# Bell's Spaceship Paradox & Length Contraction

David Lewis
You and two identical spaceships are all at rest with respect to each other. You note that the two engines start up at the same time, and the thrust curve and acceleration profile of both spaceships are identical. As the ships pick up speed, would you measure the ships to be shorter than their rest length?

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Nelli

sweet springs
For him lengths of the rockets are shortened. The distance between the ships are kept constant. The thread which is moving and accelerating is shortened so torn apart.

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As the ships pick up speed, would you measure the ships to be shorter than their rest length?
Of course (with reasonable assumptions about the ship design). But the string's length does not change until it snaps.

The thread which is moving and accelerating is shortened so torn apart.
A key point is that the thread does not shorten. In the original rest frame of the ships you said it yourself: The distance between the ships are kept constant. So the thread does not shorten in this frame - it stays the same length. In each successive momentarily comoving inertial frame of the thread (to the extent such a thing can be defined) the spacing between the ships is larger than the moment before, and the thread actually lengthens.

The point you are trying to make, I think, is that the original rest frame of the ships regards the thread as moving. Hence the thread's unstressed length would decrease over time. However, stresses build up and stretch it, countering this effect. The stresses cause the thread to break.

David Lewis and sweet springs
sweet springs
Thanks. In other words, say the thread is tightened to the front ship but loose and not tightened to the back ship, the thread is shortened as the rockets accelerate so the distance between the loose end of the thread and the back rocket increases from zero.

Ibix
Homework Helper
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You and two identical spaceships are all at rest with respect to each other. You note that the two engines start up at the same time, and the thrust curve and acceleration profile of both spaceships are identical. As the ships pick up speed, would you measure the ships to be shorter than their rest length?

You have the same issue with each spaceship. If, in your frame, the acceleration profiles of the front and rear of a spaceship are identical, then they stay the same distance apart in your frame, hence the ship must be stretching in its rest frame, such as can be defined.

If, however, the ship moves rigidly in its rest frame, then it must contract in your frame, hence the acceleration profile of the ship varies (slightly) from front to rear in your frame.

David Lewis and Ibix
David Lewis
...the acceleration profile of the ship varies (slightly) from front to rear in your frame.
This is exactly the point I was having trouble with. Many thanks.
The distance between the ships are kept constant.
Then I take it you'd measure distance a as remaining the same. Would you measure the length of the thread (b) getting shorter due to length contraction of the ships?

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sweet springs
Re:＃８ your figure shows that $$b-a$$ = length of a ship = $$l_0\sqrt{1-\frac{v^2}{c^2}}$$.
It does not matter whether there is a thread between the ships or not as far as elastic property of thread does not harm designed acceleration of rockets.

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Staff Emeritus
You and two identical spaceships are all at rest with respect to each other. You note that the two engines start up at the same time, and the thrust curve and acceleration profile of both spaceships are identical. As the ships pick up speed, would you measure the ships to be shorter than their rest length?

It's not clear to me what frame of reference "you" are in. I'm guessing that "you" are in an inertial frame of reference, and that when 'you' (David Lewis) say that both space-ships start up "at the same time", you mean that they start up the same time in said inertial frame of reference.

If I'm interpreting it correctly, and if we also assume that the spaceships are Born rigid, the answer is yes, from the inertial frame of reference one observes both spaceships to length contract.

The main point I have to make is that it matters - "you" are not being to clear about what frame of reference the other "you" is in, and this is important information to give an accurate answer.

A secondary but also important point is that one needs to know if the spaceships are rigid. The applicable notion of rigidity in special relativity is Born rigidity. If the spaceships are not rigid, then their length can change so the question cannot be answered without more information. As others have mentioned, if the spaceships are rigid (Born rigid), their bow and stern have different acceleration profiles.

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Mentor
Would you measure the length of the thread (b) getting shorter due to length contraction of the ships?

No, because the thread is being stretched, which in the frame you are using means its measured length stays the same, instead of contracting as it would if it were not connected to both ships.

Again, have you read the PF FAQ entry on the Bell Spaceship Paradox, which I linked to above? It discusses all of these points. (In particular, it compares the length of the thread connected to both ships, to the length of a thread connected only to the front ship, as the ships accelerate.) So do many other sources that discuss the paradox, including Bell's original paper.

Mentor
I'm guessing that "you" are in an inertial frame of reference, and that when 'you' (David Lewis) say that both space-ships start up "at the same time", you mean that they start up the same time in said inertial frame of reference.

This is my intepretation of the OP's intent. But you are right that in any relativity problem, it's better not to leave this up to interpretation, but to be explicit about what frame is being used.

Mentor
The thread which is moving and accelerating is shortened

No, it isn't, it stays the same length. What shortens is the unstressed length of the thread. See the FAQ entry I linked to.

Mentor
say the thread is tightened to the front ship but loose and not tightened to the back ship, the thread is shortened as the rockets accelerate so the distance between the loose end of the thread and the back rocket increases from zero.

Again, see the FAQ entry I linked to, it discusses precisely this point.

Gold Member
Is the problem equivalent to a hammock hanging at right angles to the direction of acceleration in a perfectly rigid accelerating spaceship? The hammock would eventually break if its made of breakable material?

Mentor
Is the problem equivalent to a hammock hanging at right angles to the direction of acceleration in a perfectly rigid accelerating spaceship?

No, because, as I'm understanding your description, both suspension points of the hammock would be at the same height in the ship, so they would have the same proper acceleration. The hammock would hang down in the ship just as it would in a room at rest on the surface of a planet, and would not stretch (except for a brief period when the acceleration first began, while the whole system was coming to a new equilibrium).

More generally, if the spaceship is Born rigid (which is the closest thing to "perfectly rigid" that you can have in relativity), any object inside it will not stretch, because the ship itself will not stretch. (So, for example, you could "hang" the hammock vertically, with one suspension point on the ceiling of the ship and the other on the floor, and it would not stretch, just as it wouldn't if hung vertically in a room at rest on a planet.) Actually, the usual way this is defined in relativity is to turn that around: the definition of the ship "not stretching" (and by extension everything inside the ship not stretching) is that its motion is Born rigid. There are plenty of ways to flesh this out with technical details, but that goes beyond the scope of a "B" level thread.

Grinkle
This is exactly the point I was having trouble with. Many thanks.
Then I take it you'd measure distance a as remaining the same. Would you measure the length of the thread (b) getting shorter due to length contraction of the ships?
Assuming that the ships are of identical design, then in the original rest frame equivalent points will remain at constant separation. The obvious equivalent point is the rocket exhausts on the two ships, since this is the point where the thrust is applied. Since identical ships will react identically to identical thrust, though, the noses will remain at constant separation, as will the attachment pylons for the thread. The nose-to-tail separation of the ships will vary slightly because the length of the ships varies as measured in this frame.

So, on your diagram, and assuming identical ships with identical proper acceleration profiles, and that we're working in the original rest frame of the rocket, (b) remains constant. (a) does not.

David Lewis and sweet springs
David Lewis
It's not clear to me what frame of reference "you" are in. I'm guessing that "you" are in an inertial frame of reference, and that when 'you' (David Lewis) say that both space-ships start up "at the same time", you mean that they start up the same time in said inertial frame of reference.
Correct. "You" is the spaceman depicted in the diagram attached to Post #1. I labeled him "OBSERVER".

Sorcerer
No, because the thread is being stretched, which in the frame you are using means its measured length stays the same, instead of contracting as it would if it were not connected to both ships.

Again, have you read the PF FAQ entry on the Bell Spaceship Paradox, which I linked to above? It discusses all of these points. (In particular, it compares the length of the thread connected to both ships, to the length of a thread connected only to the front ship, as the ships accelerate.) So do many other sources that discuss the paradox, including Bell's original paper.
Re: the bold (my emphasis). Am I to understand this to mean that the stretching of the thread exactly compensates for length contraction until the moment it breaks?

Homework Helper
Am I to understand this to mean that the stretching of the thread exactly compensates for length contraction until the moment it breaks?
Well, yes. It is held in place between two space craft whose positions are at a fixed separation given by the problem setup. It must remain the same length because it stretches between those two end points. Until it breaks, of course.

Gold Member
I think I am missing something. It's not clear to me how this is different from the forces within a long rocket of the same total length (from the nose of the lead rocket to the tail of the trailing rocket). Would such a rocket experience different acceleration forces at the front and back? If not, wouldn't it also be torn apart just like the thread is broken?

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I think I am missing something. It's not clear to me how this is different from the forces within a long rocket of the same total length (from the nose of the lead rocket to the tail of the trailing rocket). Would such a rocket experience different acceleration forces at the front and back? If not, wouldn't it also be torn apart just like the thread is broken?
A long rocket only has an engine at the back, so it isn't stretched. Bell's ships scenario has a rocket at front and back which can move under their own power. If you had a long rocket with engines at front and back (don't think too hard about where the exhaust from the front engine goes...) it could tear itself apart too.

From the perspective of either ship this is actually what's happening. The other ship's engine is mis-set, so it moves away and stretches the thread until it breaks.

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FactChecker
Staff Emeritus
A long rocket only has an engine at the back, so it isn't stretched. Bell's ships have rockets at front and back which can move under their own power.

I didn't think of Bell's scenario as implying that the ships themselves had rockets on both ends. He wasn't making a point about the stresses inside the ships, but about the distance between the ships. Of course, the case you're talking about, with the front and rear having independent rockets, is basically equivalent to having multiple connected ships.

FactChecker
Mentor
I didn't think of Bell's scenario as implying that the ships themselves had rockets on both ends. He wasn't making a point about the stresses inside the ships, but about the distance between the ships. Of course, the case you're talking about, with the front and rear having independent rockets, is basically equivalent to having multiple connected ships.
I was understanding @Ibix as saying, in response to @FactChecker, that Bell's setup (two independently powered ships connected) is equivalent to a single long ship with independent motors at each end.

FactChecker and Ibix
Staff Emeritus
I was understanding @Ibix as saying, in response to @FactChecker, that Bell's setup (two independently powered ships connected) is equivalent to a single long ship with independent motors at each end.

Okay, I agree with that.

I didn't think of Bell's scenario as implying that the ships themselves had rockets on both ends.
I agree - it's just my writing that was unclear. Nugatory apparently managed to puzzle out what I meant, but I've slightly edited my post to be clearer (I hope).

Nelli and FactChecker
Gold Member
So a single long rocket which accelerates with no internal stress is not experiencing the same acceleration at the front and the back. I'll buy that since the length must change. (Although I will have to think about whose reference frame I am talking about.)

So a single long rocket which accelerates with no internal stress is not experiencing the same acceleration at the front and the back. I'll buy that since the length must change. (Although I will have to think about whose reference frame I am talking about.)
Some care is needed with "no internal stress". A rocket is usually under compressive stress because of the rocket motor at the back. I suspect what you mean is under an unchanging strain, in the sense that a crewmember with a ruler will measure the rocket's length to be constant (transients when the engine is initially lit aside).

You also need to be careful about what you mean by acceleration - proper or coordinate? The proper acceleration does not change along the rocket. The coordinate acceleration does, in most frames.

The difference between the long rocket and Bell is in the startup. Bell's rockets start simultaneously in the initial rest frame. The nose of the long rocket starts moving when shock waves from the ignition of the rocket in the tail reach it - later than Bell's lead rocket starts moving. Hence the different behaviour later on, despite all proper acceleration gauges in both setups showing the same thing in steady state.

FactChecker
Staff Emeritus
You also need to be careful about what you mean by acceleration - proper or coordinate? The proper acceleration does not change along the rocket.

If we're talking about a normal rocket, propelled from the rear, the proper acceleration will vary from back to front, with the acceleration being greater in the rear.

It's a little bit misleading to attribute this to the fact that it is pushed from the rear. The same thing would be true if the rocket were pulled from the front. Immediately after launching, the rocket is compressed when pushed from the rear and stretched when pulled from the front, but that's a transient effect. After the rocket reaches an equilibrium length, the proper acceleration of the rear will be greater than the proper acceleration of the front in either case.

Ibix
Staff Emeritus
If we're talking about a normal rocket, propelled from the rear, the proper acceleration will vary from back to front, with the acceleration being greater in the rear.

It's a little bit misleading to attribute this to the fact that it is pushed from the rear. The same thing would be true if the rocket were pulled from the front. Immediately after launching, the rocket is compressed when pushed from the rear and stretched when pulled from the front, but that's a transient effect. After the rocket reaches an equilibrium length, the proper acceleration of the rear will be greater than the proper acceleration of the front in either case.

Here's a way to see that this must be true. Suppose that the rocket's length soon after launching and after it has reached its equilibrium length is ##L## according to the launch frame. Now, wait until the rocket is moving relativistically at some speed ##v##. Its length when measured from the launch frame will be ##L/\gamma##, where ##\gamma = 1/sqrt{1-\frac{v^2}{c^2})##. If you think about it, the only way for the length of an object to decrease is if the rear gets closer to the front. That's only possible if (according to the launch frame) the rear is traveling slightly faster than the front. That implies that the coordinate acceleration of the rear is always a little greater than the coordinate acceleration of the front.

If we're talking about a normal rocket, propelled from the rear, the proper acceleration will vary from back to front, with the acceleration being greater in the rear.
Ack! You're right. In order for the length to remain invariant the worldlines of the nose and tail, which are hyperbolae, must have a common focus. Which means different proper accelerations.
It's a little bit misleading to attribute this to the fact that it is pushed from the rear.
That wasn't my intention. I was pointing out that the rocket isn't unstressed, and that the two ends of the rocket don't start moving at the same time in the initial rest frame, unlike Bell's rockets. I agree that the same is true of rockets "hanging" from engines near their nose, and didn't intend to imply otherwise.

I was also claiming that the difference between Bell's spaceships and a long rocket is just the timing of the movement starts. That was wrong, as you pointed out. They are different, but the proper accelerations are also different, contrary to what I said above.

Gold Member
Here's a way to see that this must be true. Suppose that the rocket's length soon after launching and after it has reached its equilibrium length is ##L## according to the launch frame. Now, wait until the rocket is moving relativistically at some speed ##v##. Its length when measured from the launch frame will be ##L/\gamma##, where ##\gamma = 1/sqrt{1-\frac{v^2}{c^2})##. If you think about it, the only way for the length of an object to decrease is if the rear gets closer to the front. That's only possible if (according to the launch frame) the rear is traveling slightly faster than the front. That implies that the coordinate acceleration of the rear is always a little greater than the coordinate acceleration of the front.
That is how I imagined it. Which is why I don't understand why having a string in the middle changes so that the string is snapped. But this is probably where I have to put my trust more in the diagrams and figure it out from there.

Mentor
That is how I imagined it.

How you imagined which scenario?

If you have a single long rocket that doesn't stretch, that's not the same scenario as the Bell spaceship paradox. And a string attached at both ends (front and rear) of such a long rocket would not stretch, but that's because such a string is not being subjected to the same motion as the string in the Bell spaceship paradox.

To realize a scenario like the Bell spaceship paradox with a single long rocket, you would have to have rocket engines at both ends, and both providing the same proper acceleration. In that scenario (which is different from the one @stevendaryl and @Ibix have been discussing in their last few posts), the spaceship would stretch (more precisely, internal stresses in the ship would continuously increase) and eventually break, just as the thread does in the standard Bell spaceship paradox. In this scenario, the length of the spaceship as seen in the original rest frame would not decrease; it would stay the same (which is why the word "stretch" can be misleading--it's better to not use length-related terminology at all since length is frame-dependent).

Ibix
That is how I imagined it. Which is why I don't understand why having a string in the middle changes so that the string is snapped. But this is probably where I have to put my trust more in the diagrams and figure it out from there.
Draw a spacetime diagram in the initial rest frame. Bell's ships follow identical hyperbolae, one simply offset to the right. The nose and tail of the long rocket follow hyperbolae with a common focus. If you pick the back hyperbola to have its focus being the origin of coordinates, you will find that it is invariant under Lorentz transform (i.e., points on the hyperbola are mapped onto points on the same hyperbola). In the long rocket case the nose's path has the same focus so has the same property. So nothing changes in the rocket frame. But in Bell's case the front rocket's hyperbola has a different focus and it doesn't map onto itself under Lorentz transform (or, more precisely, there's no choice of origin such that both front and rear ships' hyperbolic worldlines map onto themselves). So the front ship moves compared to the back rocket.

FactChecker
Staff Emeritus
That is how I imagined it. Which is why I don't understand why having a string in the middle changes so that the string is snapped. But this is probably where I have to put my trust more in the diagrams and figure it out from there.

You're mixing up two different scenarios: Let ##F## be the initial launch frame. Let ##L## be the initial distance between the rockets. Let the two rockets accelerate for a while until the rear rocket is traveling at speed ##v## relative to ##F##. Let ##F'## be the frame in which the rear rocket is momentarily at rest. Then we have two different scenarios:
1. The distance between the rockets, as measured in frame ##F##, is still ##L##. Then as measured in frame ##F'##, the distance between the rockets will be larger than ##L##, and the string will break.
2. The distance between the rockets, as measured in frame ##F'##, is still ##L##. Then as measured in frame ##F##, the distance between the rockets will be contracted, and the string will not break.
What's important for the string to break or not is the distance between the rockets, as measured in frame ##F'##.

In scenario 1, the front rocket and rear rocket experience the same (proper) acceleration. In scenario 2, the rear rocket experiences greater proper acceleration.

FactChecker