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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?


Apr
4
comment How can I relate linear and angular motion using a single formula?
Vatsal Manot: You may find these links relevant: "4d Angular momentum as a bivector" and "Lorentz transformations intertwining 3d angular momentum L and mass moment N". (But I'd be reluctant to try to expand this comment into an answer to your specific question.)
Apr
4
answered Angular acceleration of two rods joined in the center - did I do this right?
Apr
4
comment Can we rigorously define force?
Related: How to derive or justify the expressions of momentum operator and energy operator? (PSE/q/83902). $\qquad$ p.s. Bob Dylan: "Position can be measured arbitrarily well with measuring sticks." -- $\qquad$ No: Distance of pairs of participants ("ends") can be measured (trial by trial); and if so, the applicable pair in the applicable trial is called "(ends of) a measuring stick". (Likewise: Duration can be measured ...)
Apr
3
answered Mass dimensions and weak interaction
Apr
3
comment Derivation of Kramer's equation
dexter: "link :this lecture notes explains [...]" -- Note, btw., that the linked lecture notes are nowhere referring to "Kramer's equation", but instead for instance to the "Kramers-Moyal expansion", "Klein-Kramers equation" and "Kramers' theory"; all named for H. A. Kramers.
Apr
3
revised Kramers-Kronig relations for the electron Self-Energy Σ
corrected spelling the name of H. A. Kramers (in the question title; consistent with the spelling already present in the article); cmp. http://en.wikipedia.org/wiki/Kramers%E2%80%93Kronig_relations
Apr
3
suggested approved edit on Kramers-Kronig relations for the electron Self-Energy Σ
Apr
2
comment A reference frame is any coordinate system or just a set of Cartesian axes?
bechira: "[...] why you think this is inconsistent with [...]" -- Well, to repeat the obvious: we compare "one frame" being described either as "an infinite set of point particles" (possibly with additional requirements); i.e. using terminology you suggested yourself above: as a set of (infinitely many) "timelike curves (assuming Lorentzian metric)"; or whatever you've been describing in your answer in detail, "at one single point $p$ in $M$".
Apr
2
comment A reference frame is any coordinate system or just a set of Cartesian axes?
bechira: "Ah I see [...]" -- Fine. Now: did I understand correctly that what you described as "frame" in your answer is quite different from (and even inconsistent with) the understanding of "frame" based on W. Rindler's explicit description of "inertial frame" (in distinction to "inertial coordinate system")?
Apr
2
comment A reference frame is any coordinate system or just a set of Cartesian axes?
bechira: "why would local frames at the same point on a general spacetime be related by a translation?" -- I didn't mean to suggest that; but the phrase "these frames are related by" in the final sentence of your answer could be (mis?-)interpreted as including such "frames at" various distinct points. "take a point particle to be a timelike curve $\mathbb R\rightarrow M$ (assuming Lorentzian metric)." -- Indeed. And consequently, in Rindler's sense, take "frame" to mean "timelike congruence"
Apr
2
comment A reference frame is any coordinate system or just a set of Cartesian axes?
bechira: "$p$ is a point in spacetime [...]" -- Well, then your answer is all the harder to relate to W. Rindler's notion; because, surely, any one of Rindler's "point particles" would be characterized by a (suitable) set of several "points in spacetime". Also: Shouldn't your answer therefore have referred to "Poincaré transformations", or some suitable generalization(s) of those, instead of "Lorentz transformations, as expected." ??
Apr
2
comment A reference frame is any coordinate system or just a set of Cartesian axes?
bechira: I find it very difficult to reconcile your answer with W. Rindler's dictum: "We should, strictly speaking, differentiate between an inertial frame and an inertial coordinate system {...} An inertial frame is simply an infinite set of point particles sitting still in space relative to each other.". Does "point $p$" in your answer mean "point particle $p$" (as for Rindler), or does it instead mean "element $p$ of the manifold" ??
Mar
31
comment Why are the proper time and the proper length not defined in the same frame of reference?
ACuriousMind: Given your definition of the "coordinate curve $\gamma : [t_0, t_1] \rightarrow \mathbb R^4$" it seems questionable that the "dimensional" symbol "$c$" appears, as it does, in the square root expression which you wrote to express $\tau$. Wouldn't it be interesting and challenging, especially for the purpose of establishing notation, to consider instead a path "$\Gamma : [t_0, t_1] \rightarrow \mathcal S$", where "$\mathcal S$" is the set of events of a given region, and try to express $$\tau = \int_{\Gamma} {\rm d}s = \int_{t_0}^{t_1} \frac{{\rm d}}{{\rm d}t}s~ {\rm d}t =~... $$.
Mar
30
revised Why are the proper time and the proper length not defined in the same frame of reference?
(v2.71: spelling corrected)
Mar
30
answered Why are the proper time and the proper length not defined in the same frame of reference?
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
revised What spacelike, timelike and lightlike really mean?
(v3.14159: grammar corrected. (Really only a minor edit.))
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
11
answered What spacelike, timelike and lightlike really mean?