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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?


Jul
23
comment why are the anthropometric units (which are about as big as we are) as large as they are relative to their corresponding Planck units?
@Emilio Pisanty: "Can I, sincerely, ask you to tone down the typographical emphasis? It is dizzying and distracting [...]" -- I use the available appearance-modifying features, quite consistently (I hope), in order 1.to mark quotations to which I reply, 2.to indicate and cope with the (at times almost dizzying) number of notions for which I have no satisfactory definition (but which are used by other commenters nontheless), and 3.for highlighting elements within my own writing. I heartily support User:Options for you to render my comments without any such features, if you so prefer.
Jul
23
asked Given two events such that either one of them is 'on the light cone' of the other, do they constitute a 'null interval'?
Jul
23
comment Are there more distinctive names of “null curves” with certain additional properties (absence of “chord curves”)?
ACuriousMind: Since I only now appreciate "light cones" in distinction to the notions ("of "Lorentzian geometry" as we know it today") that you described (ever so thoroughly and patiently), and considering that my OP question was foremost placed as a follow-up to "In general relativity, are light-like curves light-like geodesics?", PSE/q/76170, I should go on to ask (roughly): "Given a set of events such that any one of them is 'on the light cones' of all remaining others, do all pairs of them constitute 'null intervals'?".
Jul
23
comment Are there more distinctive names of “null curves” with certain additional properties (absence of “chord curves”)?
ACuriousMind: "In [...] "Lorentzian geometry" as we know it today [...]" -- Honestly, that's at best secondary to me. First: How, from given observational data (e.g. having "stated/sent signals", "detected/received signals"), to derive unambiguous real number values (such as "Null", or "$\pi$", etc.) The masterplan: "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."
Jul
23
comment Are there more distinctive names of “null curves” with certain additional properties (absence of “chord curves”)?
ACuriousMind: "For 1."light-like" and "null" are synonymous" -- Really!? ... I looked up what I suppose(d) to be the source of all that "-likeness" terminology: H. Minkowski, "Raum und Zeit", (1909). There appears (in translation) "space-like", and "time-like", but, curiously, no mentioning of "light-like". Instead: of certain "cones" having to do with "sending light" or "receiving light"; i.e. surely what's called "light cone". So I'll try to be more careful ...
Jul
23
comment why are the anthropometric units (which are about as big as we are) as large as they are relative to their corresponding Planck units?
@robert bristow-johnson: "user12262, so far nothing you have told me is anything i hadn't known before. i know it's a question about a dimensionless quantity" -- Then please consider changing the question title accordingly and/or asking instead about something you don't know yet. (For instance: the value of the ratio $$\frac{G~m_e^2}{\hbar~c}$$ that might be found in the next trial ...) "i think that this is a constant and any small trial-by-trial variation you might have is experimental error or "noise"." -- Something might be learned from the applicable systematic uncertainties, too.
Jul
23
revised Are there more distinctive names of “null curves” with certain additional properties (absence of “chord curves”)?
(v3.1415: defined the "cord curves" explicitly as "segments". And specified that the question about "geodesics" is in this case a question about "null geodesics".)
Jul
23
comment Are there more distinctive names of “null curves” with certain additional properties (absence of “chord curves”)?
ACuriousMind: "space-like chord curves are uninteresting [...] Thus, being a null geodesic indeed implies that, locally, there are no time-like chord curves." -- 1:Conversely, my question still is: Is there a null curve without any time-like chord curves (whether "local", or otherwise) not called a "null geodesic"? (Also, I wonder how the term "light-like curve" relates to either of these ...) 2:Could you please make explicit (and surely you've got the means to try) how to determine for a null curve with (at least) one time-like chord curve whether that's "local", or not?
Jul
23
comment Are there more distinctive names of “null curves” with certain additional properties (absence of “chord curves”)?
ACuriousMind: "You might be interested in this math.SE [MSE/q/502752] post, indicating that all points in more than two Lorentzian dimensions can be joined by a spacelike curve [...] constructed from taking a small tube around $\gamma$ [...] and letting the spacelike path wind around that tube." -- Ah, of course! (+1. That's explained so well, I put off looking at MSE/q/502752 for now ;) I'd even give +3 for your kind use of that "seat-of-my-pants" terminology "chord curve". (Did I indeed guess the 'technical term'? Hawing-Ellis is expensive! ...)
Jul
23
comment why are the anthropometric units (which are about as big as we are) as large as they are relative to their corresponding Planck units?
@robert bristow-johnson: "there is a question for physicists: why is the Bohr radius (approx the size of atoms) $10^25$ times bigger than the Planck Length?" -- Yes, that is certainly a question of great physics interest. Note that it is a question about a dimension- less quantity (i.e. plainly a real number value). And btw.: at least as far as I know it does not follow from the definition of (how to measure) this quantity that the values which would be found, trial by trial, had to be equal to each other (such that the quantity could rightfully be called "a constant").
Jul
22
revised Are there more distinctive names of “null curves” with certain additional properties (absence of “chord curves”)?
(v3.141: interchanged the index counts of the coordinate range, to better match the pattern "{ t, x, y }". Also corrected some other inconsequential oversight.)
Jul
22
comment In general relativity, are light-like curves light-like geodesics?
I've submitted the question "Are there more distinctive names of “null curves” with certain additional properties (absence of “chord curves”)?", PSE/q/127718 as a "formal" follow-up question. The purpose is foremost to establish some relation between terminology which is based on presuming manifolds, coordinates, and somesuch, and terminology describing observational content, such as signal fronts (light).
Jul
22
revised Are there more distinctive names of “null curves” with certain additional properties (absence of “chord curves”)?
(v3.141: put it the missing "t" component ot the example "chord curve".)
Jul
22
asked Are there more distinctive names of “null curves” with certain additional properties (absence of “chord curves”)?
Jul
22
comment In general relativity, are light-like curves light-like geodesics?
So, I wonder how to call a curve (if there are any) which is "everywhere null" (as is $\gamma$ in the example), and which moreover has the property that it doesn't have any such "everywhere positive chord curve" (as $\theta$ in the example) at all. (Nor any "everywhere negative chord curve" between any two "points" of the very special null curve under consideration; as far as such "negative curves" might exist at all.) How would you/we call such a "very special null curve"? (I plan to ask that question formally, too.)
Jul
22
comment In general relativity, are light-like curves light-like geodesics?
joshphysics: "A curve is null provided its tangent is everywhere null, so yes $\gamma$ is null" -- Hmm ... "provided $R \alpha=1$." -- Sure (I had neglected to specify that above); that's what I meant by curve $\gamma$. Alright, so $\gamma$ is "everywhere null". But ... you do agree, don't you, that for any two distinct real numbers $a$ and $b$ (i.e. from the domain of $\gamma$) there is (at least) one other curve $$\theta : [a,~b] \rightarrow \cal M$$ such that $$\theta[~a~] = \gamma[~a~],\qquad \theta[~b~] = \gamma[~b~],$$ but curve $\theta$ is "everywhere positive", right? [contd.]
Jul
22
comment why are the anthropometric units (which are about as big as we are) as large as they are relative to their corresponding Planck units?
user12262 wrote: "Related (though not exhaustively covered): "Understanding the “$\pi$” of a rotating disk", PSE/q/121889" -- Oops!, sorry, I had misread the title of this question as asking about "dimension- less constants", of course. Now addressing your actual question instead: I doubt that "value of a dimensionful constant" is a meaningful notion at all, since it depends on arbitrary choices of "units". Physics is done entirely in terms of dimensionless real number ratios.
Jul
22
comment In general relativity, are light-like curves light-like geodesics?
joshphysics: "consider the following curve $$\begin{align} t(\lambda)=\lambda,\qquad x(\lambda)=R\cos(\alpha\lambda),\qquad y(\lambda)=R\sin(\alpha\lambda). \end{align}$$ Let's refer to it as$$\gamma : \mathbb R\rightarrow\cal M.$$ Now, for any two different real numbers $a$ and $b$ is$$ s[ \gamma[ a ], \gamma[ b ] ] = 0~?,$$ where $s$ is (a suitable generalization of) a metric, as induced by the signals exchanged between events which are elements of the region $\cal M$ under consideration. If not, should $\gamma$ really be called "null curve"?
Jul
22
comment In terms of physics, does the phrase “time slows down” mean the same thing as “things happen more slowly?”
user3169700: Adding to the above: in typical "twin problem" setups both protagonists are presumed to conform to some particular equal proper rate; e.g. their individual proper rates of developing liver spots having been and remaining equal throughout their lifes; $\nu_A=\nu_B$. That's why we consider twins, after all. Comparing the numbers of spots over unequal durations, say say $\Delta \tau_A$ (the duration of one twin from separation to reunion) vs. $\Delta \tau_B$ (the corresponding duration of the other twin) then the corresponding increases $\Delta t_A$ vs. $\Delta t_B$ are unequal, too.
Jul
22
comment In general relativity, are light-like curves light-like geodesics?
joshphysics: "consider the following curve $$\begin{align} t(\lambda)=\lambda, \qquad x(\lambda)=R\cos(\alpha\lambda), \qquad y(\lambda)=R\sin(\alpha\lambda). \end{align}$$ -- Along with that let's also consider curve $$\begin{align} t(\lambda)=\lambda, \qquad x(\lambda)=R\cos(-\alpha\lambda), \qquad y(\lambda) = R\sin(-\alpha\lambda). \end{align}$$ Does either or both of these curves correspond to "the propagation trajectory" of a light signal (e.g. a photon)? If so, should the given region, with its parametrization "$\mathbb R^{2,1}$" be called "three-dimensional flat space with [...]"?