Why Does a Chain Snap Between Train Cars at High Speeds in Relativity?

[Moth]

I have been informed that if:

There is two train cars connected by a chain that will snap if extended by more than 1%.
The cars accelerate at exactly the same rate, so the distance between them remains constant.
As they speed up to very high velocity and special theory of relativity start making things interesting the chain will snap.

Why is this?

 

[Janus]

Well, when we say the two cars accelerate at the same rate and maintain a constant distance, we first have to ask ourselves: According to who? From the way this situation is stated we can assume that it is meant that this is according to the frame tha the cars are accelerating with respect to and not according to the cars themselves.

In this case, as seen from this frame:

As the cars gain speed, they undergo length contraction, as does the chain. Since the distance between the cars remains constant, but the chain grows shorter, the chain snaps when its contracted length is short enough.

From the car's frame. Since the cars are both in accelerated frames, the time rate in each train varies along the distance separating them. As seen from the cars, time in the lead car runs faster than time in the trailing car. Thus the lead car sees the trailing cars as falling behind, and the trailing car sees the leading car as pulling ahead. The chain length however does not change. The two cars see themselves as pulling apart (they do not maintain the same distance) and the chain snaps under this strain.

 

[Hurkyl]

As the cars gain speed, they undergo length contraction, as does the chain. Since the distance between the cars remains constant, but the chain grows shorter, the chain snaps when its contracted length is short enough.

Actually, the distance is increasing -- the distance between the tails of the trains remains constant, but not the distance between the front of the rear train and the rear of the front train. (because of length contraction)

 

[jdavel]

What am I missing here?

You have a train car of some length, followed by a chain of some length, followed by another train car. They accelerate, and all the lengths get Lorentz contracted in the stationary frame. The cars get shorter, the chain gets shorter, and the distance between the cars gets shorter.

Can't you just think about all three pieces as being one big stick with two marks on it dividing into three sections?

 

[Hurkyl]

Not if you take

"The cars accelerate at exactly the same rate"

to mean

"it is meant that this is according to the frame tha the cars are accelerating with respect to".


If the front and rear trains undergo the same acceleration, as measured in the stationary frame, then the displacement between them¹ will remain a constant, as measured in the stationary frame.


In order for the train - chain - train system to appear unperturbed to a passenger, the front of the system must always accelerate less than the rear of the system. (whether measured by the passenger or the stationary frame!)


1: Precisely, I mean the amount you would have to translate one train to put it at the exact place of the other train

 

[DB]

As the cars gain speed, they undergo length contraction, as does the chain. Since the distance between the cars remains constant, but the chain grows shorter, the chain snaps when its contracted length is short enough.

Though isn't this just a visual contraction from stationary observer? Is it an actual effect on tension or is it a visual? Would the chain really break to only length contraction? (ignoring for now the slight time dilation that both trains experiencing causing the train traveling forward to seeemingly pull ahead, the other seeming its falling behind.)

 

[Gamish]

I have been informed that if:

There is two train cars connected by a chain that will snap if extended by more than 1%.
The cars accelerate at exactly the same rate, so the distance between them remains constant.
As they speed up to very high velocity and special theory of relativity start making things interesting the chain will snap.

Why is this?

I don't see why the chain would snap, because both cars are in the same reference frame, and according to SR, it is as if they are stationary together.