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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
6
answered What's the differences between time in Physics and time in everyday use?
May
6
comment The nature of measurement
@Yogi DMT: "so measuring the particle before it hits the slits will result in an interf. pattern being produced?" -- This doesn't seem a sensible question. We can and should ask instead: How to measure the (most probable) number and shape of "slits" (vs. "potential walls") in the region between a given source and screen; wrt. specific particles having been exchanged in specific trials? Answer: from the "pattern" on the screen (if indeed there was any); in geometric relation to the source. So "measuring the particle before it hit the screen" means only to assert that it had left the source.
May
6
revised How can we define a frame of reference in general relativity?
(v3.1415926: **any** applicable timelike congruence)
May
6
answered How can we define a frame of reference in general relativity?
May
6
comment Absoluteness of Simultaneity?
Consequently "simultaneity", by Einstein's operational definition, is determined unambiguously and agreeably of (pairs of) applicable individual indications of any (pair of) participants at rest to each other; because the geometric relation "mutual rest" is sufficiently unique, and for each such pair of participants there's a "center/middle between" each other uniquely identifiable, to guarantee the required transitivity of simultaneity. But, using Einstein's definition, we cannot sensibly speak of "simultaneity of entire events". (I may eventually expand this into an answer.)
May
6
comment Absoluteness of Simultaneity?
Khushro Shahookar: "material points such as embankment points [...] How would this change any of the above?" -- At any one event there are (generally) several distinct participants passing each other (e.g. one constituent of "the embankment", one constituent of "the train" etc.). Importantly, each individual participant still has an own distinctive indication "at" any one event under consideration (or, as Einstein put it: "the position of the little hand {...}". Consequently [... contd.]
May
6
comment The nature of measurement
@Yogi DMT: "Not sure how this answers the question..." -- Let me express my answer to your first/main question concisely: No, measurement of a particle doesn't change its physical state. (Measurement means to draw a definite conclusion from given observational data; e.g. the particle having changed its state from coincidence with the source but not the screen into conicidence with the screen. Drawing conclusions from this doesn't induce further state changes of the particle). Regarding your second/minor question: Don't pretend to know number/arrangement of "slits" before measuring.
May
5
comment Absoluteness of Simultaneity?
Khushro Shahookar: "[...] switch placed at the mid-point X of two event locations A and B" -- This seems mistaken. Einstein's coordinate-free defintion of (how to determine) "simultaneity" is mentioning the "mid-point between" two suitable participants ("material points": elements of an "embankment"); not "between event locations".
May
5
answered Speed of light and perception
Apr
30
comment Can an ensemble of meta-stable systems be prepared so their survival probability drops approx. linearly right after preparation?
Emilio Pisanty: "This comes down to what you understand the initial state $| \varphi \rangle$ to be [...] hard to define the unstable states of a given system." -- At least this seems some vindication for me having been a bit skeptical of your other answer (PSE/a/178242). So: thanks again, +1. Of course it'd be nice to still have some more "specific/technical" detail on what constitutes an appropriate initial state, or what constitutes a "preparation". (I figure that this "(de-)coherence business" is well suited ...) Also, through "my" fine library, I just got a copy of RPP 41, 587 ...
Apr
29
comment Can an ensemble of meta-stable systems be prepared so their survival probability drops approx. linearly right after preparation?
These fig.s show experimental results of systems (or perhaps rather: ensembles) which had been "initialized" at "$\text{Time} = 0~\mu\text{s}$" and were then left to evolve/decay. My question (objection, nitpick) in a nutshell: Considering this very same exp. data, could it be legitimately said instead, for instance, that the preparation/initialization of the systems/ensembles had finished only at "$\text{Time} = 10~\mu\text{s}$", and the "actual decay experiment" commenced only subsequently? (If so, the survival probability was apparently not "flat", at the start of the experiment.)
Apr
29
comment Can an ensemble of meta-stable systems be prepared so their survival probability drops approx. linearly right after preparation?
@CuriousOne: "an ensemble with a Gaussian frequency distribution will decay nearly quadratically for small times, will it not?" -- Outright, I really have no idea or prejudice about that at all. (I might merely try to follow some derivitation of what you suppose, if it matters). "Is that what you mean?" -- What I'm trying to get at (namely in response and as follow-up to this answer) can perhaps be understood by looking at Figures (3) or (4) of george.ph.utexas.edu/~quantopt/papers/exp_evidence.pdf (given as reference in PSE/a/178242):
Apr
29
comment Can we find the exponential radioactive decay formula from first principles?
Emilio Pisanty: "If you're asking [...] the answer is easy: because you've initialized the system at $t=0$ to be the initial state" $| \varphi \rangle$ -- Well, but is "$| \varphi \rangle$" "just any (generic) state", or (only) some special state? ... "At later times the system is obviously no longer in that state." -- Therefore, surely, any such "later state" could also be initialized; by making an appropriate wait part of the (extended) preparation procedure. Along these lines I've now asked a follow-up question (PSE/q/179117).
Apr
29
asked Can an ensemble of meta-stable systems be prepared so their survival probability drops approx. linearly right after preparation?
Apr
29
accepted What is the most general definition of a coordinate system?
Apr
29
comment What is the most general definition of a coordinate system?
Emilio Pisanty: "In principle, yes" -- That's by itself an acceptable answer. "the more subtle notion of "useful coordinate system", which depends on what you're trying to do;" -- That's a stunningly imprecise formulation; and perhaps the main point of contention; No: "usefulness" depends apparently on 1. having some "structure", such as prototypically $(\mathcal S, s)$, given ("through measurement") and 2. the "goodness" of mapping to whatever intrinsic structure of $\mathbb R^n$ ("natural topology", "natural vector space"). "Coordinate usefulness" is never original/genuine.
Apr
28
comment Can we find the exponential radioactive decay formula from first principles?
Emilio Pisanty: "My answer is restricted to a single system, whose survival probability is $P(t)$ after being prepared in state $|\varphi\rangle$ at time $t=0.$" -- I brought up an "ensemble" because "atoms" (in plural) are mentioned in your answer; and a "population", too. "If you have $N$ systems, at time t you can expect $n(t)=NP(t)$ of them to still be around." -- Good. But how are these (which had remained so far, and which presumably continue to "decay exponentially") qualitatively different from those at the (or any) start?? (So: thanks; I'll try to ask separately tomorrow.)
Apr
28
comment What is the most general definition of a coordinate system?
@yuggib: "[... Why] putting the metric as a necessary requirement to define "coordinates"" -- Again: to define "coordinate system". Clearly there's already terminology available for "manifold" or "coordinate patch", to denote "set-isomorphic" (homeomorphic?) mapping. "at least the identification of my points with the tuples (coordinates)." -- Surely $n$-tuples of reals are useful for distinctive labelling. (By Ockham's razor, this presumes that what's being distinctly labelled was in (physical) fact established as being distinct.) But there seems not much "Co-Ordination" in this.
Apr
28
comment What is the most general definition of a coordinate system?
@yuggib: "1 [...] the least requirement possible would be a collection of points without any additional structure." -- Presumably no "structure" in addition to $$\mathfrak g : \mathbb R^n \times \mathbb R^n \rightarrow \mathbb R, \qquad \mathfrak g[~\mathbf x_a, \mathbf x_b~] \mapsto s[~\varphi^{-1}[~\mathbf x_a~], \varphi^{-1}[~\mathbf x_b~]~].$$ "[...] 3) The requirements of a coordinate system depend, in my opinion, to what you need them for" -- Well, the larger point of my question is to establish that, in Physics, there is no genuine need for coordinates.
Apr
28
comment What is the most general definition of a coordinate system?
@yuggib: "why do you want a metric space?" -- Any suitably generalized metric space. So we may speak of some sort of "system" at all. "your system seems to be a global coord. system; a unique map from all the space to all $\mathbb R^n$" -- That's not the intended meaning. I mean $\varphi$ to be a unique map from all the (non-empty) space to some (non-empty) subset of $\mathbb R^n$, or all $\mathbb R^n$. "which topological/metrical properties of the latter do you want to be preserved by the inverse map?" -- None which aren't outright required for speaking of a "coordinate system".