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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?


Mar
27
answered Experimentally Verifying a Clock's Accuracy
Mar
20
comment The choice of measurement basis on one half of an entangled state affects the other half. Can this be used to communicate faster than light?
Emilio Pisanty: "A good reference to what I mentioned is Guaranteed violation of a Bell inequality w/o aligned reference frames or calibrated devices. P. Shadbolt et al. ..." -- Thanks again; that does seem a good reference to address the "difficulty" I had tried to point out in my first comment (Mar 2 at 19:37): Namely for Alice and Bob to determine experimentally (rather than merely "by assumption or expectation") that they had indeed shared a "Bell triplet state", or a "singlet", or some particular other state.
Mar
20
comment The choice of measurement basis on one half of an entangled state affects the other half. Can this be used to communicate faster than light?
Emilio Pisanty: Thanks for the suggestions; I'm looking forward to the reference you indicated. "Why don't you post it as a separate question?" -- Depends what to make of "it" in a proper manner (free of coordinates or whatever may be scribbled on cargo boxes). I'd ask: "Is it correct that Alice and Bob determine "orientation angle $\phi$ between their detector-(pair)-settings" as $\text{ArcCos}[(N_{\alpha \, \beta} + N_{\overline{\alpha} \, \overline{\beta}} - N_{\alpha \, \overline{\beta}} - N_{\overline{\alpha} \, \beta}) / (N_{\text{all trials}})]$?" If this hasn't been asked already
Mar
19
comment The choice of measurement basis on one half of an entangled state affects the other half. Can this be used to communicate faster than light?
Emilio Pisanty: "[...] under the assumption that the cargo manifest is correct [...]" -- You had surmised already (Mar 10 at 13:01) that "[What] you're worried about should be seen as included in the 'cargo manifest' ..." I'm really doubtful (and I might have objected earlier to that). Because what am I worried about (or interested in) is rather, how some relation ought to be established at all between e.g. what Alice was supposed to refer to as "$z$ direction" and what Bob was supposed to refer to as "$z$ direction". There may be another opportunity to discuss this in more detail.
Mar
19
asked Are signal fronts in a beam not at rest to each other?
Mar
19
comment Are events in this experiment simultaneous if observed in platform's frame?
Anupam: "Are events in this experiment simultaneous if observed in platform's frame?" -- Such a formulation is awfully close to being improper. (Such as talk of "lifetime of a muon in the frame of some racetrack" instead properly of invariant "lifetime of a muon"; or "length of a train in the frame of the embankment" instead properly of invariant "length of the train" as well as invariant "distances between railway ties"; or "mass of a proton in the frame of some accelerator" instead properly of invariant "mass of a proton" ...) Unfortunately, your formulation is indeed often used.
Mar
19
comment The choice of measurement basis on one half of an entangled state affects the other half. Can this be used to communicate faster than light?
Emilio Pisanty: "Your final statement is incorrect, the resource state is the initial state" -- Thanks for the clarification. I was mislead by your above statement (Mar 10 at 20:25: "Alice and Bob can also attempt to determine the purity of their resource state") because that determination is to be made "at the source"; by participants other than "Alice and Bob". Similarly misleading seems the formulation in your answer that "they share a Bell triplet state". Instead, Alice and Bob actually share the final state (which may or may not turn out to be a "Bell triplet state", too).
Mar
12
comment The choice of measurement basis on one half of an entangled state affects the other half. Can this be used to communicate faster than light?
Emilio Pisanty: "All parties in the protocol are assumed to have full information about the protocol itself; this includes knowledge of the initial state of the entangled (resource) qubits, [as] part of their cargo manifest." -- Agreed. However, the "final state" (upon detection by Alice, and by Bob, if at all) is not included. (Btw.: the answer above doesn't seem to draw a distinction whether the "shared Bell triplet state $| \Psi \rangle$" refers to "initial" or "final".) "Alice and Bob can also attempt to determine the purity of their resource state" -- I.e. of the "final state".
Mar
10
comment The choice of measurement basis on one half of an entangled state affects the other half. Can this be used to communicate faster than light?
Emilio Pisanty: "The usual practice is for all parties in the protocol to have full information about what is going on [...]." -- "What's going on" surely includes the observations (or detector readouts) of "Alice" and of "Bob" in correlation to each other. Especially, the determination whether (or to which "purity") the sample of electron pairs which were received by Alice or by Bob had been "entangled" with each other is being evaluated from the correlations between Alice's and Bob's individual results. But such correlations are not "part of the cargo manifest" of either electron.
Mar
10
comment “String” infinity paradox
(contd. (sorry for the delay)) @anna v: "You are thinking of "phenomenological models"" -- Fair enough, in default of any other more appropriate entry in en.wikipedia.org/wiki/Model "For example Thermodynamics: it has laws that are falsified at the microscopic level, i.e. it does not apply there." -- Are the notions in terms of which those "laws of TD" are expressed even defined "at the microscopic level"? Can the corresponding quantities be evaluated at all, given such data? Are "microscopic states" eigenstates of the corresponding measurement operators? ...
Mar
8
comment “String” infinity paradox
@anna v: "In my books all "physics theories" can be falsified by data." -- In my books all "physical models" can be falsified based on observational data; but not "by (abstract or raw) data" as such, but instead by measured values derived from collected raw data. Models which haven't been falsified yet are often called "standard models" (of particle physics, of cosmology, etc.). But the definitions of quantities to be measured, i.e. of how to measure (in principle, or otherwise for assessing systematic uncertainties) are not falsifiable based on data. (to be contd.)
Mar
7
comment “String” infinity paradox
@anna v: "Physics theories are mathematical models. [...]" -- Theories are systems of axioms/postulates/definitions and the sentences/theorems which can consequently be expressed. For theories of physics, the definitions are concerned especially with quantities to be measured; and thus with the value range of measurement operators. In contrast, models of physics summarize specific values which were actually obtained, and/or which are expected. Importantly: models can be experimentally falsified (or corroborated), while theories can not.
Mar
7
answered Naive visualization of space-time curvature
Mar
6
comment Theoretically if you passed the speed of light, would there be a sonic boom equivalent?
@dmckee: "Nothing theoretical about Cerenkov radiation" -- There is plenty theoretical about Cherenkov radiation; as may be recognized for instance in the Wikipedia article about the Frank-Tamm formula (which is named for Russian physicists Ilya Frank and Igor Tamm who developed the theory of the Cherenkov effect). However: of course there's nothing hypothetical anymore about Cherenkov radiation.
Mar
5
comment Theoretically if you passed the speed of light, would there be a sonic boom equivalent?
@Nick: "[...] if it were possible [...]" -- As suggested by the reference(s) to Cherenkov radiation an "object" (such as a charged particle) possibly was "in a medium" (with refractive index different from 1), rather than "in vacuum"; and it may even be said to have "travelled in a medium with some particular speed ($v_p$)." $\therefore$: it might be worthwhile to ask, how to measure whether "a given object had been in a medium", or not; and if so, how to determine "its" $v_p$
Mar
5
comment Did a racer on a “restless” course race at some particular average velocity? (2: Starting gate and finish line being rigid to each other)
A related but different question was physics.stackexchange.com/questions/101802/… Similar to the description given in a comment there, here's a short, "non-mathematical" version corresponding to the question asked above: Starting gate $A$ and finish line $B$ are rigid (but not at rest) each other. Racer $R$ started from $A$ and arrived at $B$. How is "average velocity of racer $R$ with respect to race course $AB$" defined, in a precise sense, if at all?
Mar
5
asked Did a racer on a “restless” course race at some particular average velocity? (2: Starting gate and finish line being rigid to each other)
Mar
4
comment Did a racer on a “restless” course race at some particular average velocity? (1: Starting blocks, finish line moving uniformly wrt. each o.)
Brandon Enright: "This is really convoluted. I'm really not sure what you're getting at." -- Well: The "less-than-600-characters-version-without-any-math" goes like this: Starting blocks $A$ and finish line $B$ are moving uniformly away from each other. Racer $R$ started from $A$ and arrived at $B$. How is "average velocity of racer $R$ with respect to race course $AB$" defined, in a precise sense, if at all?. My question is expressed so the required "math" is provided in a suitably precise sense, too; I'm afraid I couldn't leave that out ...
Mar
4
revised Did a racer on a “restless” course race at some particular average velocity? (1: Starting blocks, finish line moving uniformly wrt. each o.)
(v3.14159: changed "starting block" to the correct "starting blocks" in the title. Thereby, subsequently, being forced to limit the title to 150 characters.))
Mar
4
awarded  Custodian