Ladder paradox

A paradox of the length contraction occurs when the reciprocal length contraction is considered without taking into account the relativity of simultaneity. This kind of paradox represents the counterpart of the twin paradox, which arises from the time dilation.

According to the special theory of relativity is a body, the rest in an inertial system, subject from the perspective of movable relative to the inertial effect of the length contraction. Conversely, are now also all bodies, the rest in the second inertial system, contracted from the perspective of the first inertial system. This leads to different variants of this paradox, with kinematic and dynamic components play a role. By taking into account the relativity of the concurrency problems can be all solved.

• 2.1 resolution

• 3.1 resolution

Garage paradox

A ladder and a garage move towards each other, with the head moves from the perspective of the garage in the positive x - direction. This is to be assumed that the circuit in the same state of motion being longer than the garage, that is, the idle length of the conductors is greater. But from the perspective of the inertial system in which the garage rests, is the ladder moving, and due to the length contraction, the ladder can be made so small by choosing an appropriate speed to fit in the garage (Fig. 1). In contrast, from the perspective of the system, the conductor is the garage moves and consequently contracted. From this perspective, the garage is smaller and the head can not possibly fit in the garage ( Figure 2 ).

Resolution

The fact that such situations do not lead to contradictions, lies on the relativity of simultaneity, ie which assumes an observer in the garage system as the same time, is not the same for an observer in the conductor system.

For this to a garage with two gates (A = left, B = right, Figure 3 ) are considered, which from the perspective of the garage system - ie measured with resting in this system clocks - open simultaneously before the ladder is penetrated (events A1 and B1) to close the same when the circuit is totally enclosed (A2 and B2), and to open immediately the same again in order to let through the conductors ( A3 and B3). The sequence of events is therefore A1 = A2 = B1 against B2 before A3 = B3.

In contrast, from the perspective of the conductor system (Fig. 4) go because of the relativity of simultaneity, the clocks of the garage system from left to right at ever greater distances, ie Events on the right side will take place on the left before events. First, then opens the right gate ( B1), and just before the ladder can penetrate into the garage, and the left gate (A1). Then the right-hand gate closes (B2 ) and opens up immediately again (B3 ), whereby the right head end can happen. Meanwhile, the left head end has completely passed the left gate. The fact that this gate immediately closes it again (A2) and open (A3 ), has no more meaning. In this way, the ladder comes in two inertial frames unscathed through the garage. The sequence of events is thus before A1 B1 against B2 B3 before before before A2 A3.

In connection with the original paradox ( with closed back wall ) are now two variants will be discussed:

1a) The director shall immediately come to a stop in the garage from the perspective of the garage system. This is only possible if each part of the circuit from the perspective of the garage is accelerated simultaneously. If they come to rest in this way, it also takes in the garage system back to its resting length, and since this is greater than that of the garage, they will encounter during the expansion process on both inner walls of the garage at the same time.

1b) in the pipe system is now a problem arises from the fact that the conductor is longer than the garage. How is it possible that the left head end can come across the inside of the left garage end? The resolution lies in the fact that the simultaneous acceleration in the garage system, from the perspective of the conductor system not simultaneously. First, the right part of the head is accelerated, and assumes a speed which ever approaching that of the garage. This automatically means that these sections of the conductors contract until finally the whole head is covered. Thus it is possible that here also disappears completely, the left end of the ladder in the garage, and the circuit thus on both sides of the inner walls of the garage, but can not simultaneously encounter.

2a) In contrast to the previous version, the manager will now not arrive simultaneously along its entire length in the garage system to a halt, but the stoppage is caused by the impact against the back wall. From the perspective of the garage first, the right of the ladder comes to a halt and thus expands as it now assumes its rest length. This continues up to the other end of the conductors so that after and assumes its rest length the whole conductor, and thus also to the other inside of the garage encountered during the expansion process.

2b) In the ladder system again, the problem arises how it should be possible that the left head end can come across the inside of the left garage end. The answer to this question arises from the fact that according to relativity theory can exist no rigid body. Since the acceleration in this case originates from one place only, and the coupling strength in the conductor itself can spread faster than the speed of light, it takes time until the information of the collision reached the other end of the conductor. As in 1b) of the right-hand part of the circuit is first accelerated, and assumes a speed which ever approaching that of the garage. That from the perspective of the conductor system, the conductor is compressed like an accordion due to the collision together, and the rest for the time being, left conductor end consists only in motion, when the ladder is already completely enclosed inside the garage. During the expansion process now begins ( as in the same state of motion, the conductor is longer than the garage), here too the left of the ladder will encounter the left inside of the garage.

Panzer paradox

This paradox was first introduced by the physicist Wolfgang Rindler, with several varieties of this paradox are also known as "Panzer paradox " or " skier paradox".

There was a ditch with a " resting length " given by 10 m (ie, as measured in its rest frame and the inertial frame where the trench is stationary ), and a relatively to moving rod with the same rest length. Is now the relative movement between them so high that a contraction factor of 10 is reached, this means that from the perspective of the trench, the rod is contracted to 1 m, while the trench remains undiminished. In contrast, from the perspective of staff of the trench to 1 m is contracted, and the rod is undiminished. This leads to the seemingly contradictory situation.

Assuming that the gap is closed by a trap-door, which is opened when the bar is located just above the trench, as the contracted to 1 m rod should immediately, that is, at the same time along its whole length in the 10 m wide trench fall. ( This actually happened, but an electromagnet should the trench be present, which attracts the bar - because gravity can not be treated within the framework of special relativity. ) For a co-moving with the rod observer of the trench is however contracted to 1 m, and there is for him (for now) no reason to suppose that the 10 m long "rigid" rod will fall into the only 1 m wide trench.

Resolution

The resolution of this paradox lies in the fact that the adoption of the stationary observer on the staff, namely that the staff " rigid" is, is wrong. That is, there is no " rigid body " in the special theory of relativity, as effects in the body may spread up to the speed of light, while the rigid body of the pre- relativistic physics based on the assumption that effects propagate infinitely fast in it. Similarly, the relativity of simultaneity must be considered. The corrected by the special theory of relativity point of view of the stationary observer on the staff is: The trap door will not open at all points simultaneously. And since the coupling forces in the rod due to its finite propagation speed (maximum speed of light) need time for their action, the rod can its " rigidity " is not maintained ( that is, the coupling forces in the rod react too slowly ) and is of the open positions start trapdoor immediately, due to the action of the magnet "flow" into the ditch to. In the event that no trapdoor is present, the staff would of course immediately begin to melt away when the first end of the rod above the trench.

Once the front end of the rod strikes in the grave wall, are gradually, from front to back, parts of the bar now in the rest frame of the trench until the entire rod in the forward direction at a standstill. There is then no rest frame of the rod more, in which the trench is shorter than the rod, and thus fall into the ditch of the whole staff in each case.

It might be objected that, although is now proven that the rod falls into two inertial frames in the ditch, but he would (if the angle of fall in both systems, the same would be ) from the perspective of the rest system of the trench strike much deeper on the opposite wall, as the distance to be covered and thus the flight time is much longer here, whereas the trench and thus the distance to be covered is significantly shorter from the perspective of the rod, and thus the point of impact is significantly higher. However, this assessment is wrong: For the acceleration and thus the angle of fall is (his subject or the relativistic aberration) in the system of the rod more pronounced fail, and thus, despite the shorter distance made ​​the same point on the opposite wall, as in the grave system. For it must be borne in mind that the electric Coulomb field is also subject to the contraction in the ditch from the perspective of the rod, as the trench itself, this means that the field lines of the Coulomb field normally significantly closer together pushed to the movement direction in the rest frame of the rod are, as in the rest frame of the trench.

In other variants, the scenario of a battle between two armies will be used. The now raises an army from a ditch, from which she means that he is wide enough so that the plummeting falls into enemy tanks. To be sure, it is a strong electromagnet used. On the other hand believes the other army that their armor does using ease on the contracted from their perspective ditch. The rest corresponds to the above scheme: The tank will immediately begin to melt away at the points where it floats freely over the trench (ie with trapdoor in the places where the trap door will open first, and without trapdoor immediately after reaching the edge).

Another variant includes a skier during a descent. This leads to a crevasse - from the perspective of the glacier system would fall into the contracted skis, on the other hand should be from the perspective of the skier skis come easily over the contracted column. The resolution is the same as above: The skis melt away and bang on the wall gap.

Scale paradox

A rod of length L moves with a velocity v parallel to a stationary hole, which also has the length L. At the same time the rod moves even with a small vertical speed to the hole:

( For the two preliminary figures will v so small against the accepted speed of light c, that no relativistic effects emerge. )

At an appropriate time t = 0 the rod passes through the stationary parallel hole:

• Case ( A): The rod moves parallel to the static hole.

From a frame of reference of the stationary hole appears from the fast ( here with about v = 0.94c ) moving rod through the length contraction is shortened to the length L '

• Case ( B): The hole moves towards the stationary rod

Looking at the situation, however, the co-moving system of the staff, so resting the rod and the hole moves relative to the rod with the high velocity v on the bar to: ( In the figure at t < 0, the hole still in the length L shown that it has at rest before it is accelerated to V ).

From now considered dormant respected rod that has the length L in the rest frame, the length contraction is now working on the moving hole having only the length L '

But this is a contradiction. Although the dimensions may appear shortened mutually to each other, the statement that the rod passes through the hole, but can not be made dependent on the choice of the reference system; otherwise the theory of relativity would be contradictory and wrong.

Resolution

The description of case (A) ( The rod moves parallel to the resting hole ) corresponds to reality and is displayed correctly. The description of case (B ) ( The hole moves on the stationary rod to ) was not carried out in the preceding presentation completely correct according to the laws of the Lorentz transformation, thus creating the apparent paradox that such does not exist. In the transformation to the system in case ( B ) must be taken into account that not only change the location coordinates corresponding to the Lorentz transformation and thus lead directly to the length contraction, but that the time has to be transformed. This is also the view changes on which events appear simultaneously. In case ( A) ( resting hole ) passing through the rod the hole parallel, which means that = 0 move the front and rear end of the rod simultaneously at time t through the hole. Here, the rear end of the rod through the hole in the transverse position x = 0, the front end of the rod in place. In case ( B), this passage - simultaneity remains at the two current hole edge locations but did not receive, but the time t ' in the rod - rest frame is calculated according to the Lorentz transformation:

Now, if the rear end of the rod at time t = 0 and x = 0 passes through the hole in the place, and it was done to the following time in the rod rest system:

For the front end of the rod ( at time t = 0, at the place in the hole ) takes the following time to the rod rest system:

The front end of the rod has therefore the hole much sooner pierce than the rear end of the bar (as seen in the system that moves with the rod ). The fact that the front end of the " resting" rod crosses the zoom flying hole first, clearly means that the hole is no longer parallel flies to the bar but tilted appears, so that the correct way the following transformed image for the case (B ) ( The hole moves along the stationary rod to ) gives:

Rod and fly hole in this system no longer parallel each other. It firstly follows: Simultaneous events in an inertial system running in another different fast-moving inertial frame is not also at the same time as, and second are "parallel" lines in the other inertial system no longer parallel: the angle between two directions in space is different from two relatively moving observers measured.

The argument can also be the other way round are carried out: Assuming a stationary rod, to actually have a hole to move in parallel, then creates the transformation in the system of the resting hole (case ( A)), the tilt of the bar (in the previous presentation, it was exactly the reverse, where the hole was tilted and not the rod ). The scale paradox can be so in every case consistent dissolve in the context of the theory of relativity.