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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
revised Must the product of the two complementary quantities in an uncertainty relation have unit $\text{Js}$?
Expressed the Uncertainty Principle (also) as **in**equality; and corrected spelling the surname of M. Planck.
6h
revised Must the product of the two complementary quantities in an uncertainty relation have unit $\text{Js}$?
corrected an omission (exponent) and minor spelling
6h
comment Two apparent contradictions in SR involving time dilation and length contraction
Madde Anerson: "What if we'd look from an inertial frame with the mean velocity of v(a) and v(b)?" -- O.k. But you still need to decide: Do we compare the rates of these two clocks (which each regularly ticking "good" clock has by itself, properly) to each other? (If so, it won't matter "from" which inertial frame "we look" in particular for accomplishing this comparison.) Or what (else) do you mean by "running more slowly"?? (And: calling two distinct clocks "identical" surely doesn't spare specifying how to compare their rates, especially while they were separated.)
6h
comment Must the product of the two complementary quantities in an uncertainty relation have unit $\text{Js}$?
WetSavannaAnimal aka Rod Vance: Comparing the first and last equation of your answer I wonder whether your notation is consistent, i.e. in terms of the index which only appears in the latter eq. Otherwise +1. (Also, I've done a bit of copy-editing; still struggling hard to plug my assorted reputation leaks ... &)
6h
suggested approved edit on Must the product of the two complementary quantities in an uncertainty relation have unit $\text{Js}$?
6h
suggested approved edit on Must the product of the two complementary quantities in an uncertainty relation have unit $\text{Js}$?
20h
comment Why is the momentum of a particle $\gamma mv$?
@0celo7: "This is a deterministic classical system, not a probabilistic quantum one." -- As far as I understand the OP question it is concerned with attributing and measuring momentum values, e.g. of objects (such as $A$) wrt. to suitable systems (such as $\mathcal S$). There doesn't seem to be any requirement or exclusion being made concerning some categorizazion as "classical or quantum"; whatever you might mean by that. "doesn't $\hbar =0$ [...]?" -- Perhaps you mean cases or limits in which $$\frac{m_A~c^2~\Delta \tau_A}{\hbar} \gg 1 $$ ? ...
20h
revised Accuracy and Error of Atomic Clocks
some corrections and copy-editing
1d
comment Two apparent contradictions in SR involving time dilation and length contraction
@Madde Anerson: "We wish to know which clock runs more slowly." -- If this request is understood as asking which one (of two "ticking" clocks) was ticking at a lower frequency (itself, properly), then that's a perfectly sensible and legitimate request, and SR is perfectly suited to address it consistently. (While other interpretations may lead to inconsistency.) "[...] is experiencing time dilation -- Time dilation refers to a ratio, comparing frequencies (or, foremost, durations) between participants who are not at rest to each other. It's not for only one individually "to experience".
1d
comment Why is the momentum of a particle $\gamma mv$?
@0celo7: $\Delta\tau_A$ -- the duration of object $A$ throughout the trial (as spelled out in the answer). $\Delta\tau_{\mathcal S}$ -- the duration of system $\mathcal S$ throughout the trial (I may add that to my answer ...); $\Delta\mathbf r_{\mathcal S}$ -- the (spatial) separation between the member of system $\mathcal S$ who met/passed object $A$ at the beginning of the trial, and the member of system $\mathcal S$ who met/passed object $A$ at the end of the trial (I may add that, too.) "If this actually pans out, fantastic work!" -- Hmm ... What'ya mean by "panning out"??
1d
comment Why is the momentum of a particle $\gamma mv$?
@0celo7: "where the phases and the $\hbar$ go" -- The "phases" can go into the description of states (see my above comment); and if so, the $\hbar$ goes both into the denominator of the "angle", and into the numerator of the applicable operator; so the $\hbar$ cancels (see my first comment) upon application of the operator to the state description. (So why bother inserting the $\hbar$ symbol at all? ...) "you saying special relativity follows from quantum mechanics?" -- I'd consider QM the general framework for "measurement", and GR/SR specifying particular geometric/kinematic operators.
1d
answered Accuracy and Error of Atomic Clocks
2d
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".
2d
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.)
2d
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??
2d
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.)
2d
revised Why are neutrino and antineutrino cross sections different?
Corrected spelling the surname of N. Cabibbo; together with some minor copy-editing to reach the required change of at least six characters.
2d
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."?
2d
suggested approved edit on Why are neutrino and antineutrino cross sections different?
Jun
29
revised Has anyone tried Michelson-Morley in an accelerated frame?
corrected spelling the last name of A. A. Michelson; cmp. https://en.wikipedia.org/wiki/Michelson%E2%80%93Morley_experiment (NB: also consistent with the spelling used in R. A. Mould, "Basic Relativity".)