Paradox of the twins and the Sagnac effect

The paradox of the twins and the effect Sagnac are a study of the Paradoxe of the twins of Langevin within the framework of the reference marks in rotation and utilizing the Effet Sagnac.

Here, Bernard, instead of making half-turn during his voyage carries out a circular course making it return to his starting point. Although it does not carry out a brutal half-turn, it undergoes however a centripetal acceleration and the situation is similar to the paradox of the traditional twins. However, to utilize a reference mark in rotation allows a variety of very interesting situations.

The correct analysis of these situations requires a analyzes thorough effect Sagnac.

The inertial reference mark will be noted R. Passons in review some situations.

Two twins, motionless, the other in rotation

The first case is that where Alain is motionless in the inertial reference mark R and looks at Bernard turning around him.

In fact, the Sagnac effect hardly intervenes here and one finds oneself in a traditional situation where one of the twins is inertial and the other accelerated.

The trajectory of Alain and Bernard in the reference mark R is obtained easily.

Then one carries out the integral of clean time as explained in the Paradoxe of the twins.

Here, it is particularly simple because the trajectory is tangent in any point on lines of constant slope. I.e. the speed of Bernard, in size, is constant (if rotation is uniform). One can thus use the traditional dilation of time.

What does it occur if for example Alain is at the edge of the disc (in the figure above, it is in the center)? Which is the ageing of each one after a turn of Bernard?

Thus, for a circle of radius R and an angular velocity \ omega, the time taken by Bernard (in the reference mark R) to make a turn is of:

\ frac {2 \ pi} {\ Omega}

It is the ageing of Alain.

The relativistic factor gamma \ Gamma corresponds at the tangential speed R \ omega. And ageing, weaker, of Bernard is thus:

\ frac {2 \ pi} {\ Gamma \ Omega}

They is, sums all, a completely traditional paradox of the twins!

The two twins make a turn in each direction

Let us look at the situation now where each twin makes a turn in R, at same angular velocity, but each one in a direction.

The trajectories are the same ones as above but each twin turns in a different direction.

The situation is completely symmetrical and with their second meeting (after a half-turn) or third (after a turn) they will have the same age.

For an observer in R, each twin aged of

\ frac {2 \ pi} {\ Gamma \ Omega}

after a turn and it is thus normal that the twins have the same age.

There we have again a rather traditional situation of completely symmetrical twins.

Two twins make a turn in each direction on a disc in rotation

Now let us consider two twins placed on a disc having a rotation \ omega_0. Each one leaves to make a turn of with dimensions at an angular velocity \ omega compared to the disc.

Thus, Alain turns (in R) at the angular velocity \ omega+ \ omega_0 and Bernard to \ omega_0- \ omega. One supposes by facility which \ omega is larger than \ omega_0 but it is not a need. Thus \ omega_0- \ omega is negative and Bernard turns in the other direction. R' is here the reference mark attached to the disc in rotation.

Let us note that the presence of the disc is completely superfluous, we use it only by facility of reasoning. The important thing is the fact that Alain and Bernard have different angular velocities in inertial reference mark R. For example \ omega_1 and \ omega_2 ( \ omega_1 and \ omega_2 is supposed to be positive and \ omega_1> \ omega_2, the minus sign indicating that it turns in the other direction).

That corresponds at an angular velocity of the " disque" (reality or fiction) \ omega_0= \ left (\ omega_1- \ omega_2 \ right) /2. The R' reference mark will be a reference mark attached to the disc or, if this one does not exist, a simple reference mark in rotation in R at the angular velocity given by this relation.

Let us consider the point of view of reference mark R. the situation is clearly nonsymmetrical.

The graph is similar to the precedent but each one traverses a spiral in the space time with a different angle (the step of the spiral is different). The crossing after a turn does not occur besides at the same point of R as the starting point. Each one undergoes a dilation of the time (from the point of view of R) and thus the age of the twins will be different with their meeting.

Which is the point of view in R'? This time the situation seems symmetrical since the two twins leave to make a turn at identical angular velocities in each direction.

But R' is not inertial. Space is not flat there. And we fall on the Sagnac effect.

After a full rotation in each direction the age of the twins will be different from twice Time Gap (the study of the Sagnac effect showed that it is universal, same whatever the speed of the " objets" , here the twins, who make the turn).

Detailed calculation shows obviously that the difference in age is identical according to the two points of view.

To turn around a planet

Certain alternatives of the paradox of the twins call upon the use of a planet.

One of the twins is far in space while the other continues a trajectory bringing it close to planet. It uses this planet to make half-turn and returns close to his brother.

The principal argument is that the two brothers are in inertial reference marks and thus which one falls down on the paradox of the twins.

Why these reference marks are inertial? We touch there with all that relates to the gravitation. The gravitation obeys the principle of equivalence (which is in fact a postulate). All the bodies are accelerated in a field of gravitation in the same way, whatever their masses, their volumes, their composition,… The first to have shown that is Galileo with his famous experiment with the tower of Pisa where it dropped from the top a full ball and a ball hollow and checked that they arrived at the same time on the ground. Thanks to this principle, Einstein showed that a reference mark in freefall was equivalent to an inertial reference mark. If you have the chance (or the bad luck) to find you one day in an elevator which was taken down, made the experiment, release a pen: it will float close to you. Quite simply because it falls at the same speed as you. You certainly already saw also on television this type of situation carried out during ballistic flights with an adapted plane and which is used for the drive of the astronauts. Such an elevator in freefall is thus physically identical to a cabin in weightlessness, far in space and completely open (without any force applied nor acceleration). A reference mark in freefall is thus inertial!

It is this principle of equivalence which allows a geometrical description of the gravitation! Since the gravitation depends on the environment and the place where one is and not body placed at this place, then one can geometrically describe his movement due to the gravitation. Within this framework, the gravitation is not regarded any more as a force as well as the others and a body subjected to the only gravitation (thus in freefall) is inertial.

Between bracket, the General relativity watch which the gravitation requires inevitably a description using a curved space time. The inertial reference mark in freefall is described by a space time of Minkowski which is not other than a tangent space with complete curved space. As the curve of the space time is given by the matter (through the equation of Einstein), here is the justification of the existence of the inertial reference marks.

Let us close the gravitational bracket necessary to include/understand the context and return to the assertion that the twins passing close to planets constitute a true paradox of the twins.

Of course, one can call upon two arguments against that:

  • If one of the twins uses planet to make half-turn, then it is subjected to a gravitational field different from his/her brother and the effects of gravity predicted by general relativity enter in account.
  • Comment this situation can be obtained? Thus let us suppose that the two twins are initially relatively far away from planet, one of the twins remains almost motionless while the other animated a certain initial speed towards planet approaches and makes half-turn then it. It is not so simple! It is necessary that its initial speed is low not to escape attraction from planet and from to go away without making half-turn. And if it is weak, the other twin will have time him also to be attracted!

In fact, these two arguments are not enough! Indeed, the gravitational effects are rather weak on the walk of the clocks, although considerable (we will not show it here because it is the field of general relativity). Unless calling upon extremely massive bodies like black holes or neutron stars.

And if it is difficult to find a configuration where the two twins are in freefall and meet again, it is not impossible.

For example Alain can traverse an ellipse at low speed (like a planet around the sun) and Bernard a hyperbole at high speed close to planet (like a comet). One can calculate the parameters of these two orbits so that Alain and Bernard meet at the points of crossing. This kind of situation is not so exceptional, think for example of the Comet Shoemaker-Levy 9 which, after having passed very close to the Jupiter planet, returned to strike the latter after having circumvented the Sun.

Obviously even in this case the relativistic effects will be rather weak with to less again employ an extremely massive body to obtain notable speeds.

But it is not serious because the effects of restricted relativity are, with traditional orbital velocities, more important than the gravitational effects and the paradox of the twins re-appears.

Actually, it is not no need to call upon such complicated arguments. Indeed, even if the reference marks are inertial, from the point of view of restricted relativity what is essential before very for the use of the theory, for example the transformations of Lorentz, is that the two reference marks considered (Alain and Bernard) belong to the same class of reference mark moving relative of uniform translation. If relative speed is not constant, it is a waste of time and effort. And here it is not the case.

Moreover, the study of the preceding paradoxes gave us all which we need

To solve this type of situation, two approaches are possible.

  1. One is placed entirely within the framework of general relativity. One looks at the space time corresponding to the situation, for example with a metric massive body known as of Schwartzchild (it describes the geometry with spherical symmetry around a massive body in general relativity), and one calculates the trajectories of Alain and Bernard. Then one calculates clean time by integration of the interval and the turn is joué.
    It is a general method similar to that used in the paradox of the traditional twins but the space time is not any more that of Minkowski.
    It is the surest method and most precise but it is true that it is complicated (like general relativity).

  2. If space is not curved too much, which is often the case with celestial bodies of mass moderated like the Earth, one can carry out the approximation suivante.
    One considers that the space time is of Minkowski and a reference mark attached to the geocentric reference mark like inertial. The center of the Earth is motionless in this reference mark and the Earth turns in this repère.
    In this space one has bodies in rotation: satellites and even people on the ground (the Earth turns).
    the choice of this reference mark is justified on the one hand by experimental considerations (one checks in experiments that this reference mark is quite inertial and, for example, the rotation of the Earth in this reference mark is physically measurable, for example with the method of the pendulum of Foucault, the pendulum oscillating at the same time as it rotates of 24 hours because of the rotation of the Earth) but also by theoretical considerations (the case above shows that the distribution of the masses, the Earth, the Sun,… imposes the curved geometry of general relativity and justifies the existence of the inertial reference marks as we explained higher).
    the satellites in rotation are considered to turn either because of the curve of the space time (general relativity) but simply because of the force of gravity which is anything else only one centripetal force.

Although actually the reference frame of the satellite is inertial, in this approximation one considers that it is not necessarily the case and that it is the geocentric reference mark R which is the reference.

The choice should be made: either one places oneself in the context of general relativity, the bodies are inertial, gravity is not a force but a geometrical effect, and the space time is curved, or one is placed in a Newtonian situation, the bodies are accelerated (in rotation around planet), gravity is a force and the space time in the geocentric reference mark is that of Minkowski. But the two approaches should not be mixed! They have a radically opposite design of the phenomena indeed and even if both are bound, to mix the cloths and the towels brings only errors and confusion.

One can then use the methods and calculations appropriate to the various points of sights: dilation of time, Sagnac effect,…

The possible situations are exactly the same ones as previously and are solved same manner.

This approach is definitely simpler than the preceding one but it is approximate in what it does not take into account the relativistic corrections due to the gravitation (regarded here as Newtonian) in general relativity.

A typical example is that of the GPS. One uses for those the approximation described in the second point of view.

To calculate the effects of relativistic shift of the clocks of the satellites compared to the clocks of the stations, one uses:

  • restricted relativity. Exactly as in the second point of view. One takes account of the dilation of time due at the relative speed of the satellite compared to the station. As the information exchanges are done by radio operator wave one must also take account of the distances between satellites and the speed of the signaux.
    the point of view of reference adopted is that of the stations and a very light Sagnac effect can thus intervene, but it is very weak (although measurable) and of no importance because calibrations between satellites and stations on the ground are done more quickly than a full rotation of satellite.
  • One adds a correction, much weaker than the preceding effects, due to gravity. The fact that in general relativity a clock placed in a gravitational field works more slowly. Thus a clock in altitude works more quickly.

This effect already could be checked on a height of about 100 m (by using a tower) and thus, although weak, it is necessary to hold account under penalty of having of it an important error for the calculation of position GPS.

These two effects are opposite: the dilation of time makes that the clock of the GPS seems to work more slowly compared to us while gravity makes that it goes more quickly. The second effect being definitely weaker, the Net effect is a delay of the clocks of the GPS.

There is no sophisticated calculator on board satellites GPS. Calculations concerning relativity were carried out in advance, on the ground, and the results are included in the form of simple numerical formulas in the small calculators of the satellites. A good part of the correction is already included in the rate/rhythm of the atomic clocks of the GPS, rate/rhythm modified to include most of the relativistic effects. The clocks go a little more quickly than the normal to compensate for the effects of relativity.

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