The paradox of the twins is a Expérience of thought evoked by Albert Einstein then published in detail by Paul Langevin in 1911 in order to illustrate an aspect of the restricted Relativité: the one duration measurement depends on the reference frame in which this measurement is taken. In this experiment, one of the twins remains on Earth while his/her brother makes a return ticket in the rocket at a speed close to that of the light. Because of the Dilation of time, the duration of the voyage measured by the sedentary brother is longer than that measured by the brother traveller. When they are found, the traveller is younger than his twin remained on Earth.

If the experiment of the twins is paradoxical with the direction where it unquestionably runs up against the current Intuition, it is not it with the direction where it would lead to a logical Contradiction. Such a contradiction emerges when one wrongly applies without precaution a principle of relativity of the Mouvement while asserting owing to the fact that it is indifferently one or the other of the twins who is animated an high speed and returns younger. However a twin cannot be at the same time more young and more old man but his brother at the time of the meeting again. Actually, the situation of the twins is not symmetrical: the sedentary coincides with only one Repère galiléen (that of the Earth) throughout all voyage, while the traveller carries out a half-turn and thus coincides with at least two reference marks galiléens.

Lastly, the scenario of the twins of Langevin was invented to illustrate a result of restricted relativity and of nothing utilizes the General relativity, in particular for the good one and simple reason which restricted relativity is able to deal with the problem of accelerated rocket.

## History

About the dilation of time predicted by restricted relativity, Albert Einstein indicates in 1911 If we place a living organism in a box… one could arrange itself so that this organization, after a time of flight as long as wanted, can turn over to its place of origin, hardly deteriorated, while the corresponding organizations, which remained in their initial position would for a long time have yielded the place to new generations. Because for the organization moving, the great duration of the voyage was a short moment, provided that the movement were carried out almost with speed of light. The same year, Paul Langevin develops this experiment of thought in a form which passes to the posterity: “the ball of Jules-Verne Langevin” where he reports in a realistic way temporal unfolding of the life of two twin brothers of which one travels at a speed close to the light and the other remains on Earth. This talk, at the time of the Conference to the Congress of philosophy of Bologna in 1911, makes it possible to popularize the notion of time in relativity and to illustrate the philosophical revolution which it brings. From a physical point of view, the problem is solved by using the Transformation of Lorentz relative to reference frames galiléens.

The name of “Paradoxe” however was given to the experiment of thought by the detractors of restricted relativity who saw an inconsistency of the theory there: indeed, those asserted symmetry of the problem because of the relativity of the movement and concluded that, whatever the twin chosen, this one were to measure one duration shorter than his brother and to be found young person at the time of the meeting again, from where a contradiction. This position was in particular supported by the Philosophe Herbert Dingle in the Années 1950. This reasoning is however erroneous because of dissymmetry between the two twins, like the resolutions of Langevin and Einstein proves it.

## The traveller of Langevin

Let us imagine two twin brothers, call them Pantoufle and Fidgets. Slipper remains on Earth while his/her brother very starts a voyage in space at high speed, near of that of the light (what is in addition impossible to realize concretely). When Bougeotte returns on Earth, he discovers his Pantoufle brother older than him.

That can appear astonishing, but what one calls the Dilatation of time was mentioned by the relativity restricted since 1905 and it is to illustrate it in a striking way that Langevin invented its historiette. Fidgets moving at high speed compared to Slipper, this last note with the return of the traveller whom the clock embarked in the rocket functioned more slowly than his. For Slipper, Bougeotte aged less quickly.

Can one explain what occurs during the way? Let us consider a whole of observers at rest compared to the Earth (or, if it is preferred, moving at the same speed and remainder of this fact at an invariable distance from/to each other) and laid out along the trajectory of the rocket of Fidgets, to planet that this last will join. We will call them the “  observers terrestres  ”. These observers will have as a preliminary synchronized their clocks and thus use all the same hour. The phenomenon which the restricted Relativité provides is that each terrestrial observer notes in turn, when the rocket passes opposite him, that the hour of the rocket delays compared to his (that which represents the terrestrial hour of Slipper). When Bougeotte arrives on its remote planet it will be late over this terrestrial hour (as it will be able to note it on the clock giving on this planet the hour of Slipper).

If Fidgets now make half-turn to return on Earth, it will double its delay of the outward journey and will join finally his Pantoufle brother while being found with final young person than this last.

One can wonder whether the effects of reciprocity between two inertial frames of reference, which are clean theory of relativity, will not reverse the conclusions relating to the twin traveller when one takes another reference frame that related to the Earth and to precisely lead to a “  paradoxe  ” within the meaning of logical contradiction. Because by reversing the things, Pantoufle cannot be at the same time oldest and the least old. But there is no paradox because actually there is not reciprocity between Pantoufle and Fidgets. The first preserves compared to all Référentiel galiléen a rectilinear motion and uniform . In particular Pantoufle is at rest in the system consisted all the terrestrial observers. On the contrary the Bougeotte traveller successively has (possibly apart from the phases of acceleration) two simple movements (uniform, at constant speed) distinct since contrary direction. We will detail the reasoning below.

The question of knowing if the movement of Fidgets is uniform or not, i.e. if the rocket is accelerated or on the contrary actuated by a uniform movement, does not change anything with the reasoning. Restricted relativity can treat the two cases. For proof we will give further the relativistic formulas concerning a rocket subjected to a constant acceleration.

Little time after the publication of the article of Langevin, the experiment which he had imagined involved of many comments and the sometimes erroneous analyzes. It is on this occasion that the traveller of Langevin was used at some to disparage the cogency of the relativity restricted using the false reasoning of which here substance.

### A false reasoning

It is not very teaching to show what is false but the false following reasoning is quoted so much that it is perhaps useful to state it.

Reasoning faux : restricted relativity affirms that there is no absolute reference mark. Two reference marks moving relative at constant speed are completely equivalent. What authorizes us to affirm that it is Bougeotte which moves away from Slipper and not the inverse  ? From the point of view of Fidgets, the situation must be symmetrical. It is Pantoufle which moves at high speed and which traverses a long way before finding it.

the dilation of time is perfectly reciprocal in restricted relativity. Thus Bougeotte must as note to him as the clock of Slipper functions more slowly. And at the time of their meeting again, it is thus, from its point of view, Pantoufle which should be remained young.

Pourtant, once they are again together they should agree in the comparison of their age. However Slipper cannot be at the same time more young and more old man but Bougeotte.

We would have a paradox there. When Einstein posed the Principe of relativity, it had one of the intuitions certainly among most remarkable of modern physics. And symmetry called upon above is one of the consequences. There is not there, a paradox évident ? Is this obvious contradiction the proof of a fault in the théorie ?

Actually as we have just said it, the situation is not symmetrical. It is perfectly justified to say that Pantoufle is at rest and that Bougeotte moves because this last carries out a half-turn while Pantoufle remains in place.

### Concept of reference mark

For showing the falseness of the reasoning pleading well the symmetry of the movement of the two twins, let us look further into the concept of reference mark. One speaks about the reference mark of Fidgets and the reference mark of Slipper as if a reference mark were limited to only one place. However this point of view is inaccurate. In restricted relativity, a reference mark (or frame of reference) must thus think like a whole of rockets (or buoys of space) travelling in formation the ones following the others far from any mass revolving (in a space “  plat  ”), at the same speed and without their engines exerting acceleration (or braking). These rockets contain a clock, all indicating the same hour, visible outside (it will have had to be synchronized. -->

This dilation of the durations was always checked in experiments:

• One observed that the unstable particles disintegrate more slowly from the point of view of the observer when they are driven at high speed compared to this one , in particular in the particle accelerator .
• This effect is also observed for the atmospheric Muon S products by the collision of the cosmic rays (very energy particles coming from cosmic space) and the molecules of the atmosphere. These muons, animated a speed close to that of the light, reach the ground where they are observed and this in spite of their short duration of clean life, the dilation of time giving them the time necessary to reach the detectors.
• Another case observed of temporal dilation is the shift between atomic clocks on the ground and in flight; but it becomes complicated in this last case of gravitational considerations, and we leave the framework of restricted relativity then. It is remarkable that the real experiment of the atomic clocks embarked in the plane (or in train, or car…) or which remains fixed on the ground is a realizable version, and often realized experiment of the twins, where one replaces simply the spaceship by a plane, and an human being by an atomic clock. One measures then the difference of the durations measured by the clocks.
• Signalons also that one observes also this dilation of time over the duration of the curves of luminosity of the remote supernovas.

### Reciprocity

Let us notice that it is not so immediate to establish a reciprocity with the relation which we have just established between the measurement of time in the rocket and on Earth. Indeed we saw the particular part which we made play clean reference mark of the followed object, in this case the rocket. If thus we want to symmetrize the experiment, it would be necessary to place us on Earth and (in the same way that we located the trajectory of the rocket) to locate the movement of our planet compared to a “  train of fusées  ”, each one being equipped with its clock and its beacon of position. All in all, it would not be the same experiment about which we would speak.

To visualize a symmetrical situation would have to be imagined the following experiment. Let us consider two convoys of rockets, each one constituting a inertial reference mark, moving one with the other in parallel. On a side the Land ones, other Virginides. All the Land ones are at rest the ones compared to the others, their mutual distances are invariable and their clocks are synchronized by using the Doppler effect.

One can say that the relativistic Doppler effect is the conjugation of two phenomena. There is initially the traditional Doppler effect consistent in a change of frequency of the light signal due at the relative speed of the source and receiver. According to whether the observer flees or on the contrary catches up with the luminous wave train, the frequency which he will see will be smaller (it is a Décalage towards the red) or larger than the emitted frequency. There is then this famous dilation of time by the factor $- \left(v^2/c^2\right) ^ \left\{- 1/2\right\}$ precisely constituting the object of this article.

The first effect of shift in frequency depends only on the radial component relative speed between transmitter and receiver whereas the second relativistic effect depends him on the value total speed. Moreover whereas the traditional Doppler effect changes direction according to whether radial speed is a speed of approach or a speed of distancing, the relativistic effect always corresponds to a deceleration of the clock moving.

We here will free us from the traditional Doppler effect and will preserve only the relativistic effect by imagining the following experiment.

As prédédemment Pantoufle remains on Earth while his/her twin brother Bougeotte goes on a journey towards a remote planet at a speed close to that of the light then returns on Earth under the same conditions. The twins observe both a remote Pulsar whose glare varies periodically. To free itself as we wish it traditional modifications of frequency due at the radial speeds, we suppose that the distance from pulsar observed is much larger than the distance to planet visited by the traveller of Langevin and than pulsar is located in the direction perpendicular to the way of the rocket in the reference mark of the Earth.

All the interest of this experiment resides in what the pulsar constitutes a genuine cosmic clock beating at imperturbable intervals. How Slipper and Fidgets they will perceive this pulsar during the voyage ?

Obviously continuous slipper to see the flashes of pulsar following one another the same rate/rhythm, with one period of let us say one second (it perceives 60 flashes in 1 minute). On the other hand, Bougeotte in its rocket notes on its clock that the flashes follow one another more quickly, for example at the rate/rhythm of every 0,8 seconds (he sees 60 flashes in 48 seconds, or 75 flashes in 60 seconds). All occurs as so compared to the Earth (since the pulsar continues to beat the seconde !) the clocks of the rocket had slowed down (48 seconds are passed instead of 60). This experiment is interesting in the sense that it gives a certain consistency to this concept of deceleration of the clocks, which could appear somewhat unreal. However one can say that since the pulsar does not change a rate/rhythm it is well the embarked clock which modified to it his. It would be incongruous on behalf of Bougeotte to support that the pulsar beats more quickly since its rocket sinks in space…

The other done capital it is that inevitably, since they are found at the same point at the end of the voyage, the two twins will have counted the even many flashes . How that may it be  ? Answer facile ! Fidgets will have entered with a pulsar at the faster rate/rhythm of a flash every 0,8 seconds during an amount of time the shorter (let us say 40 years) same number of flashes as Pantoufle observing during 50 years a slower pulsar, with a flash every second. The number of flashes observed by Pantoufle is proportional to 50/1 (an/seconde). It is quite equal to the number of flashes listed by Bougeotte, proportional to 40/0,8 (an/seconde). (We included in this purely numerical application, and imaginary, a speed of the rocket equalizes with (3/5  c) which gives a factor of racourcissement of the age of Fidgets of $\ sqrt \left\{- \left(9/25\right)\right\} = 4/5$.)

Fidgets and Slipper make same calculation, with their own numbers, different, and find with final the same result.

### The choice of the inertial reference mark

We explained why it is the reference mark of Slipper on Earth which constitutes a inertial reference mark whereas that of Fidgets in its rocket is not it because the latter cannot be at rest compared to only one inertial reference mark. Since Bougeotte makes half-turn it is necessary, let us repeat it, at least two inertial reference marks per report/ratio to which it will be able to have successively, to go it then to the return, a null speed. Being private engines, the reference mark galiléens cannot carry out half-turn.

One often sees written (or one intends to only say) that what causes dissymmetry between the two twins it is the acceleration which Bougeotte and which must undergo to deal with this problem would have to be called upon general relativity, able to manage accelerations. If the acceleration undergone by the rocket is well some share the cause of dissymmetry, because to make half-turn it is necessary to ignite engines, it is necessary however to demolish this false idea according to which restricted relativity could not take into account accelerations. This right handing-over in question of the generally accepted ideas is extremely well presented by Misner, Thorne and Wheeler, which shows in a convincing way qu’ one can analyze movements accelerated by making use only of restricted relativity . After all, the equations of restricted relativity are enough to describe the particles whirling in the accelerating .

It is what we will show by treating the case of a rocket accelerated without calling upon general relativity.

### Calculation of the relativistic renovation in an accelerated rocket

#### Equations of the accelerated rocket

We follow here the presentation of Taylor and Wheeler.

First of all how restricted relativity treat does the reference marks accélérés ? Réponse : as she knows only the reference marks galiléens, therefore by definition not accelerated, she at every moment brings back the movement of the mobile considered to the inertial reference mark which at this precise moment the côtoie at the same speed. In other words restricted relativity at every moment uses the inertial reference mark with which the mobile is to some extent in coincidence . In this way the theory continues to make use of reference mark galiléens and can apply the corresponding formulas transformation.

Thus let us consider a rocket (imaginary!) whose passengers would be subjected thanks to powerful engines to a constant acceleration G equal to that of gravity that we undergo on Earth. Numerically g=10 (m/s) /s = 1000 (cm/s) /s. We want to know in how long this rocket will traverse such distance, while knowing in advance which the time measured by the passengers of the rocket will be shorter than the time measured by the land ones.

Let us notice that the acceleration of the rocket is not equal to G in the terrestrial reference mark . Acceleration G is measured in the rocket for example thanks to an embarked bathroom scales. As we mentioned above, restricted relativity states that the rocket is accelerated compared to the reference mark (the train of rockets) which at the moment considered has same speed as it. This reference mark plays the part of a inertial reference mark instantaneous. In this reference mark the speed of the rocket passes from value 0 to the value FD during time $d \ tau$. And to say that acceleration is worth G means that

$dv = G D \ tau \.$

In restricted relativity one knows that speeds are not additive. So of a rocket moving at the speed v compared to the Earth is drawn forwards a ball from gun at the speed W (compared to this rocket), the speed of the ball compared to the Earth is not v + W . One watch that the relevant quantity is actually a angle $\ theta$, that one can call the angular parameter speed, defined by the relation

$\ tanh \ theta = v/c \.$

In other words

$\ theta = \ mathrm \left\{atanh\right\} \, \left(v/c\right) \.$

The principal property of this angular parameter speed is qu’ it is additive . Thus our ball drawn above from the rocket has compared to the Earth a parameter speed $\ theta$ equal to the sum of the parameters speed (I) of the rocket compared to the Earth, either $\ theta_ \left\{were\right\}$, and (II) of the ball compared to the rocket, or $\ theta\text{'}$ :

$\ theta = \ theta_ \left\{were\right\} + \ theta\text{'} \.$

Let us return to the acceleration of the rocket. In the coinciding reference mark the speed of the rocket passes from 0 to D v and at this elementary speed D v corresponds the parameter elementary speed D θ given by the formula written above

$\left(FD\right) /c = \ tanh \left(D \ theta\right) = D \ theta \.$

There is thus the relation

$D \ theta = \left(g/c\right) D \ tau \,$

who is integrated immediately like

$\ theta = \left(g/c\right) \ tau \.$

This formula gives the angular parameter speed $\ theta$ of the rocket compared to the terrestrial reference mark according to clean time $\ tau$ of the astronaut in his rocket.

Which is now the trajectory of the rocket in the reference mark terrestre ? By taking account of the relation between speed v and angular parameter speed $\ theta$ and of the traditional definition speed v=dx/dt where X is the distance covered by the rocket in the terrestrial reference mark and T terrestrial time, one has

$dx/dt = C \ tanh \ theta \ \ \ text \left\{or\right\} \ \ dx = C \ tanh \ theta \, dt$

By taking account of the dilation of time ( dt is increasingly larger than $d \ tau$)

$dt = \ cosh \ theta \, D \ tau \,$

one leads to the following formula giving the distance dx measured in the terrestrial reference mark traversed during the amount of time $d \ tau$ of the astronaute :

$dx = C \ sinh \ theta \, D \ tau \ = C \ sinh \left(g/c\right) \ tau \, D \ tau$

This expression is integrated easily to give the result

$x = \left(c^2/g\right) - 1 \.$

This formula solves the problem arising. It gives the distance X traversed by the rocket in the terrestrial reference mark for the length of time $\ tau$ measured by the astronauts.

#### Mathematical numerical application

The numerical application which follows is purely theoretical because current technology does not provide us the means of communicating to a spaceship a sufficient speed, i.e. which is close to that of the light, so that the effect of deceleration of the clocks is appreciable.

Knowing that one year makes approximately 3× 10 7 seconds, that C equalizes 3× 10 10 centimetres a second and that G is worth 10 3 centimetre-by-second a second, let us express times in years and the distances in light-years. It is found whereas (C 2 /g) is worth one light-year and that the distance in light-years traversed during time T measured by the astronauts (and counted in years) is given by the numerical formula

X (in light-years) = cosh (T) - 1

The traversed theoretical distances are given in the following table

In theory, but well in pure theory, a mathematical traveller using such an accelerated rocket could cross the whole Galaxy (which makes some a hundred and thousand light-years) in a dozen years. Its maximum speed would be practically equal to that of the light (with 10 -11 near), which is unrealizable. These calculations are thus once again only mathematical calculations on a mathematical formula.

## Summary and conclusion

One can consider it regrettable that the paradox of the twins was exploited in the direction of the description of an alleged internal contradiction of the restricted theory of relativity, a contradiction which would invalidate the theory of Einstein. We saw here that this aspect negativist is unfounded. Moreover others supposedly paradoxes were invented but none resists an analysis right of the corresponding situations.

In the paradox of the twins supposed contradiction would come owing to the fact that, all relative movements being supposed, the two twins would live symmetrical stories and consequently none could not be found young person than the other. We saw how to raise the paradoxical situation connect renovation of Fidgets compared to Slipper. In truth the two twins do not play of the symmetrical roles since Pantoufle on Earth is the only one with being remained at rest compared to only one reference frame galiléen while Bougeotte made a half-turn and was in coincidence with at least two different reference marks galiléens. The reference marks of Slipper and Fidgets are not interchangeable.