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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 Difference between Gunn Peterson trough and the Lyman Alpha Forest? Cosmological implications?
ShanZhengYang: It would have been better form, and of general interest, to explain that, as far as I understand, "IGM" abbreviates "Intergalactic Medium", and "HI" an "H I region is a cloud in the interstellar medium composed of neutral atomic hydrogen (H I) ...
Feb
22
reviewed No Action Needed Which of the following has higher resistance - milliammeter or ammeter?
Feb
22
comment Given the Wikipedia notion of “arc length”, how is its manifestly real “signed variant” to be called and denoted?
@Danu: "we [physicists] care mainly about geodesics" -- That's missing a (my) main point of interest: We physicists care foremost about how to find out whether any given set of events is (the image of) a geodesic, or not. Only subsequently we care mainly about geodesics. (Or actually: we only might; we may, or may not.) Therefore I'd certainly not be happy to (be) migrate(d) away from PSE. (Is there nevertheless a possibiliy to bring my question to the attention of a broader audience which is possibly more knowledgeable of the relevant literature? Sure: vote it up! ;) ...
Feb
22
comment Given the Wikipedia notion of “arc length”, how is its manifestly real “signed variant” to be called and denoted?
@Danu: "As is clear from the animation on top of the wikipedia page [...]" -- Are you suggesting that (the WP page claims that) every "arc" (or "curve") is straight, or geodesic? (Surely not.) Then: Are you suggesting that (the WP page claims that) for each value of "arc length" of a given "arc/curve" it should be possible to imagine a "straight/geodesic curve" of equal "arc length" value? (That's arguably accomplished by "my proposal", while the quoted "WP presentation" clearly fails in general.) Or what else are you suggesting, specificly? ... [to be continued]
Feb
22
comment What is Minkowski spacetime?
Uldreth: Thanks for your amicable comment. I rather believe in giving answers as straight and as defensible as I know; and generally attending to possible follow-up questions concerning any part of my answers. (See what that did to my "reputation"? ;). Likewise, the OP would have been smart to work out and report with which part(s) of the Wikipedia article he ("The ... guy") struggled. (It broadly reminds me on your answer, btw. There is practically no mentioning of "distance", and none of "duration" at all, but OTOH indeed quite a bit on "events". Clearly: "It has multiple isses" ...)
Feb
22
comment Given the Wikipedia notion of “arc length”, how is its manifestly real “signed variant” to be called and denoted?
0celo7: "@MBN [An arbitrary curve can ... switch from timelike to spacelike.] True." -- So: Do we agree that the "Wikipedia prescription: the sign in the square root is chosen once for a given curve" does not provide a general and satisfactory definition of "arc length"? Then I'd be surprised/shocked if others had not arrived at this conclusion before. Thus I ask: Is there an established name and notation for the particular "solution to the problem" I described? (Else let's use what I made up.)
Feb
22
comment Given the Wikipedia notion of “arc length”, how is its manifestly real “signed variant” to be called and denoted?
0celo7: "A curve cannot switch from being timelike to spacelike." -- This assertion comes as a surprise to me. (And btw., I just replied to @MBN's comment before noticing yours.) IMHO this deserves being asked as a separate question (at PSE) which I'll be happy to prepare later today.
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