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

[nakurusil]

What has a document on the clock paradox to with your statement that Lorentz transformations do not work with accelerated motion?

This is getting ridiculous and is boosted by the fact that you are claiming expertise and making continious denigrating remarks to several members on this forum here without apparently even understanding the basic scope of the Lorentz transformations with regards to boosts, rotations and reflections in flat space-time.

It gives you the application and the proper formulas of hyperbolic motion. Try learning how to apply them, it solves the Bell paradox in a few lines.

 

[nakurusil]

You're missing the point as usual. If the distance between the attachment points in the new inertial frame turns out to be larger than it was from the beginning, the stretching of that distance must have happened during the acceleration. Nothing interesting happens once the ships have turned off their engines, as we can show explicitly.

Sure, you are making as if I didn't say the same thing. Problem with your derivation is that you are deriving the distance between the ships, during the acceleration phase incorrectly. Your derivation produces the incorrect result.

This is actually correct. The point of postulating constant acceleration (which I didn't) is that it makes it possible to explicitly calculate how much the string has stretched in the accelerating frame, at any time. (Edit: Hmm, is this really true? There's an infinite number of accelerating frames here. I'm not even sure if the proper distance is the same in the two frames defined by the attachment points. I need to think about this some more).

Well, thank you, you are starting to understand.

When I said that I could have used a co-moving inertial frame, I was actually making a mistake. (Note that I just refuted your claim that I never admit mistakes).

Indeed. If you managed to see your mistake in deriving the separation between rockets using Lorentz transforms next...

It's actually not even obvious from what I wrote that I was making a mistake, so I didn't even have to admit this, but the mistake was to think that it's possible to calculate the proper distance between the attachment points during the acceleration in a co-moving inertial frame.

Actually you can. But you need to learn how to use the proper formalisms.

I wasn't wrong, and you know it. That's why this is trolling, and nothing else.

Here you go again. I was thinking about showing you how to do things correctly but now I'll just show you again that you don't have a clue:

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.

 

[nakurusil]

We don't need to calculate what length the rope would be at every moment, we are merely trying to find proof that the string actually IS stretched when the spaceships are moving.

If you measured the average velocity of the spaceships in acceleration to a specific moment, you could use the lorentz contraction for velocity.

Bad idea. The rope doesn't break due to any "lorentz contraction", it breaks due to increased separation between rockets.

 

[Fredrik]

As far as I can tell, by only looking at a few of the external sources, they all seem to be focused on the detail of constant acceleration. There's nothing wrong with that of course. When the acceleration is constant, it's possible to calculate explicitly how much the has string has stretched at the time of any given event on one of the world lines. (In this post, I will sometimes be talking about the string as if it's able to stretch without breaking, and sometimes as if it breaks at the very first moment of stretching. I hope it's obvious what I mean. If it's not, ask). But it's not necessary to calculate this explicitly. All we need to prove is that the string breaks.

Since we don't need to calculate exactly how much the string breaks, we shouldn't have to postulate that the acceleration is constant. We should be able to show that the string breaks no matter how the spaceships accelerate.

There are many different versions of this problem, for example:

a) constant acceleration for ever (this is Bell's original version, I think)
b) arbitrary acceleration until a certain proper time when the engines shut off (my version)
c) arbitrary acceleration (the most general version)

I want to prove that the string breaks in version c). I have already proven that the string breaks in version b), by explicitly calculating the proper length of the string after the engines have been shut off. (See #76 and #88).

I think it's intuitively obvious that if the string breaks in b) it must also break in c), but I'd like to find a rigorous argument.

I think my solution of b) can be used as a starting point. This is the kind of reasoning I have in mind: The proper length doesn't increase once the engines have been turned off, so it must increase during the acceleration phase. This means that the string would also break in c).

I pretty sure this line of reasoning is valid, but as it stands, I don't think it's rigorous enough to prove that the string breaks in c). I'm kind of busy today, so I'm not going to try to work this out now. Maybe I'll try to fill in the missing details tomorrow. If anyone else feels like taking a shot at it in the mean time, go ahead...

 

[cincirob]

The Rope a Rocket Problem

This is an old problem first posed by JS Bell at a gathering of physicists. A number of physicists didn't get the right answer the first time. Because of this anti-relativists have latched onto it as a proof agianst relativity because relativity physicists sometimes disagree.

In any case, the outcome is a little strange. While most (even antirelativists) will agree the rope breaks, looking at the clocks on both ships is a bit confusing.

1. Before they accelerate the clocks on the ships are synchronized.

2. If the two ships stop accelerating at the same instant in the stationary frame, they will be going the same velocity and therefore it is possible to synchronize their clocks.

3. Since the clocks underwent the same process they will read identical times when compared to clocks in the stationary frame.

4. #3 seems problematical because,if the rocke clocks are synchronized, they shoud read differently for the stationary observers; that is, a stationary observer near the rear rocket will read Tr and because the clocks are moving at some v, Tf shoul read Tr-vd/c^2(1-(v/c)^2^-.5.

So it's a bit confusing that the clocks read the same. Lately I have been working on the math to see if I can predict this outcome but I'm not finished yet. You might try it as a challenge.

 

[nakurusil]

The proper length doesn't increase once the engines have been turned off, so it must increase during the acceleration phase. This means that the string would also break in c).

Correct. So if the rope breaks it breaks during the acceleration phase. This is why you must calculate the separation distance between the rockets during the acceleration phase. This is why you must not use Lorentz transforms, they do not apply to accelerated motion.

I pretty sure this line of reasoning is valid, but as it stands, I don't think it's rigorous enough to prove that the string breaks in c).

Correct. Except that your solution does not compute the separation distance between rockets correctly. You should not be using the Lorenz transforms, you schould be using hyperbolic motion. Using Lorentz transforms is akin to using the fact that the sum of the angles is 180 degrees in a planar triangle in order to calculate the third angle of a spherical triangle when you know the first two angles. In both cases there is no justification in blindly appliying a theory derived for one instance to a totally different instance.
By applying the correct theory (hyperbolic motion) you will get the correct answer. An it is not l=l_0\gamma. If you do the calculation correctly you will get a nonlinear expression that depends on acceleration and time.

 

[Fredrik]

I don't have to calculate the separation at all. I just have to show that it increases.

My calculation is exactly right for version b) of the problem, so stop denying that or at least try to prove that you're right. That hyperbolic motion stuff is specifically for version a).

 

[nakurusil]

I don't have to calculate the separation at all. I just have to show that it increases.

Yes, you do have to calculate the separation correctly, ESPECIALLY for "your case" (b) The question that was asked is: "will the rope snap". The rope has some elasticity, so it snaps only if the distance betwen the rockets increases beyond what the rope elasticity can accommodate DURING the acceleration phase. Without a correct calculation of the separation distance betwen the rockets, you cannot find out if the rope snaps. And in "your case" (b), the distance stops increasing after you shut off your engines.

My calculation is exactly right for version b) of the problem, so stop denying that or at least try to prove that you're right. That hyperbolic motion stuff is specifically for version a).

The hyperbolic motion is the rigurous solution for ALL cases.

 

[Fredrik]

Yes, you do have to calculate the separation correctly, ESPECIALLY for "your case" (b) The question that was asked is: "will the rope snap". The rope has some elasticity, so it snaps only if the distance betwen the rockets increases beyond what the rope elasticity can accommodate DURING the acceleration phase. Without a correct calculation of the separation distance betwen the rockets, you cannot find out if the rope snaps. And in "your case" (b), the distance stops increasing after you shut off your engines.

It's hard to tell if you're being serious. The rope will certainly snap if the proper length at any time exceeds the original proper length, and my calculation is more than sufficient to prove that it does in version b).

You're still insinuating that my solution of b) is incorrect. I suggest that you either stop doing that, or prove that you're right.

The hyperbolic motion is the rigurous solution for ALL cases.

Please explain yourself. Hyperbolic motion is constant proper acceleration. So how does a calculation that takes hyperbolic motion as a starting point solve the general case?

 

[ZapperZ]

This thread has gone long enough, and it is going nowhere long enough.

I will point out to everyone involved that to re-read the PF Guidelines that you have agreed to. If you do not think we meant everything we wrote in there, think again.

Consider this as your only warning before more drastic action is taken.

Zz.