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

[pervect]

Thank you Chris and Pervect for your answers. They are both interesting.

However, I noticed that neither of you answered the question about what kind of objections have been presented against the obvious conclusion. Do they claim a) that GR is needed since acceleration is involved, b) that SR somehow implies that the rope will not break even though it's getting stretched, or c) some other kind of crank nonsense?

I'd say that c) is the closest answer. If you're really curious, check out the talk page, and wade through it:

http://en.wikipedia.org/wiki/Talk:Bell's_spaceship_paradox

I'm not really interested in rehashing these old arguments, and I think Chris is even less interested than I am - I wouldn't be surprised Chris would rather have a root canal without any anesthetic than rehash this again.

So let's forget about those old arguments as a whole (if you bother to read the talk page, and have some SPECIFIC question you want answered, go ahead and ask it though).

But, since there appears to be some interest, let's wipe the slate clean and start some new discussion, hopefully one that is more sensible and even-handed.

I think a lot of the underlying dispute is over distance measures. It seems that everyone has their own ideas on this topic, even excluding the cranks, and that even in the literature we don't see complete unanimity. For instance, their is a paper by Demystifier that talks about these issues that was discussed recently. I like my approach better than his, though :-). While my approach isn't published in any specific papers that I'm aware of, it's inspired by several common textbooks (specifically Wald and MTW).

With these caveats about a lack of complete unanimity in mind, the way I would describe the usual definition of distance would go like this. First, one needs to perform a global 3+1 split of space-time, by assigning every event in space-time a time coordinate. This can be done in many different ways. In the context of special relativity, every inertial frame of reference will have it's own 3+1 split. If two inertial observers in a flat Minkowski space-time are moving with respect to each other, they will assign different events as being simultaneous, generating a different space-time split. Only if two observers are stationary with respect to each other will they arrive at the same space-time split.

There are even more ways to perform a global 3+1 split in GR. The very first thing one must realize is that every different 3+1 split generates a different distance measure.

Given that one has this global 3+1 split, the mathematical process of defining a distance measure then becomes reasonably straightforward. Given this split, one defines a 3-d hypersurface of simultaneity as the set of points sharing the same time coordinate.

The 4-d metric, the invariant "Lorentz interval" will "induce" a 3-d spatial metric on every hypersurface of simultaneity. Any two nearby points on the 3-d hypersurface will have a space-like separation. The value of this space-like separation is just the space-like Lorentz interval between these points calculated via the 4-d metric, or physically by the Lorentz interval, which is an observer independent invariant that does not depend on any choice of coordinates.

Thus, given a global 3+1 split, we can use the Lorentz interval between nearby points to calculate the "induced" metric on the hypersurface, in terms of any convenient spatial coordinates we like.

This gives us a reasonably unambiguous notion of distance between two nearby points at any given "time", where "time" is the global time coordinate that we assigned to every event.

The process of defining a distance between two far-away points is slightly more complex. To define the distance between two far-away points, one must specify a specific curve connecting them. The length of this curve can be calculated by calculating the distance between each pair of nearby points on the curve (as above) and adding them together, i.e. via an integral.

The usual curve chosen is one which lies entirely in one particular hypersurface of simultaneity defined by the global 3+1 split, and it is a geodesic on that hypersurface - i.e. the curve generally comprises the set of points that gives the "shortest distance" between two points when the connecting curve is constrained to lie entirely within the hypersurface of simultaneity.

Here is where one must pay close attention, to make sure that this is indeed the curve being used to compute the distance by any particular author. It seems like the "obvious" choice, at least to me, but sometimes people (for whatever reason) don't make this "obvious" choice. So beware when you read a paper.

In curved geometries, you may have to worry about the fact that there can in general be more than one geodesic between any two points. For instance, on the surface of the Earth at the equator, is that coffee cup 1 meter to your west, or 40075159 meters to the east :-).

OK, this was the general approach. Now let's go to specifics. If we have an accelerated point-like observer, I suggest from my general analysis above that we have to first define some notion of simultaneity in order to be able to define distances. How do we do this? While there are many possible choices, one of the most common choices is to chose at any given instant, the notion of simultaneity of an instantaneously co-moving inertial observer as the appropriate notion of simultaneity for the global time coordinate 't'. This choice, when elaborated, ultimately winds up with the usual "Rindler" coordinates for an accelerated observer.

 

[quantum123]

I find it strange that problems can be classified as SR or GR.
Doesn't GR => SR?
If SR deals only with uniform velocity and GR deals with gravity, then acceleration in SR would be some intermediate form, which I suspect even Einstein used to arrive inductively at GR.

 

[yogi]

Quantum 123: In its initial inception, Einstein considered only Inertial motion. But the Lorentz transforms can be used to evaluate the time lost by objects which undergo acceleration so long as we take the view of the inertial unaccelerated observer. 13 years after Einstein published "On the Electrodynamics of Moving Bodies" he authored a paper explaining the twin's aging differential as consequent to an effective pseudo G field. As it turns out, most present authors take the position that, properly interpreted, SR is fully adequate to predict the correct results. While both theories give the exact same numerical results, the underlying physical causes "appear" to be different. Some have interpreted this as indicative of a deeper unity

 

[Fredrik]

OK Nakurusil, now you have proven that you're just a troll. No one could possibly have that poor reading comprehension. I don't know why I even bother to answer your increasingly absurd statements. This will probably be the last time.

No, SR handles accelerated motion...

No again, you need to understand that forces applied to an object propagate at finite speed (speed of sound).

I was asking Chris and Pervect what the cranks think are valid reasons to disagree with the physicists. Why do you pretend that I have made these objections? Seriously, what's wrong with you?

I gave it to you three times, here it is one more time:

The previous stuff proves that you're a troll. This claim proves that you're also a liar. We have not discussed the details of the spaceship scenario before. This is the first time you've made a post answering me, that makes any attempt to discuss the details.

It seems that your attempt to explain what happens in the spaceship scenario is meant to be serious though, so I will answer that as if we're actually having a discussion.

One thing you need to realize is that no one has said anything about the spaceships being "Born rigid", or in fact anything at all about the details about the spaceships. In the absence of such specifications, it is natural to treat them as point particles. If you are uncomfortable with this, then at least try to imagine the rope (or whatever) being attached to the same point on the two identical spaceships. Then it doesn't matter if the ships are Born rigid or not.

The rope will get Lorentz contracted in the launcher's frame, because its speed is changing! Its length in that frame will remain unchanged however, because the world lines of the two attachment points are identical except for their starting points in space. This means that the rope is being forcefully stretched to a proper length that when Lorentz contracted is equal to the original proper length. That's why the rope must break.

As for some of your specific claims...

1. The rear of the rocket (where the motor is) reaches the cruising speed v BEFORE the front of the rocket (due to ...Born rigidity)

If we assume that the rockets are Born rigid (and a real rocket would be, since the acceleration would not be so high that the speed has changed significantly in the time it takes a sound wave to propagate from one end of the rocket to another), then yes, this is true. However, we're talking about an extremely short time.

2. Therefore the rear of the leading rocket reaches the cruising speed v BEFORE the front of the trailing rocket.

3. Therefore the rod connecting the rear of the front rocket and the front of the rear rocket stretches

That would be part of the reason, in your version of the spaceship scenario, but if you're going to use 2 to motivate 3, then you should have mentioned that 2 also holds for any intermediate velocity u<v.

However, your 3 isn't the only reason the rope/rod/string stretches. The space between the rockets has stretched as well, and that's what this problem is really about. (If your 3 is the only reason the rope stretches in your version of the spaceship scenario, then the rope wouldn't stretch in everyone else's version of it. Everyone else thinks of the rope as being attached to the same point on both rockets, remember).

This is one thing you've missed: In the launcher's frame both rockets always have the same velocity. But in an inertial frame that's co-moving with the rocket in front, the trailing rocket will have a lower velocity during the acceleration. And if the rockets turn off their engines after a certain proper time T, the rocket in front is turning off its engine before the rocket behind it, in the co-moving frame. At this time (still in the co-moving frame), the rocket in front has reached its "cruising speed", but the rocket behind still hasn't.

4. All of the above has NOTHING to do with Lorentz contraction, contrary to your repeated claims.

That's where you're wrong. You're making a major blunder here. This has everything to do with Lorentz contraction. In fact, this is Lorentz contraction. Born rigidity was invented as a way to approximate how actual physical objects become Lorentz contracted. You have obviously completely misunderstood that.

5. All of the above shows that your claims 5-6 are physically impossible, contrary to your insistance to the contrary. You cannot "accelerate all the points in a real rigid object simultaneously" Born rigidity theory precludes this from happening.

Now you're really being a troll again. And you're wrong. What you call "all of the above" has nothing to do with my 5 and 6. And I've told you repeatedly that both 5 and 6 would require a simultaneous push to every single part of the object, something that's possible in principle. You don't seem to understand what "in principle" means, so maybe you should look it up or something.

Why don't you re-read your post #8?

No, you need to read it again, and then read the specific piece of criticism you made that started this part of the "discussion". You claimed that I had claimed that the objects in those idealized situations are rigid! I said no such thing! In fact I said the exact opposite.

 

[Fredrik]

I find it strange that problems can be classified as SR or GR.
Doesn't GR => SR?

The relationship between SR and GR is that you can obtain SR from GR by postulating that there is no matter or energy at all, anywhere in the universe. The only solution of Einstein's equation (the fundamental equation of GR) that satisfies these conditions is Minkowski space, i.e. the flat space-time of SR.

It is possible to deal with accelerated motion entirely in SR, contrary to what some people believe. It's pretty obvious really, if you think about the fact that "accelerated motion" is just a curve through Minkowski space that isn't a straight line.

 

[Fredrik]

I'd say that c) is the closest answer. If you're really curious, check out the talk page, and wade through it:

http://en.wikipedia.org/wiki/Talk:Bell's_spaceship_paradox

OK, thanks. I'll probably read some of it, but there seems to be a lot to read. I totally understand that you and Chris don't want to get into a discussion about "alternative" solutions.

...the way I would describe the usual definition of distance would go like this. First, one needs to perform a global 3+1 split of space-time

You made a very careful and very good explanation. As far as I'm concerned though, you could have saved some time by just saying that you're considering this problem in the context of GR and then skipped to this part:

While there are many possible choices, one of the most common choices is to chose at any given instant, the notion of simultaneity of an instantaneously co-moving inertial observer as the appropriate notion of simultaneity for the global time coordinate 't'. This choice, when elaborated, ultimately winds up with the usual "Rindler" coordinates for an accelerated observer.

Everything in between is more or less obvious to a Wald reader. (I'm sure that many others needed to see the details to understand what you were talking about though).

I have to admit that I hadn't even thought about this problem in the context of GR until now. I'm going to start now.

One more question though: Is everyone involved, including that Rod Ball character, in agreement about what happens in the context of SR?

 

[Boustrophedon]

No need for complications, just get the fundamentals right first !

I have to say that both Pervect and Chris Hillman are quite wrong. To show this (yet another) way, consider the following scenario...

The spaceships/string combination remains at first unlaunched while an observer accelerates up to constant velocity v in some extra 'mother ship'. From here the spaceships will appear closer by the Lorentz factor, gamma, with the string equally so, i.e.still taut.

Next launch the spaceships so as to accelerate up to join the 'mother ship' at constant v. According to incorrect Bell-type reasoning the ships will maintain constant distance, i.e. L/gamma, as they speed up to join the mother-ship but the string will un-contract back to original length L.

Now start again but reverse the launch order so that the spaceships/string first speeds up to v, where again we are supposed to believe that the string this time snaps under contraction ( let's say it's trailing end detaches from the rear ship ) while the spaceship distance stays constant. When the mother ship now accelerates up to join them at velocity v the string will regain it's original length L, but the spaceship distance will increase to gamma*L.

So taking the whole thing from the point of view of the mother-ship observer, in both cases they end up together again at the same velocity v, having accelerated identically in each case. However depending on which went first and which followed later, we get two completely different and contradictory conclusions. In one the spaceships are closer together than the string length by gamma and in the other they are further apart than the string length by gamma !

As I said before, "physical shrinkage" type contraction (as proposed by Fitzgerald & Lorentz) is outmoded by a century and plays no part in special relativity.

 

[Fredrik]

Boustrophedon, your explanation is hard to follow because it's often difficult to understand what frame you're using. Anyway, it certainly doesn't matter if the mother ship is brought to speed v before or after the other two ships.

Suppose the distance between the spaceships in the launcher's frame before they start is K. (I'm defining my own variable since I don't know what frame you used to define your L). This is what happens:

The distance between the ships will be constant (=K) in the launcher's frame, but not in the mother ship's frame. In the mother ship's frame, the distance will grow from K to gamma*K. That's why the string must break. If it breaks by detaching itself from the trailing ship at the beginning of the acceleration phase, then its length in the launcher's frame will change from K to K/gamma. In the mother ship's frame, its length will change from K/gamma to K. This is just Lorentz contraction.

Where exactly do you see a contradiction?

By the way, this is SR. It has nothing to do with the pre-SR theory that you keep mentioning.

 

[disregardthat]

Nakurusil, as far as I know, you have only discussed wether gravity happens instantly or not. That has nothing to do with this paradox.

If you think of that the spaceships are accelerated simultanously in the rest-frame, they would NOT be accelerated simultanously in the the frame of the point when they turn of their motors. In this frame, the front ship accelerates BEFORE the back ship, and this just states that the length between the ships have grown. MEaning the rope will get stretched, and snap.

 

[nakurusil]

<personal attack and ranting snipped as irrelevant to the subject>

It seems that your attempt to explain what happens in the spaceship scenario is meant to be serious though, so I will answer that as if we're actually having a discussion.

One thing you need to realize is that no one has said anything about the spaceships being "Born rigid", or in fact anything at all about the details about the spaceships.

But :

1. The ships are NOT point particles, they have dimensions
2. Born rigidity is germaine to the problem, I tried (and obviously failed) to explain to youhow it intervenes in stretching the rod.
3. Born rigidity is germaine in refuting your claims 5 and 6 as unphysical. This is how our little discussion started, with your insistance that scenarios 5 and 6 are possible. I proved you wrong but you wouldn't listen.

In the absence of such specifications, it is natural to treat them as point particles. If you are uncomfortable with this, then at least try to imagine the rope (or whatever) being attached to the same point on the two identical spaceships. Then it doesn't matter if the ships are Born rigid or not.

But the rope is not attached to the SAME point of both ships. It is attached to the front of the rear ship and to the tail of the leading ship. So you need to take Born rigidity into consideration. You are still trying to cover up for the nonsense in your claims 5 and 6.

The rope will get Lorentz contracted in the launcher's frame, because its speed is changing!
Its length in that frame will remain unchanged however, because the world lines of the two attachment points are identical except for their starting points in space. This means that the rope is being forcefully stretched to a proper length that when Lorentz contracted is equal to the original proper length. That's why the rope must break.

The correct solution to this problem has nothing to do with Lorentz contraction (even in the absence of the Born rigidity issue). It has to do with the fact that a line of simulataneity intercepts the two spacetime trajectories at a REAL (as opposed to apparent) distance that is LARGER than the length of the rod. This is your second misconception that I tried to correct but you are too stubborn to get it. It is really very simple, if you look at the wiki picture.

As for some of your specific claims...

If we assume that the rockets are Born rigid (and a real rocket would be, since the acceleration would not be so high that the speed has changed significantly in the time it takes a sound wave to propagate from one end of the rocket to another), then yes, this is true. However, we're talking about an extremely short time.

Seems that you took some time to read on Born rigidity, this is good. Now you can hopefully understand that claims 5-6 are incorrect.
What do you mean by However, we're talking about an extremely short time.? Can you quantify it? Because I can show you , mathematically, not with armwaving, how ANY amount of time taken into accelerating the ships contributes to stretching the rope. Actually, it can ve argued that the disparity in propagating the thrust forces between the rear of leading rocket and the front of the other contribute MORE to the rope stretching than the relativity of simultaneity discussed above.

That would be part of the reason, in your version of the spaceship scenario, but if you're going to use 2 to motivate 3, then you should have mentioned that 2 also holds for any intermediate velocity u<v.

However, your 3 isn't the only reason the rope/rod/string stretches. The space between the rockets have stretched as well, and that's what this problem is really about. (If your 3 is the only reason the rope stretches in your version of the spaceship scenario, then the rope wouldn't stretch in everyone else's version of it. Everyone else thinks of the rope as being attached to the same point on both rockets, remember).

I was tempted to say : "who is the troll here?". My very first post was a refutation of your claims 5 and 6 as unphysical because they contradict Born rigidity. Of course I am aware that the stretching is a superposition of BOTH relativity of simultaneity (nothing to do with any length contraction, buster) AND Born rigidity. I have shown you that you cannot ignore BORN rigidity, that's all.

This is one thing you've missed: In the launcher's frame both rockets always have the same velocity.

Not at all, I've been telling you that this is not true: during the acceleration period the rear of the leading rocket is FASTER than the front of the trailing rocket. So, do you understand Born rigidity or not? I am still not sure.

But in an inertial frame that's co-moving with the rocket in front, the trailing rocket will have a lower velocity during the acceleration.

See above, for a the complete and correct explanation.

And if the rockets turn off their engines after a certain proper time T, the rocket in front is turning off its engine before the rocket behind it,
in the co-moving frame. At this time (still in the co-moving frame), the rocket in front has reached its "cruising speed", but the rocket behind still hasn't.

Hmm, this "turning off its engine before" is a function of the way the two rockets clocks are synchronised, iyou surely knew that. If they use a light signal coming from the ground, as in the wiki example, the light signal will hit the more proximate rocket (the "rocket behind" in your text) BEFORE it hits the leading rocket, so the trailing rocket will turn off its engine BEFORE the leading rocket, further stretching the rope. So , it appears that you got it backwards.

But what is the relevance to all this in light of my refutation of your claims 5 and 6?

That's where you're wrong. You're making a major blunder here. This has everything to do with Lorentz contraction. In fact, this is Lorentz contraction. Born rigidity was invented as a way to approximate how actual physical objects become Lorentz contracted. You have obviously completely misunderstood that.

Looks like you may have made the error, see the paragraph above.

Now you're really being a troll again. And you're wrong. What you call "all of the above" has nothing to do with my 5 and 6. And I've told you repeatedly that both 5 and 6 would require a simultaneous push to every single part of the object, something that's possible in principle. You don't seem to understand what "in principle" means, so maybe you should look it up or something.

No need for personal attacks. If pigs had wings, they would fly. Born rigidity says exactly the opposite, that what you are claiming in principle, is NOT possible. IN REALITY. This IS the main disagreement between us.

No, you need to read it again, and then read the specific piece of criticism you made that started this part of the "discussion". You claimed that I had claimed that the objects in those idealized situations are rigid! I said no such thing! In fact I said the exact opposite.

Still trying to justify 5 and 6?

 

[Boustrophedon]

It doesn't matter what frame you use - I made it clear that for simplicity I used the mother-ship frame throughout. If the mother-ship takes off first to reach constant v the spaceship distance and the string length will both then be L/gamma. Then the spaceships take off to eventually join the mother ship at same velocity, during which Bell's reasoning would have the string un-contract to L while the spaceships stay L/gamma apart.

Now simply consider the same scenario again only in reverse order: The spaceships take off first and remain at constant distance L from the (unlaunched) mother-ship while the string a la Bell shrinks to L/gamma.
Now when the mother-ship speeds up to join the spaceships/string at the same constant v, the string will un-contract to L and the spaceship distance increase from L to gamma*L.

Thus we have arrived at a contradiction. The same situation ( s'ships, string & mothership back at rest w.r.t. each other ) is obtained by exactly the same acceleration processes but gives totally different comparisons depending on which order they went in. The first case ends with spaceships L/gamma apart while in the second they are gamma*L apart.

 

[quantum123]

Why must the rod obey Born rigidity? Is it just an assumption of a special case?

 

[nakurusil]

Why must the rod obey Born rigidity? Is it just an assumption of a special case?

Because all real life objects do. Forces do not propagate instantaneously in rigid or objects. (they obviously do not propagate instantaneously in semi-rigid ones). Another way of looking at this, there is no infinitely rigid material. When one pushes on a rod, the rod acts as a train, it compresses a little (because the cars are connected with spring-like devices). When one pulls a rod, it stretches, exactly like a train. The "locomotive" part gets going earlier in both cases.
Note: not only the rod but also the two rockets in the problem are affected by the Born rigidity. The rockets are NOT points, they have dimensions that need to be accounted for in solving the problem.

 

[Boustrophedon]

What rod ?, what train ?, what cars ? Do try and keep up !

 

[nakurusil]

What rod ?, what train ?, what cars ? Do try and keep up !

quantum123 asked for an explanation of Born rigidity. Do you know what it is and what role it plays in Bell's paradox? No?

 

[nakurusil]

It doesn't matter what frame you use - I made it clear that for simplicity I used the mother-ship frame throughout. If the mother-ship takes off first to reach constant v the spaceship distance and the string length will both then be L/gamma. Then the spaceships take off to eventually join the mother ship at same velocity, during which Bell's reasoning would have the string un-contract to L while the spaceships stay L/gamma apart.

Now simply consider the same scenario again only in reverse order: The spaceships take off first and remain at constant distance L from the (unlaunched) mother-ship while the string a la Bell shrinks to L/gamma.
Now when the mother-ship speeds up to join the spaceships/string at the same constant v, the string will un-contract to L and the spaceship distance increase from L to gamma*L.

Thus we have arrived at a contradiction. The same situation ( s'ships, string & mothership back at rest w.r.t. each other ) is obtained by exactly the same acceleration processes but gives totally different comparisons depending on which order they went in. The first case ends with spaceships L/gamma apart while in the second they are gamma*L apart.

You get a contradiction only if you don't understand the physics at work and you oversimplify the problem as above. There are multiple places where the problem is solved correctly . Try Wheeler and Taylor's Spacetime Physics pages 117-119 (second edition).

 

[Boustrophedon]

Yes and No. Yes I know perfectly well what it is and No it doesn't play any role in Bell's problem.

 

[disregardthat]

In the absence of such specifications, it is natural to treat them as point particles. If you are uncomfortable with this, then at least try to imagine the rope (or whatever) being attached to the same point on the two identical spaceships. Then it doesn't matter if the ships are Born rigid or not.

But the rope is not attached to the SAME point of both ships. It is attached to the front of the rear ship and to the tail of the leading ship. So you need to take Born rigidity into consideration. You are still trying to cover up for the nonsense in your claims 5 and 6.

Uhm, yes they are. I recommend you read the very paradox before you make your claims.

This is from the Wikipedia artcile Fredrik posted:

"Analysis
In the following analysis we will treat the spaceships as point masses and only consider the length of the string. We will analyze the variant case previously mentioned, where both spaceships shut of their engines after some time period T."

That makes me pretty sure at least THEY treat the spaceships as point masses

Which makes this incorrect:

1. The ships are NOT point particles, they have dimensions
2. Born rigidity is germaine to the problem, I tried (and obviously failed) to explain to youhow it intervenes in stretching the rod.

The correct solution to this problem has nothing to do with Lorentz contraction (even in the absence of the Born rigidity issue).

Do you really know what Lorentz contraction is? After reading your large post, I suspect you confuse contraction with mechanical compression.

--------------------------

I suppose we are only searching for if the string will break, and not if the space between the ships have grown after they shut their engines off. Because the moment they turn it off, depends on which frame you look at it. If they keep accelerating towards infinity, the string will eventually break.

 

[nakurusil]

Yes and No. Yes I know perfectly well what it is and No it doesn't play any role in Bell's problem.

you may want to rethink your second answer

 

[nakurusil]

Uhm, yes they are. I recommend you read the very paradox before you make your claims.

This is from the Wikipedia artcile Fredrik posted:

"Analysis
In the following analysis we will treat the spaceships as point masses and only consider the length of the string. We will analyze the variant case previously mentioned, where both spaceships shut of their engines after some time period T."

That makes me pretty sure at least THEY treat the spaceships as point masses

Can you distinguish between a pedagogical simplification and reality? Of course you do. The ships are NOT point sources. wiki shows a simplified model for the problem. I brought this fact into discussion in order to refute Fredrik's claims 5 and 6 as unphysical. His claims have to do with physical rockets, not point approximations.

BTW: The SR length contraction has nothing to do with this problem. And yes, I understand length contraction.

 

[nakurusil]

I suppose we are only searching for if the string will break, and not if the space between the ships have grown after they shut their engines off.

The way to find that out is by calculating the variation of the distance betwen the two rockets and comparing it to the length of the unstretched string (rod).

Because the moment they turn it off, depends on which frame you look at it. If they keep accelerating towards infinity, the string will eventually break.

Interestingly enough, the string will get stretched even if the two rockets accelerate only for a finite time.

 

[disregardthat]

Can you distinguish between a pedagogical simplification and reality? Of course you do. The ships are NOT point sources. wiki shows a simplified model for the problem. I brought this fact into discussion in order to refute Fredrik's claims 5 and 6 as unphysical. His claims have to do with physical rockets, not point approximations.

Ok, so you mean that in the original situation stated by the article, really mean that the string were attached to different places on each ship? Find evidence first. Here I think we are talking about the article...

And the 5. and 6. point to fredrik has absolutely zero to do with this paradox...

BTW: The SR length contraction has nothing to do with this problem. And yes, I understand length contraction.

this is nonsense, lorentz contraction has everything to do with this! Why would you say otherwise?

Interestingly enough, the string will get stretched even if the two rockets accelerate only for a finite time.

Correct, but since we know nothing of the material the string is made of, we cannot jump to conclusions when the rope will be snapped. It will happen in a finite time, but you cannot know when. That's just the reason I said "If they keep accelerating towards infinity, the string will eventually break.", somewhere along the line the string will snap...

Why are you posting so many posts after another, why not keep it to 1 post each time?

 

[nakurusil]

Ok, so you mean that in the original situation stated by the article, really mean that the string were attached to different places on each ship? Find evidence first. Here I think we are talking about the article...

And the 5. and 6. point to fredrik has absolutely zero to do with this paradox...

You still don't get it. See my first post in this thread.

Why are you posting so many posts after another, why not keep it to 1 post each time?

Because you seem unable to follow really simple stuff, so I matrying to make it easier for you to understand.

 

[nakurusil]

this is nonsense, lorentz contraction has everything to do with this! Why would you say otherwise?

There is a very good, comprehensive treatment of the problem in the external links of the wiki article. On second thoughts, someone should bite the bullet and turn that external link into an article. It shows a much higher level of detail and it includes the Born rigidity treatment as well as the hyperbolic motion. No Lorentz contraction, sorry

 

[disregardthat]

You still don't get it. See my first post in this thread.

Ok, that just shows that aren't listening to what I am saying. I said that: "And the 5. and 6. point to fredrik has absolutely zero to do with this paradox..."
Then you show a link to a post where you explain why these points are unreal. Don't you see why that is no answer? You argument to things i didn't say. That makes it impossible to discuss with you.

There is a very good, comprehensive treatment of the problem in the external links of the wiki article. On second thoughts, someone should bite the bullet and turn that external link into an article. It shows a much higher level of detail and it includes the Born rigidity treatment as well as the hyperbolic motion. No Lorentz contraction, sorry

Uhm, ok. I see we have found the root... So, you believe that there is no lorentz contraction when an object moves at higher velocities? It is contracted! That is what this very paradox is going on about.

From the wiki article:
"According to discussions by Dewan & Beran and also Bell, in the spaceship launcher's reference system the distance between the ships will remain constant while the elastic limit of the string is length contracted, so that at a certain point in time the string should break!"

That shows that contraction is the main part of the paradox...

And this quote from the article should give you proper understanding that it IS really lorentz contraction that is being talked about:

"Finally, we can say that the proper distance between spaceships A and B after the end of the acceleration phase in a comoving frame is equal to the Lorentz length of the line segment A`B``. The line A`B`` is defined to be a line of constant t', where t' is the time coordinate in the comoving frame, a time coordinate which can be computed from the coordinates in frame S via the Lorentz transform:"

Read the article before you come with statements. Because I have seen many of your statements to be wrong. And you don't even answer to your own mistakes. That is also making it very difficult to discuss with you.

 

[Fredrik]

Nakurusil,

Why don't you answer my question about why you pretended that I had made claims in post #31?

Why don't you say anything about the fact that you claimed to have discussed the details of #1 with me before, when in fact you had not?

Why don't you say anything about the fact that you claimed that I had said that the object in #8 is rigid, when in fact I had never done so?

You have no right to whine about "personal attacks" as long as you talk to me as if I have opinions that you know I don't have. If you stop doing that, I won't call you a troll again.

I'm done discussing the "in principle" vs. "in reality" issue with you. I'm just going to tell you that you are going to have a very hard time understanding physics unless you're willing to think about stuff that's only possible in principle.

I see you still claim that Lorentz contraction is irrelevant. I'm going to have to quote Penn Gillette: "You couldn't be more wrong if your name was Wrong Wrongy Wrongenstein". I don't know if I can explain it to you though. It's not that I'm not willing to explain stuff. It's that you don't seem to be able to even consider the possibility that you might be wrong about something.

1. The ships are NOT point particles, they have dimensions

Only in your version of the problem. Look at the space-time diagram in Wikipedia for example. Do you see four world-lines or two? I see two. One for each ship.

2. Born rigidity is germaine to the problem,...

Wrong. It has a small part to play in your version of the problem though.

3. Born rigidity is germaine in refuting your claims 5 and 6 as unphysical.

Wrong. If I had said that the objects in #8 are Born rigid, then you would have had a point, but I clearly said that they are not.

I proved you wrong but you wouldn't listen.

I have no doubt that you will continue to think that's what happened.

But the rope is not attached to the SAME point of both ships. It is attached to the front of the rear ship and to the tail of the leading ship.

Only in your version of this problem. This additional assumption that you introduced is just an irrelevant complication that obscures the real issue. The real issue can be seen by simply considering curves in Minkowski space, and ignoring the spaceships altogether.

You are still trying to cover up for the nonsense in your claims 5 and 6.

BS. I was trying to explain to you what the real issue is in this problem, but you obviously ignored it.

The correct solution to this problem has nothing to do with Lorentz contraction (even in the absence of the Born rigidity issue).

If you think so, then you don't know what Lorentz contraction is.

It has to do with the fact that a line of simulataneity intercepts the two spacetime trajectories at a REAL (as opposed to apparent) distance that is LARGER than the length of the rod.

This is Lorentz contraction!

Seems that you took some time to read on Born rigidity, this is good. Now you can hopefully understand that claims 5-6 are incorrect.

I understood Born rigidity a lot better than you do now a long time before I wrote #8, and 5-6 are still possible in principle.

What do you mean by However, we're talking about an extremely short time.? Can you quantify it? Because I can show you , mathematically, not with armwaving, how ANY amount of time taken into accelerating the ships contributes to stretching the rope.

Maybe it's not a small amount. I can't see this part as clearly as the rest right now, and I don't think it's relevant enough to be worth spending time on. Anyway we seem to agree about the important details about what happens to your Born rigid spaceships. But all of that stuff is irrelevant to the real problem anyway.

Not at all, I've been telling you that this is not true: during the acceleration period the rear of the leading rocket is FASTER than the front of the trailing rocket. So, do you understand Born rigidity or not? I am still not sure.

The stuff about the rear moving faster than the front is true, in your version of this problem. My claim that both rockets (not both ends of one rocket) have the same velocity in the launcher's frame is correct in both versions, so I don't know why you think I was contradicting you. But ok, let's rephrase it specifically for your version of the problem: "Consider a specific part of rocket A. In the launcher's frame, at any given time, that part of rocket A has the same velocity as the same part of rocket B".

Hmm, this "turning off its engine before" is a function of the way the two rockets clocks are synchronised, iyou surely knew that. If they use a light signal coming from the ground, as in the wiki example, the light signal will hit the more proximate rocket (the "rocket behind" in your text) BEFORE it hits the leading rocket, so the trailing rocket will turn off its engine BEFORE the leading rocket, further stretching the rope. So , it appears that you got it backwards.

Light signal from Earth?! Don't bring any more unnecessary complications into this! Just imagine two identical ships controlled by their onboard computers (identical computers, running identical programs). This will guarantee that the two world lines are identical except for their starting position in space. (If the world lines aren't identical, there must be something fundamentally different about the starting positions, and that would contradict SR).

But what is the relevance to all this in light of my refutation of your claims 5 and 6?

I have no idea why you're asking this. I have told you repeatedly that 5 and 6 have absolutely no relevance to the spaceship problem.

 

[Fredrik]

Why must the rod obey Born rigidity? Is it just an assumption of a special case?

The string/rope/rod that connects the spaceships in this problem is not Born rigid. That would contradict the description of the problem.

Because all real life objects do.

This certainly explains a lot. When I was answering you earlier I was thinking "I wonder if he thinks that all objects are Born rigid" and now I know that you do.

First of all, there's no such thing as an object that just is Born rigid. Born rigidity isn't a property of a material, it's a property of the world lines of the different parts of the object.

This is the definition:

An object is said to be going through Born rigid motion if the distance between any two points on the object, as measured by co-moving inertial observers, is constant.

A rubber band that's being pulled in two opposite directions at once certainly isn't Born rigid.

A rocket is approximately Born rigid, but it can't be exactly Born rigid through the entire acceleration phase, since the force that accelerates it is applied only to the rear of the rocket.

 

[Fredrik]

It doesn't matter what frame you use - I made it clear that for simplicity I used the mother-ship frame throughout. If the mother-ship takes off first to reach constant v the spaceship distance and the string length will both then be L/gamma. Then the spaceships take off to eventually join the mother ship at same velocity, during which Bell's reasoning would have the string un-contract to L while the spaceships stay L/gamma apart.

OK, this time I understand you. But the spaceships don't stay L/gamma apart in the mother ship's frame. The distance between them is always L in the launcher's frame, so it can't be a constant in the mother ship's frame. (This is easy to see in the space-time diagram). It will be "uncontracted" from L/gamma to L.

I will stop here. There may be other mistakes in what you said, but I'm not going to look for them, since one mistake is enough to invalidate the conclusion.

 

[nakurusil]

Nakurusil,

Why don't you answer my question about why you pretended that I had made claims in post #31?

Why don't you say anything about the fact that you claimed to have discussed the details of #1 with me before, when in fact you had not?

Why don't you say anything about the fact that you claimed that I had said that the object in #8 is rigid, when in fact I had never done so?

What's your problem? Look at the nonsense you put forward in your claims 5 and 6. I even boldened them for you and you still persist?

You have no right to whine about "personal attacks" as long as you talk to me as if I have opinions that you know I don't have. If you stop doing that, I won't call you a troll again.

You only need to look at the ridiculous claims 5,6.

I'm done discussing the "in principle" vs. "in reality" issue with you. I'm just going to tell you that you are going to have a very hard time understanding physics unless you're willing to think about stuff that's only possible in principle.

So after I called you on the silliness of those two claims you have persistently tried to justify them. They are ridiculous, they contradict physical reality.

I see you still claim that Lorentz contraction is irrelevant.

...for solving this partcular problem, yes.

Wrong. If I had said that the objects in #8 are Born rigid, then you would have had a point, but I clearly said that they are not.

Go back and re-examine your unphysical claims 5 and 6.

Only in your version of this problem. This additional assumption that you introduced is just an irrelevant complication that obscures the real issue. The real issue can be seen by simply considering curves in Minkowski space, and ignoring the spaceships altogether.

Umm, no. There is a comprehensive solution in the "External links" of the wiki page. Read it.

BS. I was trying to explain to you what the real issue is in this problem, but you obviously ignored it.

So , you still think that your claims 5,6 are physical? Once you get exposed, you seem unable to admit error.

If you think so, then you don't know what Lorentz contraction is.
This is Lorentz contraction!

Maybe where you went to school.

I understood Born rigidity a lot better than you do now a long time before I wrote #8, and 5-6 are still possible in principle.

Of course, why admit to error. When caught, do everything possiblle to cover it up, even if you are digging yourself deeper.

Maybe it's not a small amount. I can't see this part as clearly as the rest right now, and I don't think it's relevant enough to be worth spending time on. Anyway we seem to agree about the important details about what happens to your Born rigid spaceships. But all of that stuff is irrelevant to the real problem anyway.

Maybe if you tried to put it in a mathematical form you might get surprised. Try writing down the equations.

The stuff about the rear moving faster than the front is true, in your version of this problem. My claim that both rockets (not both ends of one rocket) have the same velocity in the launcher's frame is correct in both versions, so I don't know why you think I was contradicting you. But ok, let's rephrase it specifically for your version of the problem: "Consider a specific part of rocket A. In the launcher's frame, at any given time, that part of rocket A has the same velocity as the same part of rocket B".

But this is exactly the version that disagrees with your claims 5,6

Light signal from Earth?! Don't bring any more unnecessary complications into this! Just imagine two identical ships controlled by their onboard computers (identical computers, running identical programs). This will guarantee that the two world lines are identical except for their starting position in space.

Umm, no. You can't do that. The two rockets have DIFFERENT speeds , therefore their onboard clocks, computers, oscillators would be DESYNCHRONIZED. You need a common reference. Try googling "Cassini".

(If the world lines aren't identical, there must be something fundamentally different about the starting positions, and that would contradict SR).

Don't you think that the two pilots need a signal to tell them when to shut off the engines? This is exactly what happens in the picture of wiki version, you know, the one that you keep quoting. I pointed that out to you but you snipped it. Whatever you do, don't admit to error. Ever.

I have no idea why you're asking this. I have told you repeatedly that 5 and 6 have absolutely no relevance to the spaceship problem.

5,6 have no relevance whatsoever since they violate physical reality. This was my point all along. Finally we agree on something.

 

[nakurusil]

The string/rope/rod that connects the spaceships in this problem is not Born rigid. That would contradict the description of the problem.


This certainly explains a lot. When I was answering you earlier I was thinking "I wonder if he thinks that all objects are Born rigid" and now I know that you do.

Umm, no. You are twisting my words. I went on to explain that there is no such thing as an ifinitely rigid object, this is why I objected so strongly to your claims 5,6.

A rocket is approximately Born rigid, but it can't be exactly Born rigid through the entire acceleration phase, since the force that accelerates it is applied only to the rear of the rocket.

And this is exactly why I objected repeatedly to 5,6. Finally you caught up on reading on the subject, congratulations.

 

[Fredrik]

OK, that does it. I'm done talking to you Nakurusil. You're a troll. (Definition here).

 

[nakurusil]

OK, that does it. I'm done talking to you Nakurusil. You're a troll. (Definition here).

You insult me because I exposed your erroneous claims and you ran out of logical and pertinent arguments?

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.

...and you would want people to believe that you understand Born's theory of rigidity? Let's be serious.

 

[nakurusil]

OK, this time I understand you. But the spaceships don't stay L/gamma apart in the mother ship's frame. The distance between them is always L in the launcher's frame, so it can't be a constant in the mother ship's frame. (This is easy to see in the space-time diagram). It will be "uncontracted" from L/gamma to L.

I will stop here. There may be other mistakes in what you said, but I'm not going to look for them, since one mistake is enough to invalidate the conclusion.

Lorentz transforms do not apply to accelerated motion. Didn't you know that?
Now I understand your insistance in claiming that "The distance between them is always L in the launcher's frame, so it can't be a constant in the mother ship's frame. (This is easy to see in the space-time diagram). It will be "uncontracted" from L/gamma to L."

You can't simply brute force the problem and apply length contraction, you need to use the equations of hyperbolic motion in order to calculate the proper distance. Didn't you know that? Apparently not.

 

[MeJennifer]

Lorentz transforms do not apply to accelerated motion.

That is incorrect, they apply to all relative motion.
An accelerated user is at rest with a sequence of comoving inertial frames that each have a different relative speed.

 

[nakurusil]

That is incorrect, they apply to all relative motion.
An accelerated user is at rest with a sequence of comoving inertial frames that each have a different relative speed.

Don't think so. Think about how the Lorentz transforms have been derived.
You are talking about the fact that you can apply a sequence of Lorentz transforms over a sequence of infinitesimal changes in speed in order to derive the equations of hyperbolic motion. This doesn't mean that you can replace the equations of hyperbolic motion with a single Lorentz transform. I think that this is what is done on the wiki page of the Bell's paradox and it most probably led to an incorrect solution. All this bickering with Fredrick about his misunderstanding of the Born theory of rigidity may have uncovered something interesting after all.
Either way, the safe way is to use the equations of hyperbolic motion. I don't think that you get the same results using the correct eqiuations (hyperbolic motion) as using the Lorentz transforms. Try it on the wiki solution of the Bell paradox, do the calculations.
I also think that Chris Hillman had a link to his solution in the wiki archives. It treats the complete case correctly, I think that it should be put back in.