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Hey Crazies.
You’ve probably heard of length contraction and time dilation before.
It’s the idea that your length shrinks and your time slows the faster you move.
But, it’s not just an idea. It’s real. It’s measurable. We’ve tested it.
We’ll spend this entire video discussing how and why it happens.
And we’ll do it visually so it’s as intuitive as possible.
No ridiculous hypotheticals. No fluid analogies. No historical fluff. Just geometry.
Let’s do this!
This episode was made possible by generous supporters on Patreon.
Sure sure. Welcome to relativity,
the show where everything is made up and the measurements don’t matter.
Alright, stop right there.
This model is real! It’s been experimentally tested!
Atomic clocks up on the ISS. Particle accelerators. Muons from space.
Muons from Space!
The universe looks different to different observers.
And, actually, “looks” is an understatement.
The universe doesn’t just look different to different observers. It is different to different observers.
That's why we call it relativity.
Observers disagreeing about measurements doesn’t make those measurements fake,
it just makes them relative.
Say we’re looking out the window of my space-station at a couple of identical rockets.
One is at rest relative to us.
The other is being flown by Rocket Clone.
They’re both identical, but the moving one is shorter.
That’s length contraction.
So are they, like, physically compressed?
Well, not really.
Stress compression is when the atoms get closer together.
That’s not what I mean.
Length contraction is just when two observers disagree about how to measure length.
It’s not an optical illusion though. The rocket is shorter for one of the observers.
But rule number one of relativity says:
No frame of reference gets priority over any other frame of reference.
If Rocket Clone measures the length of their own rocket, it would appear normal.
For them, it’s my space-station that’s moving and, therefore, my space-station that’s length contracted.
Measurements like length are relative to the observer and so is time.
It would look to us as though Rocket Clone’s clock is ticking slower.
We call that kinetic time dilation, time dilation caused by motion.
According to Rocket Clone, it’s our clock that ticks slower.
The ideas in relativity are weird. I’ll give you that.
But they’re very real.
And they actually make a lot more sense if you just suck it up and do the geometry.
Doing dynamics in space is fine,
but you miss a lot of intuition that way.
Adding a time axis changes everything!
This is a spacetime diagram.
The rocket is still moving directly to the right in space.
That upward motion you’re seeing is only through time.
If you’ve seen a lot of my videos, you’ve seen diagrams like this before.
They’re very powerful.
Powerful!
Spacetime is what we’d call a hyperbolic geometry.
Anyone’s first exposure to hyperbolas is usually with conic sections.
It’s what you get when you slice a cone parallel to its height.
On an xy-plane, the most basic hyperbola would look something like this:
x-squared minus y-squared equals one.
If we include scaling factors for each direction, we can tweak it a little,
but the basic shape is always the same.
Any geometry that has that specific of pattern is called a hyperbolic geometry.
The Minkowski spacetime in special relativity is a hyperbolic geometry.
The x-axis is space and the y-axis is time.
This minus sign makes it hyperbolic.
In fact, these hyperbolas would be the paths of something accelerating.
Just remember, we’ve suppressed the other two space dimensions.
These accelerating rockets are only moving left and right, not up and down.
The up and down direction is time, not space.
You still with me?
Ok, let’s keep going.
Anything stationary follows a path that lines up with the time axis.
Anything moving steadily follows a straight path.
Anything accelerating follows a hyperbolic path because "hyperbolic geometry."
The axes are not absolute though.
These belong to our space-station, but these belong to Rocket Clone.
Notice how the time axis lines up with his path now.
Different observers can and will disagree about lots of measurements
because they take those measurements in their own frame of reference.
The angle any path makes with the time axis tells you the speed of that thing in that reference frame.
I repeat: Speeds are measured as angles.
Small angles are slow.
Large angles are fast.
Angles close to 45-degrees are near the speed of light.
Fast fast!
Wasn’t this video supposed to be about length contraction?
Yes yes, we’re almost there.
Every individual point on this diagram is called an event.
Grid lines that line up with the space axis are all the events that happen at the same time.
Grid lines that line up with the time axis are all the events that happen at the same place.
That last bit is going to be very important because,
before we can see how length contraction happens, we need to define “length” first.
Length is the spacetime separation between two events that occur at the same time.
What?!
Think about any length you’ve ever measured.
You put some kind of ruler or tape measure down and you look at the markings.
The reason that works is because you have two events,
one on each end of the ruler, and you’re measuring them simultaneously.
You do essentially the same thing if you're measuring the length of a rocket.
It’s just that not everyone agrees on which events happen at the same time.
Remember, grid lines that line up with the space axis are all the events that happen at the same time.
If two observers have different space axes, they’ll disagree on which events are connected.
Since the events that define length have to be simultaneous, they’ll disagree about length too.
And spacetime diagrams allow us to predict what those lengths will be.
If we measure the length of the rocket, we would use these two events
because, for us, those happen at the same time.
If Rocket Clone measures the length, he would use these two events
because, for him, those events happen at the same time.
You can see right away the lengths are different.
The real question is: "How different?"
Well, remember that speeds are measured as angles.
This angle is the relative speed between us and the rocket.
Do some quick trigonometry and BAM!
You’ve got yourself a ratio.
If we see Rocket Clone moving at 55% the speed of light, he might measure a length of 20 meters,
but we’ll measure 16.7 meters.
We see a contracted length.
Oh! Oh! What about time dilation?
The geometry works for that too.
A time interval is the spacetime separation between two events that occur at the same place.
Grid lines that line up with the time axis are all the events that fit that description.
If two observers have different time axes,
they’ll disagree on which events are connected.
Based on this definition of time intervals, that means they’ll disagree about clocks too.
So, let’s say my clone’s clock is at the center of his rocket.
We need two events. These two occur at the same place for him, but not for us.
We’ll need to use these two events.
Again, these separation are obviously different.
The question is: "How different?"
Using the relative speed and some trig. BAM! You’ve got a ratio.
If we see Rocket Clone moving at 55% the speed of light, he might measure a time interval of 1 minute.
but we’ll measure 1.2 minutes.
That’s kinetic time dilation.
So, you can see now that weird relativistic effects are just the result of geometry;
a hyperbolic geometry where speeds are measured as angles, but a geometry nonetheless.
If different observers disagree on what events happen at the same place
and what events happen at the same time,
then they’re going to disagree on just about everything.
They’ll disagree on how long a rocket is.
They’ll disagree on how much time has passed for that rocket.
But it all makes sense in the geometry of spacetime.
Frankly, it couldn’t be any other way.
So did that help you understand length contraction a little better? Or was it too deep?
Let us know in the comments.
Thanks for liking and sharing this video.
Don’t forget to subscribe if you’d like to keep up with us.
And until next time, remember, it’s ok to be a little crazy.
If you want to learn more about special relativity, Brilliant is a good place to start.
The first section of their course on special relativity has a quiz on length contraction and time dilation
where you’ll use the Lorentz transformation to solve problems.
Later in the course, you’ll get a chance to see the full power of spacetime diagrams in action.
It’s just one of over 60 interactive courses in math, science, and computer science that Brilliant has to offer.
Their thought-provoking content helps guide you to mastery
by taking complex concepts and breaking them up into bite-sized understandable chunks.
You'll start by having fun with their interactive explorations
and, over time, you'll be amazed at what you can accomplish.
If this sounds like a service you’d like to use, go to brilliant dot org slash Science Asylum today.
The first 200 subscribers will get 20% off an annual subscription.
In our last video, we talked about the CMB and Urbano Quiroz said:
Wait a minute, isn't orange the new black?
Nope!
Technically speaking, black is the new orange.
I’m such a dork.
Anyway, thanks for watching!