Thanks to Skillshare for helping support this episode.
Hey Crazies.
Consider the following scenario:
Two spaceships connected by a taut string are accelerating in exactly the same fashion.
What happens to the string? Does it snap?
If the acceleration is the same, then the distance between them should also remain the same.
So the string should not snap,
right?
But if they accelerate anywhere near the speed of light, length contraction happens.
So snap?
But the spaceships don't observe the length contraction and neither does the string.
So which is it? Does it snap or not?
We can solve this, but it's going to test the limits of special relativity.
[Intro Music]
So which is it then? Snapped or not? Oh, it definitely snaps.
How do you know?
This one took me longer to understand than I expected, so buckle up Crazies.
The problem we have here is called a "paradox."
According to Merriam-Webster,
that's an argument that apparently derives self-contradictory conclusions
by valid deductions from acceptable premises.
Okay, that's a lot of $10 words.
Essentially, if you start with ideas that you know to be true
and through different trains of thought, you can arrive at two opposing conclusions,
then you've got yourself a paradox.
Now, paradoxes aren't a problem with the universe.
They're a problem with us.
In other words, one of our premises is faulty.
We either have a preconception were unaware of
or we've stepped beyond the scope of the model.
This spaceship paradox is no exception,
though it's actually one of the more recent paradoxes in special relativity.
To the timeline!
Special Relativity was published in 1905.
General was ten years later.
Edmond Dewan and Michael Beran wouldn't start this paradox until 1959.
But, as you can see here, their solution wasn't very thorough.
This is the entire publication.
It did spark a lot of discussion though.
So Dewan revisited the topic in 1963,
where he considered it in tandem with the famous pole barn paradox.
The debate centered around material stress and how that should be treated relativistic.
In 1976, John Stewart Bell, of quantum entanglement fame,
finally fleshed out the nuances by adding a third observer
to represent the previously abstract inertial reference frame.
We now refer to this as "Bell's spaceship paradox."
I'm not sure why it ended up named after him instead of Dewan and Beran.
Maybe because he was more famous? I don't know.
Anyway, let's work out some of these nuances.
Say I've made three clones: Arthur, Bernard, and Charles.
Arthur and Bernard's rockets are connected by a string exactly the length of the gap between them.
If both rockets begin accelerating at exactly the same moment, at exactly the same rate,
then what happens to the string?
To be clear, all observers should agree on whether or not the string snaps.
Observers can disagree about when events occur,
how far apart they are, and even sometimes their order.
But they must all agree on which events occur.
Since Charles is stationary, he would observe relativistic length contraction.
The rockets contract in length,
but the string can't because it's attached to the rockets.
Therefore, the string snaps under the stress.
So paradox solved? Bingo Bango?
Not so fast. We haven't considered the other two observers yet.
From the point of view of Arthur and Bernard, the string should be stationary.
That means no length contraction, which means no snapping of the string.
And there is our contradiction, our paradox.
The string can't be both broken and unbroken.
All observers must agree on which events occur.
The string either snaps for everyone
or it doesn't... for everyone.
Charles represents what we call an "inertial frame of reference."
For special relativity, "inertial" just means stationary or moving steadily.
In a space time diagram, those are always straight paths.
This means we can trust our conclusions about what Charles sees.
He observes length contraction, so the string snaps.
I suppose you could stop there.
It's the only obvious inertial reference frame in this scenario
and we know all observers must agree on which events occur.
So we know Arthur and Bernard will also see the string snap.
But where's the fun in that?
Besides, this is the Sience Asylum.
If we're going to call this paradox solved,
I think our aspirations should be a bit higher than sufficient.
We need to ask a deeper question.
It may be true that the string also breaks for both Arthur and Bernard,
but why?
They don't observe the length contraction, so what's the cause of the snap?
Well, as I said before, there are two possibilities.
We either have a preconception were unaware of
or we've stepped beyond the scope of the model.
It's the second one, isn't it? Huh?
Special relativity can't handle accelerations, right?
Actually, no. That's a common misconception.
Special relativity can handle acceleration just fine.
The equations don't look particularly elegant, but they exist and they're valid.
It's accelerated frames of reference that are problematic, but even those are workable
if we're sneaky about it.
In a spacetime diagram, accelerated paths are hyperbolas
because flat spacetime is hyperbolic.
Switching between inertial frames of reference is easy.
You just rotate the axes.
To shift to an accelerated frame of reference, you'd have to curve the axes,
which officially jumps out of special and into general relativity.
We don't want to do that if we don't have to.
General relativity is nasty.
What we're going to do instead is imagine this hyperbola is just a bunch of little straight paths.
If there are enough of those tiny segments, we can't tell the difference.
It's mathematically equivalent to the hyperbola.
Now, if that sounds too much like math hand-waving to you,
there is a more tangible way to visualize this.
Imagine there's a big set of inertial pilots all watching an accelerated rocket.
Each of these new ships will be traveling at a steady speed in a straight line.
No acceleration.
They're just positioned in such a way
that they're going the exact same speed as the accelerating ship
when that accelerating ship catches up with them.
We call these new reference frames momentarily comoving inertial frames
or MCIFs.
They're momentarily in the same frame of reference as the accelerating ship.
It's the same as splitting the hyperbola into a bunch of tiny straight segments,
each segment being one of those MCIF's.
The point is, special relativity can handle accelerated reference frames just fine
if you're careful.
There's no need to invoke general relativity.
We haven't stepped beyond the scope of our model,
so we must have a hidden preconception.
What preconceived notion do we have?
It has to do with the distance between the rockets.
It changes? It changes.
When we first described the scenario,
we said the rockets must begin accelerating at the same moment
and must have the same acceleration rate.
At the beginning, everyone is stationary relative to each other,
so they're all in the same frame of reference.
Having the rockets start at the same moment isn't the problem.
It's the acceleration rate that's the problem.
There's only one observer that measures those rates to be the same:
Charles.
For him, the spacetime paths or "worldlines" for Arthur and Bernard look like this.
Since they accelerate at the same rate, their paths have the same shape.
The rockets maintain a steady distance from each other while contracting in length.
But the string is attached to both ships, so it can't contract even though it's trying to.
The extra tension makes it snap.
It also snaps for Arthur and Bernard, but for an entirely different reason:
because their rockets move apart.
You always measure length or distance parallel with your own space axis.
Hence these distance measurements Charles made from his own frame of reference.
But Bernard's axes change over time.
If his space axis changes, then how he measures distance also changes.
Bernard sees Arthur accelerate too quickly, which pulls on the string making it snap.
Arthur sees something similar.
In his frame, Bernard is lagging behind.
He's accelerating too slowly, which pulls on the string making it snap.
There are three frames of reference and the string snaps in all three of them.
It just happens for different reasons.
But... when does it break? Ooooh, now we're cooking with gas!
(Laughing)
Yet another assumption we've made here is that the string is incredibly weak.
It has no effect on how the rockets move.
Technically, this string breaks immediately.
You could easily choose something stronger, like a thick metal rod or something.
But the stronger it is, the more it will affect how the rockets move.
If it's strong enough, the rockets won't even be independent anymore.
Arthur will just be towing Bernard.
They'll act as one object.
So does the string brake, given our initial setup?
Yes, in every frame of reference.
For Charles, the string breaks due to length contraction.
For Arthur and Bernard, the string breaks because the distance increases.
Could we make the attachment strong enough so that it doesn't ever break?
Sure, but then you've changed the scenario.
A different scenario means a potentially different outcome.
When you're working with special relativity, you've got to be careful.
Your personal instincts are your own worst enemy.
They can lead you down the road toward unrealistic paradoxes.
And, if you wander too far, you might end up answering a different question than you wanted.
Speaking of questions, a lot of people ask me what I use to make my animations.
It's almost exclusively After Effects,
which you could learn how to use by joining Skillshare.
Skillshare is an online learning community with thousands of inspiring classes
for anyone who wants to explore their creativity or learn new skills.
Many of my animations aren't even really animations. They're simulations.
I code physics directly into After Effects using what they call "expressions."
Skillshare has a class on this called Advanced Adobe After Effects by Jake Bartlett.
I watched this class myself recently and it was great.
It'll walk you through how to make a fancy digital name badge like this,
which honestly is the perfect launching point for learning After Effects expressions.
But, if that's not for you,
Skillshare also has classes on photography, graphic design, productivity, and more.
You should be able to find classes that match your goals and interests.
If that sounds like something you might be interested in, check out Skillshare.
The first 1000 people to use my special link will get a one month free trial.
It'll also let Skillshare know you heard about them for me, which helps out the channel.
So are you satisfied with my solution to the paradox?
Please share in the comments.
Thanks for liking and sharing this video.
A special thanks goes out to all my Patreon patrons and YouTube members.
Don't forget to subscribe if you'd like to keep up with us.
And until next time, remember, it's okay to be a little crazy.
A couple of you thought it was really cool that the universe still surprises us,
and I completely agree.
It could just mean that our technology is limited,
but it could also mean that we're going to learn something new.
Personally, I'm excited to see what the James Webb Space Telescope has to offer.
Anyway, thanks for watching.