Paradox of the twins in compact spaces

The paradox of the twins in compact spaces is an extension of the Paradoxe of the twins of Langevin within the framework of a space time of which the space part has a topology " fermée". More correctly expressed, the space part of these space times is a compact space.

Here, Bernard is satisfied to make the " turn of the univers" at constant speed while following a trajectory " fermée" space whereas Alain remains motionless there. Like Bernard never makes half-turn, it does not undergo any acceleration. The situation seems completely symmetrical between Alain and Bernard but they is false because in a space time of space part compacts (even if it is flat and static as it is the case of the static space time hypertoric for example) Restricted Relativity does not apply. Indeed, relativistic symmetry is violated there overall. In these space times, in spite of the absence of acceleration of the twin moving, this violation (at the origin of a family of privileged reference frames inertial that one can describe as motionless) is enough to raise the " paradoxe" dissymmetry of ageing of the twin in uniform translatory movement compared to the ageing of its motionless twin.

In fact, the dissymmetry of the relativistic effects appearing in the space times of compact space part, presents a strong analogy with the dissymmetry of these same effects in the referential in rotation (effect Sagnac, nonreciprocal temporal dilation of Lorentz applicable to the revolving clocks and nonreciprocal circumferential contraction of Lorentz applicable to the meters revolving). The comprehension of the dissymmetry of ageing of the twins, of the contraction of Lorentz of the meter of the twin moving and the analog of the Sagnac effect in compact spaces (anisotropy the relative speed of the light in the inertial reference frame of rest of the twin moving) is facilitated by this correspondence. These nonreciprocal effects were indeed already analyzed in the case of the referential in rotation.

There is however a difficulty. In the static space time hypertoric, for example, in a vicinity of any event, the metric one is of Minkowski. One could be tried to believe that the symmetry of the relativistic effects thus remains applicable. That can return the paradox of diverting Langevin there if one did not include/understand the difference between local symmetry and total symmetry. Indeed, the topology of this space time is the cause of the existence of privileged inertial reference frames. There is evil to admit it (and to admit the dissymmetry of the relativistic effects which results from this naturally) if one is a little too much accustomed to the absence of inertial reference frames privileged people clean to the space time of Minkowski.

This difficulty of comprehension is easily overcome if it were included/understood how to highlight, within this framework, a class of privileged inertial reference frames which one can describe as motionless inertial reference frames. One reaches that point while carrying out a synchronization of the distant clocks by transmission of light signals on the basis of two transmitters located halfway of these two clocks on each of the two portions of trajectory making the turn of the universe and passing by these two clocks (when the trajectory which passes by these two clocks closes again on itself, these two clocks moving at constant speed on this trajectory). The motionless inertial reference frames in this space time are characterized by the fact that the double synchronization of these clocks systematically gives place to a single definition of their simultaneity (not of conflict between two synchronizations, conflict of synchronization similar to the Sagnac effect when these clocks are moving at the same nonnull speed v). This method of double synchronization identifies the motionless inertial reference frames in the static space time hypertoric and at the same time makes it possible to define a universal simultaneity in it (simultaneity having course in the motionless inertial reference frames in this space time).

We will look at only one situation a little more closely.

Alain is motionless in a Espace-temps static hypertoric but it does not know anything of it. As long as it has access only to local effects, it believes that its space time is a space time of Minkowski. Indeed, locally, it is not informed of the total effects (violating relativistic invariance) which would enable him to carry out its error. Conscious of the only local effects, he thus believes naively that the concept of immobility does not have a physical direction in its space time (the absence of objective concept of immobility expresses invariance known as of Lorentz i.e. the relativity of the translatory movement at constant speed).

Bernard makes a turn of universe. He leaves Alain and returns close to him after a certain time (which can be very long if dimensions characteristic of the universe are very large, it is not the kind of experiment in experiments easy to realize, contrary to the other situations under consideration within the framework of the revolving reference frames).

The two reference frames are inertial and it seems that one can apply the reciprocity of the dilation of time. Thus Alain, indicator Bernard moving, notes (indirectly by information exchange with colleagues located in the same inertial reference frame) that the clocks of Bernard go less quickly than them his. When Bernard returns at Alain, Alain notes that Bernard is indeed younger than him. It is thus well Bernard who travels says itself it rightly.

Bernard being in an inertial reference frame, he believes capacity to apply the same reasoning. He sees Alain moving away at high speed. When he sees it returning near him (on other side), he must however go obviously. Alain aged much more than him. It is thus Alain (and not him) which remained motionless.

We fall down on the Paradoxe of the twins of Langevin and it is solved in a similar way. Relativistic symmetry does not apply to this situation. Indeed, the static space time hypertoric does not respect relativistic symmetry overall.

This situation is more difficult to justify when it was badly included/understood. The explanation below itself is based on the study of compact spaces as on a detailed effect Sagnac analyzes. It is not the kind of thing which one explains in five minutes (at least if one makes a point of initially giving heaps of small details without much interest). The study of the Sagnac effect shows the traps of the complicated reasoning clearly, not very useful complication to include/understand what is essential because likely to make appear complicated things finally rather simple. This way complicated to present simple things often results from a bad comprehension. It rises from the erroneous attribution of a character of general information to the respect (validates only within the framework of the space time of Minkowski) of a total invariance of the laws of physics with respect to the actions of the group of Poincaré.

The paradox of Langevin was sometimes used by certain amateurs of relativity to affirm that in compact spaces Restricted Relativity leads to a paradox… It is, of course, completely normal since Restricted Relativity (in particular the symmetry of the relativistic effects) does not apply to it.

Certain amateurs not having included/understood the bases of relativity, seem to have let escape the analogy between nonthe reciprocity from the relativistic effects in the space times of compact space part and in the revolving reference frames. More important still, some did not understand that the metric one of Minkowski is valid everywhere in the static space time hypertoric (a particular case of space time of space part compacts), but that does not make it possible therefore to apply Restricted Relativity to it (see the references).

Let us return to the explanation of the paradox of Langevin in the static space time hypertoric.

In the case of Alain, not of problem. Its reasoning is correct because it is motionless in a space time of space part compacts whose metric one is the metric one of Minkowski. It is in fact identical to the explanation of the traditional paradox except that Bernard returns towards Alain without making half-turn. Instead of having the traditional diagram:

One a:

Calculations are the same ones.

Let us not forget that the space time considered is not a space time of Minkowski. The dilation of time is not reciprocal there. In fact, it is there that the difficulty is, the question is posed: as long as Alain moves away a little Bernard one can apply the reciprocity of the dilation of time. From when isn't it valid any more?

The answer is simple. The apparent reciprocity of the dilation of time is an illusion due to a lack of information on the total effects violating invariance of Lorentz (the relativity of the movement). The turn of universe has a crucial role: it allows Alain and Bernard to compare their respective ageings directly (without recourse to an arbitrary choice of inertial reference frame of synchronization). That revealed dissymmetry due to the privileged character of the motionless inertial reference frame of Alain. Alain remaining motionless, it is of course older than Bernard at the time of their meeting again.

Knowing the speed v of Bernard and the length L of the trajectory " fermée" traversed by Bernard, you can have fun with " calculer" (the term is a little strong for " calculs" also simple) the clean duration T=L/v of the voyage of Bernard, then more important ageing T0= \ frac {T} {(\ frac {1-v^2} {c^2}) ^ {(1/2)}} of Alain during the voyage of Bernard.

You can as amuse you to analyze this situation under the angle of the Doppler effect by not forgetting as the universe is " fermé". If Bernard moves on geodetic whose space projection (the trajectory) is " fermée" from length L and if it sends light signals in the two directions along its trajectory, it receives them both but not at the same time. The signals follow geodetic lights in a space time whose topology is not that of IR^4 (topology of the space time of Minkoski). The trajectories of the luminous rays are connected, of course, with helicoids traced on a tire tube. The shift delta T of the moments of reception of the light signals emitted at the same time by Bernard (when he traverses a trajectory closed length L at the speed v in the static space time hypertoric) is similar to that of the Sagnac effect. One has, obviously:
delta T = (\ frac {L} {(c-v)}- \ frac {L} {(c+v)}) (\ frac {1-v^2} {c^2}) ^ {(1/2)}

Lastly, you can have fun to make small simple calculations allowing to see whether you included/understood well. Why not calculate the tensile stress induced by the contraction of Lorentz for a snap ring without mass turning in this space time along a closed trajectory minimal length in the static space time hypertoric (it is a very simple calculation which takes less than one line)?

See too

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