Visual physics and mathematics/Relativity

The paradox of the race after the light

Light travels away from us at 300,000 km/s. If a rocket goes at 200,000 km/s to chase the light, the light still moves away from the rocket at the speed of 300,000 km/s, and not at 100,000 km/s as one might expect. We can never catch up with light or change its escape speed. The speed of light in a vacuum is a universal constant: it is the same for all observers, whether they are on Earth or in a rocket.

The contraction of lengths, the slowing down of clocks and the relativity of simultaneity are enough to show that the paradox of the race after light does not lead to a contradiction.

Relativistic wheels

All moving solids are contracted in the direction of their movement. This effect is appreciable when their speed approaches that of light. This is the contraction of Fitzgerald. It is a consequence of the geometry of space-time, discovered by Einstein with his theory of relativity.

A hoop which rotates on itself so quickly that the speed of its circumference approaches that of light therefore has a smaller diameter than the same hoop at rest. The same goes for a wheel, if its spokes are much softer than its circumference. If the wheel turns on itself, its spokes do not undergo Fitzgerald contraction, because their movement is perpendicular to their length, but they can be compressed by the pressure of the circumference.

If a wheel rolls on the ground with a speed close to light, the base of the wheel is not contracted, because its speed is zero relative to the ground. On the other hand, the top of the wheel is strongly contracted:

Remark: Light is not the only way to form images. The imprint of a foot on wet sand is an image of the foot. A printing house produces images daily by contact. In the animation above, it has been assumed that the images of the wheels are obtained by contact (for example by projecting their shadow on a wall of photodetectors in contact with them), not through an eye or another optical system.

Spatio-temporal measurement devices

A rigid ruler, or a compass, makes it possible to measure the distance between its extremities by comparing it to other distances. A rigid ruler is therefore a device for measuring a length, or a spatial interval. The important point is that the ruler is rigid, so that the measured distance is always the same. Spatio-temporal measurement devices can be designed on the same model. A simple clock is such a device. It allows to point two events which are always separated by the same duration. One can also use two clocks fixed to the same rigid support, each being used to point a certain event. If the device that triggers the two clocks is regular, the spatio-temporal interval between the two pointed events can always be the same. Such devices make it possible to measure spacetime in the same way that rigid rulers make it possible to measure space.

An interval is timelike if it is on the path of a massive material point. It is lightlike if it is on the path of a ray of light in vacuum. All other intervals are spacelike. When an interval is spacelike, there is always an inertial frame for which its extremities are simultaneous events. Spatio-temporal measurement devices measure the three kinds of intervals. A single clock is sufficient to measure timelike intervals. Two clocks fixed on a rigid ruler and suitably synchronized make it possible to measure spacelike or lightlike intervals.

All lightlike intervals are equal

The theory makes it possible to assign a real number to all spatio-temporal intervals, timelike, lightlike and spacelike, or more precisely to their square. Timelike intervals in particular are all equal to zero and therefore all equal to each other. This is a priori surprising. This means, for example, that the spatio-temporal interval between the emission of a photon and its reception three meters away is equal to the interval between its emission and its reception three light-years away, as if the advancing photon never departed from its starting point. Isn't it wonderland?

Spatio-temporal measurement devices and the slowing down of clocks allow to understand this counterintuitive result. Imagine a rocket launched towards a star three light-years from Earth. A photon is emitted at the rear of the rocket and received at the front, three meters away, from the point of view of the rocket, so a tiny fraction of a second later. But because of the slowing down of clocks, what lasts a tiny fraction of a second from the point of view of the rocket can last three years from the point of view of the Earth, provided that the rocket is fast enough. The same spatio-temporal device measures both a lightlike interval of three light-years and a lightlike interval of three meters. It thus establishes their equality.

The paradox of double contraction of lengths

Can a long moving beam enter a barn shorter than it, since its length is contracted in the direction of its movement ?

But the barn is also contracted in the direction of its movement :

If Y is in motion relative to X, it is contracted in the direction of its motion from X's point of view. But if Y is in motion relative to X, X is also in motion relative to Y, and it is also contracted in the direction of its motion from the Y's point of of view. If for example AB and CD are two rigid rules of the same length when they are at rest and which slide over each other, CD is shorter than AB from AB's point of view:

A---B

C--D

and AB is shorter than CD from CD's point of view:

A--B

C---D

This is the paradox of the double contraction of lengths. To understand why this paradox does not lead to a contradiction, we must take into account the relativity of simultaneity:

Let E, F and G be three immobile points, such that F is the midpoint between E and G and EF = 3 m. Let P, Q and R be three points with the speed v with respect to EFG. We assume that at time t= 0, P, Q and R coincide with E, F and G respectively:

E---F---G

P---Q---R

Suppose that a light flash is emitted at F at time t= 0. At time t = 10^-8 s, it arrives simultaneously at E and G, from the point of view of EFG , because EF = FG. At this instant P, Q and R have moved:

E---F---G

-P---Q---R

From the point of view of PQR, the flash emitted at Q at time t = 0 arrives simultaneously at P and R, because PQ = QR, but it does not arrive simultaneously at E and G, because at the instant when it arrives at G, QE is larger than QG, and the flash has therefore not yet arrived at E. The simultaneity of two events therefore depends on the point of view. What is simultaneous from the point of view of an observer is not necessarily so from the point of view of another moving observer. Simultaneity is not absolute but relative. This is why this theory is called the theory of relativity.

Let AB and CD be two rigid rules of the same length when they are at rest and which slide over each other. From the point of view of AB, at the instant when C passes through A, D has not yet arrived at B. This is why CD is contracted with respect to AB from AB's point of view:

A---B

C--D

From CD's point of view, at the instant C passes through A, D has already passed through B. This is why AB is contracted with respect to CD from CD's point of view:

A--B

C---D

The relativity of simultaneity therefore resolves the paradox of the double contraction of lengths :

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The paradox of the double slowing down of clocks is similar to the previous one: a clock I moving in relation to a clock H is slowed down from the point of view of H. But from the point of view of I, it is the clock H which is slowed down with respect to I. The relativity of simultaneity is sufficient to show that this paradox does not lead to a contradiction.

If b is a clock is at the periphery of a rotating disk, it is slowed down compared to a clock a which remains at the center:

But the situation is not symmetrical between a and b, because there is an inertial reference frame where a is always at rest, but there is none for b. This is why an astronaut who makes a long, very fast trip could return to Earth several centuries after her twin has died.

The same goes for relativistic hoops and wheels. There is no inertial frame of reference where their periphery is always at rest. The situation is not symmetrical between an observer at rest in relation to the center of the hoop and who does not rotate on himself, and an observer who rotates on himself with the hoop, because the frame of reference of the second is not not inertial. This is why a hoop that rotates on itself is really contracted compared to the same hoop that does not rotate on itself.

The contraction of lengths, the slowing down of clocks and the relativity of simultaneity are enough to show that the paradox of the race after light does not lead to a contradiction, because the measurement of the speed of light depends on the measurement of lengths , durations and simultaneity. The measuring instruments on board the rocket are contracted and slowed down as the rocket accelerates, and the measurement of simultaneity is modified. This is why light can always maintain the same speed relative to the rocket:

All the lights on a single train flash simultaneously from the perspective of an observer resting on the train, but they do not flash simultaneously to an observer watching the train pass.

The propagation of light is represented by the movement of the orange lines. If the length of a train is the unit of length and the period of the flashing light, the unit of time, then the speed of light c = 1.

One can see that moving trains are contracted, that moving clocks are slowed down, and that light always has the same speed equal to one, regardless of the observer's point of view, at rest, or in a moving train.

Space-time diagrams of an encounter between two trains of flashing lights

On these space-time diagrams, space is represented by the horizontal direction, time by the vertical direction. Points on the same vertical are events that occur at the same location. Time flows upwards. Points on the same horizontal are events that occur at the same time. If c = 1, the propagation of light is represented by lines inclined at 45° (here the orange lines).

All these diagrams can be obtained from just one of them by a Lorentz transformation:

$x'=\gamma (x-vt)$

$t'=\gamma (t-vx/c^{2})$

where $\gamma =1/(1-v^{2}/c^{2})^{1/2}$ is the coefficient of expansion of durations. $1/\gamma$ is the coefficient of contraction of lengths. $v$ is the relative speed of the two frames of reference which determine a Lorentz transformation.

The addition of speeds

If B is going at speed u relative to A and if C is going in the same direction at speed v relative to B, then C is going at speed u+v relative to A.

This theorem of addition of speeds is a theorem of Newtonian physics, which ignores the contraction of lengths, the slowing down of clocks and the relativity of simultaneity. If we take these three relativistic effects into account, the speed addition theorem becomes:

If B goes at speed u with respect to A and if C goes in the same direction at speed v with respect to B, then C goes with speed (u + v)/(1 + uv/c²) relative to A, where c is the speed of light.

If v = c, (u + v)/(1 + uv/c²) = (u + c)/(1 + u/c) = c, the speed of light relative to at A is therefore always equal to the speed of light relative to B, whatever the speed of A relative to B.