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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?


Jan
30
comment What does the statement “the laws of physics are invariant” mean?
Stan Shunpike: "I might know what an inertial frame is depending on the context." -- Fair enough. But then you ought to make damn sure you know how to recognize and to communicate the context which you want (and which you want anyone else, too) to consider.
Jan
30
comment Why do things slow down when you move faster, rather than speed up?
WetSavannaAnimal aka Rod Vance: p.s. Well, since that what I've been specificly asking about (i.e. including the inequality above), and what you've more or less been getting at, too, is apparently not yet listed in the Wikipedia table of available generalizations of the "metric space" notion, may I suggest (after a quick check with Google, and with a nod to J. L. Synge): "chrono-metric space".
Jan
30
answered What does the statement “the laws of physics are invariant” mean?
Jan
30
comment Why do things slow down when you move faster, rather than speed up?
WetSavannaAnimal aka Rod Vance: "it grates a bit at first to see a "metric" stripped" -- Yes, this exactly reflects my experience and present state of understanding: Looking at the "Wikipedia table of available generalizations" and missing the appropriate entry "xyz-metric space". "Well they are all billinear functionals of a pair of members of the set or billinear forms" -- I'm still skeptical about the hint of "linear" in the name ... en.wikipedia.org/wiki/Degenerate_form -- Cool. News to me. Thx.
Jan
29
comment Why do things slow down when you move faster, rather than speed up?
WetSavannaAnimal aka Rod Vance:"[...] Minkowski "metric". Its name "metric" is a bit misleading [...]: it does not fulfil [...]" -- non-negativity, pos. definiteness (identity of indiscernibles), and subadditivity (triangle inequality). Right. So: How do mathematicians call such general "functions of pairs of a set" in the literature ??; without requiring "smoothness", or "flatness", but possibly: $$\forall x,y,z:0\gt s^2[~x,y~]\ge s^2[~y,z~]\gt s^2[~x,z~]\implies$$ $$\sqrt{-s^2[~x,y~]}+\sqrt{-s^2[~y,z~]}\le\sqrt{-s^2[~x,z~]}$$.
Jan
29
comment Are some events simultaneous in all reference frames? (Einstein goes drinkin')
WillO: "The collision is a single event. It makes no sense to ask whether that event is "simultaneous"." -- That's correct and an important insight; +1. Other aspects of your answer I find less agreeable, cmp. my own answer here.
Jan
29
answered Are some events simultaneous in all reference frames? (Einstein goes drinkin')
Jan
29
comment Basic Relativistic Question - length measurement
Clever: I agree with your result value, for the exact OP problem statement: $$\text{Jack (frame U') flew over a house ... It } {\mathbf{ \text{ took him } } 100~\text{ns}.}$$ However, the problem could be expressed even more consistently from the perspective of Jack (frame U') as: $$\text{A house flew by (underneath) Jack (frame U') ... It }{\mathbf{\text{ took him }}100~\text{ns}.}$$ Or, interestingly, "the opposite problem" could stated just as consistently: $$\text{Jack flew over a house ... It } {\mathbf{\text{ took the house }}} 167~\text{ns to have Jack fly from one end to the other.}$$
Jan
28
comment Basic Relativistic Question - length measurement
Clever: "Unfortunately I don't know how to draw here in SE.." -- That's my complaint as well. Until mathjax is up to allowing us to draw here in PSE I practise with the "LaTeX Previewer by Troy Henderson".
Jan
27
answered Times at relativistic speeds
Jan
26
answered Does coordinate time have physical meaning?
Dec
7
revised Consistent answers in special relativity
(v3.141592: minor copy-editing.)
Dec
7
answered Consistent answers in special relativity
Dec
6
comment Proof in physics
George Smyridis: If you follow the suggestion by @ACuriousMind of reading the Wikipedia article on the Scientific method you may note especially that 1: there is (presently) no (explicit) mentioning of "proving theories" or "falsifying theories" or "testing theories", but rather: hypotheses. But 2: It's still claimed there quite explicitly that definitions (as part of "characterizations"), too, are subject of experimental tests.
Dec
6
comment Proof in physics
anna v: "premises [...] for example in classical physics all variables identified with measurable quantities are continuous" -- The definitions of (how to measure) "quantities", i.e. the main contents and objective of any physics-related theory, include the definitions of their range from the outset. Any "new datum/observation" about some defined quantity cannot falsify its defined range. (The reason to reject/replace "classical physics" is rather its, in hindsight, glaring lack of any careful definitions of "how to measure").
Dec
5
comment Formal definition of an observer?
John O: "two ideas that need to be cleanly separated [...] a grid of sensors, and a single sensor" -- Correct: these are two separate notions. Now, in order to separate them cleanly suitable distinct verbiage (terminology) must be available and used consistently. The key phrase here is "grid of" (or "system") which implies and refers to several distinct elements, possibly in some particular relation(s) between each other. Consequently there must be verbiage/terminology to denote such elements, namely as "sensors", or "material points", or: "observers".
Dec
4
comment Proof in physics
anna v: I appreciate your being responsive. Now: "If [...] a new datum/observation falsifies not only the model but a basic premise/postulate of the theory on which the model is based, [...]" -- Yes, this clarifies your answer: it's now more clearly wrong! Because: "a premise is an assumption that something is true.". Therefore any premise belongs to a particular (falsifiable) model, while the applicable theory is (only) concerned with defining the applicable "something" in the first place, regardless of whether that's then held "true", or "false".
Dec
3
comment Proof in physics
anna v: Your answer carefully describes and distinguishes the notions "theory" vs. "model"; therefore I find it largely very agreeable. But there's one important exception (i.e. one inconsistency in your otherwise great answer): "If the observations and experiments are correctly described by a model based on a theory," -- ... yes ... "and a new datum/observation falsifies the theory, [...]" -- No: if new/additional observations or (derived) measured values are not correctly described by some particular model then they falsify that model; without affecting the underlying theory.
Dec
3
comment Is operator $\hat{O}_{\alpha}:|\phi,\psi\rangle\mapsto |e^{i~\alpha[\phi,\psi]}~\phi,e^{-i~\alpha[\phi,\psi]}~\psi\rangle$ unitary?
@Sofia: "[...] unitary if its action can be reversed" -- Well, then: does the suggested operator $\hat O_{\alpha}$ satisfy that? For arbitrary "phase functions $\alpha[ \phi, \psi ]$"? Or only for some? Also: note that the operator is defined for a composite state, as I've tried to describe ... (It'd be nice if you could demonstrate and discuss this as an answer. My own focus had rather been on the property of unitary operators to "preserve the inner product".)
Dec
3
comment Is operator $\hat{O}_{\alpha}:|\phi,\psi\rangle\mapsto |e^{i~\alpha[\phi,\psi]}~\phi,e^{-i~\alpha[\phi,\psi]}~\psi\rangle$ unitary?
ACuriousMind: "What is a "state with two explicit components"?" -- Well, my question is motivated by (and aiming to dispute) this answer; especially its EDIT. Correspondingly, the two components would be Bob's positron state (say $|\psi\rangle$) together with some suitable separate "blank" state (say $|\phi\rangle$, for instance the state of some particular proton). Together, formally: $|\phi,\psi\rangle$. "Do you mean an element of the tensor product?" -- You tell me, please ...