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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?


Jul
21
revised Clebsch-Gordan Coefficients for two spin-1 particles - Why is there a ∣0⟩∣0⟩ ket?
corrected spelling the name of P. Gordan; cmp. http://en.wikipedia.org/wiki/Clebsch%E2%80%93Gordan_coefficients
Jul
21
suggested suggested edit on Shouldn't the addition of angular momentum be commutative?
Jul
21
suggested suggested edit on Labelling representations using isospin and hypercharge
Jul
21
suggested suggested edit on Triangle inequality Clebsch-Gordan coeffcients
Jul
21
suggested suggested edit on Clebsch-Gordan Coefficients for two spin-1 particles - Why is there a ∣0⟩∣0⟩ ket?
Jul
21
answered Why Light and Observers have different laws of physics
Jul
21
comment Twins Paradox Paradox
queueoverflow: "The perceived time is $\Delta t_T$ [...]" -- ?? "perceived time" ?? Let me specify what I see in your sketch (and you've used "to see" in your answer text yourself, btw.): There are nine "tick marks" drawn on the path of the "Earth twin", but just one on the "outward section" of the traveller (and one more would fit on the "return section"). "with $\beta=4/5$ you can read off $\Delta s_T\approx 1/3~\Delta s_E$" -- Well, with $\beta=4/5$ and $\tau_T = \sqrt{1 - \beta^2}~\tau_E = 3/5~\tau_E$ I'd expect to be shown $\approx$ five "tick marks" on the entire traveller's path
Jul
21
comment Metric tensor in special and general relativity
@ACuriousMind: "It seems to me that you're dissatisfied that I have not given a complete course in (pseudo-)Riemannian [DG ...]" -- Not at all; your intro ("In the beginning ... we have a manifold $\mathcal M$") is as complete as can be expected. I'm dissatisfied because that's not the beginning of physics, nor of (G)TR in particular. (But James Machin first has a course to pass, to earn the leisure of such considerations ...) "path length metric [...] requires [...] geodesics and the exponential map." -- Such overhead seems superfluous for certain curves $\lambda$ with $L[\lambda]=0$.
Jul
21
comment Metric tensor in special and general relativity
@James Machin: "I'm trying to understand how your [ACuriousMind's] definition of the metric [tensor]" -- i.e.$$L[\gamma]:=\int_{\gamma}\mathrm d s$$together with$$\gamma : [a,b]\rightarrow \mathcal M$$and$$L[\gamma]:=\int_a^b \sqrt{g_{jk}~\mathrm d x^j~\mathrm d x^k}$$ "turns into that" -- i.e.$$L[\gamma]=\int_a^b~\mathrm d~t~\gamma'~\lim_{\Gamma \rightarrow \mathcal X} \{\frac{\int_a^{\gamma^{-1}[\mathcal \Gamma]} \mathrm d s-\int_a^{\gamma^{-1}[\mathcal X]} \mathrm d s }{\gamma^{-1}[ \Gamma]-\gamma^{-1}[ \mathcal X ]} \}.$$ -- Me, too. (But that's a bit beyond "exam questions" ...)
Jul
21
revised Metric tensor in special and general relativity
The question is apparently about http://en.wikipedia.org/wiki/Metric_tensor_%28general_relativity%29 and not about http://en.wikipedia.org/wiki/Metric_%28mathematics%29. One more instance of "metric" corrected to "metric tensor".
Jul
21
suggested suggested edit on Metric tensor in special and general relativity
Jul
20
comment Metric tensor in special and general relativity
@ACuriousMind: -- Well, I've certainly expressed the reasons for my dissatisfaction; and a fitting quote about how to do better. Also, I'll wait another day for the OP (James Machin) to edit the question title (at least) before suggesting the corresponding edit(s) myself.
Jul
20
comment Metric tensor in special and general relativity
@ACuriousMind: "if $\gamma$ is the worldline of something travelling at the speed of light, then $g(\gamma'(t),\gamma'(t))=0$ at every point along the path." -- How so? What do you mean by "speed" (i.e. a notion which apparently has not been mentioned in your answer, as it presently stands)? ...
Jul
20
comment Metric tensor in special and general relativity
@ACuriousMind: "[1] [...] an ordinary function between (subsets of) cartesian spaces, [...] has a derivative" -- Surely not every function $$\mathbb R \rightarrow \mathbb R^n$$ has a derivative. However, perhaps functions such as $$(x \circ \gamma) : [a, b] \rightarrow \mathbb R^n,$$ i.e. as considered in the answer, may have additional (stronger) properties which may (or may not) imply the existance of a derivative.
Jul
20
comment Metric tensor in special and general relativity
@ACuriousMind: -- Arguably "the beginning" for deriving the indicated "reasons" is that "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".
Jul
20
comment Metric tensor in special and general relativity
@ACuriousMind: "Let's start at the beginning: The setting for relativity [...] is that spacetime is a manifold $\mathcal M$ [...] that the components of the metric [tensor] are, for chosen basis vectors $\delta_i$ of $T_p \mathcal M$, defined by $g_{ij}=g(\delta_i,\delta_j)$. [...] it can be zero. Vectors for which it is zero are usually called lightlike." -- Is there some reason (having to do with participants, observations, physics) to associate the case of "zero" component values with "light" ? If so, are there corresponding reasons concerning cases of other component values?
Jul
20
comment Metric tensor in special and general relativity
@ACuriousMind: "You're free to suggest that edit" -- I would certainly prefer a decision by the OP (James Machin); along with a corresponding action, as I had requested in my first comment here. "(as is everyone else)" -- (I forgot whether there was some minimum "reputation" score required.) "I think it is unnecessary." -- I think I'll try to research that "on meta", and ask about that, if I find it necessary. "I use "they" as the neutral pronoun in the singular" -- That is curious, to me; but, hey ...
Jul
20
comment Metric tensor in special and general relativity
@ACuriousMind: "It is obvious that in this context, the OP is talking about the metric tensor, as they write [$g_{ab}$]." -- ("They"??) In any case, if it is obvious (to some) that the OP's question(s) concern(s) "metric tensor", rather than "metric space", then I suggest (for the benefit of all others) that the question title should express that explicitly. "However, as the metric tensor naturally induces a metric on the spacetime manifold via the path length [...]" -- You seem to presume "(values of) metric tensor" as given. It's not obvious to me whether the OP does so, too.
Jul
20
comment Metric tensor in special and general relativity
James Machin: "I'm having trouble understanding the metric in general relativity." -- For the benefit of readers who rely on correct, complete and unambiguous terminology being used consistently, would you please explicitly state (and also express in the title of your question) whether you're referring to "metric tensor", or "metric space".
Jul
20
comment Twins Paradox Paradox
queueoverflow: "If you look at a space time diagram of this, [...] When you draw in the planes of simultaneity [...] you can see that the earth time intervals are much longer for the traveler." -- If I read that drawing correctly (please correct me otherwise) the relevant value $\beta$ is $$\beta \approx \frac{4}{5}.$$ Correspondingly I'd expect that the duration of the "travelling twin", from separation of reunion, is $$\approx \sqrt{1 - \left( \frac{4}{5} \right)^2} = \frac{3}{5}$$ of the duration of the "earth twin", from separation of reunion. Can this ratio be seen in your drawing?