Bell’s Spaceship Paradox. Does the string break?

In my original post, I said:

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"The LCE seems to require that the two curves get closer together as time increases. Does the upper curve slowly get closer to the lower curve? Or does the lower curve approach the upper curve?"

I've realized that the bottom curve doesn't move upward, because it already has speeds that approach the speed of light "c", and so it's speeds can't be increased any. So all of the decrease in their separation has to come from a lowering of the upper curve.

So I suppose that is enough information to allow the correct upper curve to be plotted ... just subtract the amount of length contraction (using the LCE) from each point of the upper curve.

A new question: Do the two observers who are doing the accelerating agree that their separation is decreasing?
(Inertial observers don't ever think the yardsticks between them contract, so maybe accelerating observers don't think the yardsticks between them contract either.)

 

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The answer is: No, they don't. They say their separation doesn't change. In fact, the above diagram (without the diagonal lines) shows THEIR perspective.

One thing that diagram DOESN'T show is how the ages of those two observers compare. They are NOT ageing at the same rate. Einstein (in his 1907 paper) said the leading person ages exp(D * A) times faster than the trailing person, but I showed in an earlier paper ( https://vixra.org/abs/2109.0076 ) that that exponential equation is incorrect. I derived the correct equation in https://vixra.org/abs/2201.0015 .

 

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No, the LCE does not require the two curves get closer together. The yardsticks that are moving get shorter according to the ICF in your chart, and as the yardsticks get shorter, more of them will fit in between the two curves. More yardsticks fitting between the curves at the right compared to at the left is an indication that the distance between the two spaceships is increasing in the reference frames of the spaceships. The increased space between them is also why the string breaks in Bell's spaceship scenario.

No, the curves remain a fixed distance apart on your chart.

You should have paid more attention to what Bell's spaceship scenario was supposed to teach you.

 

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I don't know what you meant by "ICF" ... maybe you meant "IRF" (inertial reference frame)?

The scenario can be viewed as two rockets connected by some specific number of end-to-end yardsticks. You can glue the junctions of the yardsticks together, if you'd like. Each yardstick can have it's own rocket and accelerometer to insure that both spaceships and all yardsticks ALL accelerate with exactly the same acceleration. The length contraction equation (LCE) says that an inertial observer, stationary with respect to the two spaceships and connecting yardsticks immediately before the acceleration begins, will conclude that that whole assembly gets shorter (in the direction of the motion) by the gamma factor. The number of yardsticks between the two spaceships is constant.

 

Yes, sorry, I meant IRF.

No you can't glue them together, as that changes the outcome. This is the reason Bell's spaceship scenario specifies "a very weak string", or a "a delicate thread". You cannot just connect them with an infinitely strong beam, as that would require changing the initial parameters that are given in the scenario.

You should leave the yardsticks unglued and then you should conclude that their ends will come apart from each other as each yardstick gets shorter due to LC in the IRF of your diagram.

All of the proper accelerations shown on the accelerometers will only be equal if the yardsticks are not glued together.

In that case (where the yard sticks are glued together) then you must redraw your diagram to show the spaceships getting closer together in the IRF. And of course that means the rear accelerometers must show greater proper acceleration than the front accelerometers.

This is what you were supposed to learn from Bell's spaceship paradox. How much more explaining do you think you will require before you get it? This is why I gave up replying to you in the other thread. It seems you are blind to your own errors.

 

A more accurate way to say that last sentence is:

"So I suppose that is enough information to allow the correct upper curve to be plotted ... just divide the distance between the two curves by the gamma factor, and add that to the lower curve to get the corrected upper curve."

 

Neddy, your understanding of the array of clocks and yardsticks in inertial reference frames that Einstein described is better than anyone else's that I've ever known. You know that it makes the conclusions of an inertial observer, about the current age of a distant person, fully meaningful and real for that inertial observer. But the mystery to me is why you are so resistant to the corresponding idea of an array of ACCELERATED clocks and yardsticks, providing a corresponding meaningful and real answer about the current age of a distant person, according to an ACCELERATING observer. Einstein, in his 1907 paper, showed us how to construct that accelerating array. He got the equation wrong (the exponential), but his basic idea was correct.

 

Thanks, Mike, I think you are good at SR also, except you seem to be manifestly misunderstanding Bells' spaceship scenario.

I am not resistant to the idea of using accelerated clocks & yardsticks in SR. In SR, accelerated clocks should be able to use the same simultaneity convention as if their instantaneous velocity relative to whatever were constant in that moment of time.

But the inertial frame which watches Bell's scenario isn't even accelerating. So naturally length contraction of the string is supposed to occur, even though the parameters of the scenario do not allow the string to contract. Which is why it breaks

 

Neddy, if you look at the entire history of the Bell Spaceship "paradox", you will find that it's resolution has been far from unanimous. At one point, the theoretical group at Cern concluded that the thread does NOT break. But even more important is that Einstein himself concluded that the two spaceships, according to the accelerating people on the spaceships themselves, say that the separation of the spaceships is constant, and therefore the thread doesn't break.

In his 1907 paper, Einstein gave the equation for the ratio of the clock tic rates (for the leading clock relative to the trailing clock) as

R = exp(L * A),

where "L" is the separation of the two clocks, and "A" is the constant acceleration. If Einstein had believed that "L" is a function of time, he would have needed to specify how "L" varies with time. He did NOT do that.

Einstein's exponential equation WAS wrong, but his conclusion that the separation "L" is constant was NOT wrong.

 

"To attempt to resolve the dispute, an informal and non-systematic survey of opinion at CERN was held. According to Bell, there was "clear consensus" which asserted, incorrectly, that the string would not break."

So the correct answer is given there. The string breaks.

Einstein can talk about a situation where two clocks are accelerating in such a way that they remain a constant distance apart in their own rest frames, but that is not the same situation as Bell's spaceships, which get farther away from each other in their own rest frames.

 

Bell said there was a clear consensus (among the Cern theoretitions) that the string wouldn't break. (And it wasn't "informal" and "non-systematic", at least not in Cern's opinion.) Bell said the Cern theoretions were wrong, and the string breaks. So the Cern theoretitions disagreed with Bell. YOU assume that Bell was right, and Cern was wrong. Why?

In each of those scenarios, the rockets are characterized as being identical. To ensure that the two rockets perform as claimed, each rocket can have an attached accelerometer that guarantees that the accelerations are identical. That is true for BOTH the Bell scenario AND for Einstein's scenario. The two scenarios are identical. In both scenarios, the separation between the two rockets is constant, and identical.

You are mistaking Bell's erronious analysis for differences in the two scenarios. The scenarios are identical. If the two scenarios weren't identical, there wouldn't be any point in comparing them.

 

Neddy, when you give what you believe to be Bell's conclusions, what diagram are you basing Bell's conclusions on? I THINK the diagram he (and you) are using is the same diagram I showed at the beginning of this thread. It shows two curves that are equally spaced (in the vertical direction, which gives the conclusions of the inertial observers who are stationary wrt the spaceships immediately before they start their accelerations). The diagram also shows two sloped straight lines, that show the conclusions of another set of inertial observers, who are momentarily stationary wrt the rockets later in the acceleration.

IF that is indeed what you and Bell are basing your conclusions on, it shows where you are going wrong. As I said earlier in this thread, that diagram (which I had believed was correct for MANY years) is INCORRECT. The initial inertial observers MUST conclude that the two rockets get closer together as the speed of the rockets increases ... this follows directly from the length contraction equations (LCE) of special relativity: inertial observers say that yardsticks moving wrt them shrink by the factor gamma = 1 / sqrt (1 - v*v), compared with their own yardsticks, which they think have a constant length. There can be no doubt about the validity of the length contraction equation. So that initial diagram I gave MUST be modified as I described. The result is that the initial inertial observers must conclude that the two rockets get closer together as the acceleration progresses. The lower curve remains the same, but the upper curve curves downward and approaches the lower curve from above as time goes to infinity.

Anyone using the original diagram to analyze the Bell scenario will get an incorrect answer.

 

mike;


You cited Rindler and Einstein, who analyzed gravitational time dilation.
That is not the same as acceleration in translational motion.
Clock synchronization was done in an inertial frame, not an accelerated frame.
There is no such thing as an instantaneous co-moving inertial frame. An accelerating frame passing an inertial frame cannot acquire an equivalent synchronization which requires a finite time for round trip light signals.
It's another one of those mystical/magical figures of speech. When reading this stuff, be sure to wear your waders!

This plot shows the change in simultaneity (green Ax and Bx axes), which is a result of measurement (no magical telescope), done after achieving a constant speed. As A and B continue accelerating, their green x axes will approach the blue light velocity profile. U would observe their clock rates slowing toward 0.

Initially I don't see any error in Einstein's td equation for clocks in a g-field.

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Scenarios of gravitational fields (with no accelerations) and scenarios of accelerations (with no gravitational fields) are related by Einstein's equivalence principle. In 1907, Einstein KNEW how to analyze the case where there are two observers, each accelerating with the same acceleration "A", and separated by a constant distance "L" (in the direction of the acceleration), with no gravitational fields anywhere. He was interested in using that KNOWN information to help him understand gravitation, which he DIDN'T understand at that time.

 

mike;

This graphic may show the difference between gravitational time dilation and the Bell spaceship example in a simpler way.
left:
Two clocks c1 and c2 at different altitudes in a non uniform g-field, with clock rate of c1 less than c2.
right:
Same clocks in space craft in tandem accelerating in x direction. They could be launched from anywhere in space, outside earth g-field. A flexible cord would not form a single rigid object.

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Attached below is the corrected diagram.

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Yes, the diagram Bell and I are using is the same as the one shown in your post #1. Bells' scenario defines the problem completely in the inertial frame of the initial rest state of the spaceships. The wikipedia article states, "A delicate thread hangs between two spaceships headed in the same direction. They start accelerating simultaneously and equally as measured in the inertial frame S, thus having the same velocity at all times as viewed from S."

No, there is nothing wrong with that diagram. The scenario which Bell defined requires that the spaceships maintain a constant distance apart as measured by the inertial frame S. Just because you do not like it or do not understand the implications does not mean you get to change it.

That is a different scenario. Both are possible. The one shown in post # 1 has the spaceships accelerating "simultaneously and equally as measured in the inertial frame S, thus having the same velocity at all times as viewed from S." which is Bell's defined scenario. Your new one doesn't have them accelerating equally, so you have changed it.

Yes, accelerating yardsticks would become more and more length-contracted according to inertial frame S, but BY DEFINITION OF THE SCENARIO, the space between the spaceships is constant in frame S. So the correct conclusion is that more and more yardsticks will fit in that constant space as the yardsticks become smaller. That means the proper distance between the spaceships is increasing, which is why the thread breaks.

I have explained this to you before, but you refuse to understand. That is not my problem, and not Bells' problem. That is your problem.

Wrong, again. You have changed the scenario to one in which the spaceships do not accelerate equally in frame S. That would be correct if Bell started with an infinitely strong string and then only stipulated that the front spaceship accelerates constantly. Then the strong string would pull the rear spaceship closer and closer to the front spaceship according to frame S, causing the two spaceship accelerations to be unequal in frame S. But that is precisely why the problem states a very weak string, and both spaceships accelerating equally in frame S! Come on, man, those are two different scenarios, and the one Bell had in mind is the post # 1 diagram, not post #14 diagram! (although both are possible, only one matches Bell's)

 

If they do that, the acceleration shown on accelerometers attached to the rockets will NOT show the same value. That's not a scenario that I have any interest in.

If a yardstick is moving away from the inertial observers, (regardless of whether or not it is accelerating of not), the inertial observers will say it is shorter than their own yardsticks.

 

Here's another posting I put on usenet:

In the scenario that I am interested in, and which I have analyzed, the two rockets (immediately after they are ignited) always have the same constant acceleration, as reported by accelerometers attached to the two rockets.
In that scenario, the separation of the rockets, according to the accelerating traveler in the trailing rocket, is constant, and the string doesn't break.

According to the initial inertial observers who are stationary wrt the rockets immediately before they are fired, the separation of the two rockets decreases as the acceleration proceeds, as required by the famous length contraction equation (LCE) of special relativity. So they also conclude that the string doesn't break.

The scenario, as given in Bell's Paradox, may be a completely different scenario from the above scenario. As far as I know, there is no mention of rocket accelerometer readings in that Wiki article on Bell's Paradox. If, in Bell's Paradox, the initial inertial observers correctly conclude that the rocket separation doesn't decrease, then the rockets CAN'T be accelerating at the same rate (according to accelerometers attached to the rockets) ... the leading rocket must have a greater acceleration than the trailing rocket. If so, I have no interest in that different scenario.

 

The idea that the two scenarios (Bell's and mine) are different is appealing to me, because it means that both my answer to my scenario, and Bell's answer to Bell's scenario can both be correct ... everyone wins. But I am still troubled, because I don't understand why Bell's scenario wouldn't properly compute the true acceleration (consistent with the readings on accelerometers). The initial inertial observers compute those accelerations using what seems to me to be a correct procedure:

(1) At each time, compute the rapidity theta = (A * t).

(2) Compute the velocity v = tanh(theta).

(3) Integrate the velocity v with respect to time t to get the distance traveled.

(4) That produces the curves in my original diagram.

(5) From each of those curves, the change in velocity in a very short time gives the acceleration.

WHY does that calculated acceleration not match the reading on the accelerometer? It seems to me that it should, which would mean that the Bell scenario is NOT different from my scenario.

But if so, how is my contention, that the upper curve violates the length contraction equation, resolved?

(I'm not "trolling" here ... I really am perplexed. I was happier thinking that the two scenarios are different, but the above arguments seem to show that they aren't different.)