Clocks Within Each Ship in Bell Spaceship Paradox

[1977ub]

I believe length contraction always makes more sense when integrated with reminders of relativity of Simultaneity.

Let's say the engines are at the back end of each rocket. For the viewer "A" in the initial frame, they begin moving and continue accelerating simultaneously, and clocks next to the engines are seen in synch.

However right away, "A" will measure that for each rocket, a clock by the engine is running faster than the clock by the head of that same rocket. I have never seen this pointed out in a discussion of this paradox, and I think that this may be one reason people so often are mistaken about this situation. The idea that "both spaceships accelerate and keep their clocks synchronized" to "A" distracts the reader and puts one in the mind of a situation without SR. The reminder that within each rocket, the clocks appear out of synch to "A" might snap the view back to SR.

We might attempt to apply this same logic to the rope, but right away we run into a problem, since the back end of the rope is connected to the front of the back rocket, and the front of the rope is connected to the back of the front rocket. If the back end of the rope was accelerating in synch with the backs of the rockets, the rope back clock could not be connected to the back rocket front clock, since they are out of synch. And the similar applies to the front of the rope being attached to the front ship back clock. We are asking of a suggested nonstressed rope that it fulfill contradictory requirements.

 

[Dale]

We are asking of a suggested nonstressed rope that it fulfill contradictory requirements.

And everybody knows that fulfilling contradictory requirements is highly stressful.

 

[harrylin]

Rearranging:

I believe length contraction always makes more sense when integrated with reminders of relativity of Simultaneity.

Relativity of simultaneity must certainly be dealt with when using less convenient inertial reference systems, and I agree that it is useful. When this is properly done it emerges that although length contraction is "relative", it plays a role in any inertial frame.

Let's say the engines are at the back end of each rocket. For the viewer "A" in the initial frame, they begin moving and continue accelerating simultaneously, and clocks next to the engines are seen in synch.
However right away, "A" will measure that for each rocket, a clock by the engine is running faster than the clock by the head of that same rocket. I have never seen this pointed out in a discussion of this paradox, and I think that this may be one reason people so often are mistaken about this situation. [...] The reminder that within each rocket, the clocks appear out of synch to "A" might snap the view back to SR.

I'm not sure if I simply misunderstand what you mean, but it sounds like a wrong argument. "A" would measure if it were technically feasible that for each rocket, a clock by the engine is running extremely slightly slower than the clock by the head of that same rocket due to the rocket's length contraction. However that's not spectacular at all.
Or perhaps you mean that an observer "B" inside one rocket, when verifying clock synchronization, will discover that the rocket has accelerated from the fact that the synchronization of the instantaneously co-moving inertial frame at approximately that time does not correspond anymore with that of "A"'s reference frame (a clock by the engine will now appear to be behind compared to the one near the head).
Adding such useless detail is more than likely to distract from Bell's striking example of a breaking string.

[...] The idea that "both spaceships accelerate and keep their clocks synchronized" to "A" distracts the reader and puts one in the mind of a situation without SR.

Certainly not! It's key to Bell's clear and straightforward argument about the physical consequences of SR's length contraction. His case is perfectly SR and it's a simple scenario with identical rockets.

We might attempt to apply this same logic to the rope, but right away we run into a problem, since the back end of the rope is connected to the front of the back rocket, and the front of the rope is connected to the back of the front rocket. If the back end of the rope was accelerating in synch with the backs of the rockets, the rope back clock could not be connected to the back rocket front clock, since they are out of synch.[..]

Clocks that are out of sync can certainly be connected.. I'm afraid that I can't follow your logic at all.

Note that "Bell's spaceship paradox" was only paradoxical for his colleagues who misunderstood SR; it was his example to drive home their misconception. How do you think that your discussion better clarifies Bell's example than he did himself?

 

[1977ub]

I'm comparing this scenario to the more familiar one of the single train/ship which travels by, and which was accelerated at some unknown time in the past, all of its clocks now in synch. There are no stresses there, and there is no "real" length contraction which can be divorced from RoS. In a frame where the clocks at both ends are in synch, there is no length contraction. In a different frame, the clocks at both ends are now out of synch, and there is measured a length contraction.

The ladder paradox, too, shows that very starkly.

Would the rope break in Bell's example if the ships did not contract (in the initial frame)? No. Would the ships contract (in the initial frame) without the clocks at their front and back ends being out of synch (in the initial frame)? No.

The alternative is to add a detail to Bell's example whereby each ship has an engine at the front and at the back, and so all 4 engines fire on the same program, in synch from the initial frame. In that example, the ships themselves would stretch and break apart.

But that is not the situation in Bell's standard paradox. I didn't think about this until I was looking at the wiki page, and the contracted ships.

https://en.wikipedia.org/wiki/Bell's_spaceship_paradox

 

[harrylin]

[..] Would the rope break in Bell's example if the ships did not contract (in the initial frame)? No.

The ships could be made (artificially) to not contract in the initial frame; the string would still break.

Would the ships contract (in the initial frame) without the clocks at their front and back ends being out of synch (in the initial frame)? No.

I think that I already clarified that the clocks inside the rocket are only very slightly out of synch according to "A", and that this is caused by the rocket's contraction (due to the contraction, the front clock's linear speed is all the time very slightly less than that of the rear clock). The inverse is incorrect: the rate at which clocks tick or the way they are synchronized has no effect on the length of the rocket - that is unphysical!

The alternative is to add a detail to Bell's example whereby each ship has an engine at the front and at the back, and so all 4 engines fire on the same program, in synch from the initial frame. In that example, the ships themselves would stretch and break apart.

And then, I'm afraid that someone else will propose to put clocks in the 4 engines in order to clarify that the engines contract.
It's easy to make the string much longer than the rockets, so that their contraction can be ignored; no need to complicate it with 4 engines or breaking rockets. And see next!

But that is not the situation in Bell's standard paradox. I didn't think about this until I was looking at the wiki page, and the contracted ships.
https://en.wikipedia.org/wiki/Bell's_spaceship_paradox

That drawing nicely illustrates that the contraction of the rockets can be made totally irrelevant. If the rockets as drawn there are made with some technical means to have a "proper stretching" such that they do not length contract in the original rest frame, it will make no difference at all for the breaking of the string.

 

[1977ub]

The ships could be made (artificially) to not contract in the initial frame; the string would still break.

Agreed. The string would break and the ships would break.

I think that I already clarified that the clocks inside the rocket are only very slightly out of synch according to "A", and that this is caused by the rocket's contraction (due to the contraction, the front clock's linear speed is all the time very slightly less than that of the rear clock).

All of these effects are "only slight" depending upon the actual velocities in play. Also by out of synch I should clarify that I don't mean simply not ticking at the same *rate* but more importantly in the ladder/barn way, i.e. the back clock of each ship is presumably made set earlier due to being at the back. Surely this applies? I'm not up to doing the math for the accelerating case, but in a non-accelerating case, a moving ship is seen to be shortened, and with the back clock set later than the front - by viewers in the "platform" frame. This is the "out of synch" I refer to. The RoS out-of-synch which accompanies length contraction.

The inverse is incorrect: the rate at which clocks tick or the way they are synchronized has no effect on the length of the rocket - that is unphysical!

Of course. I was grasping for an inference, a comparison with the non-stressed familiar 2nd-frame-train, not implying direct cause and effect.

And then, I'm afraid that someone else will propose to put clocks in the 4 engines in order to clarify that the engines contract.

Not at all. I would think pointing out that the clocks within the rockets are not in synch would shatter the sleight-of-hand of the way the "two simultaneously accelerating" rockets scenario is set up.

It's easy to make the string much longer than the rockets, so that their contraction can be ignored; no need to complicate it with 4 engines or breaking rockets. And see next!

It doesn't matter how long the rockets are compared to the rope/space between them. If they contract, it will put stress on the rope. Why does this not happen with the 2nd-frame-train? Everything contracts together. The scenario here has been designed so that they whole getup can't contract *together*.

That drawing nicely illustrates that the contraction of the rockets can be made totally irrelevant. If the rockets as drawn there are made with some technical means to have a "proper stretching" such that they do not length contract in the original rest frame, it will make no difference at all for the breaking of the string.

If we set the string up for "proper stretching" as well, it won't break. My real point is that for platform observers, an unaccelerated moving flotilla of ships is expected to be shortened and with rear clock set ahead of front clock. I'm trying to find the differences here. I realize that on an unaccelerated moving ship, clocks can be synchronized, such that they appear unsynchronized for platform viewer. I'm not quite sure what can be said analogously for the case of a single accelerating ship, its clocks, and comparison with platform viewer.

 

[harrylin]

[..] by out of synch I should clarify that I don't mean simply not ticking at the same *rate* but more importantly in the ladder/barn way, i.e. the back clock of each ship is presumably made set earlier due to being at the back. Surely this applies?

Certainly not! Once more, your statement in your first post, that "right away, "A" will measure that for each rocket, a clock by the engine is running faster than the clock by the head of that same rocket", is totally wrong.
The back clock of each ship is just like the clocks of the two ships set in synch to the original rest frame before take-off, and if their travel histories are identical, that cannot change.
Let's take the example of your 4 clocks that are made to accelerate identically in "A"'s rest frame thanks to a slight artificial stretching of the rockets. For that case they will remain in synch with each other according to the original frame, the rest frame of "A".

I'm not up to doing the math for the accelerating case

That is - happily - not required to understand Bell's spaceship example.

[..] in a non-accelerating case, a moving ship is seen to be shortened, and with the back clock set later than the front - by viewers in the "platform" frame. This is the "out of synch" I refer to. The RoS out-of-synch which accompanies length contraction.

It accompanies length contraction if the operator performed a so-called Einstein synchronization at that velocity.

I would think pointing out that the clocks within the rockets are not in synch would shatter the sleight-of-hand of the way the "two simultaneously accelerating" rockets scenario is set up.

Not at all: "simultaneously" refers to the launch pad reference system. It's as much a "sleight-of-hand" as synchronous clocks in the GPS system that you may use in your car.

It doesn't matter how long the rockets are compared to the rope/space between them. If they contract, it will put stress on the rope.

Please look again carefully at the Wikipedia sketch. Once more: as pictured there, Bell's example is insensitive to length contraction by the rockets.

Everything contracts together. The scenario here has been designed so that they whole getup can't contract *together*.

It sounds as if you are bugged by the bug that bugged Bell's colleagues. Material objects length contract with their increase of speed, but their speed increase cannot affect the space between them.

If we set the string up for "proper stretching" as well, it won't break.

The point of Bell's Spaceship example is that the string undergoes "proper stretching" so that it breaks...

My real point is that for platform observers, an unaccelerated moving flotilla of ships is expected to be shortened and with rear clock set ahead of front clock.

Here you are getting to the essential point: shortened as compared to a measurement with instruments of the flotilla, if those have first been synchronized according to convention.

I'm trying to find the differences here. I realize that on an unaccelerated moving ship, clocks can be synchronized, such that they appear unsynchronized for platform viewer. I'm not quite sure what can be said analogously for the case of a single accelerating ship, its clocks, and comparison with platform viewer.

The main difference here is that Bell discusses physical effects due to a change in velocity, while the other examples merely examine differences in measurements between two independent inertial reference systems.

A similar issue arises with time dilation: the discussion of effects of change in velocity ("twin paradox") is quite different from the discussion about how two systems in inertial motion measure each other (mutual time dilation).

Observational symmetry if broken with a change of velocity.

Does that help?

 

[1977ub]

I have to start again. :) One rocket, with acceleration = 0, moving with positive velocity V wrt ref frame S, is measured by observers in S to be shortened (compared to an identical rocket sitting next to the platform) in the direction of motion, and also to have a clock at the back which is set later than the clock in the front (assuming that the denizens have Einstein sync'd them). Now, give that rocket a slight acceleration, and all of those effects still hold, and their extent is slightly changing with time. For observers in S, the clock at the back is getting increasingly ahead compared with a clock at the nose, and the rocket is getting progressively shorter. Any problem there? (The denizens won't be able to do exact Einstein sync if accelerating though, right?)

 

[harrylin]

I have to start again. :) One rocket, with acceleration = 0, moving with positive velocity V wrt ref frame S, is measured by observers in S to be shortened (compared to an identical rocket sitting next to the platform) in the direction of motion, and also to have a clock at the back which is set later than the clock in the front (assuming that the denizens have Einstein sync'd them). Now, give that rocket a slight acceleration, and all of those effects still hold, and their extent is slightly changing with time. For observers in S, the clock at the back is getting increasingly ahead compared with a clock at the nose, and the rocket is getting progressively shorter. Any problem there? (The denizens won't be able to do exact Einstein sync if accelerating though, right?)

I immediately noticed two problems with that:

1. Once more: according to S the clock at the back is not getting increasingly ahead compared with a clock at the nose (I explained why it's even getting slightly less ahead).
By what physical means, do you think, would S propose that the clock at the back should tick faster than the clock at the nose??

2. You seem to try to show something completely different from what Bell's spaceship example shows!
How does your example demonstrate that one should not confound the length contraction of material objects with the space between them? It was apparently the mistaken notion of space contraction that made Bell's example a "paradox" for many of Bell's colleagues (this is also referred to in the intro in Wikipedia).

 

[stevendaryl]

I have to start again. :) One rocket, with acceleration = 0, moving with positive velocity V wrt ref frame S, is measured by observers in S to be shortened (compared to an identical rocket sitting next to the platform) in the direction of motion, and also to have a clock at the back which is set later than the clock in the front (assuming that the denizens have Einstein sync'd them). Now, give that rocket a slight acceleration, and all of those effects still hold, and their extent is slightly changing with time. For observers in S, the clock at the back is getting increasingly ahead compared with a clock at the nose, and the rocket is getting progressively shorter. Any problem there? (The denizens won't be able to do exact Einstein sync if accelerating though, right?)

What you're saying isn't correct.

Let's assume that the rocket is "Born-rigid". What that means is that it has the same length $L$ in any frame in which it is (momentarily) at rest. Then it follows, by the mathematics of relativity, that as the rocket accelerates, it keeps getting shorter and shorter, as viewed in the original rest frame S.

Now, think about what "getting shorter" means. It means that the rear end of the rocket is getting closer to the front end of the rocket. Which means that the rear end is traveling (slightly) faster than the front end. Which means (by time dilation) that the rear clock is running slightly slower than the front clock.

So what you're saying is exactly backwards. The rear clock gets farther and farther behind the front clock.

 

[stevendaryl]

I think a point of confusion is clock synchronization in different frames.

According to the Lorentz transformations, if (1) a rocket is moving at constant velocity v relative to frame S, and (2) the clocks at the front and the rear are synchronized, according to the rocket's reference frame (call it S'), then according to frame S the front clock will be behind the rear clock by an amount

$\delta t' = \frac{v L'}{c^2}$

where $L'$ is the length of the rocket in its own rest frame (by length contraction, $L = \frac{L'}{\gamma}$ is the length in frame S).

Note the phrase: if the clocks are synchronized in frame S'. That's not going to happen naturally; the people on board the rocket have to adjust the clocks to make that happen. They have to SET the front clock so that it's synchronized with the back clock.

So you can imagine a constantly accelerating rocket to be approximated by the following discrete process:

1. The rocket is initially at rest in some frame $S_0$. The clocks at the front and rear are synchronized in that frame.
2. At time $t=0$, the rocket accelerates instantaneously to speed $\delta v$ relative to $S_0$. So it's at rest in a new frame, $S_1$.
3. The front clock must be set back by an amount $\delta t' = \frac{\delta v L}{c^2}$ in order for the two clocks to be in synch in frame $S_1$.
4. At time $t = t_1$, the rocket again accelerates to speed $\delta v$ relative to $S_1$.
5. Again, the front clock must be set back.
6. etc.

Every time the rocket accelerates, the front clock must be set back. If you didn't continually adjust the front clock, then the front clock would get farther and farther ahead of the rear clock.

 

[1977ub]

What you're saying isn't correct.

Let's assume that the rocket is "Born-rigid". What that means is that it has the same length $L$ in any frame in which it is (momentarily) at rest. Then it follows, by the mathematics of relativity, that as the rocket accelerates, it keeps getting shorter and shorter, as viewed in the original rest frame S.

Now, think about what "getting shorter" means. It means that the rear end of the rocket is getting closer to the front end of the rocket. Which means that the rear end is traveling (slightly) faster than the front end. Which means (by time dilation) that the rear clock is running slightly slower than the front clock.

So what you're saying is exactly backwards. The rear clock gets farther and farther behind the front clock.

Instead of acceleration, let us consider a sequence of rockets, each moving with constant velocity, only progressively faster. One way to visualize the clocks being out of synch to the platform viewer is that there is a light beacon in the center of the rocket. For observers in that vehicle's frame, the ping guarantees that the end clocks are in synch. For a viewer in initial frame S, the light signal rushes backward to the rear of the vehicle, triggering the clock to tick forward, before the corresponding signal from the beacon reaches the (receeding) front clock. Therefore the clock at the rear is set to a later time than the one in the front. Now let's move on to a faster moving vehicle. The effect is more pronounced. Observers in S now measure that the rear of the 2nd vehicle is set even later with regard to the front clock than in the initial vehicle's case. etc. No?

 

[stevendaryl]

Instead of acceleration, let us consider a sequence of rockets, each moving with constant velocity, only progressively faster. One way to visualize the clocks being out of synch to the platform viewer is that there is a light beacon in the center of the rocket. For observers in that vehicle's frame, the ping guarantees that the end clocks are in synch. For a viewer in initial frame S, the light signal rushes backward to the rear of the vehicle, triggering the clock to tick forward, before the corresponding signal from the beacon reaches the (receeding) front clock. Therefore the clock at the rear is set to a later time than the one in the front. Now let's move on to a faster moving vehicle. The effect is more pronounced. Observers in S now measure that the rear of the 2nd vehicle is set even later with regard to the front clock than in the initial vehicle's case. etc. No?

When people say that the front clock on a rocket runs faster than the rear clock, they are NOT talking about synchronization. Forget about synchronization of front and rear clocks, and instead, consider the following thought experiment:

Take two clocks in the rear of the rocket. Set them both to $t=0$. Leave one clock in the rear, and bring the other clock to the front. Let both clocks continue to run for one year. Then bring the front clock back to the rear. Then the clock that had been in the front of the rocket will show more elapsed time than the clock that had been in the rear of the rocket the whole time.

 

[stevendaryl]

When people say that the front clock on a rocket runs faster than the rear clock, they are NOT talking about synchronization. Forget about synchronization of front and rear clocks, and instead, consider the following thought experiment:

Take two clocks in the rear of the rocket. Set them both to $t=0$. Leave one clock in the rear, and bring the other clock to the front. Let both clocks continue to run for one year. Then bring the front clock back to the rear. Then the clock that had been in the front of the rocket will show more elapsed time than the clock that had been in the rear of the rocket the whole time.

You cannot keep the clocks in the front and rear synchronized in an accelerating rocket, if they are identical clocks.

 

[1977ub]

When people say that the front clock on a rocket runs faster than the rear clock, they are NOT talking about synchronization. Forget about synchronization of front and rear clocks, and instead, consider the following thought experiment:

Take two clocks in the rear of the rocket. Set them both to $t=0$. Leave one clock in the rear, and bring the other clock to the front. Let both clocks continue to run for one year. Then bring the front clock back to the rear. Then the clock that had been in the front of the rocket will show more elapsed time than the clock that had been in the rear of the rocket the whole time.

Yes, this is precisely the effect I wish to ignore. I am attempting to find a way to bring the insights of the ladder/barn paradox - in which there is no "real" shortening apart from the one related to relativity of simultaneity - into the situation of the Bell's Spaceship paradox. I'm not quite there yet, and am hoping to get there, a step at a time.

 

[1977ub]

Perhaps if someone can confirm this for me - this other (very similar to other cases being discussed here) situation. Just like Bell's, except that the programmed identical acceleration goes on for a finite amount of time, then the engines shut off (the times at which they do so only being the same in the platform frame) and both ships coast after that. The platform observers now find that even though the backs of the ships are the same distance from one another as they began, and the fronts of the ships are the same distance from one another as they began, that both now coasting ships are now both shortened. This is correct?

 

[stevendaryl]

Yes, this is precisely the effect I wish to ignore. I am attempting to find a way to bring the insights of the ladder/barn paradox - in which there is no "real" shortening apart from the one related to relativity of simultaneity - into the situation of the Bell's Spaceship paradox. I'm not quite there yet, and am hoping to get there, a step at a time.

Hmm. What I described is a real effect. It's not a coordinate effect.

 

[stevendaryl]

Perhaps if someone can confirm this for me - this other (very similar to other cases being discussed here) situation. Just like Bell's, except that the programmed identical acceleration goes on for a finite amount of time, then the engines shut off (the times at which they do so only being the same in the platform frame) and both ships coast after that. The platform observers now find that even though the backs of the ships are the same distance from one another as they began, and the fronts of the ships are the same distance from one another as they began, that both now coasting ships are now both shortened. This is correct?

That is correct, as I understand it.

 

[1977ub]

I'm trying to figure out if this is similar enough to Bell's Spaceship paradox to be illustrative. The ships are shortened, and while the distance between the backs of the ships remains the same to original observer, the ships are shortened to him, and so the distance from the back of the front ship to the front of the back ship is shortened >>although not by as great a percentage as either ship<<. Observers on the ships agree that their ships are now farther apart than they were before their acceleration phase. They will not be surprised to see the rope has broken. Is there something extra, something mysterious, about the "always accelerating" detail of Bell's Spaceship paradox which negates or calls into question the relevance of this particular narrative?

 

[A.T.]

The ships are shortened, and while the distance between the backs of the ships remains the same to original observer, the ships are shortened to him, and so the distance from the back of the front ship to the front of the back ship is shortened

The breaking of the rope has nothing to with the contraction of the ships in the original rest frame. The rope is attached to points on the ships which have a constant distance in the original rest frame, but it still will break.

 

[1977ub]

The breaking of the rope has nothing to with the contraction of the ships in the original rest frame. The rope is attached to points on the ships which have a constant distance in the original rest frame, but it still will break.

Ok. I see. I'm still trying to connect this to the relativity of simultaneity somehow, since length contraction and RoS seem to always be two sides of the same coin.

 

[stevendaryl]

Ok. I see. I'm still trying to connect this to the relativity of simultaneity somehow, since length contraction and RoS seem to always be two sides of the same coin.

It's definitely related to relativity of simultaneity. As I suggested, instead of doing continuous acceleration, you can imagine the rockets having a schedule:

1. At t=0, the rockets are all at rest in frame $S_0$
2. At t=1 second, according your own rocket's clock, turn on your engines briefly, and quickly accelerate to speed $\delta v$ relative to $S_0$. The new rest frame is $S_1$
3. At t=2 seconds, turn on your engines, and accelerate to speed $\delta v$ relative to $S_1$. This new frame is $S_2$
   
4. etc.

From the point of view of frame $S_0$, the rockets are perfectly in synch, each accelerating at the same time. But from the point of view of frame $S_1$, the forward rocket makes the jump to frame $S_2$ slightly before the rear rocket. In frame $S_2$, the forward rocket makes the jump to frame $S_3$ before the rear rocket. Etc.

So in every frame except the initial rest frame, the front rocket is accelerating more often than the rear rocket. So in every frame other than the initial rest frame, the front rocket is pulling away from the rear rocket.

(Note: this is the case in which the rockets accelerate in such a way that the distance between them stays constant, as viewed in the initial rest frame, $S_0$.)

 

[1977ub]

It's definitely related to relativity of simultaneity. As I suggested, instead of doing continuous acceleration, you can imagine the rockets having a schedule:

1. At t=0, the rockets are all at rest in frame $S_0$
2. At t=1 second, according your own rocket's clock, turn on your engines briefly, and quickly accelerate to speed $\delta v$ relative to $S_0$. The new rest frame is $S_1$
3. At t=2 seconds, turn on your engines, and accelerate to speed $\delta v$ relative to $S_1$. This new frame is $S_2$
   
4. etc.

From the point of view of frame $S_0$, the rockets are perfectly in synch, each accelerating at the same time. But from the point of view of frame $S_1$, the forward rocket makes the jump to frame $S_2$ slightly before the rear rocket. In frame $S_2$, the forward rocket makes the jump to frame $S_3$ before the rear rocket. Etc.

So in every frame except the initial rest frame, the front rocket is accelerating more often than the rear rocket. So in every frame other than the initial rest frame, the front rocket is pulling away from the rear rocket.

(Note: this is the case in which the rockets accelerate in such a way that the distance between them stays constant, as viewed in the initial rest frame, $S_0$.)

Yes thanks. I also think that

It's definitely related to relativity of simultaneity. As I suggested, instead of doing continuous acceleration, you can imagine the rockets having a schedule:

1. At t=0, the rockets are all at rest in frame $S_0$
2. At t=1 second, according your own rocket's clock, turn on your engines briefly, and quickly accelerate to speed $\delta v$ relative to $S_0$. The new rest frame is $S_1$
3. At t=2 seconds, turn on your engines, and accelerate to speed $\delta v$ relative to $S_1$. This new frame is $S_2$
   
4. etc.

From the point of view of frame $S_0$, the rockets are perfectly in synch, each accelerating at the same time. But from the point of view of frame $S_1$, the forward rocket makes the jump to frame $S_2$ slightly before the rear rocket. In frame $S_2$, the forward rocket makes the jump to frame $S_3$ before the rear rocket. Etc.

So in every frame except the initial rest frame, the front rocket is accelerating more often than the rear rocket. So in every frame other than the initial rest frame, the front rocket is pulling away from the rear rocket.

(Note: this is the case in which the rockets accelerate in such a way that the distance between them stays constant, as viewed in the initial rest frame, $S_0$.)

Yes, thanks. That makes sense.

I'm still mulling this over:
https://en.wikipedia.org/wiki/Bell's_spaceship_paradox#Importance_of_length_contraction

Interesting that there could be disagreement between such experts over whether length contraction has "physical reality."

 

[harrylin]

Yes thanks. I also think that

It is true for every rocket rest frame other than the initial rest frame. However, the point of view of inertial reference systems that move in the opposite direction is equally valid in SR.

In such systems, RoS works contrary to your wish: the rear rocket appears to take off before the front rocket so that the distance between the rockets decreases. And still the string will break.

Length contraction plays a role in all inertial reference systems, but the description of the role it plays depends on the reference system of choice.

 

[harrylin]

[..]I'm still mulling this over:
https://en.wikipedia.org/wiki/Bell's_spaceship_paradox#Importance_of_length_contraction

Interesting that there could be disagreement between such experts over whether length contraction has "physical reality."

While I have not read those publications, the Wikipedia comment sounds like a philosophical debate that boils down to the question if SR is a physical theory or not.
And a quick look at one summary suggests to me that confusion over two meanings of "contraction" may be involved (see also the dictionary).

Length contraction in this context can either refer to:
1. a Lorentz transformation between two inertial reference systems, representing different perspectives on a stressed string.
That is not what Bell discusses.
2. a change of state, such as the breaking of a string - which is definitely a physical effect.
And that is what Bell discusses.

 

[A.T.]

since length contraction and RoS seem to always be two sides of the same coin.

Length contraction, relativity of simultaneity and time dilation are all consequences of the Lorentz Transformation. This is nicely visualized here at 1:00

 

[stevendaryl]

It is true for every rocket rest frame other than the initial rest frame. However, the point of view of inertial reference systems that move in the opposite direction is equally valid in SR

Every reference frame is equally valid, but our intuitions about questions such as "will the string break" is formed based on nonrelativistic physics. From nonrelativistic physics, we would expect, if the front rocket takes off before the rear rocket, that that would put stress on the string connecting them. But the point of frame $S_1$ is that the rockets are moving nonrelativistically in that frame, so we would expect that nonrelativistic physics to give a good approximation to the answer.

 

[1977ub]

It is true for every rocket rest frame other than the initial rest frame. However, the point of view of inertial reference systems that move in the opposite direction is equally valid in SR.

In such systems, RoS works contrary to your wish: the rear rocket appears to take off before the front rocket so that the distance between the rockets decreases. And still the string will break.

Length contraction plays a role in all inertial reference systems, but the description of the role it plays depends on the reference system of choice.

Are you saying there are frames in which the accelerating rockets continually move closer together? I don't think so.

For an observer on the platform (no X axis movement) the ships begin to accelerate simultaneously, with their clocks reading the same time as each other and the same as his.

For an observer moving in the negative X direction, the back ship starts moving first, its clock set later than the clock on the front ship. In fact, the back of the back ship starts moving before the front of the back ship. For this viewer, the whole thing starts out contracted and out of synch, and so the intuition that the platform observer might have regarding the rope - that physics should work as expected in one's rest frame - is already subverted.

 

[1977ub]

While I have not read those publications, the Wikipedia comment sounds like a philosophical debate that boils down to the question if SR is a physical theory or not.
And a quick look at one summary suggests to me that confusion over two meanings of "contraction" may be involved (see also the dictionary).

Length contraction in this context can either refer to:
1. a Lorentz transformation between two inertial reference systems, representing different perspectives on a stressed string.
That is not what Bell discusses.
2. a change of state, such as the breaking of a string - which is definitely a physical effect.
And that is what Bell discusses.

It is clear that in the ladder/barn paradox, contraction is not a "physical effect" and very much a frame dependent facet which involves RoS.

 

[stevendaryl]

Are you saying there are frames in which the accelerating rockets continually move closer together? I don't think so.

There is no frame in which the rockets are continually moving closer together, but there is a frame in which they move closer together for a time, then move farther apart.

 

[harrylin]

Every reference frame is equally valid, but our intuitions about questions such as "will the string break" is formed based on nonrelativistic physics. From nonrelativistic physics, we would expect, if the front rocket takes off before the rear rocket, that that would put stress on the string connecting them. But the point of frame $S_1$ is that the rockets are moving nonrelativistically in that frame, so we would expect that nonrelativistic physics to give a good approximation to the answer.

Perhaps I was not clear enough. I made the OP aware of inertial reference frames going in the opposite direction, which is the group of frames that you did not consider. In that group of reference systems, RoS works against breaking the string as the front rocket takes off after the rear rocket.

 

[stevendaryl]

It is clear that in the ladder/barn paradox, contraction is not a "physical effect" and very much a frame dependent facet which involves RoS.

Well, it's a little ambiguous what it means for something to be "physical". Certainly an outcome that can be verified using any frame is physically meaningful. But the reasoning used to explain/predict the outcome is often frame-dependent. Different frames would explain the same result in different ways. But being a little more loose, you could consider frame-dependent (or more generally, coordinate-dependent) effects to be "physical" if they can reliably be used in predicting physical effects.

The difficulty in reasoning about thought experiments such as Bell's spaceships, or the barn and pole, or the twin paradox, or whatever is that we don't actually have any Lorentz-invariant equations of motion to apply. If we did, there would be no ambiguity: Just pick a frame, and apply the equations of motion. If the equations are Lorentz-invariant, then you get the same result, no matter what frame you pick. But if someone just tells you "I have a pole moving at close to the speed of light" or "I have a clock moving at close to the speed of light" or "I have a string connecting two rockets moving at a significant fraction of the speed of light", you don't have equations of motion to derive the result. Instead, you have to use rules of thumb and intuition about how poles, clocks, strings work. The rules of thumb that we are most comfortable with are Newtonian physics, which are only applicable when things are moving slowly relative to the speed of light. So you can try to analyze things locally, in a frame where things locally are moving non-relativistically, and hopefully piece together the local pictures into a global picture.

But Bell's argument with his spaceship paradox is that we can augment purely Newtonian reasoning with intuitions specially developed for SR. He claims that we should start to think of Lorentz contraction as a physical thing, to get an intuition about things moving relativistically. The rule of thumb is: If you take an extended object, and accelerate it to relativistic speed, then it will tend to contract. To prevent it from contracting requires applying stresses on the object. Whether you call such reasoning "physical" or not is a matter of definitions, but Bell's point was that, at least in many circumstances, such reasoning gives you a quick intuition about what the right answer is.

I don't think that the ladder/pole paradox contradicts Bell's contractionistic reasoning. From the point of view of the barn frame, the pole is contracted, and it is possible to close both doors simultaneously. And if the barn doors are really strong, compared to the pole, it IS possible to get the entire pole into the barn at once. Of course, using "contractionistic" reasoning, you would conclude that AFTER the barn doors are closed, the pole would expand to its normal length, and would get smashed to pieces by the strong barn doors.

You get the same conclusion from the point of view of the pole. From that point of view, it's the barn that is contracted, not the pole. But it's STILL possible to fit the pole into the barn: You smash the barn into the pole at relativistic speed, and the pole will be crushed to a size that fits inside the barn.

 

[stevendaryl]

Perhaps I was not clear enough. I made the OP aware of inertial reference frames going in the opposite direction, which is the group of frames that you did not consider. In that group of reference systems, RoS works against breaking the string the front rocket takes off after the rear rocket.

I understood your point, but I'm saying that that point of view doesn't really lead to the opposite conclusion, if you think through the details. Yes, in that frame, the rear rocket takes off before the front rocket, but the front rocket takes off before the information that the back rocket has taken off can propagate to the front. If the back rocket takes off at time $t_1$, and the distance between the rockets is $L$, then it will take at least until $t_1 + L/c$ before the loosening of the string can propagate all the way to the front of the rocket. If the front rocket takes off at time $t_2 < t_1 + L/c$, then the stress on the string near the front rocket will be the same as if the front rocket took off first.

 

[harrylin]

[..] For an observer moving in the negative X direction, the back ship starts moving first, its clock set later than the clock on the front ship. In fact, the back of the back ship starts moving before the front of the back ship.

Exactly

For this viewer, the whole thing starts out contracted and out of synch, and so the intuition that the platform observer might have regarding the rope - that physics should work as expected in one's rest frame - is already subverted.

Physics must also work for that viewer. The point of Bell's exercise - as well of the point of this extension of it - is to enhance intuition of relativistic physics. No matter which frame you chose, length contraction in the sense of Bell (meaning no.2 in post #25 ) plays a role.

It is clear that in the ladder/barn paradox, contraction is not a "physical effect" and very much a frame dependent facet which involves RoS.

That is exactly the confusion that I addressed - here you talk about meaning no.1.

 

[harrylin]

I understood your point, but I'm saying that that point of view doesn't really lead to the opposite conclusion, if you think through the details. Yes, in that frame, the rear rocket takes off before the front rocket, but the front rocket takes off before the information that the back rocket has taken off can propagate to the front. If the back rocket takes off at time $t_1$, and the distance between the rockets is $L$, then it will take at least until $t_1 + L/c$ before the loosening of the string can propagate all the way to the front of the rocket. If the front rocket takes off at time $t_2 < t_1 + L/c$, then the stress on the string near the front rocket will be the same as if the front rocket took off first.

I gave that additional example in order to prevent that the OP concludes that somehow RoS and not length contraction "explains" the breaking of the string. Instead, it's the departure times of the rockets as well as length contraction that must be taken in account, no matter what inertial reference system one chooses.