I finally solved the paradox that split smart people 50/50! (It's wild)
Two identical spaceships joined by a delicate string accelerate to near light speed at the same instant and rate. Does the string snap? This is Bell's spaceship paradox, which split CERN's theorists in the 1980s. Mahesh Shenoy of FloatHeadPhysics shows the string does snap, then argues that Bell and most popular explanations get the right answer for the wrong reason by blaming length contraction, which is only geometry and cannot cause stress. The real cause is that equal acceleration at both ends grows the string's proper length, an objective fact that literally stretches it until it tears, a picture that scales up to model black holes and the Penrose diagram.
Published Feb 25, 202623:58 video20 min readAdded Jun 16, 2026Open on YouTube →
At a glance
Two identical spaceships sit at rest, joined by a delicate string. They fire identical engines at the identical instant and accelerate toward the speed of light. Does the string survive or snap? This is Bell's spaceship paradox, and in the 1980s it split the theory division at CERN almost evenly, with the eventual consensus landing against the people who first felt sure. Mahesh Shenoy of FloatHeadPhysics shows that the string does snap, then makes the sharper claim that almost every popular explanation, including John Bell's own, gets the right answer for the wrong reason.
The wrong reason is length contraction. Borrowing from Lewis Epstein's Relativity Visualized, Mahesh rebuilds contraction as nothing but geometry, a rotation in spacetime seen from an angle, which means it cannot pull on anything and cannot break a string. The real culprit is acceleration, and the key fact is that a rigid ship needs its tail to accelerate harder than its nose. When both ends of the string instead accelerate equally, its proper length genuinely grows, an objective fact in every frame, so the string is literally stretched until it tears. The same picture, scaled up, turns accelerating ships into a model of a black hole and points straight at the Penrose diagram.
The page below walks the same path Mahesh walks, in order: the puzzle, both camps and who won, the pencil that teaches what contraction really is, the derivation that falls out for free, the clocks that drift, and the acceleration gradient that is the true answer.
The puzzle, stated cleanly
Two identical spaceships, connected by a delicate string. They begin accelerating very close to the speed of light, at exactly the same time and at exactly the same rate. As they speed up, does the string stay intact or eventually snap?
The naive read is airtight on its face. Both ships accelerate together at the same rate, so their separation should stay the same. If the separation never changes, the string is never pulled. If the string is never pulled, it never snaps. So why on earth would it ever break? That clean argument is exactly the trap, and roughly 60% of Mahesh's own audience fell into it in his community poll, voting that the string survives.
Figure 1. The setup. Two identical ships, one delicate string, identical engines lit at the identical moment. The "obvious" answer says the gap is constant, so the string is never pulled, so it cannot snap. Holding that thought is what makes the resolution surprising.
Length contraction enters, and the room divides
Near the speed of light, Einstein's relativity kicks in, and its most famous prediction is length contraction: a moving object is shorter along its direction of motion than it is at rest. Everyday speeds are so far below light speed (in relativity we usually set the speed of light to one) that we never notice. But close to light speed the effect is huge. A ship moving at 87% of light speed measures half its rest length. So as the ships and the string speed up, they tend to shrink, and the question becomes what that shrinking does to the string.
John Bell, the same Bell of Bell's theorem that demolished Einstein's hidden variable hopes for quantum mechanics, argued the string must snap. His reasoning: because the ships start accelerating at the same time, the distance between them always stays the same. But the ships and the string each want to contract. The string is tied at both ends, so it cannot freely contract, stress builds, and it eventually breaks.
His colleagues at CERN pushed back hard. In Bell's own words, "A distinguished experimental physicist refused to accept that the thread would break, and regarded my assertion that indeed it would as a personal misinterpretation of special relativity." Their counterargument has real force. If the whole system accelerates together, why would it not contract together, leaving no stress at all? And motion is relative, so from the ships' own frame they are always at rest relative to each other, with no contraction between them; it is the rest of the universe that rushes backward and contracts. By that account there is no stress and the string survives, and the reasoning seems consistent in every frame.
So CERN's theory division held a vote. Bell again: "We decided to appeal to the CERN theory for arbitration and made a not very systematic canvas of opinion in it. There emerged a clear consensus that the thread would not break." A clear majority of professional physicists said it survives.
They were wrong. Bell, convinced he was right, popularized the puzzle in a chapter titled "How to teach special relativity," walking through the snap in excruciating detail down to atoms and electromagnetic fields. The modern consensus agrees with him: the string snaps. That is why we now call it Bell's spaceship paradox. But the deeper question stands. Why does length contraction work this way and not the other, what does it look like from inside the ship, and how could anyone reason it out from scratch?
The pencil that breaks the spell
Bell's chapter is correct but, by Mahesh's lights, too technical to build any intuition. The breakthrough came from Lewis Epstein's Relativity Visualized. Voiced as a dialogue, Epstein's first move is to attack the premise. Asked what length contraction is, Mahesh gives the textbook line, that moving things are shorter than resting things. Epstein's reply: no, they are not. That is a misconception.
Take a pencil. Held straight on, it has a certain length. Rotate it and it looks shorter. The pencil did not shrink; you are simply viewing it from an angle. Looked at head on, the length you measure equals its true length. As it rotates, the measured length shrinks while the true length never changes, because all you are ever measuring is the pencil's projection onto your axis. That projection is the coordinate length. The pencil's actual length is its proper length. The proper length never moved. The pencil just rotated, and the measured length fell. That, says Epstein, is length contraction. It is geometry, not shrinking.
The obvious objection: a spaceship does not rotate, it accelerates. Epstein's answer is the hinge of the whole video. Acceleration is rotation, not in space, but in spacetime.
Here is why. Even a ship sitting still in space is moving through its own proper time, and the speed at which everything travels through spacetime is fixed: it is the speed of light. You, me, donkeys, every object moves through spacetime at exactly one speed, the speed of light. At rest, all of that speed points along the time axis, so spatial velocity is zero and your clock ticks at full rate. When the ship accelerates, its total spacetime speed cannot change, but its direction can. The velocity vector tilts toward the space axis, spatial velocity grows, and time slows to compensate. Tilting the velocity vector is rotation. So a ship at rest is measured head on, coordinate length equals proper length; a moving ship is measured from an angle, coordinate length falls below proper length. The faster it goes, the more it rotates, the more the measured length contracts. Nothing shrank. It rotated in spacetime, and we measured it from an angle.
Figure 2. The reframe at the heart of the video. Length contraction is a projection, not a shrinking. A rotated pencil casts a shorter shadow on your axis while its true length is fixed. Acceleration tilts a ship's spacetime velocity vector off the time axis, which is the same rotation, so the measured length drops while the proper length does not. Because nothing physically shrank, this geometry cannot, by itself, stress a string.
The math falls out for free
Mahesh wants the equation, not just a picture. It drops out in a few steps. Connect the proper length and the measured length by completing the triangles in the spacetime rotation, and the two triangles are similar. So the measured length divided by the true length equals one side of the triangle divided by another. Set the speed of light to one, as relativity usually does. Then the relevant side is, by Pythagoras, the square root of 1 minus v squared. That is exactly the Lorentz factor term: the famous root of 1 minus v squared in the length contraction formula, derived in a few elegant strokes from a rotated triangle rather than memorized.
Figure 3. The length contraction law as a curve. Measured length over proper length is the square root of 1 minus v squared. It is nearly flat at everyday speeds, which is why we never notice contraction, then plunges near light speed. The marked point reproduces the video's example: at 0.87 c a ship measures half its rest length. The whole shape is just the projection of a rotated object, the cosine of its spacetime tilt.
The clocks that drift, and why simultaneity is relative
One critique remains. When the pencil rotates, its horizontal length contracts but its vertical length grows. So if a ship rotates in spacetime, its spatial length should contract while its temporal length grows. What is that temporal length?
Epstein's answer ties the picture to clocks. Put two perfectly synchronized clocks on the ship, one at the front and one at the back. In the ship's own frame they stay in sync even as it accelerates, so the temporal distance between them is zero. In our frame they drift apart: the back clock runs ahead in proper time compared to the front clock. The temporal length is exactly how far out of sync the clocks are. As the ship accelerates, the spatial length shrinks while the clocks fall more and more out of sync, so the temporal length grows, just as the rotated pencil's vertical projection grows.
Put another way: in the ship's frame, 1 second at the front is simultaneous with 1 second at the back. In our frame, 1 second at the front is simultaneous with 2 seconds at the back. That is the relativity of simultaneity, the fact that two events called "at the same time" by one observer are not simultaneous for another. We knew it was true; here we see it fall out of the same rotation.
The blow to Bell's reasoning
Now the punch lands on Bell himself. His story was: the string and ships try to contract, the tied ends forbid it, stress builds, the string snaps. But the video has just shown that length contraction is only the artifact of measuring a rotated object from an angle. It is pure geometry. Geometry cannot generate stress. So Bell's stated mechanism is wrong.
To be unmistakable: the string does snap, the stress is real, that part is true. But blaming the stress on length contraction is the error, and it is the same error baked into the most popular explanations of this puzzle online today, which, like Bell, attribute the stress to contraction in our frame. If contraction is not the cause, something else is. Epstein says the answer was in front of us the whole time. It is acceleration. But how can identical acceleration at both ends pull a string apart?
The accelerometers, and the gradient nobody expects
Forget the two ships for a moment and study one. Mount an accelerometer at the front and one at the back. An accelerometer reads acceleration through a physical effect, a spring compressing, more acceleration meaning more compression. As the single ship accelerates rigidly, do the two read the same value or different values?
Instinct says identical: the whole ship accelerates together, so the acceleration is one number everywhere. But recall that acceleration is rotation in four dimensional spacetime, and on anything that rotates, like a door, the outer edge sweeps faster than the inner edge. So the back of the ship should at every moment carry slightly more speed than the front. More speed at the back means the back gains velocity faster, which means the back has a higher acceleration than the front.
That seems to break the ship. If the back genuinely moves faster than the front at every instant, shouldn't the ship physically shrink? It does not, and the reason is simultaneity again. What counts as "at the same moment" for us is not the same moment for the ship. In our frame, at a given instant, the back is faster than the front. But pick a single moment in the ship's own frame and the back and front have the same velocity. The ship sees itself moving rigidly, all parts matched, while we see a back that outruns the front. So a ship that wants to stay rigid as it accelerates must have its tail accelerate harder than its nose. Deeply counterintuitive, and yet forced the instant you accept that acceleration is rotation.
Now flip the question. What if the front and back accelerate with exactly equal magnitude? Then in our frame, at any moment, front and back carry the same speed, so the measured length never contracts. But the ship is still rotating in spacetime, and a rotating object whose measured length refuses to shrink can only be doing one thing: physically stretching. Its proper length is growing. And growing proper length under tied ends is real, frame independent stress.
Figure 4. The real resolution. Left: to stay rigid, a ship must accelerate its tail harder than its nose; then only the measured length contracts, the proper length holds, and there is no stress. Right: when both ends accelerate equally, the proper length genuinely grows while the measured length stays put, so stress builds. The string in the paradox lives in the right hand case.
Bringing the paradox home
With that, the puzzle solves itself. Both ships start accelerating at exactly the same rate at exactly the same time, so the two ends of the string have equal acceleration. By the result just established, the proper length of the string and the proper distance between the ships increase. This is not a statement about measured lengths, which depend on your reference frame. It is a statement about proper length and proper distance, an objective fact that every observer agrees on. The string is being pulled, stress builds, and a delicate string snaps. Paradox resolved.
One honest loose end. Shouldn't the same stretching tear the spaceship too, since its ends also begin with equal acceleration? Yes, the stress appears there as well, but the ship's hull is strong. As stress pulls on its ends, the front's acceleration drops slightly and the back's rises slightly until the ship settles into the correct acceleration gradient, tail harder than nose. At that point the stress stops growing and the ship coasts rigidly at constant proper length. The string is too delicate to negotiate that truce, so it stretches past its limit and breaks.
And what would we actually measure from outside? As the ships and string accelerate, the proper length and proper distance grow, an objective fact, but the whole thing is also rotating in spacetime as it stretches, and those two effects exactly cancel in the measured length. So our ruler reads no change at all even while the string is being torn. That is why the snap feels mysterious: nothing looks like it is stretching, yet the proper length, the thing that matters, has been climbing the entire time.
Where this goes next: black holes and Penrose diagrams
Mahesh closes by scaling the idea up. Take a whole fleet of ships, all at rest, and ask them to accelerate at once so that the proper distance between every pair stays constant, each ship always seeing the others at rest relative to itself. Instinct says give them all identical acceleration. The paradox just proved that is wrong: the ships in front must accelerate gently, the ships in back must accelerate hard, a smooth gradient.
Now the twist that opens the next video. If all those ships hold a fixed proper distance from one another, could they tell whether they are accelerating through empty flat spacetime or hovering motionless near a black hole? No. Near a black hole, the ship nearer the event horizon must fire its engines harder than the ones farther out, the very same gradient. By the equivalence principle, the two situations are physically identical. That means a fleet of accelerating observers in flat spacetime is a working model of a black hole, the spacetime diagram it traces out predicts real black hole physics, and zooming out far enough reveals the Penrose diagram. That is the promised sequel.
Key takeaways
The string in Bell's spaceship paradox does snap. In the 1980s most CERN theorists voted that it would not, and they were wrong; Bell was right that it breaks.
Bell, and most popular explanations, give the right answer for the wrong reason. They blame length contraction. Length contraction is pure geometry and cannot cause stress.
Length contraction is not shrinking. It is the projection of an object rotated in spacetime, measured from an angle. Proper length, the true length, never changes; only the measured coordinate length falls.
Acceleration is rotation in spacetime. Everything moves through spacetime at the speed of light; accelerating tilts that velocity vector from the time axis toward the space axis.
The Lorentz term root of 1 minus v squared drops straight out of similar triangles in that rotation, with no memorization.
A rigid accelerating ship must accelerate its tail harder than its nose. Equal acceleration at both ends instead grows the object's proper length, which is the real, frame independent source of stress.
Because the string's ends accelerate equally, its proper length genuinely increases while its measured length stays constant, so it is literally stretched until it snaps.
Scaled up, a fleet holding constant proper distance needs a thrust gradient identical to hovering near a black hole. By the equivalence principle this models black holes and leads to the Penrose diagram.
Chapters
Timestamps are clickable. Click one and the player jumps there and keeps playing while you read. This video has no creator set chapters, so these are estimated from position in the talk.
0:00 The puzzle: two ships, one string, does it snap?
0:55 Length contraction enters the picture
1:20 John Bell says it snaps
1:35 CERN disagrees, and the vote
2:20 Bell's book and the modern consensus
2:35 Epstein's Relativity Visualized
2:45 The pencil: contraction is just a projection
3:25 Acceleration is rotation in spacetime
4:00 Why measured length contracts but nothing shrinks
4:10 Deriving root of 1 minus v squared from triangles
4:30 The drifting clocks and relative simultaneity
5:10 The blow to Bell's reasoning: geometry cannot cause stress
5:40 The real cause is acceleration
6:00 Brilliant (sponsor)
6:40 Accelerometers front and back
7:00 Why the back must accelerate harder than the front
7:40 Equal acceleration grows the proper length
8:20 Solving the paradox: the string is truly stretched
8:50 Would the ship snap too? The acceleration gradient
9:10 What we measure from outside: no visible change
9:30 Scaling up: a fleet that models a black hole
9:50 Toward the Penrose diagram (next video)
Notable quotes
Even the person who got it right, turns out that he got it right for the wrong reasons.
Mahesh Shenoy, 0:25
A distinguished experimental physicist refused to accept that the thread would break, and regarded my assertion that indeed it would as a personal misinterpretation of special relativity.
John Bell, quoted by Mahesh, 1:35
There emerged a clear consensus that the thread would not break.
John Bell on the CERN vote, quoted by Mahesh, 2:05
Moving things are shorter compared to when they're at rest. And Epstein says, no, they are not. That's a misconception.
Mahesh Shenoy and Lewis Epstein, 2:45
Acceleration is rotation. Not in space, but in space-time.
Lewis Epstein, voiced by Mahesh, 3:25
Length contraction is just an artifact of measuring something from an angle. It's just geometry. So, it cannot cause stress.
Mahesh Shenoy, 5:10
For a ship to maintain its rigidity as it accelerates, the back should indeed have a higher acceleration compared to the front.
Mahesh Shenoy, 7:30
We are talking about proper length and proper distances, which is an objective fact that everybody agrees on.
Mahesh Shenoy, 8:30
This is a teaching video, and on the physics it is solid. The modern consensus does hold that the string snaps, the result is standard in textbooks on Bell's spaceship paradox, and the proper distance between two equally accelerating ships really does grow in their instantaneous rest frames, which is what tears the string. Mahesh's central polemic, that length contraction is the wrong mechanism to cite, is a defensible and increasingly common pedagogical position: the honest invariant is the rise in proper length under Born rigidity failure, not the frame dependent measured length.
Two fair caveats. First, "acceleration is rotation in spacetime" is a genuine and beautiful idea (hyperbolic rotation, or rapidity), but the rotation is hyperbolic, not the ordinary circular rotation the pencil analogy pictures, so the analogy is a ladder to throw away once you are up it, not a literal equivalence. Second, the claim that Bell got the right answer "for the wrong reason" is partly a matter of framing; Bell's atom by atom field argument is rigorous and correct, and reasonable physicists describe the mechanism in more than one valid way. None of that dents the resolution. It is a clean, intuition first rebuild of a problem that embarrassed a roomful of experts, and it sets up the black hole sequel honestly.
Full transcript
Two identical spaceships are connected by a delicate string. They start accelerating very close to the speed of light at exactly the same time and at exactly the same rate. The question is, as they accelerate, will the string stay intact or will it eventually snap?
This puzzle is so counterintuitive that in the '80s even the top physicists couldn't agree on what the right answer was. And in fact, even the person who got it right, turns out that he got it right for the wrong reasons. And that's why even today some of the most popular videos on YouTube on this topic still miss what's actually going on. But I think I've finally cracked it. And the reason I'm sharing this video is not just because the real solution is truly elegant, but because of its far-reaching consequences. It led me to eventually think about black holes in a way that I had never before, the way physicists actually do. And it eventually pushed me into rediscovering Penrose diagrams from scratch.
But I'm getting ahead of myself. Let's go back to the puzzle. If both ships accelerate together at the same rate, their separation should stay the same, right? And if the separation between them stays the same, the string is never pulled. So, it should just stay intact. So, why on earth would it ever snap?
Well, remember when you go close to speed of light, Einstein's relativity starts kicking in. One famous prediction of relativity is that moving objects are shorter in the direction of motion compared to when they are at rest. This is called length contraction. Now, since everyday speeds are much much much much much slower than that of light, which, you know, whose speed we usually choose to be one in relativity, we just don't notice this at all. But, of course, close to speed of light is huge. For example, if this ship was traveling at, say, 87% the speed of light, then its moving length would be half when it's at rest. This means as the ships and the string speed up, they tend to shrink. So, the big question is, what does that do to the string? How does length contraction affect the outcome of the puzzle?
In the '80s, the physicist John Bell, yes, the same John Bell of the Bell's theorem, who disproved Einstein's hidden variable assumptions of quantum mechanics, he argued that the string must snap. He says, "Look, Mahesh, since the ships start accelerating at the same time, the distance between the ships should always always stay the same. But, of course, the ships and the string contract. However, since the string is tied at the ends, it can't freely contract. So, stress starts building up, and as a result, it eventually snaps." But, many of his colleagues at CERN disagreed. He says, and I quote, "A distinguished experimental physicist refused to accept that the thread would break, and regarded my assertion that indeed it would as a personal misinterpretation of special relativity." But, what was their argument against it?
Well, they argued that if the whole system is accelerating together, why wouldn't the whole system contract together? Then, look, there will be no stress in the string, and so it wouldn't snap. And remember, motion is relative. So, from the ships' perspective, they're always at rest relative to each other. So, there is no length contraction between them at all. It's the rest of the universe that's moving backwards and contracting. So again, no stress on that string and it stays intact. Which means this reasoning seems consistent in all frames. And so for me, this reasoning sounds super intuitive. And I'm guessing the same is true for most of you. About 60% of you answered that the string wouldn't snap in the community poll.
But what's the consensus? Well, again, Bell says, "We decided to appeal to the CERN theory for arbitration and made a not very systematic canvas of opinion in it. There emerged a clear consensus that the thread would not break." So that's it. End of story. This is the correct answer and Bell is wrong, right?
Well, Bell was actually pretty convinced that he was right. And so he popularized this puzzle by writing a chapter in his book called how to teach special relativity. In there he explained in excruciating detail, down to the level of atoms and electromagnetic fields, why the string should snap. And today, the consensus is that Bell was right. The string indeed snaps. That's why we call this today as the Bell's spaceship paradox.
But why? Why does length contraction work this way and not this way? And what does it look like from the perspective of the people inside the ship? And most importantly, how can we figure this all out ourselves? Well, at first I thought I should look at Bell's original argument, but like I said, it's way too technical. It defeats the purpose of gaining some intuition. A breakthrough came for me when I went through the book Relativity Visualized by Lewis Epstein. Oh my god, that totally blew my mind.
So Epstein, why does length contraction work this way and not this way? And Epstein says, "Mahesh, what exactly is length contraction according to you?" Well, moving things shorter compared to when they're at rest. And Epstein says, "No, they are not. That's a misconception." I'm like, "What?"
Look at this pen. This is how long it looks right now. But what if I rotate it? It looks shorter. Not because it actually got shorter. The pen did not get shorter. It just looks shorter because now you're looking at it from an angle. So, when you're looking at this pen directly, the length that you measure is exactly the same as its true length. But watch what happens as the pen rotates. The measured length gets shorter. Not because the pen actually got shorter. It didn't. Its true length is still the same. But because you're now measuring it from an angle. It's all about geometry.
So, the key lesson over here is that we were never measuring the pencil's length in the first place. All we are measuring is its projection onto our axis. That's why this is also called the coordinate length. And the pencil's true length, we also call it the proper length. So, look. The proper length didn't change. The pencil did not get shorter. It just rotated and that's why the measured length got shorter. This is length contraction.
This sounds incredibly amazing. But Epstein, there's a problem. Our space shuttle is not rotating. It's accelerating, right? So, how do you explain what's going on over here? And Epstein says, "But Mahesh, acceleration is rotation. Not in space, but in space-time."
When a spaceship is at rest in space, it's still moving through its own time. Its proper time. And this speed with which it moves through space-time is the speed of light. In fact, everything in this universe, including you, me, donkeys, we all move through space-time at only one speed, the speed of light. It's just that when we are at rest, all of that speed is directed through time. That's why the spatial velocity is zero and our time ticks at full speed. But what happens when the space shuttle accelerates? Well, as it accelerates, the speed through space-time cannot change. It should still be the same, but the direction does. The velocity vector tilts. And as a result, its spatial velocity increases.
So look, acceleration is rotation in space-time. And that's why when the spaceship is at rest, you measure its length head-on. So the measured length equals its proper length. But when it's moving, look, the length you measure becomes shorter than the proper length. Not because the ship physically shrank, not because it got shorter, but because it rotated. This is length contraction. So the faster it moves, the more it rotates and more the length contracts.
This is insane. So length contraction is not that moving things are getting shorter. No, no, no. Instead, it's just that moving things are rotated in space-time and so we're just measuring it from an angle. It's an artifact of measuring something from an angle. That's amazing.
But wait, I am not fully sold yet. Einstein, does the math work out over here? Turns out it actually works out beautifully. See, all we need to do is now find a connection between its true length and the measured length, right? How do we do that? Well, just complete the triangles and look, these two triangles are now similar. So we can say the measured length divided by the true length should equal this side divided by this side. Now, we usually take the speed of light to be one in relativity, so let's call it one. Then, what exactly is this side? Well, from Pythagoras, we can say that is the square root of 1 minus v squared. And so, that's where that root of 1 minus v squared comes in the length contraction formula. Isn't it amazing? We just derived it in a few steps, and isn't this so elegant? The math works out. Wow.
But, wait. I'm still not convinced yet. I have one last critique, okay? When the pencil rotates, look, in the horizontal, its length contracts, but the vertical length increases, right? So, shouldn't the same thing happen to our space shuttle? As it rotates, the spatial length contracted, but shouldn't the temporal length increase? What does this temporal length signify?
Epstein says, "Well, in the ship's frame, two perfectly synced clocks stay in sync even as the ship accelerates. So, the temporal distance between the clocks in the ship's frame is zero. But, in our frame, they go out of sync. The clock at the back will be ahead in proper time compared to the clock at the front. The temporal length equals how much the clocks are out of sync. So, look, as the ship accelerates, the spatial length decreases, but the clocks go more and more out of sync, so indeed the temporal length increases.
Another way to think about this is that in the ship's frame, 1 second in the front is simultaneous with 1 second at the back. But, in our frame, 1 second in the front of the ship is simultaneous with 2 seconds in the back. So, this shows that simultaneity is relative. That's amazing. We know that to be true, but now we can see it in another way. That's amazing.
But now this brings up the biggest problem with Bell's original explanation. Bell says, as the string and the ships accelerate, they try to contract. However, the string's ends are tied, therefore it can't contract, and that produces the stress, and that snaps the string. But we just saw length contraction is just an artifact of measuring something from an angle. It's just geometry. So, it cannot cause stress. And that's what makes this puzzle so puzzling. Bell's original explanation is wrong.
I mean, just to be clear that we're on the same page, the string does snap. There is stress. That part is true. But saying that length contraction causes the string to snap, that is wrong, in my opinion. Because length contraction is just geometry, nothing more. And that's why I think even the most popular explanations on YouTube today about this topic is misleading, because they too, like Bell, attribute that stress to length contraction in our frame.
So, if length contraction is not the cause for the stress, what is actually generating it? What truly makes the string snap? And Epstein says the answer has been in front of us all along. It's acceleration. But how? I mean, both the ships are accelerating at exactly the same rate. How does that produce stress in the string?
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So again, if both the ends of the string are accelerating at exactly the same rate, how exactly does that produce stress in the string? And Epstein says, "Mahesh, for a moment, forget about both the shuttles. Just consider a single space shuttle." His question is, "If we were to keep a couple of accelerometers, remember an accelerometer measures acceleration using a physical effect like a spring compressing. The greater the acceleration, the more the spring compresses. So, if you place one at the back and one at the front, will they read the same value or different values?"
Well, my instinct says since the whole ship is accelerating together, they should read the same value. Both of the accelerations must be exactly the same. The whole ship should have the same acceleration value. But, if we remember that acceleration is rotation in four-dimensional space-time, and remember that when something rotates like a door, the outer edge moves faster than the inner one, right? So, the same thing must happen here. The back should be moving faster than the front. So, the back of the ship should at any given moment have a higher speed compared to the front of the ship, which means the back of the ship should gain velocity quicker than the front, which means the back of the ship should have a higher acceleration compared to the front.
Oh, wait. That doesn't sound right. So, let's think a little bit about this. At any given moment, the back of the ship is having more speed compared to the front of the ship. And that explains why the measured length contracts. So, that makes sense. Okay. But, I'm thinking if the back of the ship is literally moving faster than the front, shouldn't the ship literally shrink? Like, shouldn't it physically shrink? Why doesn't that happen?
Well, that's because remember what's simultaneous for us is not simultaneous for them. Now, for us at any given moment, the back of the ship is moving faster than the front. But, if I consider a particular moment in their frame, look, the back and the front have the same velocity. Again, if I consider a particular moment in their frame, the back has the same velocity as the front. Oh my god, this is so wild.
So, for a ship to maintain its rigidity as it accelerates, the back should indeed have a higher acceleration compared to the front. This is so counterintuitive. Of course, it makes sense if you think about acceleration as rotations, but still, man.
Okay. So, the now obvious question could be what if the back and the front had exactly the same acceleration? What would happen now? Well, now, at any given moment in our frame, the front and the back would have the same speed. That means the measured length would remain the same. It wouldn't contract. But, how can that be? How can the measured length stay the same while the shuttle is rotating in space-time? It cannot unless the shuttle physically stretches as it rotates.
Oh my god. Now we see the full picture. If the front of the ship has a smaller acceleration compared to the back, we might expect the ship to physically shrink, but it doesn't. Its proper length stays the same. Of course, the measured length gets shorter, but that's just an artifact of the whole thing rotating. So, there'll be no stresses. There'll be no shrinking. None of that is happening.
In contrast, if the front and the back of the ship had the same acceleration, then we might expect the proper length of the ship to stay put, but it doesn't. In fact, the proper length increases, and there will now be stresses generated inside. Of course, the measured length stays the same because as the ship is rotating, it's also stretching. This is what's going on. So, this is the true key to the puzzle.
So, moment of truth, folks. Now that we understand exactly how acceleration works in space-time, can you pause the video and solve the Bell's spaceship paradox once and for all? Pause and bring it home. Give it a try.
All right, here we go. Since both the ships start accelerating at exactly the same rate and exactly the same time, the ends of the string have the same acceleration. Therefore, the proper length of the string and the proper distance between the ships increase. This is what's truly happening. And remember, this statement that we have made is an absolute fact. We're not talking about measured lengths, which are dependent on reference frames anymore. We are talking about proper length and proper distances, which is an objective fact that everybody agrees on. So, this is truly what's going on. And so, as the string is being pulled, that produces the stress.
But woah woah woah, wait a second. Wait a second. One last question we could be having is, shouldn't the same thing happen to the spaceship itself? Even for the spaceship though, accelerations are the same to begin with, probably. And so, shouldn't there also be stresses generated inside? Yes, it happens. But we are assuming that the material that makes up the spaceship itself is pretty strong and it can handle that stress. And so, look, as the stress pulls on the ends, the acceleration of the front will decrease slightly and the acceleration of the back will increase slightly. Eventually, the ship settles into the correct acceleration gradient needed, and when that happens, the stress stops further increasing. And from now onwards, the ship maintains its proper length and continues accelerating rigidly. But of course, we're assuming the string to be quite delicate because it's a string. So, it won't be able to handle that stress, and that's why it snaps.
But of course, what would we measure in our frame? Well, as the ships and the string accelerate, the proper length and the proper distance is increasing. Remember, that's an objective fact. But because it's also rotating as it stretches, look, the measured length stays the same. And that's why this stress generated seems mysterious because we don't measure a change in the length, but its proper length is increasing. So, the mystery disappears once we realize that the proper length and the proper distance between the ship has been increasing all this while. And so, the string is literally being stretched, and that's why eventually the whole thing snaps. Paradox resolved.
But now, let's take this to the next level. Ready? Suppose we have a bunch of ships at rest, and we want them to start accelerating at the same time in such a way that the proper distance between them stays a constant. In other words, each ship should always see the others at rest relative to itself. How should they all accelerate? Again, our instincts might suggest that they should all have the same acceleration, but now we know that's wrong. The ships in the front must have a smaller acceleration than the ships behind.
But wait, if all of them are at rest relative to each other, could they tell whether they are accelerating in empty flat space-time or hovering near a black hole? No. Because near a black hole, the situation is quite similar. The ship closer to the horizon must fire its engines harder than the ones farther away. So, these two situations are pretty much identical. That means we can use accelerated observers in flat space-time to model black holes. And when you do that, a beautiful space-time diagram emerges. One that lets us predict the true physics of black holes. And when you zoom out far enough, you discover the Penrose diagram. All of that in the next video. So, stay tuned.
