A$ differentiable wrt. $\tau_A$".
Mar
28
comment Experimentally Verifying a Clock's Accuracy
Chris Mueller: "How to determine that it has a particular frequency: beat it against another identical one." -- Suppose you had two clocks; distinguishable, but co-located (to avoid other "complications"). Suppose they'd not tick (or, as required, oscillate) "in unison", so there could be some non-zero "beat" at all. Might they then be called "identical"?? "common non-stationarity [...] you have to rely on theory" -- I'm not sure which or how much such "theory" the OP would tolerate; but note the discussion of "ideal clocks" in MTW §16.4
Mar
27
comment Experimentally Verifying a Clock's Accuracy
Chris Mueller: "If the noise of your clock is [...]" -- Please define "noise of a clock" (as a quantity to be measured). "Comparing two oscillators is as simple as summing the two waves. This produces a beat note which tells you the difference in frequency between the two" -- How should be determined whether any one of those oscillators had been oscillating at some particular "frequency" at all, i.e. such that its "oscillation periods" were all equal, and not of varying durations?
Mar
27
revised Experimentally Verifying a Clock's Accuracy
(v3.1415: Ockham's razor swing-back.)
Mar
27
comment Experimentally Verifying a Clock's Accuracy
Kvothe: "[...] compare your clock with other clocks. If yours differs significantly from the average then you know your clock is inaccurate." -- No, that's not a measure of accuracy (or rather trueness) of the one clock in question, but (merely) of its precision wrt. the other clocks that might be available or considered for comparison.