Black Hole Enigma - Script (Patreon)

Black Holes seem like they should have no entropy, but they in fact hold most of the entropy in the universe. Let’s figure this out. 

For a while it seemed that black holes were so simple they should have no entropy. It turns out they hold most of the entropy in the universe. Let’s see why, because this fact may also be telling us that the universe is a hologram.

Let me tell you the story of how the casual observation of a link between black holes and entropy led many great physicists 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’re 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 by their merger.

And yet if black holes exist – which, apparently, they do – they contradict other theories in physics as sacred as general relativity.  They cause all sorts of problems with quantum theory, which we’ve talked about and will 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 we’re almost there. We just need to understand string theory first and to understand that, you’re going need to know about why Black Holes contain most of the universe’s entropy.

But I’m getting WAY ahead of myself. Let’s actually rewind back to those episodes where we laid the black hole information paradox because they’re going 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 3 properties– mass, spin, and electric charge. This is the no-hair theorem. A fundamental tenet of quantum mechanics is that quantum information can never be destroyed. But if black holes evaporate, as Hawking discovered and we also covered, then 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 in-falling 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 was thinking about black hole entropy.

OK, first: entropy. Yeah, we’ve talked about that a lot recently also. Why so many throwbacks to previous episodes today? It’s almost like all of that work is starting to come together. ALMOST like we planned this! Go watch that background stuff either now or afterwards if you’re behind. But of course I’ll give you a quick too-long-didn’t-watch on entropy also. 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, etc. 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, etc.

OK, so the 2nd law of thermodynamics states that the 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. That’s a high entropy beast – super hot and full of randomly moving particles. We have no information about the individual particles, but at 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 the 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 … yes, the resolution to the information paradox also saves the 2nd law of thermodynamics. Pfff, that was easy. I thought physics was supposed to be hard. OK, hang on, let’s think a bit more about this.

It was this seeming violation of the 2nd 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’s 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 now? That should mean that black holes can only grow; they can never shrink in mass or in 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 spinning black holes. And by you, I don’t mean you-you. I mean a super-advanced far-future civilization.

Gravitational radiation and the Penrose process reduce black hole mass and radius – or the sum of masses and radii of merging 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 in the work done when you extract energy from a black hole and it looks the same as the work extracted from a thermodynamic system.

Bekenstein had just discovered black hole thermodynamics, but this didn’t give him exact definition of black hole entropy. For that, Bekenstein turned to Ludwig Boltzmann’s  informational definition of entropy. Entropy can be defined as the information hidden in a system’s microscopic 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 contain 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 also 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 heat-glow of a particular temperature that depends on 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 to figure out the total entropy contained by 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 surface area. How … bizarrely consistent. I’d say that means it’s right. The 2nd 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 in 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 3-D volume of the universe is just a projection of information encoded on a 2-D surface surrounding that universe. You just need to add a little string theory. It’s a hell of a conceptual leap given it started with Jacob Bekenstein noticing a peculiar similarity between some formulae. It also might be true. And, obviously, we’ll be back before long to talk about string theory and the holographic nature of space-time.