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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?


Nov
6
comment How to determine “timelike”-ness without using a coordinate system?
Incnis Mrsi: "It is, generally, not true in a curved spacetime." -- Also correct. But would that invalidate my examples 2 and 3 ?
Nov
6
comment How to determine “timelike”-ness without using a coordinate system?
Incnis Mrsi: "in Minkowski’s world, if interval A–B is time-like, then: • future light cones of A and B never intersect; • past light cones of A and B do not intersect. _" -- Correct (where "_A" and "B" denote distinct events). However: the future light cone of one of them (without loss of generality, say the future light cone of "A") and the past light cone of the other ("B") do intersect. (And, thereby, the past light cone of "A" and the future light cone of "B" do not intersect.) That's pretty much the point of my examples 2 and 3.
Nov
6
comment Is it theoretically possible to have a universe where sound travels faster than light $c$?
@Brandon Enright: "[...] electromagnetism [...]" -- The OP didn't ask specificly about electromagnetism, or about specificly (the speed of) quanta of the electromagnetic field; but about (the speed of) "light"; presumably in the sense in which this term is used when discussing the theory of relativity. (My answer below is based on that presumtion, spelling it out more explicitly.) Otherwise the OP may have to specify the question and terminology (and I may correspondingly review my answer).
Nov
6
comment Is it theoretically possible to have a universe where sound travels faster than light $c$?
Nathaniel: "the speed at which light travels [...]" -- You should define this notion more specificly; helpful may be the Wikipedia page on "front velocity" ... "and the maximum speed at which anything can possibly travel" -- Again: define this notion more specificly; refer to "front velocity" ... "In our universe, in a vacuum, [they are equal] as far as we know." -- You need to prove that those are distinct notions at all, to even consider their equality or inequality.
Nov
6
comment Is it theoretically possible to have a universe where sound travels faster than light $c$?
@Ben Crowell: "There is no way to answer a question about what laws of physics could exist in alternate universes." -- Except: Such "laws" would use the same definitions and definite notions which we use to express "laws" of our universe. And that's already enough to address the OP's question.
Nov
6
answered Is it theoretically possible to have a universe where sound travels faster than light $c$?
Nov
6
revised Is there a physical reason for colors to be located in a very narrow band of the EM spectrum?
(some copy-editing. The order of appearance of answers on a page may of course change due to votes being cast.))
Nov
6
reviewed Reviewed Is there a physical reason for colors to be located in a very narrow band of the EM spectrum?
Nov
6
suggested approved edit on Is there a physical reason for colors to be located in a very narrow band of the EM spectrum?
Nov
5
comment Formal definition of an observer?
FenderLesPaul: "[...] to lowest non-vanishing order the extent of the rods or meter sticks is negligible" -- To lowest non-vanishing order ... of what ?? There is a principal distinction of whether two distinguishable participants (a.k.a. "material points", "principal identifiable points" (MTW, Box 13.1), "observers" ...) were coincident, or not. A "rod" ("stick") means a pair of distinguishable "ends" which were not coincident; and the distinction to having been coincident is not negligible as a matter of principle.
Nov
5
comment Formal definition of an observer?
FenderLesPaul: "An observer is an orthonormal tetrad of the above kind, there is no distinction between them." -- MTW certainly draw a distinction: writing (e.g. in the title of §6.4) of "a tetrad carried by an observer", instead of "a tetrad being (the same as) an observer". "[...] replace the notion of an observer with that of a local Lorentz frame" -- Does this suggested replacement apply in Einstein's prescription: "If the observer perceives the two flashes of lightning at the same time, then ..." ??
Nov
5
comment Formal definition of an observer?
Ben Crowell: "[...] there is a one-to-one correspondence between normalized velocity vectors and Minkowski coordinate systems" -- Rather, there is a one-to-one correspondence between "normalized velocity vectors" (i.e. with respect to one particular choice of inertial system as reference) and certain equivalence classes of "Minkowski coordinate systems"; where each equivalence class contains all "Minkowski coordinate systems" which are equivalent "modulo" translations, rotations, and being either "spherical", or "Cartesian", or any other "spatial" coordinate assignment.
Nov
5
comment Formal definition of an observer?
FenderLesPaul: "Physically the Lorentz frame represents a local set of three orthogonal meter sticks or [...]" -- That's apparently represented by at least four distinct, non-intersecting, time-like worldlines (i.e. one for each distinct physical "end" of such a stick); not just one.
Nov
5
comment Formal definition of an observer?
FenderLesPaul: "An observer is a timelike worldline with 4-velocity $u^{\mu}$ and an orthonormal basis [...] C.f. chapter 6 of MTW, section 13.6 of MTW" -- Comparing with MTW sec. 13.6 ("The proper reference frame of an accelerated observer") and chap. 6, especially sec. 6.4 ("The tetrad carried by a uniformly accelerated observer") it seems that your answer is dealing with "What's an orthonormal tetrad?", or "What's a proper reference frame of an accelerated observer?", instead of the OP question itself.
Nov
4
revised Formal definition of an observer?
(v3.141592: Depreciation of coordinates expressed even more strongly. Also: some formatting.)
Nov
4
answered Formal definition of an observer?
Nov
4
asked Can a characterization of “inertial motion” be expressed in terms of interval ratios?
Nov
4
comment Is there a rigorous, explicitly geometric, general characterization for whether a given clock had been “good”, or not?
@Ben Crowell: "[...] in the quote from MTW, they discuss inertial motion" -- The quote mentions "time coordinate of a local inertial frame", thus apparently dealing with, and being restricted to, clocks "in inertial motion". My question aims at a generalization to clocks "in any (time-like) motion". "the notation given is clearly inadequate to talk about this." -- Hmm ... At least, the suggested notation allows to express interval ratios as real numbers, such as $$\frac{s^2[~\varepsilon_{C K}, \varepsilon_{C P}~]}{s^2[~\varepsilon_{C J}, \varepsilon_{C Q}~]}$$, etc.
Nov
4
comment Is there a rigorous, explicitly geometric, general characterization for whether a given clock had been “good”, or not?
@Ben Crowell: "The question asks for answers in a specific notation, [...]" -- If answers have been prepared using a different notation (or terminology), a map should be included (or may be added as comment), mapping any applicable symbol (or notion) used in the answer to precisely one symbol (or notion) I suggested.
Nov
4
comment Is there a rigorous, explicitly geometric, general characterization for whether a given clock had been “good”, or not?
@Ben Crowell: "[...] the point of listing indices like J, K, P, and Q" -- These examplify how to denote distinct events in which $C$ took part. Since, according to Einstein: "All our well-substantiated space-time propositions amount to the determination of space-time coincidences {such as} encounters between two or more recognizable material points", $\varepsilon_{C J}$ is for instance meant to denote the coincidence event in which $C$ and $J$ took part, but neither $K$, nor $P$, nor $Q$.