
#37
Feb1413, 08:05 PM

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PF Gold
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If what you mean by "the twin paradox" is the question of what's wrong with the calculation that finds that the Earth twin is younger, then there are a number of final definitive resolutions, for example the observation that the time dilation formula doesn't apply since the astronaut twin's world line doesn't coincide with the time axis of any inertial coordinate system. I like to supplement this with a diagram that shows the simultaneity lines of the two inertial coordinate systems that are comoving with the astronaut twin before and after the turnaround. 



#38
Feb1413, 08:48 PM

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It is an interesting pdf. It says that negative alpha is inconsistent with the group that they have defined. So I guess that a grouptheoretic description allowing negativealpha would need a different group to the one they define. I wonder if there is an easy adaptation to get such a group. Maybe I'll think about it later when I am more awake :)




#39
Feb1413, 09:07 PM

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http://arxiv.org/abs/astroph/9909311 (which is also in Phys Rev D), and http://arxiv.org/abs/astroph/0111306 (which appears not to have been published in a peerreviewed journal). Your "open interval around v=0" hypothesis is (essentially) equivalent to the above: there must be a continuous set of physical velocities containing v=0 or the theory is useless. 



#40
Feb1413, 09:15 PM

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So if we want to allow K<0 for some reason, we either have to take the set of allowed velocities to be full of holes, or we allow transformations with infinite speed, i.e. ##\Lambda\in G## such that ##(\Lambda^{1})_{00}=0##. Buuuut...here's something I learned very recently: That would imply that the zerovelocity subgroup of the proper and orthochronous subgroup is not the group of all rotations of space. That sounds undesirable too. 



#41
Feb1413, 09:23 PM

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#42
Feb1413, 09:29 PM

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http://en.wikipedia.org/wiki/Gyrovector http://en.wikipedia.org/wiki/Beltram...%93Klein_model 



#43
Feb1513, 05:32 AM

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Stepanov's paper is really weird, it states as its goal to demonstrate that the second postulate is not necessary only to actually proof in sections 5 and 6 that additional assumptions are needed, that ultimately come down to choose a positive alpha by empirical means. This looks suspiciously similar to the second postulate wich even though it was introduced as a postulate, it's now closer to an empirical fact that leads us to reject Galilean transformations(alpha=0), and alpha<0 is similarly straightforward to discard empirically.




#44
Feb1513, 11:57 AM

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#45
Feb1513, 12:41 PM

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$$u\oplus v=\frac{u+v}{1\frac{uv}{c^2}}.$$ This set must be such that the righthand side of this formula is welldefined for all u,v in the set, i.e. it can't contain u,v such that uv=c^{2}. In particular, it can't contain c or c. If S is such a set, then $$\left\{\left.\frac{1}{\sqrt{1+\frac{v^2}{c^2}}} \begin{pmatrix}1 & v/c\\ v & 1\end{pmatrix}\rightv\in\mathcal S\right\}$$ would be a group that could in principle be used in a theory of physics. But I think any such theory can easily be ruled out by experiments. 



#46
Feb1513, 01:23 PM

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#47
Feb1513, 01:27 PM

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#48
Feb1513, 03:34 PM

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Roughly, the axioms are 1. Defines the domain of the maths. (Looks like the field R^{4}) 2. For any inertial observer, the speed of light is the same everywhere and in every direction, and it is finite. Furthermore, it is possible to send out a light signal in any direction. 3. All inertial observers coordinatize the same set of events. 'Subaxioms' 4a. Any inertial observer sees himself as standing still at the origin of his coordinates. 4b. Any two inertial observers agree as to the spatial distance between two events if these two events are simultaneous for both of them; furthermore, the speed of light is 1 for all observers. The first theorem derived from these five axioms is : no inertial observer will 'coordinatize' another inertial observer as travelling at v >= c. The proof is clunky and inelegant, however ( just my opinion). After this they adopt the Poincare group of transformation as the change of coords between inertial frames. But if the Poincare group is adopted as an axiom, the first theorem (and possibly axiom 4b) is redundant and the theory is more elegant. So, it doesn't live up to my expectations, but still has some interesting points. I may finish reading it sometime. 



#49
Feb1513, 06:31 PM

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I.e., it only makes mathematical sense in a situation where all observers are at rest relative to each other. Thus, it is indeed "pretty much useless" for physics. Edit: In more detail... Consider the velocity addition formulas in Fredrik's treatment for both cases. 1) First the "negative alpha" case. In units where c=1, the velocity addition formula is $$ u' ~=~ \frac{u + v}{1  uv} ~=~ \frac{\tan\eta_u + \tan\eta_v}{1  \tan\eta_u \, \tan\eta_v} $$where ##\eta_u, \eta_v## can take any real value. (In the +ve alpha case, we'd call them "rapidities".) We now ask: on what domain of ##\eta## is this formula welldefined? Clearly, if ##\eta_v=0##, the denominator is always 1, so that's ok but rather boring since it just means ##u'=u##. But now suppose ##\eta_v\ne 0##. In that case, for any value of ##\eta_v## I can find a value of ##\eta_u## such that the denominator becomes 0. This is because the range of the tan function is ##\pm\infty##. We conclude that this formula is only welldefined on the trivial domain consisting of the single value ##v=0##. 2) Now the "positive alpha" case. Staying with units where c=1, the velocity addition formula is $$ u' ~=~ \frac{u + v}{1 + uv} ~=~ \frac{\tanh\eta_u + \tanh\eta_v}{1 + \tanh\eta_u \, \tanh\eta_v} $$where, as before, ##\eta_u, \eta_v## can take any real value. In this case, we can validly call them "rapidities". In this case, even if ##\tanh\eta_v < 0##, there is no value of ##\eta_u## which makes the denominator 0. This is because the range of the tanh function is bounded by ##\pm 1##, and it approaches these limits only asymptotically. Therefore, on the (open) domain of velocities such that ##v<1## the transformation is welldefined. With closer analysis, one can also show that the limit as ##v\to 1## remains sensible  in that the result of the "addition" always approaches 1 in that limit. So we can complete this to a closed domain in various situations by taking limits carefully. 



#50
Feb1513, 08:23 PM

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There's a sign error in the "positive alpha" case. The velocity addition formula is
$$u'=\frac{u+v}{1+\alpha uv}$$ in both cases. So the denominator is 1+uv when ##\alpha=1##, and 1uv when ##\alpha=1##. $$(\theta,\eta)\mapsto\begin{cases}\theta+\eta &\text{if }\pi/2<\theta+\eta<\pi/2\\ \theta+\eta\pi &\text{if }\theta+\eta>\pi/2\\ \theta+\eta+\pi &\text{if }\theta+\eta<\pi/2\end{cases}$$ Maybe there's a nicer way to state this definition. Edit: Hey, isn't ##(\pi/2,\pi/2)\cap\mathbb Q## such a set? Two rational rapidities can't add up to a forbidden value like ##\pi/2## (infinite speed) or ##\pi/4## (speed =c=1), because the forbidden rapidities are all irrational. 



#51
Feb1513, 09:05 PM

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Consider: $$ e^{\eta K} e^{\eta K} ~=~ e^{2\eta K} $$ so by composing transformations I can get arbitrarily large rapidities. Any restriction on the values of ##\eta## must also be compatible with arbitrarily many such compositions, else we don't have a group. The only such valid restriction (afaict) is the restriction to ##\eta=0##, i.e., a group consisting trivially of the identity and nothing else. 



#52
Feb1513, 09:58 PM

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There is of course nothing wrong with such assumptions, but I'd like to point out two things. 1. This assumption implies my 1b, and is much stronger than my 1b. 2. This assumption is not one of the statements that turns the principle of relativity into a mathematically precise statement. Rather, this assumption should be thought of as making "boost invariance" mathematically precise. This is when we are talking specifically about rapidities. If the parameter had been a position or an Euler angle, it would have been part of making the principles of "translation invariance" (="spatial homogeneity") or "rotation invariance" (="spatial isotropy") precise. So I think we have to consider ##\alpha<0## with a nontrivial set of allowed rapidities to be consistent with the principle of relativity, but not consistent with these other principles. 



#53
Feb1513, 10:41 PM

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