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Jan
4
comment Comparing durations for two simply described motions in Schwarzschild geometry
John Rennie: "[...] watching from afar (which makes you a Schwarzschild observer)" ... surely there's an appropriate more specific (and coordinate-free) description of what you mean by "Schwarzschild observer (wrt. object $M$)" ... "if you start your stopwatch when [you saw that] $A$ passe[d] $B$ then the next time [you saw that] $A$ passe[d] $B$ your stopwatch will show a time $t$. The equations for the time measured by $A$'s clock and $B$'s clock are given [as ...]" -- No, no, no! Instead: If these "t" values of these three clocks were related as stated then they ran equally.
Jan
4
comment Comparing durations for two simply described motions in Schwarzschild geometry
John Rennie: "An object is following a geodesic if it is falling freely. [...]" Alright, thanks. Sorry for responding so late; partly due to the holiday break, partly because my question is quite shallow and rather an expression of my surprise about ... the remarkable inequality of durations arising in such a seemingly simple setting. I'll rather reward (and/or question) your more detailed derivation elsewhere. Here's just a general objection: [continued]
Jan
2
awarded  Popular Question
Dec
22
asked Comparing durations for two simply described motions in Schwarzschild geometry
Dec
4
answered Is it possible to learn about an event that occurred outside of your observable universe?
Dec
4
reviewed Approve Guess the wave function in a given potential
Dec
3
revised Questions arising from the presentation of “Schild's ladder” in “Gravitation” (Misner, Thorne, Wheeler)
(v3.14159: reference corrected)
Dec
3
revised Questions arising from the presentation of “Schild's ladder” in “Gravitation” (Misner, Thorne, Wheeler)
(v3.1415: formatting)
Dec
3
asked Questions arising from the presentation of “Schild's ladder” in “Gravitation” (Misner, Thorne, Wheeler)
Nov
27
comment Angular velocity from orientational displacement
@Markus Fjellheim: "I found a solution where" -- Thanks for letting me know. I plan to look and possibly comment or vote on your answer later today. Without having done so, for now just some thoughts: "the difference vectors can be [...] zero" -- Interesting idea. If only one difference vector is zero, but not the other, then I suppose/believe that the constant vector gives the rotation axis. If both difference vectors are zero then angular speed seems ambiguous by some multiple of $2~\pi / \tau$, and the direction of rotation is completely undetermined.
Nov
26
comment Is max speed of causality (light) proven experimentally?
@Benito Ciaro: "signaling mechanism [...] Why stop there, just because we don't yet [...]?" -- We're considering thought experiments where each participant is supposed to be able to judge whether some signal had been observed (first) regardless of whatever "mechanism"; as a prerequisit for even being able to define various "mechanisms" and (possibly) attributing the signal front to one particular such "signaling mechanism". The remaining difficult practical questions: How sensitive is some actual detector to some definite "sig. mechanism"?, and: What's its actual index of refraction?
Nov
26
comment Is max speed of causality (light) proven experimentally?
Benito Ciaro: "if $c_0$ were speed of sound or the speed of carrier pigeons." -- $c_0$ is signal front speed (cmp. the link given in the answer). In the hypothetical case that a carrier pigeon has landed on your shoulder and that only on this occasion you're learning for the very first time that it had taken off from "somewhere else" at all, then the speed of this carrier pigeon (wrt. yourself and the suitable "starting pad") is correctly evaluated as $c_0$. (More typically you can see a pigeon already as it approached you, thus learning that it had started before it lands on you.)
Nov
26
comment Is max speed of causality (light) proven experimentally?
@Benito Ciaro: "your argument says nothing about a maximum." -- I wrote already: "the signal front speed $c_0$ constitutes a maximum: any signals having been exchanged, as far as a speed value can be attributed at all, had at most the speed of the corresponding signal front." I can add: the symbol $c_0$ is not "infinity" either, because a finite ping duration should define a finite distance value; and zero ping duration should define zero distance. "only talks about the speed of communication between participants." -- Sure. What else? [continued]
Nov
26
comment Angular velocity from orientational displacement
@Floris: "presumably in OP's case the vectors are "real life" and therefore have some relationship" -- The OP is specificly asking to consider an applicable case ("a,b") ("A 3-D object is rotating around an unknown 3-D axis [...]"); rather than asking "How do we determine whether two vector pairs arose from a rotation, or not?". If answers to the actual OP question may serve to characterize inapplicable cases ("p,q") as well, that's just a bonus. "Add measurement error [...]" -- IMHO that's deserving of a separate question. (Which I'd find interesting and might get around asking.)
Nov
25
comment Angular velocity from orientational displacement
@Floris: "But [...]" -- As far as I (am beginning to) understand what you meant by "the problem being overconstrained", I agree: there are examples of two vector pairs, say $\vec p_0,\vec p$, and $\vec q_0,\vec q$, with non-parallel vector differences and even with $$\|\vec p\|=\|\vec p_0\|,\qquad\|\vec q\|=\|\vec q_0\|,$$ which are not related by the same one rotation (described though angular velocity $\vec\omega$, and duration $\tau$). They fail to have equal "corresponding expressions" for determining a consistent value of angular speed; in contrast to the case considered in my answer.
Nov
25
comment Angular velocity from orientational displacement
@Floris: "I'm not convinced there is a unique solution for any combination of $a$ and $b$. Is there?" -- If the vector differences $\vec a - \vec a_0$ and $\vec b - \vec b_0$ are parallel to each other then the direction of the angular velocity $\vec \omega$ is not uniquely determined. (This seems obvious in "my approach", and should therefore apply to "your approach", too.) Also, there's an ambiguity in the magnitude; related to the periodicity/multiplicity of $\text{ArcCos}$. But (... obviously ... &) trials may be restricted so that $\| \vec a - \vec a_0 \|$ only increased.
Nov
25
comment Angular velocity from orientational displacement
@Floris: "This may turn out to be simpler than my approach..." -- Yeah. At least, by "some algebra", I find that our answers agree in determining the direction of $\vec \omega$. If "my approach" should be not immediately obvious, I could add that, obviously, $\vec a$'s "component along" $\vec \omega$ should remain constant; thus $$\frac{\vec \omega~(\vec a \cdot \vec \omega)}{(\| \vec \omega \|)^2} = \frac{\vec\omega~(\vec a_0 \cdot \vec \omega)}{(\| \vec\omega \|)^2}. $$ So $$(\vec a \cdot\vec\omega) = (\vec a_0 \cdot\vec\omega)$$ and immediately $$((\vec a - \vec a_0)\cdot\vec\omega)=0.$$
Nov
25
answered Is max speed of causality (light) proven experimentally?
Nov
24
revised Angular velocity from orientational displacement
(v3.1415926: too, too.)
Nov
24
answered Angular velocity from orientational displacement