Behind everything that has ever happened, is happening, or ever will happen, there is a gradient.
We are slaves to the gradient.
This episode was made possible by generous supporters on Patreon.
Ok, crazies.
Before we get too nihilistic here, let's ask a proper question:
Um, what’s a gradient?
Yes! This is going to be fun.
Let’s start with the definition.
The gradient represents the greatest increase in a number across space.
Hmm, greatest increase, what can we do with this.
Say you’re standing on a hill.
You could move north, south, east, west, or some combination those.
At each location, you have an elevation measured vertically from the bottom of the hill.
So, now we've got a bunch of numbers assigned to different points in space.
Wait a minute!
That's what we call a field.
A number or set of numbers assigned to each point in space.
In this case, the number is elevation so we'll call this an elevation field.
Any direction you move on the hill will change that elevation.
The direction of the gradient is the direction you can walk
that will give you the quickest change in elevation.
But this gradient also depends on location, so it's also a field.
Just like the elevation is a number assigned to each point in space,
the gradient is an arrow assigned to each point in space.
The direction of each of those arrows is the direction of greatest increase in elevation
and their size tells us how quickly that increase is happening.
At this point though, some of you might be wondering:
But, like why does any of this matter?
Because it tells us what's going to happen.
Elevation isn't the only number we assign to points in space.
Say you've got a tank full of water.
Let's imagine the water is made of very tiny water cubes.
Each of those cubes has a pressure exerted on it from the surrounding cubes.
We'll measure that pressure as a number and assign it to the cube
and we'll do the same for all the other cubes in the tank.
This is something we might call a pressure field.
Just like we did with elevation, we have a pressure assigned to points in space.
It's not perfectly uniform throughout the tank though.
It's higher at the bottom than it is at the top.
That's what we call a pressure gradient
and it points in the direction of greatest increase.
In this case, straight downward.
That gradient is important because pressure can control what happens.
The gradient of a pressure is a force!
Actually, when you work out the units, you get a force density not a force.
Close enough!
Also, force isn't the positive gradient.
It's the negative gradient.
Remind me again why you had to come back from vacation.
I missed you.
Ok, ok, back to the tank!
Say we put something else into the tank, like a bowling ball.
It's going to experience the pressures from the surrounding water cubes,
but those pressures are larger on the bottom than on the top.
This results in an upward force
per unit volume.
Much better.
That bowling ball is displacing a particular volume of water.
The pressure gradient across that volume gives us a force:
Something we call a buoyant force.
If that buoyant force is strong enough to overpower gravity,
the bowling ball will float.
Yes, some bowling balls float.
The heavier ones sink, but the lighter ones can float
if they're under 12 pounds or so.
The reason that happens is because of a pressure gradient.
For that matter, it's why a lot of things happen.
It's what allows you to breathe.
It's what creates wind.
And it’s why a portal between Denver and New York would be a very bad idea.
But that bowling ball example had another force in it:
Gravity, which is also the result of a gradient.
An energy gradient.
Technically speaking, a pressure is an energy density,
so a pressure gradient is an energy gradient.
I'm not sure that insight is helpful here, but thanks?
So energy gradients.
At every possible location inside this tank,
the bowling ball has an energy given to it by the Earth.
Those energies increase as you move up in the tank,
so the energy gradient is straight upward.
This gradient causes a force and, as Nerd Clone mentioned,
that force will be the negative or opposite of the gradient.
It'll point downward which is exactly what you’d expect for gravity.
Mathematically, we'd write the gradient like this:
The change in energy over the distance in the upward direction.
To get the force of gravity, we just take the negative or opposite of that.
That means gravity points in the direction of greatest decrease in energy, which is downward.
This is a really simple example though.
The energy only changes in one direction.
In some other example, it might change in two directions or even three.
We do live in a three dimensional space, after all.
If we add a term for each of the other directions, we get this.
Of course, that looks kind of nasty,
so we relabel it with a single symbol:
The Nabla.
It's an old Greek word for harp apparently.
I guess it kinda looks like a harp,
but it's really just an upside down delta for change.
A change in some number across space.
Lots of measurements can be different in different places
and those differences are what cause things to happen.
Weather patterns on the Earth are caused by pressure and temperature gradients.
Buoyancy is caused a pressure gradient.
Newtonian Gravity is caused an energy gradient.
Yeah yeah, I know, gravity is really the curvature of spacetime, but there's still a gradient.
It's just a little beyond this video.
Concentration gradients are a way cells exchange molecules with their surroundings.
We call it passive transport.
Fluid motion? Gradient.
Brain function? Gradient.
Electromagnetic interactions? Gradient!
The eventual heat death of the universe is literally the end of all gradients.
If there's no more gradients, that's it.
That's the end.
Nothing will ever happen again ever.
Probably.
Behind everything that has ever happened, is happening, or ever will happen, there is a gradient.
We are slaves to the gradient.
All hail the gradient!
So how do you feel about this?
Are you ok being a slave to the gradient?
Or do you think we have the ability to rise above it?
Please share your thoughts in the comments.
Thanks for liking and sharing this video.
A special thanks goes out to Patreon patrons like Ken Davis
who help keep this show going with their generous support.
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.
To everyone who commented about perpetual motion machines:
No, those don't suddenly become possible.
Even though more dark energy is being created over time,
it remains perfectly uniform throughout space,
which means there's no gradient,
which means we can't use it.
Anyway, thanks for watching.