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Bell’s Spaceship Paradox
Background
So the “paradox” is outlined on Wikipedia pretty nicely: link, but I still find myself somewhat unsatisfied by the explanations. I only recently became aware of this problem from a YouTube video by Mahesh, here. He’s great and I can’t imagine anything is really properly wrong with his explanation either, but it remains somewhat unintuitive to me. It’s especially confusing because he has a subtle disagreement with Bell’s own explanation, and I think that’s fair. But now I’ve got a couple explanations that technically disagree with eachother, and neither really matches my own intuition. Now there’s not real problem here, just that I think there is an easy explanation that isn’t being articulated, so I’m going to outline that here. The big caveat is that I am not an expert, and nobody else is explaining it this way so I assume it is also subtly wrong, lol. This has been bugging me for a while, so I better get it out of my head so I can stop thinking about it.
Just to Recap
The actual problem is, roughly, imagine there are two spaceships connected by a fragile string. They start off at rest to each other and to an inertial reference frame S, they accelerate instantly to some relativistic speed to a new reference frame S1 (say whatever speed it takes to make an observer in S see a length contraction of 1/2 in S1). Will the string break?
Because I’m not trying to arrive at the result independently or hold everyone in suspense, the answer is yes. Bell more or less directly attributes this to length contraction, but that’s the part that bothers me the most. I’ve always understood length contraction to be something that is essentially unknowable to the party being contracted, after all we are all length contracted in all different directions at all times. I know that’s not exactly what he’s saying, but attributing a force to this contraction seems like at best a misleading shortcut to the answer.
Mahesh points out that the acceleration as described by the problem would be overcompensating for the length contraction, meaning the string and space itself between the ships will have contracted, but the observed distance between the two ships will still be the same. I think this is correct, I’m not trying to argue with him, I just think the explanation gets really fuzzy and I’m not at all sure why it has to.
My Attempt
Naturally, I have to start off my explanation with an entirely different problem, one that my high school physics teacher brought up.
The Small Barn
I’m sure this has some real origin, but I didn’t look it up, but imagine we have some sort of ship with length l and a barn with length .9l. The barn has doors on either side and the spaceship is moving fast enough that it is length contracted to some l' < .9l, and it is moving in the same direction as the length of the barn, straight through the doors. Ignoring the impracticality associated with opening and closing barn doors at relativistic speeds, we can keep the far door closed, until the moment before the ship tries to leave out of it, and close the rear door the instant the ship clears it, and we will have, for a brief moment, put a spaceship inside a barn that is physically too small to hold it, and closed both doors at the same time.
The not-a-paradox here is that special relativity is symmetric, in the sense that all observers are at rest relative to themselves, and perceive everyone else as doing the moving. So from the perspective of the ship, the barn is moving at relativistic speeds, and the barn is shorter than it usually would be at rest. The barn starts off at .9l and contracts down to some l'' < (.9)(.9)l even shorter than the already-too-short length it has at rest. How, then, does the barn hold the spaceship with both doors closed at the same time? It’s impossible?
The answer is that it is impossible, and that’s fine. From the perspective of the spaceship, the barn doors are never both closed at the same time. The lesson this problem is meant to teach you is that simultaneity is a feature of each reference frame, which is to say it is relative. This should come as no surprise, the whole point of special relativity is that spacetime will bend and flex to make sure that nothing goes faster than the speed of causality. If time is relative, then surely simultaneity is also relative, no disrespect here, it’s not like I immediately understood this in high school either, but duh…
Back to the problem
With this result in mind, I think it becomes a lot clearer what is happening in Bell’s Spaceship Paradox. The instantaneous, simultaneous acceleration all the way from S to S1 is hiding a very important detail. The whole of that acceleration is perceived from S as simultaneous. This is kind of annoying to me, because the question is more or less a question of one’s intuition about special relativity, but the complexity is hiding inside an unintuitive, and rather impractical, humongous acceleration that occurs instantaneously. Not clear that the universe even needs to be able to answer this problem.
I feel like this is technically enough to know that the string will break, but I’ll keep going to try and make it a bit more obvious. What worked for me is imagining a situation where the same initial and final conditions are reached, but we make sure that the string does not break. To do this, I’m going to ditch the big single acceleration, and instead imagine a series of incredibly mundane accelerations that will eventually bring us up to 87% the speed of light or whatever. So let’s say we start in S_0 at rest with some initial observer, and we will wind up in S_N at rest with some final observer. We will be advancing from S_0 to S_1 with a synchronous acceleration that is mild enough not to break the string. Importantly, the burn is synchronous in S_0, but the next burn will be synchronous from the perspective of our ship/string system in S_1. Again, this mundane acceleration will not break the string. This continues all the way to S_N. Since the string is always in the same reference frame that the acceleration is synchronous in, the string never experiences any meaningful strain and will not be broken. Someone can email me if I’m wrong about this, I don’t mind being wrong. But I believe this guarantees that the string does not break.
The question is, in my contrived example where we know the string doesn’t break, what does our original observer back in S_0 see? The first acceleration is synchronous, the second is pretty close to the first so it is still effectively synchronous. However, at some point the system will be moving at relativistic speeds and, for the same reason as our barn example, the initial observer in S_0 will start to observe a difference. Maybe we pretend each acceleration is a burn that he can see the light from. I believe (without doing any math, lol) that the front burn simply starts and finishes a little bit later than the rear burn. It will achieve the same change in speed as the rear burn, and both ships will still be moving at the same speed as each other after the burn is over. But, the rear will have started earlier, ended earlier and spent a little bit of extra time at the higher speed. The result being the rear ship will be closing the distance between them, only from the perspective of the observer in S_0. This means that, after all the accelerations are complete, our initial observer will see two ships separated by the same distance as the contracted length of the string.
So, in the conditions we have described that do not break the string, the acceleration cannot be wholly simultaneous from the original reference frame. If the acceleration is simultaneous in the original reference frame, then the relativity of simultaneity cannot allow the ships to close the distance and compensate for the length contraction. This means the problem is less of a paradox and more of an aliasing issue. An instantaneous acceleration means there are only two reference frames, so the acceleration is simultaneous for every reference frame that the system experiences and simultaneous from the original reference frame. Allowing for the intermediate frames makes the problem much clearer, but the whole question comes down to which of the two analogues is appropriate, the one where the acceleration is simultaneous in S_0 or the one where the acceleration is simultaneous in S_i where i is just whatever frame the system happens to be in along the path. I’m basically just arguing that the aliasing problem isn’t really there either. The fact that the acceleration is simultaneous from S_0 prevents the necessary desynchronization that would allow the ships to close the gap. Doesn’t matter that the ridiculous acceleration prevents the system from actually seeing the desynchronized accelerations from its frame of reference, after all, we do not need to explain states that the problem prevents from occurring. We can’t tell you where the string broke, because it had to happen in the same instant that everything else happened, nothing inside that instant is well defined by the problem. The only point that matters, is that we must see the ships close the gap from S_0 in order for the string to stay intact.
Conclusions
I dunno what I’m doing here really, just think this problem doesn’t need to be as hard as it sounds in either the video or in Wikipedia. I do think it, and really the barn example too, both highlight how much harder it is to think about bending spacetime than bending space. And, I think it is a useful problem to wrestle with, because it helps show how all of these measurements we can make are collaborating to enforce the universal speed limit. It is easy to think about length contraction as something that is happening by itself, but it always comes with a frame of reference that moves through time differently as well. The “now” for a contracted entity spans many moments from the perspective of any observer who sees it contracted. And the universe is not responsible for reporting to you about the goings-on of events that occurred inside a single moment from your perspective, what you can’t see can’t hurt you.