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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?


Feb
22
comment Given the Wikipedia notion of “arc length”, how is its manifestly real “signed variant” to be called and denoted?
@MBN: "What would be the purpose for defining arc length for such curves [i.e. presumably: such as any "curious curve $\gamma_c$" described above] ?" -- Well, foremost, in mathematical physics: If there is no proof (yet) that it cannot be done, then let's try to do it. Some incidental insights or applications might thereby be gained, too: for instance, an argument why such curves cannot be called "straight". (p.s. It seems that you accept that "objects" such as $\gamma_c$ are legitimately being called "curves" to begin with ...)
Feb
22
comment What is Minkowski spacetime?
Uldreth: "Basically everything in theoretical physics are mathematical models and abstractation." -- True. But when discussing "spacetime" the principal abstract(ed) notion is "event", not "real number tuple". "quite confident that the /definition/ of Minkowski spacetime is that of a four dimensional real inner product space with the metric I stated above." -- I claim that the /definition/ of Minkowski spacetime is (at least equivalent to) that of a four dimensional real inner product space with (generalized) metric relations as sketched in my answer. No need to involve coordinates.
Feb
22
comment Bending of space vs. bending of a body - which is larger, and by what factor?
Hans973: "This does not answer the question, does it?" -- Correct. My response is meant to reject your OP question, as not properly and in good conscience answerable; and to sketch how to ask such a question instead, in the given context (namely: "Is there a way to quantify material properties?"). Hope this helps.
Feb
22
comment Given the Wikipedia notion of “arc length”, how is its manifestly real “signed variant” to be called and denoted?
p.s. Ocelo7: "I have never seen that second definition before." -- That was directly addressing my question (or at least part of it); +1.
Feb
22
comment Given the Wikipedia notion of “arc length”, how is its manifestly real “signed variant” to be called and denoted?
Ocelo7: "[...] that $g(\dot\gamma,\dot\gamma)<0$ for a timelike curve in $(-+++)$." -- OK. So how about curves $$\gamma_c : [0, 1] \rightarrow M,$$ (index "$c$ for "curious") for which $$\exists~ a, b \in [0, 1] \text{ such that } g(\dot\gamma_c[~a~],\dot\gamma_c[~a~])<0, \text{ while } g(\dot\gamma_c[~b~],\dot\gamma_c[~b~])>0$$ ?? Are there such "curves $\gamma_c$" at all? ... "What purpose would an overall sign serve?" -- The (resulting) overall sign would seem to serve a similar purpose as any particular "sign convention", such as "$(-+++)$". What exactly is the purpose of that?
Feb
21
comment What is Minkowski spacetime?
@ACuriousMind: "This is not what Minkowski space is in mainstream physics." -- Note that the OPs question was about "Minkowski spacetime", and my attempt was to answer this question. (It may be debated whether the manifold that you indicated is a model of that particular spacetime.) p.s. Thanks for not "drive-by voting"; I noticed your comment only after giving a comment to the answer by Uldreth.
Feb
21
comment What is Minkowski spacetime?
Uldreth: "What we call Minkowski space(time) then, is the pair $(\mathbb R^4, \eta)$, where $\eta$ is the indefinite Minkowski scalar product" -- Accordingly, what you call "Minkowski space(time)" is apparently a (Lorentzian) manifold. However, the OPs question was about "Minkowski spacetime", and spacetime is not a manifold. (At most, the hypothesis stands that, generally, spacetime may be modelled as a manifold; or specificly, that "Minkowski spacetime" may be modelled as the manifold you explained.)
Feb
21
answered What is Minkowski spacetime?
Feb
21
answered What observations would be needed to falsify the law of conservation of energy?
Feb
21
answered Bending of space vs. bending of a body - which is larger, and by what factor?
Feb
20
asked Given the Wikipedia notion of “arc length”, how is its manifestly real “signed variant” to be called and denoted?
Feb
20
revised Questions about “geodesic path” and “arc length of a geodesic path” in the context of GTR and “Lorentzian manifolds”
(v3.14159265: tl;dr)
Feb
20
comment Questions about “geodesic path” and “arc length of a geodesic path” in the context of GTR and “Lorentzian manifolds”
@ACuriousMind: "There're no space-like geodesics because [...]" -- Wikipedia says otherwise, FWIW: "For a space-like geodesic through two events, there are always nearby curves {of} either a longer or a shorter proper length than the geodesic". (And I'm stickin' to it. ;) "smooth functions from an interval onto a (smooth) manifold." -- Then "$[a,b]$" is read as some closed "real line interval"; go on to "$\gamma([0, 1])$" and the important specific events $A$, $B$ are hidden from view
Feb
20
comment Questions about “geodesic path” and “arc length of a geodesic path” in the context of GTR and “Lorentzian manifolds”
But: of course that's rather an indication of my poor grasp at the appropriate terminology; or, possibly, the present lack of any appropriate terminology to be grasped. "it turns (non-trivially) out that the maximizing "causal" geodesics are more relevant." -- Fair enough; but then how to talk about "acausal" (spacelike) geodesics, or best of all how to be "transparent to these issues"?. "Mather theory and other things" -- News to me (+1 ?) ...
Feb
20
comment Questions about “geodesic path” and “arc length of a geodesic path” in the context of GTR and “Lorentzian manifolds”
@ACuriousMind: "1. For f:A->B, the set of points of B which are images of points of A is very commonly denoted im(f) or f(A)." -- True, but my objection is a bit more subtle: Was I correct to interpret the symbol $a$ (or likewise $b$) in your above comment as one real number tuple? If so, what exactly did you mean there by "$[a,b]$"? (Or else? -- go figure.) 2. [...] "minimal length" would be a geodesic that is as closest as possible to a null curve" -- That's a particularly "Sheldon-esque" hang-up. Meteorologists don't seem to have it when reporting temperature extremes [to be contd.]
Feb
20
comment Questions about “geodesic path” and “arc length of a geodesic path” in the context of GTR and “Lorentzian manifolds”
@ACuriousMind: p.s., returning to "2. [...] (sign issues)" -- In the second ("non-random", "interesting") part of the OP I went to some pains not having to spell out a particular "sign", or correspondingly, not to distinguish "minimal" from "maximal". Similarly, perhaps I should not necessarily have to ask about "(a) geodesic(s) of minimal arclength", but rather about "(a) geodesic(s) of extremal arclength (in a/any consistent sense)".
Feb
20
comment Questions about “geodesic path” and “arc length of a geodesic path” in the context of GTR and “Lorentzian manifolds”
@ACuriousMind: "1. The notation used is unusual [...] $\gamma([a,b])$ corresponds to what you consider a geodesic." -- "$\gamma([a,b])$" seems to be a stretch (abuse?) of notation: doesn't function $\gamma$ take only one real-number tuple, e.g. $a$, as argument? I'd compromise:$\mathcal G\equiv\gamma_{\mathcal S}[~A,B~]$. "2. "Min. arc length" is a difficult term" -- Is "geodesic of min. arclength" not definite enough? "3. Why?" -- Not least: physics.stackexchange.com/questions/162170/…
Feb
19
asked Questions about “geodesic path” and “arc length of a geodesic path” in the context of GTR and “Lorentzian manifolds”
Feb
18
comment Why is the cross product between two vectors calculated by the determinant of a matrix
@Stan Shunpike: "What do the braces terms (eg $\{ab\}_i$) stand for?" -- This was just some ad hoc ("seat-of-my-pants") notation for expressing one particular "component coefficient" (real or complex number) of the cross product $\vec a \times \vec b$, referring to the (chosen) basis vector $\vec i$. It was meant to formally express that this "component coefficient" depends on vectors $\vec a$, and $\vec b$, and $\vec i$; without already writing down an explicit expression. Actually, for orthogonal basis vectors: $$\{ab\}_i := (a_j~b_k - a_k~b_j)~\frac{|\vec j|~|\vec k|}{|\vec i|},$$ etc.
Feb
5
answered Causality and Simultaneity in special relativity