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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
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 $Q$.
Nov
4
comment How can we measure time?
Qmechanic: "By repeating the synchronization procedure many times, we can test SR. Any inconsistency (that can't be explained by faulty equipment etc)" -- This parenthetical remark has profound implications: we always only test hypotheses concerning our models/expectations/prejudices about the "equipment etc" involved having been and remained "not faulty"; we cannot experimentally test our definitions of what we mean by "inconsistent" or "faulty" in the first place. Any theory serving this purpose, such as (S)RT, must be selected beforehand, and stuck to, based on comprehension alone
Nov
4
asked Is there a rigorous, explicitly geometric, general characterization for whether a given clock had been “good”, or not?
Nov
3
comment How to determine “timelike”-ness without using a coordinate system?
Incnis Mrsi:"Strictly speaking, two endpoints of a time-like interval are not necessarily causally related" -- Alright; causal relations, as considered in geometry are not quite the same as causation, e.g. due to some "material trace propagating". "[...] situations 2 and 3 are not possible in Special Relativity." -- But they certainly are (read the descriptions carefully again ...)
Oct
28
comment How to determine “timelike”-ness without using a coordinate system?
[... continued] Or 3: if there exists at least one "material trace" who to part in distinct events $\mathcal{A, B, P}$ (in that order) where $\mathcal{A, Q}$ as well as $\mathcal{B, Q}$ are lightlike?
Oct
28
comment How to determine “timelike”-ness without using a coordinate system?
Incnis Mrsi: "[...] reflected or scattered light, or by their material traces" -- So, applying this terminology to what I asked specificly at the end of my question text: Do you agree that two events ($\mathcal{P, Q}$) are related by a timelike interval 1: if there exists at least one "material trace" which took part in both of these events; Or 2: if there exists at least one "material trace" who to part in distinct events $\mathcal{A, B, Q}$ (in that order) where $\mathcal{A, P}$ as well as $\mathcal{B, P}$ are lightlike; [... to be continued]
Oct
28
comment Which nucleus is the most resilient against gamma-induced fission?
@Ben Crowell: "Generally we don't call it fission unless both of the products are heavier than alpha particles." -- Note, however, the "neutron-induced fission of Uranium-235", for example ...
Oct
28
comment Which nucleus is the most resilient against gamma-induced fission?
@Ben Crowell: "[...] would include processes that I don't think any nuclear physicist would call fission, such as knocking out a single neutron." -- It'd be helpful to learn how nuclear physicists, if any, who wouldn't call the (gamma-induced) knocking out of a single neutron a "(gamma-induced) fission" might call such a process instead. Speaking perhaps of "(gamma-induced) nuclear disintegration" is not (at present) viable either, since Wikipedia (presently) claims that's synonymous to "(gamma-induced) nuclear decay chain".
Sep
30
awarded  Explainer
Sep
24
awarded  Autobiographer
Sep
24
revised When has the speed of light been measured, recently?
(v3.1415926: tiny typos corrected.)
Sep
23
answered When has the speed of light been measured, recently?
Sep
15
awarded  Yearling
Sep
15
revised Can isotropy (or anisotropy) be expressed in terms of intervals ($s^2$) between pairs of events?
(v3.1: corrected question title.)
Sep
15
revised Can isotropy (or anisotropy) be expressed in terms of intervals ($s^2$) between pairs of events?
(v3: named the quantity "s^2"; and some formatting.)
Sep
15
revised Can isotropy (or anisotropy) be expressed in terms of intervals ($s^2$) between pairs of events?
rolled back to a previous revision