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 solution is the real
solution is truly elegant, but because
of its farreaching 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 slower than that of light
which you know whose speed we usually
choose to be one in relativity we just
don't notice this effect at all. But of
course close to speed of light it's
huge. For example if the 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 ship's perspective, they're always
at rest relative to each other. So there
is no wink 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, Belle
says, "We decided to appeal to the CERN
theory division 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 Belle is wrong.
Right? Well, Belell 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 Belle was
right. The string indeed snaps. That's
why we call this today as the Bell's
spaceship paradox.
But why? Why does link 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 do how can we figure
this all out ourselves? Well, at first I
thought I should look at Belle'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 Louis Epstein. Oh my god, that
totally blew my mind.
So Epstein, why does link contraction
work this way and not this way? And
Epstein says, Mahesh, what exactly is
link contraction according to you? Well,
moving things are shorter compared to
when they're at rest. And Epson 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
at 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
Epson 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 spaceime 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 takes at full speed. But
what happens when this space shuttle
accelerates?
Well, as it accelerates, the speed
through spaceime 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
spaceime. And that's why when the
spaceship is at rest, you measure its
length headon. 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, you know, 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
spaceime 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. Epstein, 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 divide by
the true length should equal this side
by this side. Now, we usually take the
speed of light to be one in relativity.
So, let's call that one. Then, what
exactly is this side? Well, from
Pythagoras, we can say that is the
square root of 1 - v².
And so that's where that root of 1 - v
square comes in the link 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?
Epson 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 two
seconds in the back. So this shows that
simultaneity is relative. That's
amazing. We know that is to be true. But
now we can see it in another way. That's
amazing. But now this brings up the
biggest problem with Belle's original
explanation.
Belle says as the string and the ships
accelerate, they try to contract.
However, the strings ends are tied.
Therefore, it can't contract. and that
produces the stress and that snaps the
string.
But we just saw link 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.
Belle's original explanation is wrong. I
mean just to be clear that we are 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. H 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 wing
contraction in our frame. So if length
contraction is not the cause for this
stress, what is actually generating it?
What truly makes the string snap? And
Epston 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? It's kind
Okay, so happy learning. Now, let's get back to the video. 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
Epson says, Mahes, for a moment, forget
about both the shuttles. Just consider a
single space shuttle accelerating very
close to the speed of light. 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 the the behind 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.
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 contract. 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 spaceime? 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 will be no stresses.
There will 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 this, you know, 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.
Oh my god.
This is 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 spaceime, 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 whoa whoa whoa wait a second wait a
second. One last question we could be
having is shouldn't the same thing
happen to the space shuttle itself? Even
for the space shuttle the accelerations
are the same to begin with probably. And
so shouldn't there also be stresses
generated inside?
Yes, it happens. But we're assuming that
the material that makes up the space
shuttle 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 spacetime 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 accelerate
observers in flat spacetime to model
black holes. And when you do that, a
beautiful spac-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.