Why is the Wikipedia article about Bell's spaceship "paradox" disputed at all?

Fredrik

Link to the article

This problem is ridiculously simple. The condition that the spaceships experience the same acceleration implies that their world lines will have the same shape. (The acceleration doesn't have to be constant. It's sufficient that both spaceships accelerate the same way). This implies that the length of the rope will remain constant in the launcher's frame. Think about that for a second. After a while, the rope is moving at a high velocity in the launcher's frame, and must therefore be Lorentz contracted, but it's still the same length! That means that it must have been stretched. If it was already stretched to its maximum length when the acceleration began, it must break. It's as simple as that.

This is all very basic stuff that belongs in an introductory level class about special relativity. So why is this article disputed at all?

Is it because of the common (but silly) misunderstanding that you can't solve a problem involving any kind of acceleration entirely in SR? (It's really weird how many people who have studied SR still believe that you need GR for problems like this).

Or is it because some people who understand that the rope gets stretched are arguing that SR somehow also implies that the rope gets stronger, so it can handle getting stretched?

I can't think of a third reason.

I know that some people here have been working on this article. Perhaps one of you can explain this to me.

quantum123

Perhaps as one poster said, there is no stretching at all - just a matter of siumultaneity.
To measure length, you need to know two points at the same time.

Boustrophedon

You're right about the problem being simple but it is rather important to use special relativity and not Lorentz's pre-1905 theory, which postulates actual physical contraction of solid lengths in the direction of motion. It should be remembered that Einstein's SR is a fundamentally different theory even though it leads to the same Lorentz transformations.

Einstein's theory involves a purely kinematical approach involving no physical "shrinkage" but achieving contracted measurements by means of the relativity of simultaneity. That is to say, the shift in simultaneity causes the front end to be measured first with respect to the rear end a moment later, resulting in a reduced measurement.

What this means is that a rod initially at rest with respect to an observer does not, in SR, change its length with respect to that same observer, as it is accelerated to some fraction of c. What happens is that the length of the rod defined by another observer moving with it will appear to get longer with respect to the "stationary" observer who sees the rear end marked increasingly before the front end, as the moving observer's simultaneity shifts.

Einstein's 1905 paper only concludes that a length defined in K' appears shorter in K by the Lorentz factor, and vice versa, where K' and K are in relative uniform motion. It does not say nor suggest that a body would change its physical length during acceleration.

The idea of a rod "shrinking" as it accelerates is an unfortunate anachronism - a "hang-over" from Lorentz's earlier theory that still lingers on a century later and even finds its way into textbooks now and then.

If you read Bell's original article you will see that he makes it clear throughout that he is using Lorentz's theory in preference to SR. He was a quantum physicist with no track record in relativity and it's not clear whether he disliked Einstein's SR or whether he didn't realise the significant difference between the two approaches.

Fredrik

Boustrophedon, you're wrong. I don't know how you got the idea that everyone who's making correct claims about SR is actually wrong because they're using some pre-SR theory that we've barely heard about, but I can assure you that's not what I'm doing, and it's not what the people in the other thread are doing either.

Fredrik

If someone said that, he or she was either talking about Lorentz contraction in general, or hasn't understood SR at all. Lorentz contraction can be said to be "just a matter of simultaneity", but in this case the rope is clearly being forcefully stretched. Otherwise its length wouldn't remain the same when it gets Lorentz contracted.

That obvious. What's your point?

Ich

The problem has nothing do with whether you choose Einstein or Lorentz. When people here are speaking of actual stretching/compression, they mean actual stretching.
There is confusion about what exactly "accelerating a rod" means. I can think of 4 substantial different scenarios:
1. The rod is being pulled. It will first get stretched, stay so during acceleration, and finally come to "rest" at its initial proper length when the acceleration ceases. There are no internal stresses then.
2. The same with a rod being pushed; just replace "stretched" with "compressed".
3. The rod is being pushed and pulled in a way that the proper accelerations of both ends are the same. It will experience stretching and compression (depending on the position) during acceleration. It will come to "rest" with a greater proper length, actually stretched.
4. Every single point of the rod is being accelerated with a carefully chosen proper acceleration such that no internal stresses occur. Its length, as measured in in a suitable comoving frame, will stay constant. It will come to rest with its original proper length an no stresses.[

quantum123

OK.
I see the problem here.
There is a rigid rod to the co-moving observers.
But the lab observers see a contracting rod.
So shall I say that the spring constant has also got be relativistically transformed.

Fredrik

Just adding to the list...

5. Every point of the rod is instantaneously (or near instantaneously) boosted to a new velocity, all at the same time in the frame where the rod was at rest before the boost. This stretches the rod to a longer proper length.
6. Every point of the rod is instantaneously (or near instantaneously) boosted to a new velocity, all at the same time in the frame where the rod will be at rest after the boost. This compresses the rod to a shorter proper length.

Ich

If you measure the rod contracted according to Length Contraction, you know that it has not changed at all. No need to consider spring constants.

Fredrik

I think he's trying to argue that the string/rope/rod in the Bell's spaceship scenario won't break even though it's getting stretched. Any such argument would have to say something about the properties of the material.

nakurusil

Both 5 and 6 are not possible. Forces travel at finite speed (the speed of sound) in rigid materials. This is a pretty slow speed , so you cannot have "instantaneous (or near instantaneous) velocity boost". This is what Born rigidity is all about.
So, the rear of the rocket , where the engine is, is boosted earlier than the front.
So, the rear of the leading rocket is boosted earlier than the front of the trailing rocket.
So, the rod anchored between the rear of the leading rocket and the front of the trailing rocket gets STRETCHED (if the two rockets motors exhibit the same uniform acceleration) .
If you would like the mathematical tratment that goes with it, you can check wiki on the "Bell's paradox" or I can add a reference to a college course notes on hyperbolic motion/Born rigidity. They show the conditions under which the rod breaks.

Now, why is the wiki article disputed? I am sure CH can explain this a lot better, the short of it is that it takes one kook (Rod Ball in this case) to slap the "NPOV disputed" on any wiki article. If you click on "discussion", you will find the never ending argument with Rod Ball.

Fredrik

They're possible. You just have to find a way to push every atom at the same time.

I'm not saying it's easy. 5 and 6 are not reasonable ways to accelerate objects, but they have a pedagogical value.

nakurusil

Physics says that the above is not possible. (unless you decide to attach a miniature rocket motor to each atom)


If you use absurd premises don't be surprised to get absurd conclusions.

Fredrik

So it is possible. What's interesting is what's possible in principle, not what's easy.

5 and 6 are interesting mainly because thinking about those only for a few seconds is by far the easiest way to understand the claim that there are no rigid bodies in SR.

nakurusil

Sounds like an attempt to justify why you got the solution wrong at post #1

Fredrik

Are you one of those guys who just tries to deliberately misunderstand everything?

nakurusil

No, not at all. I tried to help you see your own mistakes. Look at your initial post and at your persistence that all atoms in a rigid object can be accelerated in sync.