What’s a Singularity?

Well we say someone is singular when they are… one.

A single person.

And not just any individual, someone odd.

But that’s not quite right.

A singularity is more like the answer to a question for which there is no answer.

Confusing?

Here.

Take this very simple function.

When x is 1, what’s the value of f(x)?

OK, let’s substitute.

1 over 1 is… 1.

Now what happens when x is zero?

Well that’s easy.

Substitute, and…

The function has no solution, because it doesn’t make any sense,

because 1 divided by 0 doesn’t make any sense.

Unless you use a Riemann sphere in the complex plane which allows for division by zero in

some circumstances, in a way that makes expressions such as this one well-behaved.

The point is that these are just constructs.

They don’t exist in the real world.

They are more like ideas.

See, when we built mathematics, we decided the rules, and according to those rules this

equation doesn’t make any sense.

It has no answer.

So what’s the problem?

It all goes backs to this guy.

Einstein’s field equations.

You can easily derive their exact solution

for a single spherical non-rotating mass if your name is Karl Schwarzschild and you’re

a genius physicist with a badass mustache

and then you end up with this, simple-looking formula.

Let’s unpack this.

c is the speed of light G is the gravitational constant

M is the mass of the object and rs is the Schwarzschild radius

c and G are constants.

Say you have an ball of mass M.

If you squeeze it, the mass stays the same but its volume gets smaller.

If you keep squeezing the ball, it will become tiny, but stay just as massive as before,

all the molecules compressed close to each other.

Until it reaches a critical size.

And that’s the Schwarzschild radius.

It is the radius of a sphere such that, if all the mass of an object were to be compressed

within that sphere, the escape velocity from the surface of the sphere would equal the

speed of light.

This means that in order to escape the gravitational pull of the object, whatever tries to get

away from it would have to move faster than the speed of light.

But because the speed of light is constant and nothing can ever go faster, nothing can

escape.

Including light.

It’s essentially a black pit where anything you put in doesn’t come back.

And that’s what we call the congressional budget.

I mean, a black hole.

Because the speed of light is such a gargantuan number.

Its square is f**king gianormous.

Which means unless you have an unbelievably big mass, the Schwarzschild radius is very

tiny.

Take the Earth, which is pretty massive.

It has a radius of over 6,000 km.

Its Schwarzschild radius is only 8mm.

I don’t think I can even begin to explain the scale of things.

It would be like turning this gigantic thing where all of humanity lives cozily and people

take years to travel around by foot.

Into something the size of a penny.

Let’s go back to our Schwarzschild radius.

If the object’s radius is less than that, it becomes a black hole.

But what happens at the center?

In other words, what happens if the object's radius IS zero?

This equation defines the shape of spacetime near a spherical mass.

Now you don’t need to understand all of it.

Just have a quick look and see if you can spot the problem.

Congratulations, you just broke physics.

By creating a singularity.

An imaginary point in spacetime where all the laws we know don’t make any sense.

Now let’s make something clear.

The black hole isn’t the singularity.

This is the event horizon, sort of a “beyond this none shall return” threshold.

That’s the black hole.

And in the middle it's where you divided by zero.

The singularity isn’t really a place, because in theory it occupies no space, and it has

infinite density.

It’s like the concept of point in geometry.

It’s hypothetical.

Except we have experimentally discovered black holes, exactly how Einstein and Schwarzschild

had predicted, and there is reason to believe that singularities do exists.

There is copious amount of speculation on what might happen at a singularity, including

wormholes, time travel, and portals to other universes.

If you’re curious about that please let me know in the comments.

We just don’t know for sure what happens at a singularity, and probably never will.

At least not until we get to “THE SINGULARITY”, which I’ll explain in a later video.

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Thank you for being curious.