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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
16
comment What exactly is meant by saying that two events had been “simultaneous in an inertial frame”?
Emilio Pisanty: "you could say that $M$ passes through the middle of $J$ and $K$" -- Right (a general consenquence in the given example, too): $M$ and $N$ met/passed each other (event $\varepsilon_{MN}$). But does this make $N$ strictly being "placed at the mid-point of the distance AB {... as observer who} perceives the two flashes of lightning at the same time", to use Einstein's exact wording? Does it make $M$ "placed at the mid-point of the distance JK" ??
Apr
15
asked What exactly is meant by saying that two events had been “simultaneous in an inertial frame”?
Apr
13
comment Lorentz contraction in continuously accelerating rod
If the rod is not to stretch not to be compressed, but if its two ends ($A$ and $B$) maintain constant ping durations between each other (which are of course not equal but $\tau A_{BA}$ and $\tau B_{AB}$, respectively) and if $A$ maintains hyperbolic motion with constant acceleration $a$ then $B$ also maintains hyperbolic motion with constant acceleration $b := a~\text{Exp}[ \frac{a}{2~c}~\tau A_{BA} ]$, where vice versa $a := b~/~\text{Exp}[ \frac{b}{2~c}~\tau B_{AB} ]$. Cmp. physics.stackexchange.com/questions/38377/…
Apr
13
answered The nature of measurement
Apr
10
comment Average acceleration versus instantaneous acceleration
The Dark Side: +1 (Clearly the best answer given here at the moment. Still consider correcting the spelling of "anything as afar as", replacing "time period" or "time interval" by "duration", and drawing a picture with the vector quantities denoted as $\vec v$ and $\vec a$, respectively (i.e. as in your text already) if and when MathJax drawing capability becomes available here at PSE.
Apr
9
comment The nature of measurement
Yogi DMT: "To gain information about a system you need to interact/affect it in some way." -- That's flexible enough to include so-called "interaction-free measurements" (even "with high efficiency", as sketched in sect. III of this article). The point I wish to make is (merely): "gaining information" ("observational data") is not to be confused with "carrying out (concluding) the measurement by drawing a particular conlcusion or deriving a particular result value"
Apr
8
comment The nature of measurement
Yogi DMT: "Does the measurement of the particle change it's physical state?" -- No. To measure means to derive real or Boolean values from given observational data. (The only change of state involved in measurement is the state of the experimenter carrying out these calculations, or the state of anyone to whom the result values are communicated.) Of course, often there are several observational data concerning the same particle (e.g. "having been issued by a source" and "having reached the screen"). p.s. I'm planning and looking forward to expanding this comment into an answer.
Apr
8
answered How can the contact point of rolling body have zero velocity?
Apr
8
revised Seeking a coordinate-free expression of the orbital period for uniform circular motion around a non-rotating point-like mass
(v3.14159: EDIT: considering the case that the problem, as it had been stated, was underdetermined. Also some copy-editing. )
Apr
7
comment Would Special Relativity Predict Time Dilation of a Geostationary Satellite Compared to an Observer on Earth?
Pulsar: "both special relativity and general relativity have to be taken into account." -- That's certainly preferable instead of saying "SR and/or GR predict". However, your formulation gives the impression as if SR should be taken into account in addition to GR. "The total time dilation is given by [...]" -- I wonder (especially) about the meaning of $v$ in your formula. (The OP wrote of "linear velocity" ... this question makes my doubt more precise). "This enables you to calculate [...]" -- Consider giving a value of $\frac{G~M}{c^2}$.
Apr
7
answered What experimental proof has been found of Einstein's theory?
Apr
7
revised Seeking a coordinate-free expression of the orbital period for uniform circular motion around a non-rotating point-like mass
rolled back to a previous revision
Apr
7
revised Seeking a coordinate-free expression of the orbital period for uniform circular motion around a non-rotating point-like mass
(v3.14159: Corrected the exponent on dimensional symbol ... again.)
Apr
7
revised Seeking a coordinate-free expression of the orbital period for uniform circular motion around a non-rotating point-like mass
(v3.1415: Corrected the exponent on dimensional symbol.)
Apr
7
asked Seeking a coordinate-free expression of the orbital period for uniform circular motion around a non-rotating point-like mass
Apr
6
comment Angular acceleration of two rods joined in the center - did I do this right?
user47989: Yeah, great stuff.
Apr
4
comment How can I relate linear and angular motion using a single formula?
Vatsal Manot: "Thanks for the tip" -- You're welcome. (Notice that, as original poster, you're being notified of comments to your post in any case.) "but do you know of any formals related to my big bold question at the bottom?" -- Honestly: not directly. Anyways, I had found the question title that you had chosen far more inspiring and insightful than your post and big bold question turned out to be. (+1 at least for that title. ;) Besides: the remainder of the linked Wikipedia page calls itself explicitly "a summary from MTW"; so if you're serious: get a hold of "MTW" ("Gravitation").
Apr
4
comment How can I relate linear and angular motion using a single formula?
Vatsal Manot: "There's no mention of linear motion whatsoever in the provided links." -- Right ... There, specificly, is instead (equivalently) mentioning of "COM boost". Well, you know, on the very same web page (but admittedly a different section) there is even mentioning of "motion" and "that the system's COM moves in a straight line" ... (p.s. Btw., I found your latest comment merely "by accident", or rather by "abundant care". In order to put a notification in a user's inbox, add "@<user name>".)
Apr
4
comment Average acceleration versus instantaneous acceleration
John Rennie: "The instantaneous acceleration is the time derivative of the velocity vector:" -- Good enough, for the problem at hand. (How to determine and to compare "velocity vector" values as function of "time" is sure to come up elsewhere.) "If the velocity is changing then the acceleration will be non-zero." -- Not quite: If the values of (instantaneous) velocity are changing (are different) at different "instances" then the value of acceleration at some particular instance is not necessarily non-zero, nor even defined. (Or are you considering "v changing at an instant"??)
Apr
4
revised Angular acceleration of two rods joined in the center - did I do this right?
(v2.7181: corrected expression of moment of inertia of attached mass by adding the necessary "square".)