[MUSIC PLAYING] Black holes seem like they should have no entropy, but in fact they hold most of the entropy in the universe.

Let's figure this out.

[MUSIC PLAYING] At first it seemed that black holes were so simple they should have no entropy.

Well, it turns out they contain most of the entropy in the universe.

Let's see why because this fact may force us to conclude that the universe is a hologram.

Black holes are a problem.

They are the inevitable result of extreme gravitational collapse.

At least they are inevitable according to the equations of Einstein's general theory of relativity.

That theory is one of the most thoroughly tested in all of physics, which means we should probably believe in black holes.

Also, we've seen them in their gravitational effects on their surrounding space and in the gravitational waves caused when they merge.

And yet if black holes exist, which apparently they do, they contradict other theories in physics that are as sacred as general relativity.

They cause all sorts of problems with quantum theory, which we've talked about before and we'll review in a sec.

But they also present an apparent conflict with the notion of entropy and the second law of thermodynamics.

It was while pondering that conflict that Jacob Bekenstein realized an incredible connection between black holes and thermodynamics.

His insight launched an entire new way of thinking about the universe in terms of information theory and ultimately led to the holographic principle, which I promise we're getting to and are almost there.

But first, you are going to need to know more about why black holes contain most of the universe's entropy.

OK, I'm getting way ahead of myself.

Let's actually rewind back to those episodes where we laid out the black hole information paradox because they're going to be critical to a proper understanding.

We're also rewinding to the late '60s, early '70s when physicists realized something odd about black holes.

What they realized is that it doesn't matter what material goes into one.

From the point of view of the outside universe, black holes can only have three properties-- mass, spin, and electric charge.

This is the so-called no-hair theorem, and it suggests that most of the information about anything that falls into a black hole is lost to the outside universe.

But a fundamental tenet of quantum mechanics is that quantum information can never be destroyed.

So if black holes evaporate, as Hawking discovered and we also covered, this evaporation should destroy a black hole's internal quantum information, giving us the black hole information paradox.

Eventually, a possible resolution to this paradox was found by Gerard 't Hooft.

He described a mechanism by which the information contained by infalling particles could be preserved on the event horizon of the black hole.

From there it could be imprinted on the outgoing Hawking radiation, allowing the information to escape back into the universe.

OK, problem solved.

But in our previous episodes we skipped the key insight that started all of this.

It all began with Jacob Bekenstein thinking about black-hole entropy.

OK, first, entropy.

Yeah, we talked about that a lot recently also.

You know, it's almost like all those episodes are starting to come together, almost like we planned this.

Go and watch that background stuff if you're behind, but of course for now I'll give you a quick TLDW on entropy.

So we can think of entropy in two ways.

One, it's a measure of how evenly energy is spread out.

High entropy means thermal equilibrium.

So energy is very evenly distributed and can't be extracted in a useful way.

And two, entropy measures the amount of unknown information that you would need to perfectly describe the system's internal state like all the particle positions, velocities, et cetera.

The higher the entropy, the more randomly distributed its particles and the more possible configurations lead to the same macroscopic state.

The higher the entropy, the less you can guess about the properties of individual particles based on the global properties like temperature, volume, pressure, et cetera.

OK, so the second law of thermodynamics states that entropy of an isolated system must always increase, which means energy tends to spread out evenly and particles tend to randomize, reducing our information about their microscopic states.

How does this relate to black holes?

Let's make a black hole and see what happens to entropy.

We start, as usual, by collapsing the core of a dead star.

Now that's a high entropy based, super hot and full of randomly moving particles.

We have almost no information about the individual particles, but that information still exists in the universe.

Like, I guess, the particles know where they are.

At the instant the star collapses far enough to form an event horizon, it becomes a black hole.

We go from knowing next to nothing about the object to knowing everything.

We can easily measure its mass, spin, and electric charge, and according to the no-hair theorem that's all there is to know.

The region of space in which the black hole formed appears to have gone from high entropy to zero entropy in an instant, shattering the second law in the process, which, to put it mildly, is a problem.

But if you paid attention to the whole information paradox bit, you might be able to think of a solution.

If quantum information is stored on the surface of the black hole, can't we store entropy there also?

And then why not radiate the entropy back into the universe as Hawking radiation?

Actually, yeah, the resolution to the information paradox also saves the second law of thermodynamics.

That was easy.

I thought physics was supposed to be hard?

OK, let's think about this a little bit more.

It was this seeming violation of the second law that got Jacob Bekenstein thinking about the connection between black holes and information in the first place.

The breakthrough insight was this simple observation.

The surface area of a black-hole event horizon can never decrease, at least not according to general relativity.

So you know how nothing can escape black holes, ignoring Hawking radiation for the moment.

That should mean that black holes can only grow.

They can never shrink in mass or radius.

Well, that's not quite true.

If you merge two black holes, some of their mass gets converted to the energy radiated away in gravitational waves.

There's also the Penrose process in which you can extract rotational energy of a spinning black hole.

And by you I mean not you you.

I mean super-advanced, far-future civilizations.

Gravitational radiation and the Penrose process reduce black-hole mass and radius or the sum of masses and radii of emerging black holes.

But there's one property of black holes that no process other than Hawking radiation can decrease.

That's the surface area of the event horizon.

Do anything to black holes and their total surface area can only grow or stay constant.

Bekenstein saw a close correspondence between the always-increasing event-horizon surface area and the always-increasing nature of entropy.

He also realized that the equation relating the change in black-hole surface area to the change in its mass closely resembles the original definition of thermodynamic entropy.

Just replace change in entropy and internal thermal energy with change in black-hole surface area and black-hole mass, respectively.

You can also add the work done when you extract energy from the black hole, and it looks the same as the equation for the work extracted from a thermodynamic system.

Bekenstein had just discovered black-hole thermodynamics, but that didn't give him the exact definition for black-hole entropy.

For that, he turned to Ludwig Boltzmann's informational definition for entropy.

So entropy can be defined as the information hidden in a system's macroscopic configuration times the Boltzmann constant.

Bekenstein estimated the amount of information that would be lost into a black hole as it grew.

Essentially he built a black hole out of idealized elementary particles that each contained a single bit of information.

And guess what?

The information content of a black hole is proportional not to its mass or radius or volume.

It's proportional to its surface area.

In fact, the information content is very close to that surface area divided by the number of Planck areas.

It's as though each of these minimum-possible quanta of area each contain a single bit of information.

Now just multiply that information content by the Boltzmann constant and you have the entropy of a black hole, which is going to be directly proportional to the surface area of the event horizon.

Bekenstein's connection between surface area and entropy could have been a coincidence, at least until Stephen Hawking came along.

In 1974, a year after Bekenstein's first paper on black-hole thermodynamics, Hawking published his first Hawking radiation paper.

He showed that black holes radiate random particles exactly as though they have a peak glow for a particular temperature that depends on their mass.

So if black holes have a temperature, then they also have entropy.

Good old-fashioned thermodynamic entropy tells us that change in entropy is change in internal thermal energy divided by temperature.

So Hawking just plugged his Hawking temperature into that equation along with black-hole mass for internal energy and figured out the total entropy contained in a black hole.

He got an expression almost identical to Bekenstein's with just a slightly different constant of proportionality.

So you get the same result for black-hole entropy whether you figure it out from the amount of information that gets trapped building a black hole or the amount of heat that leaks as it evaporates, and it's proportional to the surface area.

How bizarrely consistent.

I'd say that means it's right.

The second law of thermodynamics is saved because black holes do have entropy.

In fact, they have enormous entropies, the maximum possible, so much that black holes are now believed to contain most of the entropy in the universe.

But the real importance of this work wasn't the solution to some obscure conundrum.

It changed our thinking about the informational content of the universe.

Bekenstein's formula was derived for black holes, but it also gives the maximum amount of information that can be fit into any volume of space.

In this respect, it's called the Bekenstein bound, and it's proportional to the surface area of that space.

This is unexpected.

Surely the maximum amount of information you can fit into some patch of space depends on the volume of that space as in one bit per tiny volume element inside that space.

But in fact the rule is one bit per tiny area element on the surface of that space.

That also means that the information needed to describe any volume of space, no matter its contents, is proportional to the area bounding that space.

I've hinted once or twice that this simple idea led to the holographic principle, the idea that the entire 3D volume of the universe is just a projection of information encoded on a 2D surface surrounding the universe.

You just need to add a little bit of string theory.

It's a hell of a conceptual leap given it started with Jacob Bekenstein noticing a peculiar similarity between some formulae.

It might also be true, and obviously we'll be back before too long to talk about string theory and the holographic nature of spacetime.