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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?
@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 [...]"?
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?
Related (though not exhaustively covered): "Understanding the “$\pi$” of a rotating disk", PSE/q/121889
Jul
22
comment In terms of physics, does the phrase “time slows down” mean the same thing as “things happen more slowly?”
user3169700: "Ok [...]" -- Glad to hear. "but what do you mean by "under the condition of equal (proper) rate?"" -- Given the ordered set $\cal A$ of indications of one participant, and considering one specific ("way of") parametrizing these indications: $$t := \cal A \rightarrow \mathbb R,$$ then the corresponding proper rate (or frequency) is $$\nu := \frac{d}{d\tau}[~t~]$$ (if the derivative exists; otherwise consider "average proper rate"). The parametrization $t$ may for instance be given as "number of ticks", or "number of liver spots", etc.
Jul
22
comment Why Light and Observers have different laws of physics
@ACuriousMind: "I'm voting to close [...]" -- I agree with your vote (FWIW while I'm not (yet) permitted to vote on such matters) because the OP has greatly "moved the target" of the question, and not dealt with the substance of answers provided so far. However: "because it is about nothing well-defined. You keep throwing words around like "the laws of physics", [...]" -- Well, we can hardly blame the OP when he (Derek Roberts) and the larger public is confronted with ill-defined assertions such as "A law of physics is just some set of equations that we use to predict what happens."
Jul
22
answered In terms of physics, does the phrase “time slows down” mean the same thing as “things happen more slowly?”
Jul
21
revised The Euler equations as a RNG fixed point
corrected spelling the name of L. Prandtl; cmp. http://en.wikipedia.org/wiki/Prandtl_number
Jul
21
suggested suggested edit on The Euler equations as a RNG fixed point
Jul
21
revised Saturation of the Cauchy-Schwarz Inequality
corrected spelling the name of H. A. Schwarz, cmp. http://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality; also some copy-editing.
Jul
21
suggested suggested edit on Saturation of the Cauchy-Schwarz Inequality
Jul
21
revised Coupling constant in electroweak theory
corrected spelling the name of S. Weinberg; cmp. http://en.wikipedia.org/wiki/Weinberg_angle