The singularity.

The point of infinite density at the core of a black hole, but also so much more.

In mathematics, singularities come in wild and wonderful varieties.

The black hole itself contains more than one.

[MUSIC PLAYING] 10 00:00:22,370 --> 00:00:24,950 Isaac Newton's universal law of gravitation was an incredible insight when he figured it out in the late 1600s.

In fact, we still use it to fly spacecraft around the solar system today.

However, it has its problems.

Let's look at the math.

Newton's equation gives you the gravitational force exerted between two masses, m1 and m2 that are a distance R apart.

Straightforward enough, but that R squared in the denominator spells trouble.

It means the force gets larger the closer the masses are to each other.

That makes sense, but what about when R gets really close to 0?

Then the result of the equation, the force, becomes extremely large and is infinite when R becomes equal to 0.

That doesn't really make a lot of sense.

Infinite force means infinite acceleration, which means-- well, physics breaks.

According to Newton's law, in order to fuel that infinite gravitational acceleration you need to get zero distance from an object's center of mass.

That means all of that object's mass would need to be concentrated at that center, a single point of 0 size, which means infinite density.

And that, of course, would make it a black hole.

We often use the word singularity to describe the hypothetically infinitely dense core of a black hole, but in math the meaning of this word is much more general.

You know what?

Instead of me trying to explain mathematical singularities, how about we get a real mathematician to do this properly.

Guys, meet Kelsey Houston Edwards of the new PBS show, "Infinite Series."

Hey, Kelsey.

Hey, Matt.

Thanks for having me on.

Kelsey, the math for black holes goes to infinity for different properties and in different locations.

What does this mathematical weirdness tell us?

Well, mathematicians use the word singularity pretty broadly.

It's really just any point that causes problems.

Commonly these problematic points are where quantities become bigger and bigger approaching infinity as they do near a black hole.

Some singularities come about from your choice of reference frame or coordinate system.

An example of a frame dependent singularity that might be familiar to space time viewers is the event horizon of the black hole.

I'll leave that to you to explain.

Here on earth, the North and South pole are examples of coordinate singularities.

It's possible to pass their time zones infinitely quickly, but only because of your choice of spherical coordinates.

All right.

That makes sense.

But the gravitational singularity at the center of a black hole is a so-called real singularity, right?

I mean, the curvature and the density are infinite from any frame of reference.

Right, and there's no way to avoid a horrible crushing death just by switching coordinate systems.

But the reality of the black hole singularity may give reason to doubt the theory that predicts such a thing.

In fact, it's happened many times before.

From models of the movement of water to human population growth, mathematics predicts a physical singularity, and we've been forced to reject the corresponding theory.

So you're saying Einstein is wrong?

Blasphemy.

Actually, Einstein himself agreed on this point.

Guys, you should check out Kelsey's show "infinite Series," where she goes into much more depth on the nature of singularities.

It's a math show, by the way, so it's sometimes about real stuff.

Mathematicians are lucky.

Being limited by reality is so boring.

So, does the fact that it includes a singularity mean there's something fundamentally wrong with Newton's law of gravitation?

Well, we already know the law isn't really so universal.

When the gravitational field is too strong-- say, near a star or a black hole-- Newton's law gives the wrong answers, and we need Einstein's general theory of relativity, which is the far more complete theory of gravity.

So does general relativity rid us of Newton's pesky singularity?

No.

In fact, it gives us even more singularities.

To understand this, we need to look at something called the Schwarzschild metric.

It's what you get when you solve the delightfully complicated Einstein field equations for the simple case of a spherically symmetric mass in an otherwise empty universe.

We're going to simplify it to only allow movement directly towards or away from our massive object.

In that case, it looks like this.

OK, that sure is some math.

Hey, this is "Space Time."

We can deal.

Actually, it's really easy to see the singularities in this equation, but let me first walk you through what it tells us.

The Schwarzschild metric allows us to compare two points or events in space time around a massive object from the perspective of different observers.

For example, a short space time path of some object, so it's world line, might move an object a distance delta r over a short time-step delta t. That motion is towards or away from the massive object, which is a distance r away.

That delta s squared thing is the space time interval, and it's a strange and interesting quantity.

Every inertial, so non-accelerating observer, will agree on the same space time interval for every pair of events and for every world line.

We talk about this in a lot more detail in our relativity playlist.

Today, we going to keep it simple.

As long as our objects world line doesn't require faster than light motion, then the square root of the space time interval is equal to the amount of time that the object itself feels over that interval.

We call that the object's proper time.

Oh, and r subscript s is a measure of the mass of the massive object.

In fact, it's 2 times the gravitational constant times the mass.

There would have been some speed of lights through the equation, but we set them equal to 1 because we're that cool.

Now the first thing to notice is that the singularity is still present in the Schwarzschild metric.

r, the distance to the center of mass, remains in the denominator just as it was in Newton's law.

When you use the Schwarzschild metric to calculate the curvature at r equals 0, that curvature is infinite.

This gives us the same infinite gravitational pull as the Newtonian singularity.

And just as with the Newtonian case, this gravitational singularity can only exist if infinite densities are possible.

But unlike Newton's laws of gravity, the Schwarzschild metric actually tells us whether or not that infinite density is expected.

To see how, we need to look at the second singularity in this equation, a singularity that Newton's law does not contain.

See, when distance to the center of mass is exactly equal to this rs thing, then rs over r is equal to 1.

At that point, the entire equation starts behaving very badly.

It's as much a mathematical singularity as the one in the center of the black hole.

If you haven't guessed, this bad behavior corresponds to the event horizon.

An rs is the Schwarzschild radius.

Imagine an object sitting at the event horizon but not moving, so its delta r would be 0.

But this bracket is 0 also, because 1 minus 1.

The entire space time interval for a non-moving point at the event horizon is 0.

But remember, for sub-light speed world lines, the space time interval tells us the rate of flow of proper time.

So does that mean time doesn't pass for an object hovering at the event horizon?

Not quite.

Time certainly doesn't pass at the event horizon.

No clock ticks can't ever happen there.

But the prohibition against objects experiencing time at the event horizon is actually a prohibition against objects spending time at the event horizon.

No temporal thing, nothing that normally experiences the passage of time, can have a space time interval of 0.

At the event horizon the only way to get a non-zero space time interval is to have a non-zero delta r. An object at the event horizon has to change its distance from the black hole to keep its clock ticking.

That means falling below the event horizon.

And once inside, inward spatial movement continues to be the only way to fuel the ticking of an object's proper time clock.

We'll come back to that bit of awesome weirdness in a future episode.

There is one thing that can have a space time interval of 0, light.

Actually, anything capable of traveling at light speed can only have a space time interval of 0.

From its perspective, a photon exists in a single instant, and so it can hang out at the event horizon, which also only exists at 1 infinitely stretched out instant.

The act of crossing the event horizon is where this singularity really starts to behave badly.

At the moment of crossing, the denominator here in the Schwarzschild metric is 0, and the whole equation leads up to infinity.

But what is actually infinite here?

It's nothing physical.

It's the fact that even an outgoing light ray takes infinite time to move any distance, so using boring old time and distance, delta t and delta r, doesn't let us trace a world line smoothly across the event horizon.

That horizon is a coordinate singularity, just like Kelsey talked about.

But that means we can fix it.

There are ways to construct our space time axes so this singularity just evaporates.

For example, Eddington-Finkelstein Tortoise Coordinates that compactify with the stretching of space time to cancel out the infinities.

That's a bit much for right now, but Google away, my friends.

Anyway the upshot is that it's really a breeze to drop through the event horizon, both physically and mathematically.

Of course, once inside the event horizon, we still have that central singularity to deal with.

Unfortunately, that one can't be done away with by a simple change in coordinates.

But can that point of infinite density really exist?

Actually, Einstein's theory and the Schwarzschild solution that is derived from it suggests it must exist.

The apparent inevitability of this singularity may be evidence that general relativity is incomplete.

But to better understand why the central infinity is unavoidable in Einstein's theory, we have to go back to that coordinate shift at the event horizon.

There, the causal roles of space and time switch places, and the central singularity becomes not so much a location in space but an inevitable future.

Actually, to really get this we're going to need another entire episode.

Standby to explore what happens when you switch the causal rolls of time versus space to space time.