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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?


May
12
revised Does the trial index of typical CHSH experiments constitute a “hidden variable”?
(v3.1415: expressions of separate constraints corrected.)
May
12
asked Does the trial index of typical CHSH experiments constitute a “hidden variable”?
May
11
comment Is there a specific name for the highest energy state in quantum mechanics?
EverettYou: I'd suggest calling such a state the "roof state" of the system (while "ceiling" seems a better match for "floor"). But since there's no established (short) name yet at all, you should generally add the description "the highest energy state" at least in parentheses. Btw., a related, already established notion is that of the/an "asymptotically free state" of a suitable system (i.e. having the least energy as to not being "bound" anymore; if it exists). However, there isn't necessarily some "state with the highest energy still less than that of an asymptotically free state".
May
10
comment Questions about MTW's “thousand” tests of the Einstein principle
@magma: "Free fall is defined and explained in paragraph 1.3 pag. 13 [MTW]" -- Defined ?? Even looking beyond the model-dependent drivel of pag. 13 ("free fall of an object" as "its normal track through spacetime"): Pag. 16 gives a kinematic desciption: "it moves {...} in a straight line with uniform velocity." But this still leaves open how to find out whether a given object had so "moved", in a trial under consideration, before and without having an "(ideal) clock" and an "(ideal) rod" in this trial available, which by §16.4 require "free objects" having been identified.
May
10
comment Proving the conservation of 4-momentum for a particle collision $A+B\to C+D$
Noting that "$\mu$" and "$\nu$" are distinctive reference base indices, corresponding to the two reference systems under consideration, you could write more effectively: $$p_A^{\nu} + p_B^{\nu} - p_C^{\nu} - p_D^{\nu} = C^{\nu} \tag{3~},$$ with the goal of proving "$C^{\nu} = 0^{\nu}$"; and $$p^{\nu} = \Lambda^{\nu}_{\mu}~p^{\mu}. \tag{4~}$$
May
10
comment Proving the conservation of 4-momentum for a particle collision $A+B\to C+D$
Henrikas: Re-writing your eq. (1) a little more careful than you did, I get instead: $$p_A^{\mu}+p_B^{\mu}-p_C^{\mu}-p_D^{\mu} = 0^{\mu}\tag{2~},$$ and consequently your eq. (6) evaluates rather more consistently: $$\Lambda^{\nu}_{\mu}~(p_A^{\mu}+p_B^{\mu}-p_C^{\mu}-p_D^{\mu}) = \Lambda^{\nu}_{\mu}~0^{\mu} = 0^{\nu}. \tag{6~ }$$"am I doing something ineffectively?" -- Yes: you could obviously leave off the "primes" in your eqs. (3) and (4) [... contd.]
May
9
comment What is the most general definition of a coordinate system?
p.s. @Timaeus: "[...] often you want something more general than a metric space." -- I always wrote "suitably generalized metric space"; and "$(\mathcal S, s)$", rather than e.g. "$(\mathcal S, d)$". Now, you may find my concrete OP suggestion of "$s$" ... undercomplex. But it can of course be further generalized, e.g. as $$s : \text{Powerset}[~\mathcal S~] \times \text{Powerset}[~\mathcal S~] \rightarrow \mathbb R,$$ (which can include "topological space" as special case), etc. Are there even (infinite dimensional, complex ...) coordinates "to cooperate with this much structure" ??
May
8
comment What's the differences between time in Physics and time in everyday use?
robert bristow-johnson: "the statement from Einstein that "Time is what clocks measure."" -- Actually, Einstein is on record with: "{for} ''time'' we substitute ''the position of the little hand of my watch''". That's decidedly not referring to a measurement of duration, but (merely) a pronouncement of appearance, or indication. This understanding of "time" by Einstein is thereby actually quite close to that of Wheeler ("You don't necessarily make all your observations at once"; IIHMS ...)
May
8
comment What is the most general definition of a coordinate system?
Namely to demonstrate that in order to consider and evaluate any such possible "cooperation" we, as physicists, must be able to consider and evaluate the relevant (intrinsically meaningful) "$\text{structure}$" separately, in the first place; beginning with the sheer distinctiveness of all the elements of set $\mathcal S$. Such intrinsic "$\text{structure}$" must not be confused with whatever "incidental coordination" might be super-imposed by any particular assignment of coordinates, or the other.
May
8
comment What is the most general definition of a coordinate system?
@Timaeus: "You asked for generality, and [...] Usually you want your coordinates to cooperate with some structure" -- I admit that my question suffers from this ambiguity: On one hand asking for "the most general structure" by which to characterize some (given) set $\mathcal S$ as constituting a "space $(\mathcal S, \text{ structure})$", intrinsically. (That's interesting in its own right.) But on the other hand: asking what the minimal requirements (if any, apart from sheer distinctive labelling) are on the "cooperation" you mentioned. My focus/intention is the latter aspect:[...contd.]
May
7
comment What is the most general definition of a coordinate system?
@Timaeus: "Can the suitably general metric space be degenerate (e.g. [...])?" -- Interesting and even relevant case (e.g. "events on one light ray"). Does the mere distinctive labelling of elements of such a degenerate space lead to a "coordinate system"? (Or a "degenerate coordinate system"?) Hence my question. "$n$ is finite?" -- Does it matter? Even $n = 1$ seems to gives us plenty of distinctive labels ... "[...] topology." -- Well, I like to re-examine why I presumed the underlying reference system as (generalized) "metric space", rather than "topological space". (May take a while.)
May
7
comment What is the most general definition of a coordinate system?
@Timaeus: "Should we assume that the inverse is a left inverse, a right inverse, both? [...]" -- My intention was to ask about exactly one "coordinate system"; therefore, AFAIU, about one suitable set (or "patch"?) $\mathcal S$ constituting a suitable reference system $(\mathcal S, s)$, and a suitable set of (real-valued) n-tuples $C_n \subseteq \mathbb R^n$ as coordinates, where $$\phi : \mathcal S \leftrightarrow C_n, \qquad \phi^{-1} : C_n \leftrightarrow \mathcal S,$$ $$\phi^{-1} \circ \phi \equiv \mathbf I_{(\mathcal S)}, \qquad \phi \circ \phi^{-1} \equiv \mathbf I_{(C_n)}$$.
May
7
comment How can we define a frame of reference in general relativity?
Timaeus: "when I say metric I mean invertible ranks 2 tensor." -- I prefer to call tensors "tensor", and to call spaces (in the mathematical sense) "space"; thus expressing and appreciating their distinctiveness, and relations between them. Btw., definitions of "quasi-metric space", or "semi-metric space", or "pseudo-metric space" always have that pesky $$d[~x,y~]\ge 0.$$Therefore I suggested to call the obvious generalization a "chrono-metric space".
May
7
revised Time taken by a stone to reach the ground is independent of point of dropping
(v4: minor spelling correction.)
May
7
comment How can we define a frame of reference in general relativity?
2: You seem to suggest that if you're given a coordinate system then you're not (necessarily, by definition) already given all geometric relations (such as distances, or durations, or intervals); so there was indeed occasion and need for measuring geometric relations in addition to what's given already. How, then, would you distinguish a coordinate system from any plain labelling with n-tuples of real numbers?? (See also "What is the most general definition of a coordinate system?" (PSE/q/178907).)
May
7
comment How can we define a frame of reference in general relativity?
Timaeus: "Thank you, I've edited my answer." -- Thank you, in turn. You've substituted the phrase: "You can't measure distances between events with a coordinate system alone. But [...]" -- Two remarks: 1: It's still incorrect and puzzling that you refer to "distances between events". If you like to avoid mentioning the measurement of intervals between events, or of related quantities, and you like to mention the measurement of distances, then you should refer to "measurement of distances (or more generally: separations) between participants (as grid constituents), or of durations".
May
6
comment Why more Fe-56 than Ni-62 as fusion product in heavy stars?
Possible duplicate of physics.stackexchange.com/q/15943
May
6
comment How can we define a frame of reference in general relativity?
Timaeus: "If you want to measure distances between events" ... or rather: intervals $s^2$ between events, or related quantities such as $\text{sgn}[~s^2~]~\sqrt{s^2~\text{sgn}[~s^2~]}$ ... "you'll need the coordinate system and the metric [tensor]" -- No: we certainly don't need coordinates for measuring geometric relations. Instead, we may (as an afterthought) derive the metric tensor from measured intervals, for any suitable (differentiable) assignment of coordinates to events.
May
6
answered Time taken by a stone to reach the ground is independent of point of dropping
May
6
revised What's the differences between time in Physics and time in everyday use?
v3.141: spelling corrected. (This is really a minor edit.))