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In the Wikipedia article on the twin pararadox, there is an interesting chapter which calculates the difference of age for the twins, with steps of accelerated movement, and steps with constant speed, for the traveller twin.

We simplify here the Wikipedia example in deleting the constant speed steps (so $T_c=0$), we have :

$$ \Delta \tau = \frac{4c}{a} {\rm arsinh} (\frac{aT_a}{c}) \qquad, \Delta t = 4 T_a\tag{1}$$ where $a$ is the proper acceleration of the traveller twin, relatively to the sedentary twin, $T_a$ is the (sedentary twin) time necessary for the traveller twin to go from a speed zero to a maximum speed (and vice-versa), $ \Delta \tau$ is the elapsed time for the traveller twin, $ \Delta t$ is the elapsed time for the sedentary twin. We have : $\Delta \tau < \Delta t$

Now, certainly, there must be an asymetry about proper acceleration, that is : the proper acceleration of the sedentary twin, seen by the traveller twin, has to be different (in absolute value) with $a$.

I am looking for a rigourous argument, and I have only some hints. I think there is a symmetry between relative speeds $\vec v_{S/T} = -\vec v_{T/S}$, and this should extend to the spatial part of the $4$-velocities $\vec u_{S/T} = -\vec u_{T/S}$, while the time part of the $4$-velocities $u_{S/T}^0 =u_{T/S}^0$should be equal. The problem, I think, seems to come from the definition of the proper acceleration $ \vec a = \frac {d \vec u}{dT}$, where $T$ is the time of the observer. The symmetry would be broken, because, in one case (proper acceleration of the traveller twin), one hase to use $dT=dt$, while in the other case (proper acceleration of the sedentary twin), one has to use $dT=d \tau$, so, there would not exist a symmetry between $\vec a_{T/S} = \frac {d \vec u_{T/S}}{dt}$ and $\vec a_{S/T} = \frac {d \vec u_{S/T}}{d\tau}$

Is this correct ?

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[...] $a$ is the proper acceleration of the traveller twin, relatively to the sedentary twin

Surely the "sedentary twin" is therefore supposed to be (and to remain) a member of some particular inertial system; the "sedentary twin" is supposed to be (and to remain) at rest wrt. other participants (which may be thought-experimentally assumed) throughout the experiment.

The "sedentary twin" and other members of this inertial system are therefore capable of measuring relevant distance ratios and duration ratios (such as ratios in comparison to $T_a$) between each other, and consequently capable of measuring the speed $v_{S/T}[ t ]$ at which the "traveller twin" passed various members of the inertial frame,
and of measuring whether the "traveller twin" passed various members of the inertial frame corresponding to hyperbolic motion, characterized by some particular non-zero number $(a/c) \, T_a$.

If so then the "traveller twin" is not a member any particular inertial system throughout the experiment. The motions of the two twins under consideration are thus explicitly asymmetric.

I think there is a symmetry between relative speeds

Certainly not in terms of explicit, direct measurement; in particular because the "traveller twin" is unable to determine any (non-zero) values of distance ratios since he is not at rest wrt. any other participants throughout the experiment. (At most the "traveller twin" and suitable other participants may find constant ratios of ping durations between each other, i.e. determine that they had been rigid to each other, in this sense. However, such ratios of ping durations are not symmetric and thus not distance ratios, but at most ratios of quasi-distances.)

The relation

$\vec v_{T/S}[ t ] = -\vec v_{S/T}[ t ]$

is therefore at best an assignment,
defining "$\vec v_{T/S}[ t ]$" in terms of the actually measurable velocity $\vec v_{S/T}[ t ]$.

Consequently ...

[...] in the other case (proper acceleration of the sedentary twin) [...]

... attributing some non-zero value of "proper acceleration" to the "sedentary twin" (who is and remains at rest wrt. all other members of "his" inertial system) is nonsense at best unphysical.

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a) Each observer, inertial or not, may send or receive signals. So, the ability of measuring does not depends on the inertial status of the observer. b) Maybe the relative speeds are not opposite, so what is their relation ? - There is certainly something asymetric, I agree with that, but I am not really convincing by your arguments. It is very possible that there is a link between the possibility of defining a proper acceleration and the inertial status, but you need to add precisions. – Trimok Sep 11 '13 at 7:12
@Trimok: "(a) Each observer, inertial or not, may send or receive signals." -- Right; that's the usual convenient tought-experimental assumption. For this discussion we may even grant outright that each observer can therefore measure (compare) ping durations to all others. "So, the ability of measuring does not depend on the inertial status of the observer." -- But measuring what, explicitly? "Sedentary" members of the same IS will obtain distance ratios (from the ping duration ratios) among each other; and in turn values of $\beta_{T/S}$ and $a$ for other participants. --> – user12262 Sep 11 '13 at 18:04
(Contd.) In contrast, consider the (one) "traveller" (who was found having moved hyperbolically by the IS members): What can he conclude specificly from his observations and (granted) ratios of ping durations? For starters, does he obtain distance ratios to any other participants? (To whom?, in terms of motion characterized by the IS members.) If not, then this "traveller" can't obtain explicit measurements of $\beta_{S/T}$ either. – user12262 Sep 11 '13 at 18:05
This previous post is clear. Proper acceleration is a Lorentz invariant quantity, so the only meaning of a proper acceleration is relatively to a inertial frame (that is, a standard Minkowski metrics). My hypothesis, in the question, is wrong (there are not symmetry between speeds). However, I still think that the movement of a inertial point from a non-inertial frame could be considered, but it is more complex, it does not involve a Minkowki metrics, etc... I have some ideas about it, and this is perfectly coherent. – Trimok Sep 13 '13 at 8:09
@Trimok: "This previous post is clear." -- Sorry, I myself consider attempts at explanations in terms of "feeling an acceleration", coordinates and somesuch utterly unclear and unacceptable. "My hypothesis [...] is wrong (there are not symmetry between speeds)." -- My point was: it is not even wrong (and much less right). The quantity $-v_{T/S}$ of your question is not a "speed" relating your setup in the first place. Also please note… Thanks for the +25 rep, though (I may yet begin to enjoy being social ...) – user12262 Sep 13 '13 at 15:19

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