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Hey Crazies.
There's a lot of hype over grand unified theories and theories of everything,
but no one ever seems to be asking the following question:
Why are we even trying to unify everything?
To answer this, we need to take a look at the original unified theory:
The inverse square law!
Yes, I'm aware that many of you may have heard of it before,
but barely anyone ever explains why it's such a big deal.
So why is it such a big deal?
It's a pattern that appears over and over again.
And it all started in 1604.
To the timeline!
There was a guy by the name of Johannes Kepler, who you probably know for his laws of planetary motion,
which he published in 1609 and 1619.
But, in 1604, he was proposing rules for light what he called "The Optical Part of Astronomy."
Those were simpler times in science.
Reading his actual words can be a bit confusing these days because, well, you know,
people talked differently back then, but here's the basic idea.
Say you've got a spherical source of light, maybe a light bulb.
Kepler's interest was the Sun, which is also a spherical source of light,
but we're going to use light bulb.
Say you turn the bulb on and then immediately off again.
You've created a spherical shell of light that spreads equally in all directions.
Now, that shell only has so much energy in it.
As the shell grows, that energy has to spread thinner and thinner and thinner,
which makes the light dimmer and dimmer and dimmer.
It's actually pretty relevant for studio lighting,
which is why half the videos that come up when you search “inverse square law”
are lighting tutorials.
Anyway, Kepler used the idea of this spherical shell
to come up with a rule for how much the light dims as it moves outward.
Since the light was spread over the area of the shell and that area is 4 pi r-squared,
the brightness of the light should drop off like one over r-squared,
the inverse square of the distance.
That's why it’s called an inverse square law.
If you double the distance, you quadruple the area.
If you triple the distance, you nonuple the area, and so on.
The bigger the area, the dimmer the light, so we flip those multiples over.
But this isn’t just limited to light.
It works for sound too and, frankly, anything we can imagine spreading out uniformly.
You might call it a unified law of spherical flows.
Didn’t you say this was supposed to be profound or something?
Yes, yes, it’s so much more than this!
Back to the timeline!
Several decades later, around 1665, Isaac Newton was studying motion
and thought up another inverse square law, this one for gravity.
It actually explained Kepler’s laws of planetary motion from the beginning of that century.
It’s called Newton’s law of gravity.
It should really be called the Newton-Hooke law of gravity,
but history can be cruel and lazy sometimes.
Anyway, notice the square in the bottom, the square of distance just like before.
Does that mean gravity flows away from a source?
Nope! It does nothing of the kind.
Then why is it an inverse square law?!
Exactly!
In Newtonian mechanics, gravity is a force.
Yes, we now know that its space-time curvature, but don’t get ahead of me in the timeline.
Newton's force of gravity is described by this law, an inverse square law.
If you double the distance between the Earth and, say, a satellite,
you divide the force between them by two-squared or four.
If you triple the distance, you divide the force by three-squared or nine, and so on.
Almost as if there are spherical shells involved.
I imagine this seemed pretty odd at the time,
then again maybe they didn't think about it that deeply,
at least not for another century.
In 1784, Charles Coulomb came up with another inverse square law.
This one was for electric charges and we now call it Coulomb’s law.
Actually, he was French, so his name is pronounced.
Stop. Stop right there.
I clearly can’t even pronounce German names.
I'm not even going to try French.
OK, but I would have liked to try.
Coulomb's force law looks like this.
It's another inverse square law.
They just keep appearing!
I mean, look at how similar these two laws are.
A constant out front, a property of two different objects on top
and the distance between those objects squared in the bottom.
It's almost like there's some fundamental reason.
Almost like you could unify them into a single law.
But that law wouldn't come until we see Carl Friedrich Gauss in 1835,
who gave us what we now call Gauss's law.
Keep in mind though, Gauss was a mathematician and we all know what they do.
They generalize things to death!
He wrote up his law using an abstract vector field.
A field is a value or set of values assigned to every point in space.
So a vector field is just vector assigned to every point in space,
a vector being a value and a direction.
Remember when we thought the inverse square pattern
was a unified theory of spherical flows?
Well, in 1835, Gauss realized it was actually pattern of 3-dimensional fields.
As long as that field has a spherical shape, what we'd call spherical symmetry,
then Gauss's law spits out an inverse square, which is a big deal!
Light moving away from a bulb?
Inverse square law!
Sound moving away from a speaker?
Inverse square law!
Even some spherical fluid flows obey inverse square laws.
In this case, the vector is velocity because it's about motion.
But velocity isn't the only vector that exists.
The gravitational field around the Earth is also a vector field.
It might not be the flow of anything, but it's still a vector field and it has spherical symmetry.
That means you get an inverse square law.
The same goes for the electric field around a charge.
It's not the flow of anything, but it's still a vector field with spherical symmetry.
That means you get an inverse square law.
But this poses a limitation.
Gauss's law tells us they're not really inverse square laws.
They're inverse area laws.
That r-squared is really a 4 pi r-squared.
These two laws should really be written with a 4 pi r-squared in the bottom.
We just clump the 4 pi together with the constant in front for simplicity.
Both of these laws require there be spherical symmetry.
If your fluid flow isn't spherical, there's no inverse square law.
If instead of a charge we had a charged rod, the symmetries are no longer spherical,
which means no more inverse square.
It's approximately an inverse square from a great distance, but not in the details.
So, as it turns out, while inverse square laws are a profoundly common pattern,
they aren't quite as profound as we'd hoped.
That didn't stop physicists from trying to unify things anyway.
In 1885, Oliver Heaviside unified electricity and magnetism
by pretty much inventing vector calculus.
But then in 1915, Einstein ruined unification by replacing
Newton's law of gravity with general relativity.
It had to be done, but it's unfortunate.
So why was the inverse square law such a big deal?
It's a big deal historically because it hinted that everything was connected.
Physicists got a taste and they just can't let it go.
Maybe we'll unify everything in the near future.
Maybe we won't.
Maybe relativistic gravity is so fundamentally unique; it can't be unified with anything.
One thing is for sure though.
Newtonian gravity and electricity are incredibly similar.
So similar that it's convenient to use gravity as an analogy for electricity,
which is exactly what we're going to do in some upcoming videos.
So, do you think we'll find a theory of everything some day?
Please discuss your opinion in the comments.
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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.
By far, the most common comment from the last video was:
How do gravitons escape a black hole?!
One!
Graviton models never work, hence the struggle.
Two!
Even if they found a model that did work, it would still have to consistent
with everything we know about black holes.
Nothing escapes a black hole!
Anyway, thanks for watching!