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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
11
comment How can a Physical law not be invariant?
user1620696: "On differential geometry everything is defined so that things don't depend on coordinates." -- I wonder ... "So for example: vectors are defined as certain differential operators" -- So what constitutes a "differential operator" wrt. some given set $M$? Required is apparently (at least) a certain "topology" assigned to set $M$; and the debate (in which I'm interested) is whether and how it can be derived from the elements of $M$ itself, and not only by its (possible) representation as "real n-tuple topology" of certain equivalence classes of coordinates.
Mar
11
comment What spacelike, timelike and lightlike really mean?
@MBN: "This [...] is discussed in every relativity book!" -- If the "physical intuition" which the OP is asking about were indeed discussed "in every relativity book", or at least in one "relativity book" available to you, then your comment could and should be expanded into an answer.
Mar
11
comment What spacelike, timelike and lightlike really mean?
SirElderberry: "Spacelike separation means that there exists a reference frame where the two events occur simultaneously, but in different places." -- That's overstretching the meaning of "simultaneous" in two ways: "simultaneity" is not defined for entire events, and determination of "simultaneity" requires participants at rest wrt. each other (as joint members of an inertial frame) while events may be attributed "spacelike separation" even in regions where inertial frames cannot be found at all.
Mar
4
comment Proper length in GR
@Max Lock: "And is there any alternative way of measuring a distance in GR other than chronometry?" -- Well, the question is rather whether and how "chronometry" is feasable in the first place; i.e. how to determine whether any given clock had been "good" (or how to quantify it having deviated from "being good"), at least in principle.
Mar
4
comment Proper length in GR
@Max Lock: "[...] the phys. interpretation [...]" -- AFAIU in order to be attributed a "proper length" value, an everywhere spacelike path would also have to be "straight", i.e. any three of its elements, $a,b,c$, should satisfy $$\sigma[~a,b~]^2+\sigma[~a,c~]^2+\sigma[~a,c~]^2=2~\sigma[~a,b~]~\sigma[~a,c~]+‌​2~\sigma[~a,b~]~\sigma[~b,c~]+2~\sigma[~a,c~]~\sigma[~b,c~],$$ where "$\sigma$" denotes Synge's world function. For two suitable participants, it could then be asked for the "straight path" of max. "proper length" between them.
Mar
3
comment Given the Wikipedia notion of “arc length”, how is its manifestly real “signed variant” to be called and denoted?
@Qmechanic: A quote from http://physics.stackexchange.com/help/bounty, "How is a bounty awarded?": If the bounty was started by the question owner, and the question owner accepts an answer posted during the bounty period, and the bounty expires without an explicit award then we assume the bounty owner liked the answer they accepted and award it the full bounty amount at the time of bounty expiration.. 'nuf said.
Mar
1
comment What does $\frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$ mean with respect to special relativity?
@Kyle Kanos: "[...] w/o first-principles motivation" -- Hardly: Einsteins original example formulation reads (in translation): "the exact pointing of the little hand of my watch to 7". "do all this in like 4 lines using standard notation" -- It is one thing to appeal to whatever is deemed "standard notation", but quite another to explain and clarify what's being denoted. Besides: Appealing to coordinates while trying to address a physics question ... sucks.
Mar
1
comment What does $\frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$ mean with respect to special relativity?
@Kyle Kanos: "[Do we agree that the notion is meaningful: "the duration of ... from ... until ..."?] No" -- (Hmm ... Makes we wonder what you imagine NeuroFuzzy's coordinate symbol "$t$" is supposed to mean either.) Do we at least agree that the notion "indication of one specific participant" is meaningful?, such as "the large hand of my watch pointing to 10", or "$A$'s indication of having been met and passed by $K$"; and to be distinguished (also in notation) from "$K$'s indication of having been met and passed by $A$"?
Feb
28
comment What does $\frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$ mean with respect to special relativity?
@Kyle Kanos: A related and arguably even more basic question is how to denote just one particular indication of one specific participant; such as "the large hand of my watch pointing to number 10", or "$A$'s indication of having been met and passed by $K$". I'd write the latter explicitly as "$A_K$". (Then the appearance for expressing a duration value may be rather a matter of taste.)
Feb
28
comment What does $\frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$ mean with respect to special relativity?
@Kyle Kanos: "$\tau \! A[\circ_K, \circ_H]$ ... I am not aware of any such notation" -- Fair enough. Do we agree first of all that the notion to be expressed symbolically is meaningful: "the duration of $A$ from ($A$'s indication of) having been met and passed by $K$ until ($A$'s indication of) having been met and passed by $H$)"? "but I'd think $\tau \! A(K-H)$ (since $K-H$ is a duration [...])" -- I prefer to symbolize duration values using the letter "$\tau$" (with applicable arguments: whose duration, and the two specific indications "from" and "until"). But "$K-H$" by itself ??...
Feb
28
comment What does $\frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$ mean with respect to special relativity?
@Kyle Kanos: "Google Docs Drawing is freely available, try that?" -- ??? To repeat: I won't decorate my answer with some illustration. Someone who genuinely finds my answer (as it stands, itself) "quite incomprehensible" wouldn't comprehend any accompanying illustration either. (Instead, such a hypothetical someone should ask about specifics of my answer.) Besides: to create drawings for my PSE contributions I have used, and I'd use again, ASCII or the "LaTeX Previewer by Troy Henderson" in the expectation that this is supported here anytime now, too.
Feb
28
comment What does $\frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$ mean with respect to special relativity?
... and "duration" of a participant from one indication to another (such as "$\tau \! A[\circ_K, \circ_H]$", i.e. the duration of $A$ from having been met and passed by $K$ until having been met and passed by $H$). If there is a PSE standard for the notation of any or all of these notions would you please point that out, for me to check my compliance. Apart from that, since I have no desire to impose a particular notation on anyone, I'd welcome the possibility of setting notational rendering as individual [[user:option]].
Feb
28
comment What does $\frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$ mean with respect to special relativity?
Let's then not even contemplate trying to draw (some particular example instance of) $A,B,F$ and $H,J,K,Q$ all together in relation to each other ... Also: Which drawing commands are available in the here available Mathjax version anyways? "(b) more common notation for what on earth you're trying to say." -- The notions in my answer which are represented in some particular notation are "participants" (such as "$A$"), "distance" of pairs of participants who were at rest to each other (such as "$AB$") and [to be continued]
Feb
28
comment What does $\frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$ mean with respect to special relativity?
@Kyle Kanos: "more useful if you included (a) an image depicting what on earth you're picturing" -- You need a drawing of $A,B$ and $F$ having been at rest to each other, with distance ratios $$\frac{AB}{AF}+\frac{BF}{AF}=1, \qquad \frac{AB}{AF}=\beta,$$ for some particular real number $0 \lt \beta \lt$, and/or of $H,J,K$ and $Q$ having been at rest to each other, with $$\frac{JK}{JQ}+\frac{KQ}{JQ}=1, \qquad \frac{JK}{KQ}=\beta$$ etc. ?? That'll not be helpful if you don't understand these prescriptions first by themselves; it would be "leading" and thus harmful, IMHO. [to be continued]
Feb
28
comment Do tachyons move faster than light?
@Leandro M.: "if you ask that as question I can give you some more mathematical detail" -- Well, the one specific self-contained question which I asked already in commenting on your answer is: "How to determine the momentum of a tachyon, at least in principle?". (Perhaps that's already addressed at PSE; otherwise I may get around asking directly.) Apart from that, Qmechanic's response here makes me wonder whether in these cases it is justified at all to treat signal front velocity as a definite real quantity, "$c = 1$".
Feb
27
comment Given the Wikipedia notion of “arc length”, how is its manifestly real “signed variant” to be called and denoted?
(eq. 11.59) "Math. Tools f. Phys." reads: $$(\hat x')^T(x-\hat x)+(\hat\mu')^T(\mu-\hat\mu)=\Delta,$$ where $\hat x$, $\hat\mu$ are previous solutions, and $()'$ denotes diff. wrt. arc length. Then $$x-\hat x:=\int_{t[~\hat x~]}^{t[~x~]} {\rm d}s\text{ with } {\rm d}s=\left\{\begin{array}~ \sqrt{g}{\rm~d}t\text{ if }g\ge 0\cr\text{else }0\end{array}\right\},$$ $$\mu-\hat\mu:=\int_{t[~\hat\mu~]}^{t[~\mu~]}{\rm d}s\text{ with }{\rm d}s=\left\{\begin{array}~ -\sqrt{-g}{\rm~d}t\text{ if }g\le 0\cr\text{else }0\end{array}\right\},$$ $$()':= \frac{d}{d\Delta}()$$ gives the sought funct. as $\Delta$.
Feb
27
comment Do tachyons move faster than light?
Qmechanic: "basically repeat Leandro M.'s good answer using formulas. [...] spinless relativistic complex scalar field [...]" -- 1: Would you please relate the formulas and results of your answer to @Leandro M.'s remark that "people typically just calculate the effective potential and see if it has an imaginary part, which sidesteps the issue of talking about "particle" states." 2: "1a) [...] The phase velocity is [...] The group velocity is [...]" -- What about calculating signal front velocity? And what about cases "1b" and "2"?
Feb
27
comment Given the Wikipedia notion of “arc length”, how is its manifestly real “signed variant” to be called and denoted?
user12262: ""pseudo-(arc)-length", (eq. 11.59) itself" -- Would you please sketch more explicitly which interpretation might be given to (eq. 11.59) of M. Grinfeld, "Math. Tools for Physicists" to include the proposed functional?
Feb
27
comment Given the Wikipedia notion of “arc length”, how is its manifestly real “signed variant” to be called and denoted?
user12262: If this answer would be accepted as satisfactory, and I thereby might gain some number of reputation points, then I hereby undertake: $\hskip 23em$ If a substantive claim were being submitted here as an answer afterwards by someone else ("the claimant"), then I'll ask a follow-up question about notation and terminology of such a possible claim which should be easy for the claimant to answer acceptably, and with a bounty attached equal to the number of reputation points I would have gained (rounded up, if applicable, to the next multiple of 50).
Feb
24
comment EPR-type experiments and faster-than-light communication using interference effects as signaling mechanism
user1247: "but there ["Dopfer EPR", PSE/q/79427] the top voted answerer claims [...]" -- And there I've given my answer, too, which may well be understood as counter-claim. So, as mostly, you have to try and figure that out for yourself, too. Do you have any specific follow-up question(s) on my answer (or ensuing comments) there? Meanwhile I'll take another look at your answer here I try to come up with a specific follow-up question here (it may take a few days) ...