The Ehrenfest Paradox and the Wall of Death

Question

In another question evaluating the reality of length contraction, the circular motion was involved and some answers argued that centrifugal force would negate any possible length contraction. A famous paradox called "The Ehrenfest Paradox" analyses a similar situation with relativistic motion in a circle and it suffers from the same criticism.

The effects due to centrifugal motion in a circle (as far as length contraction is concerned) can be removed by considering the following thought experiment. A large circular wall is constructed and a railway track is built on the inside vertical surface of the wall. In fact, the wall can be built in a pit dug in solid granite, to support the wall and the wall is made of the toughest materials known to man. The idea is that the radius of the track is not allowed to increase to any significant extent.

Initially, trains are placed on the track one at a time, each linked to the next by a short spring, until the track is completely filled with connected trains.

Now the trains are accelerated to relativistic speed. SR seems to predict the trains will length contract according to the observers at rest with the track and the connecting springs (made of lower tensile strength material) will be stretched.

One of the answers in the other thread contradicts this conclusion, saying from the point of view of the passengers on the trains the track is length contracting. He implies that since both cannot happen at the same time, neither the track nor the train length contracts and the gaps between the trains will not increase.

How is this paradox resolved? What is the explanation for what is going on and what will actually be observed? How do the observers on the trains explain why the gaps between the trains increased and the springs were stretched if that is what happens?

Please don't say SR cannot analyse this and GR is required. The train is operating in essentially flat spacetime and SR can cope with circular motion and acceleration.

In summary, what actually is measured to happen and how is it explained?

Additional note: For clarity, I intended the trains to be accelerated to a final constant velocity in a Born rigid manner, such that the trains maintain a constant proper length as measured by observers at rest in the trains.

Edit: This part is transferred and paraphrased from the comments to avoid an extended discussion there: Consider a slight modification, this time without any connectors. Lets say the trains are accelerated to a velocity such that they are measured to contract by about 50% and then maintain that constant speed. On the roofs of the trains are poles that are the same length as train.

When the train is at constant speed, the passengers can feed these poles into the gaps between the trains so that they fit snugly without any (parallel to the track) stress or strain on the poles or the trains. Now we could have a collection of objects with a total rest length equal to twice the rest length of the track, yet fitting into a circumference equal to the rest circumference of the track. This would seem to indicate that something really has contracted relative to the other. (Let me know if this second part should be moved to its own separate thread.)

Answer 1

To gain insight, replace your track with a long thin rectangle with slightly rounded corners. That way every car is inertial most of the time.

Now the trains are accelerated to relativistic speed. You don't specify exactly how the acceleration takes place, but let's suppose for concreteness that every part of every train car is accelerated in the same way, as measured in the track frame. Therefore no car's length changes in the track frame.

Now suppose you're in a car on one of the long sides of the track. Here is what you will say:

1. The front of my car began to accelerate before the back of my car. Therefore my car got stretched. The same thing happened to all the other cars on my side of the track, and to all the springs connecting us. Therefore either the cars or the springs or both are now longer than they used to be.

2. As for those cars on the opposite side of the track, exactly the opposite thing happened. Their backs began to accelerate before their fronts did. Therefore either those cars or the springs between them or both are now shorter than they used to be.

3. There are now more cars on the far side of the track than there are on my side (because the two sides of the track are equally long but the cars [and/or the springs] have different lengths).

Now once you've digested that, go back to the case of a circular track. Riding on a given car, and describing the train in your instantaneous inertial frame, you'll say that each car accelerated differently, and you'll assign different lengths to different cars. Those near you (including yours) will have gotten longer; those more-or-less diametrically opposite will have gotten shorter, and each by a different amount. But they'll still all fit on the track.

Answer 2

The other thread is a more complicated scenario than this one, so I wouldn’t say that you should expect the answers of that one to be exactly similar to this one.

SR seams to predict the trains will length contract according to the observers at rest with the track and the connecting springs (made of lower tensile strength material) will be stretched.

You are correct that the springs will stretch, but it is incorrect to call this length-contraction. Length contraction is a disagreement between two inertial frames regarding the distance between two points which are at rest in one frame. There is no material strain involved in length contraction.

In this case the trains are not inertial. And angular acceleration is not a rigid motion so it is always associated with material strain. So yes, what you describe does happen, but no it is not length contraction.

from the point of view of the passengers on the trains the track is length contracting

This is a very “fraught” statement. The tricky issue is that the passenger’s frame is non-inertial. So there is no standard meaning to their point of view.

Because they are non-inertial they are not symmetric with non-rotating observers. Only the congruence of observers that underwent angular acceleration experience a measurable material strain, and both the train and wall observers agree that is the train observers.

What is the explanation for what is going on and what will actually be observed. How do the observers on the trains explain why the gaps between the trains increased and the springs are stretched if that is what happens?

If strain gauges are placed on the springs then all observers will see that there is a measurable strain in the springs. The observers on the train attribute the measured stretching of the springs to the forces that produces the angular acceleration. These forces include the external forces on the train cars as well as the internal forces keeping the train cars intact and the forces exerted on the springs. This is a mechanical strain and it is caused by mechanical forces.

This would seem to indicate that something really has contracted relative to the other.

I would interpret “really has contracted” as referring to a measurable strain. So yes, there is real measurable strain in this scenario. The contraction is therefore not “length contraction” which is strain-free.

Consider an inertially moving train car on a straight track adjacent to the curved track. As it passes the curved track the two cars (one on each track) are momentarily at rest and immediately adjacent to each other.

In the ground frame the inertial car is shortened. There is no material strain: this is length contraction.

The non-inertial car is strained, so its shape is distorted compared to the inertial car. This material strain is unavoidable, physically measurable, asymmetric, caused by real forces, and is not length contraction.

Answer 3

Uniform circular motion along the circumference and motion along the diameter while it is spinning are two different reference frames:

1. Motion along the circumference can be treated as an inertial frame.

The tangential velocity along the circumference is constant and there is no tangential acceleration. All the acceleration is centripetal and is uniform at all points of equal radius. If half a circumference C is stretched into a straight line then the centripetal acceleration will resemble a constant force along the the straight length (like gravity) If the stretched line exerts a constant normal force that counters the centripetal acceleration, then the sum of the acceration is zero.

Now all the requirements for an inertial frame have been met so for relativistic tangential velocity $$C' = \frac{C_o}{\gamma}$$

2. Motion along the diameter of a circle exhibiting uniform circular motion is a non-inertial frame of reference.

Any motion along the diameter has a constantly changing centripetal acceleration parallel to the direction of motion which is maximum at R and approaches a minimum (0) as one approaches the center. The motion along the diameter is identical to the motion of the amplitude of a sine wave (Simple harmonic motion) and can be modeled as rotational motion by looking at the shadow of a peg on a wheel rotating at a constant angular frequency.

The motion along the radius can be expressed as $x(t)=Acos(ωt)$

If there is a full special relativity solution to the Ehrenfest paradox, it is in the proper treatment of the measurement of length or motion along the diameter such that the problem can be expressed in an inertial frame of reference.

Answer 4

Special relativity can be used to analyze simple rotations from the point of view of the non-rotating frame. However, the rotating frame is not equivalent to the non-rotating frame. While, there is no absolute inertial frame, there is an absolute frame of rotation, it is the one in which the universe is not rotating. In the rotating frame, space is non-Euclidean, time is dilated, extra forces exist, and light does not travel in straight lines. Using just SR is not enough. It is best, if one is using special relativity, to examine what things look like from the non-rotating frame. Notice that all of these paradoxes are paradoxes in special relativity and involve accelerations which special relativity was not meant for.

For example, one might, from using just SR, conclude that passengers on the train would see the track contract. But since they are in a rotating frame, they measure the circumference of the circular track to be increased by $\gamma$ because space has become non-Euclidean and the circumference of such circles is $\gamma 2 \pi r$.

In the non-rotating frame, the centrifugal forces on the train will, for real materials, squeeze it down and lengthen it. The springs would also be crushed and lengthened. So, presumably, no further spring extension would be needed.

But note, if we could ignore gravity (easy, just do this in space) and centrifugal force (it is a thought experiment), then it is reasonable to think that the train would lift off the tracks and make a smaller circle. The springs would be contracted but not stretched, so just static forces are involved (ie., no energy). The passengers on the train would see the radius magically shrink. Only a physicist on the train could explain what is happening to them, with a little differential geometry. Now, invoke centrifugal force and the springs expand. For a moment you have the simple configuration described in the question. But, after the train hits the wall, it get smashed like a bug on the windshield.

We can, also for the sake of the thought experiment, suspend reality by imagining that the train is composed of impossibly rigid materials. If one alters the order of events, one can better understand where the forces and energy are coming from. Disconnect one spring and then accelerate the train to relativistic speed. One gets a shortened train and a large gap. Now slow down the back car and speed up the front car until you can connect the two by a spring. Now, one has the same final configuration. However, one finds that it required extra energy and force on the part of the train engines to stretch the springs and bring the total length of springs and cars back to their inertial total length. The order of coming to this configuration should not matter, so the same must be true if the springs had all been connected from the beginning.

As in the Bell's Spaceship Paradox, it can require force, and hence energy, to undo some of the effects of a Lorentz contraction; for instance, returning the length of an object now moving to its length when it was at rest. In essence, the Bell's Spaceship Paradox and this Ehrenfest Paradox are the same. The rotating band and rotating disk problems are more interesting because they look like there is potential to extract energy from length contraction. But, in reality, there is always a net expansion and no way to extract energy.

So why do I focus on where the energy for the forces comes from? Since relativistic length contraction is not one of the forces of nature, it follows that one can not extract energy from it. You can only expend energy to make it appear that it is not happening.