I Some thoughts about Bell's string paradox alternatives

[member 728827]

I think that depends on what Bell meant by "identical acceleration programmes". I think he meant "same F/M ratio". In other words, he was specifying what the spaceships' engines were programmed to do: put out the same constant F/M.

Bell writes: "Then, as reckoned by an observer in A - the rest observer at the starting position -, they will have at every moment the same velocity, and so remain displaced one from the other by a fixed distance". This is a constant acceleration motion description.

Wikipedia comments: "Bell's spaceship paradox is not about preserving the rest length between objects (as in Born rigidity), but about preserving the distance in an inertial frame relative to which the objects are in motion...".

Bell scenario is in the Chapter 9 of the book "Speakable and unspeakable in Quantum Mechanics". The title of the chapter is "How to teach Special Relativity". So, is an example chosen to teach SR. As such, is not a very precise description, and needed to put some drama in the plot.

I think that some pages after, Bell's says something like "actually, the thread will pull the spaceships and break if it can't change the acceleration of the heavy masses of the spaceships" (I don't have now the exact quote, but I remember more or less). All very imprecise.

But, of course, what seems to happen is that there's really no need for the string to be "weak", because the relativistic acceleration due to the time dilation in accelerated motion is so tiny, that at 1g, a spaceship of 10⁹ Kg, would not be able to break a 0,02 N ultimate tensile strength thread, and so the students would probably say: bah, and what's the point then? It introduced some dramatic effects for the string.

 

[PeterDonis]

This is a constant acceleration motion description.

Yes, I understand that. But, as has already been noted, he also specified a "fragile" string, which means that the string exerts no force on the spaceships, so "constant acceleration" is the same as "the same F/M ratio".

As I remarked earlier, Bell's scenario is ambiguous about how to handle extensions to cases where the string is not "fragile".

what seems to happen is that there's really no need for the string to be "weak", because the relativistic acceleration due to the time dilation in accelerated motion is so tiny

As I commented before, that's only because you picked a very small proper acceleration. With a much larger proper acceleration, even a real, not "fragile" string would break quickly. You would need a string made of a very special material (I suggested carbon nanotubes in an earlier post) to withstand much larger acceleration forces long enough to have an effect on the motion of the ships.

Bell, of course, did not specify the magnitude of the proper acceleration in his scenario either. So this is yet another ambiguity in how to extend the scenario to cover more cases.

 

[PeterDonis]

Bell's scenario is ambiguous about how to handle extensions to cases where the string is not "fragile".

Bell, of course, did not specify the magnitude of the proper acceleration in his scenario either. So this is yet another ambiguity in how to extend the scenario to cover more cases.

@Lluis Olle I should emphasize that I am not saying your proposed scenarios are not valid scenarios. They are. I am only pointing out that, given what Bell left out in his specification of his scenario, the range of possible ways to extend it is extremely wide.

 

[PeterDonis]

So, is an example chosen to teach SR. As such, is not a very precise description, and needed to put some drama in the plot.

It might be worth looking at why Bell proposed this scenario as an example of how to teach SR. He did it because he wanted to emphasize that using frame-dependent concepts like "length contraction" to reason about what would happen in a particular scenario was perfectly valid. In other words, he was protesting, in a way, against claims by other physicists that said you should avoid using such frame-dependent concepts at all and only look at invariants. He was simply giving an example of a scenario where a simple use of the concept of "length contraction" gives the right answer, whereas Bell was able to confuse many other physicists who refused to use such concepts into giving the wrong answer when he posed his scenario to them.

Given the above, it's easy to see why Bell included the "fragile" string in his scenario: he wanted to rule out any kind of objection that the string might exert a force that would affect the motion of the spaceships. And even with that very strong constraint, he was able to confuse other physicists into giving the wrong answer, that the "fragile" string would not break.

You appear to be using Bell's scenario as a basis for illustrating something different, so it is to be expected that there will be information that is simply not specified in Bell's scenario that is important to your scenario.

 

[member 728827]

Is just for fun. I'm not going to discover the onion soup.

 

[member 728827]

Does time dilation have to be taken into account here? I think for an accelerating ship clocks at the front of the ship will tick faster than the back, so presumably, clocks will tick faster on the leading ship.

Yes absolutely, I change my original answer a little. Time dilation between the trailing and leading spaceships is the key concept to understand why the string in the paradox "breaks" (if it does so... conditions apply^(*)).

Why the string "breaks"^(*) from the viewpoint of a non-inertial observer that travels in the trailing spaceship (for example)? Because for him, the leading spaceship has a relative speed, is physically moving away from this observer. Then, if the string can't cope stretching, and can't change the accelerations of the spaceships with its elastic tension, breaks.

And why is the leading spaceship moving away? Because if the observer with its own t₁ proper time, watches a rear display of the leading spaceship, that with big characters displays continually its t₂ proper time, then he can do a simple reasoning, and say that leading spaceship has a relative speed such as:

$v_{21}=g*(t_2-t_1)$

This is not quite so, because in the accelerated frame there's also an acceleration that depends on the position of the point considered with respect to the observer $\frac{g^2}{c^2} \cdot x$. An approximate correction is:

$\int_{t_1}^{t_2}\frac{g^2}{c^2} \cdot x \cdot dt \simeq \int_{t_1}^{t_2}\frac{g^2}{c^2} \cdot \left[\frac{1}{2} \cdot g \cdot (t-t_1)^2 \right] \cdot dt$

So

$v_{21} \simeq g*(t_2-t_1)+\frac{\left[g*(t_2-t_1)\right]^3}{6 \cdot c^2}$

 

[PeterDonis]

Time dilation between the trailing and leading spaceships is the key concept to understand why the string in the paradox "breaks"

No, it isn't. The key concept to understand why the string breaks (under conditions in which it does) is that, in SR, unlike in Newtonian mechanics, a congruence of worldlines all having the same proper acceleration in the same direction is expanding (in more technical language, its expansion scalar is positive). This is the invariant that explains why the string breaks.

I really think you need to stop speculating about such things and learn the correct concepts from SR.

 

[member 728827]

No, it isn't. The key concept to understand why the string breaks (under conditions in which it does) is that, in SR, unlike in Newtonian mechanics, a congruence of worldlines all having the same proper acceleration in the same direction is expanding (in more technical language, its expansion scalar is positive). This is the invariant that explains why the string breaks.

At least, it sounds very impressive.

But then, it's not true that $g*(t_2-t_1)$ gives a very close value to the relative speed of the spaceships in the Bell's paradox ?, and $g*(t_2-t_1)+\frac{\left[ g \cdot (t_2-t_1) \right]^3}{6 \cdot c^2}$ is a better approximation ?, because numerically gives the correct value up to 15 decimal places.

I really think you need to stop speculating about such things and learn the correct concepts from SR.

Then, think also about not being so impolite with people. I'm absolutely insensitive to such comments, but is just because I think you would be happier in life.

 

[Ibix]

But then, it's not true that $g*(t_2-t_1)$ gives a very close value to the relative speed of the spaceships in the Bell's paradox ?

It may well do. But it's a coordinate dependent claim and probably specific to this scenario, and these are precisely the kind of things that make Bell's paradox a paradox in the first place. The lesson we learn from Bell's paradox, where a lot of physicists made wrong predictions from quick coordinate-based reasoning, is that it's a mistake because it very rarely generalises well.

It's like the people who say "acceleration causes the traveling twin to be younger", which is a bad generalisation of a rule that only applies to some versions of the scenario. It may give correct predictions in certain cases, but it's hard to determine what cases those are and when the rule fails completely. Does your rule work if the accelerations vary? If I choose to work in Milne coordinates?

The way that works in general is to study invariants - the expansion scalar is the relevant one here. Then you can derive approximations and rules of thumb as much as you like, with the added bonus of understanding when they will go wrong.

Then, think also about not being so impolite with people. I'm absolutely insensitive to such comments, but is just because I think you would be happier in life.

Peter (and I am sure he will correct me if Ixm wrong) will tell you that he's not unhappy. He's just giving you his opinion (and on the second page of your third (fourth?) thread on the topic, so not without evidence available. That is, an expert in a topic is telling you that you are exploring blind alleys. Being insensitive to such comments is probably a mistake. There's nothing wrong with exploring blind alleys if that's what makes you happy, but you should be aware you're doing it.

 

[member 728827]

It may well do. But it's a coordinate dependent claim and probably specific to this scenario, and these are

It's a claim from the point of view of the tip of the string. And if you feel more confortable, is the viewpoint of an observer in the trailing spaceship, or do you mean that's not a valid because is "coordinate dependent", or it's not applicable talking about the Bell's paradox?

Peter (and I am sure he will correct me if Ixm wrong) will tell you that he's not unhappy

I said "happier". If you're not confortable with this thread, close it and that's it.

 

[Ibix]

It's a claim from the point of view of the tip of the string.

No - it's a claim about interpretation of measurements made from the point of view of the tip of the string in a particular choice of coordinate system. The parts in italics are important, and neglecting them is a trap a lot of people fall in to. Finding statements that are true independent of arbitrary choices of coordinates is a more generally useful approach.

I said "happier".

Ironically, given the photo, you rather seem to have missed the point I was making in the paragraph you quoted. As I said, someone who knows what he's talking about was telling you that there are better ways to study relativity than the way you are going at it.

You may, of course, ignore the advice.

 

[member 728827]

I'm not studying S-R, and much less G-R. I just watch this YT video, I express some thoughts. Of course, you and many others in this Forum have a theoretical background that I don't. I studied some S-R and Differential Geometry some 40 years ago, but I'm now retired.

Nevertheless, I got not very convincing arguments so far. I would apply the photo I attached to myself then. As someone said, "You'll win, but you're not convincing me". At my age, I'll not go lying and saying "hey, what a great argument", when I believe is not.

And, you gave me an idea - thanks. Don't miss the post about Twin paradox!

 

[Dale]

I got not very convincing arguments so far.

Well, that is a little untrue. You got better discussion here than you did at any of the other places you posted, right? And the arguments were at least convincing enough that if you post on another forum you probably will change how you describe it, right? And I would hazard a guess that you probably have no immediate plans to post this on another forum, indicating at least some small sense of resolution to the argument.

 

[member 728827]

@Dale,

I'm talking about my last post, saying that the time dilation is key to understand the Bell's paradox, that was commented with an absolute "no, it isn't right" by @PeterDonis, not the previous ones.

You did give very informative and instructive arguments. If you read the post, you'll see I use the acceleration formula you obtained from Christoffel symbols, and that response is what someone would expect from this forum.

You put out of context whatever I say.

Close the thread.

 

[Dale]

You put out of context whatever I say.

My mistake, sorry about that.

 

[PeterDonis]

Close the thread.

Done.