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30: The Abyss (Part One; Black Holes)
517 Plays6 years ago
The idea of something that is inescapable, at first glance, seems to violate our sense of freedom. This sense of freedom, for many, seems so intrinsic to our way of seeing the universe that it seems as though such an idea would only beget horror in the human mind. And black holes, being objects from which not even light can escape, for many do beget that same existential horror. But these objects are not exotic: they form regularly in our universe, and their role in the intricate web of existence that is our universe is as valid as the laws that result in our own humanity. So what are black holes? How can they have information? And how does this relate to the edge of the universe?
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Transcript
Introduction to Black Holes
00:00:00
Speaker
The idea of something that is inescapable, at first glance, seems to violate our sense of freedom. This sense of freedom for many seems so intrinsic to our way of seeing the universe that it seems as though such an idea would only beget horror in the human mind. And black holes, being objects from which not even light can escape, for many do beget the same sense of existential horror.
00:00:20
Speaker
But these objects are not exotic. They form regularly in our universe, and the role in the intricate web of existence that is our universe is as valid as the laws that result in our own humanity. So what are black holes? How can they have information? And how does this relate to the edge of the universe? All of this and more on this first part of a series on black holes, episode 30, The Abyss.
Meet the Hosts & Episode Structure
00:00:48
Speaker
I'm Jonathan. And I'm Gabriel. And you're listening to Breaking Math. Today, we're doing one of Gabriel's and one of mine favorite subjects, black holes. Yeah, this is an episode that I've been looking forward to since before we began the Breaking Math podcast. It's one that I've never felt quite adequately prepared to do. And right now, as you all have heard, this is a three-parter. It may even be a four-parter eventually, but we'll see.
00:01:13
Speaker
Yeah, and the thing is about planning an episode that's more than one part long is that you have to make sure that each part gives you something, gives you a beginning, middle, and end. It's a very ordered way of doing something, and with something like black holes, sometimes there seems to be so much chaos since there's still an object of exploration that you don't know where to begin and where to end.
00:01:36
Speaker
Yeah, I'd like to think that this episode is for anybody who may not know anything at all about black holes or may know something about black holes.
Why Are Black Holes Fascinating?
00:01:46
Speaker
They are considered one of the most fascinating objects in our universe by a lot of people. They are quite popular.
00:01:53
Speaker
What we hope to do in this episode is to talk about, number one, why are they so popular? Why are they so intriguing? But also, we'd like to talk about the significant way that our understanding and our models of black holes has changed, especially during the previous century. It's a considerable amount.
00:02:15
Speaker
Yeah, and one thing that really defines the 20th century in terms of physics is how non-intuitive stuff seemed to get, how much we had to rely on analogy for understanding things. And because from the beginning people had a fear of black holes, fear became part of the analogies. You had these exotic analogies, and black holes, although they are not exotic to the universe, are exotic to humans because they deal with the speed of light.
00:02:42
Speaker
Anything that deals with the speed of light or too small of timescales is exotic to humans because we live in a very temperate kind of zoned kind of atmosphere. I mean, even the coldest place on earth isn't as cold as space and so on and so forth. And using things that we usually do when we're trying to learn about the universe, our own ability to model, our own intuition, those are absolutely challenged when trying to understand black holes.
Early Concepts of Black Holes
00:03:08
Speaker
Before we begin, we'd like to give you a little preview on what you can expect to learn in this episode. We're going to start off with the very, very first ideas about black holes that first occurred perhaps before you realized, even in the 1700s, the late 1700s, there were some ideas about black holes from some people who considered the implications of Newton's equations for gravity.
00:03:33
Speaker
And then we're going to talk a little bit about black holes and how they developed from the centuries onward, including the specialty amount of stuff in the 20th century. Yeah, quite a bit there, quite a bit there. We're going to talk about, of course, the contributions or the implications of Einstein's field equations and black holes and how they made a splash in the 20th century. We're also going to talk about assumptions about black holes and things like entropy.
00:03:57
Speaker
or temperature. We're going to bring in Shannon's information theory. And we are going to bring in a little bit of math, but not too much. So it should be pretty enjoyable and pretty listenable for all audiences. So without further ado, here is the abyss.
00:04:15
Speaker
So during the Enlightenment in the late 1700s, the idea of black holes was first proposed. And they actually weren't called black holes until the 1960s or so. But at the time, they were called dark stars.
00:04:33
Speaker
And yeah, it was Laplace who we've talked about before on Navier-Stokes and English cleric John Mitchell who first hypothesized that dark stars being something from which light could not escape could exist.
00:04:49
Speaker
And I think this came about from ponderings about some of the implications of Isaac Newton's theories of gravity and his mathematical equations. And I think that the key component of Newton's equations was the effect of the relationship between mass and gravity.
00:05:06
Speaker
And as you had mentioned, Newton was the first person who had the idea that light actually traveled in little packets, like little bodies called corpuscles,
Understanding Escape Velocity
00:05:17
Speaker
I believe, yes, which is what we now know of as photons. So I think the French physicist Pierre Simon du Laplace, and I'm sorry if I mispronounced that. I think it's du. Yeah, du Laplace.
00:05:30
Speaker
And John Michelle, they took his idea that gravity will pull on any object of mass and they applied that to light. It's kind of interesting because it makes you wonder, did they think that light had mass?
00:05:44
Speaker
And before we go into that, we do have to talk about escape velocity and what it is, essentially. And escape velocity is the velocity that something has to be thrown from the surface in order to escape the pull of gravity. On Earth, it's about 11.186 kilometers per second.
00:06:06
Speaker
which is pretty fast. I mean, if you're in the United States, it's almost seven miles per second. And that's, that's how fast you'd have to throw something. So if you had a rock and you wanted to throw the rock out of orbit, so it doesn't come back toward the earth, but just goes off into space indefinitely, it would have to be, as you said, what was it? Seven kilometers per second.
00:06:27
Speaker
Yeah. And we'll seven, about seven miles per second, 11.1 86 kilometers. Thank you. Okay. I stand corrected. That's fast. That's ridiculously fast. So then if you, if something does not reach escape velocity, then ultimately gravity will overtake it and it'll just come right back down toward the earth.
00:06:45
Speaker
Yeah, and the escape velocity is dependent on two different factors and one universal constant. It's dependent on the mass of the object. The higher the mass, the higher the escape velocity, but the higher the radius, like how far you are from the center of mass, the lower the escape velocity.
00:07:05
Speaker
And so if you were on the Sun, the escape velocity probably would be higher, just given what we know about how dense the Sun is. But it could be lower, even if it had more mass than the Earth, because you are farther away from the center of mass of the Sun, which is the center of the point of the Sun. Isn't there, there's an equation for the escape velocity, isn't there?
00:07:26
Speaker
Yeah, it's square root of 2GM over R. So M being the mass of the object, R being how far you are from the center of mass of the object, and G being the universal gravitational constant, which we've covered on the show. What's worth pointing out is these early physicists who contemplated the idea of dark stars, or as we now know them, black holes.
00:07:45
Speaker
did not know anything about Einstein's general relativity. They did not know about the bending of space and time. They didn't know anything about time dilation or even the idea that the speed of light was the fastest conceivable speed in this universe.
Physicists' Early Understanding
00:08:01
Speaker
But they did know that light had to do with transmitting data about at least objects. So they knew for a fact that it would be, quote unquote, invisible. And of course, black holes have exhibited a property called gravitational lensing, meaning they act like lenses in the middle of space. They bend light around them.
00:08:23
Speaker
Yeah, you know, I got to say it still absolutely blows my mind that at that time people thought that there could be objects that are invisible simply due to their mass. I mean, how do you think about that? That's just to me, that's just amazing. You know, I often wonder not only about what people thought of, but what inspired them. I mean, I don't think in all the time in the world I ever would have thought of that.
00:08:47
Speaker
Well, one of Newton's contributions to physics was his book on optics. And his book on optics seemed to talk a lot about that for light to travel, it has to come from the object and be bounced off. I believe that was part of it. I know he didn't subscribe to the idea that Aristotle had, or was it Plato? That the eye generated things that felt the environment around them. Interesting. But that's something completely different.
00:09:16
Speaker
Yeah, yeah. So the only characteristic of these black holes at the time would just be that they can't be seen and also that they are very, very massive.
00:09:24
Speaker
Yeah, and what's funny is that even though they conceived of these things, because the physics was so like, not the concept of physics was so mismatched for the reality of physics having to do with these things that there wasn't much fruitful stuff that came from it. They didn't have, for example, Riemannian geometry, which is the geometry of curved surfaces. They were still working with Euclidean geometry until the mid 19th century.
00:09:52
Speaker
And then in the mid-19th century, Maxwell had a wave description of light, which seemed to contradict the corpuscle theory of light until the double slit experiments done by Jung in, I think, the 20s, right? Oh gosh, I think it was before that, 1920s? I think that was actually done in 1800s, wasn't it? Something.
00:10:10
Speaker
I don't have the timeline from the 19th to 20th century, very clear. But yeah, the wave description of light seems like it would have put a dampener on the idea of black holes, even though they weren't that big of a thing. So what I'm trying to get to is that black holes were a very exotic-seeming object to humans for a very long time.
00:10:34
Speaker
And as you just said, there wasn't really a whole lot that was done in terms of thinking about them, any sort of contemplation or modeling or any other thought experiments, pretty much until the 20th
Schwarzschild & the Event Horizon
00:10:44
Speaker
century. And I think the next time that they were considered at all was after Einstein had published, was it just the special theory or was it also his general theory of relativity? Well, I know that he publishes general theory of relativity that had to do with the curving of space-time.
00:10:57
Speaker
It wasn't until 1916 that German astronomer Karl Schwarzschild used Einstein's field equations to calculate what we now know of as the Schwarzschild radius. And how would you explain what the Schwarzschild radius is?
00:11:10
Speaker
The Schwarzschild radius is the radius of the event horizon of a black hole. An event horizon, meaning that if an event happens within that radius, there's a horizon, you know, like if a horizon on Earth, meaning you can't see the sun behind it, and our event horizon is a horizon that you cannot see events behind. It's a place where not even light can escape, right?
00:11:33
Speaker
Exactly, exactly. Now this has huge significance following Einstein's field equations. Einstein's field equations allow for the existence of black holes. And this actually deeply disturbed Einstein. A lot of people now have seen really great visual diagrams or visual analogies for the formation of a black hole.
00:11:57
Speaker
I think a very common analogy is the rubber sheet analogy, and I know that analogy does have its problems. It's not entirely accurate, but when you think about a black hole... Real quick, the rubber sheet analogy is the analogy that if you look at gravitational bodies as though they were put on a rubber sheet or something like a trampoline,
00:12:19
Speaker
that they curve space-time in a way that, so let's say you have a big trampoline, you have a very heavy weight in the middle of it, it'll pull everything down kind of around it, so that if you take something very light, like a golf ball, and you toss it into the deformation made by the massive object, it'll go around it, and that's essentially how gravity works.
00:12:43
Speaker
You know, at many, many museums that I've been to, there is a fundraiser where there is something where you drop a quarter into and it's sort of like a funnel. And as you drop the quarter, it rolls down this little groove and then it's dropped into the edge of this funnel and you watch it spin around. I don't know how many times it seems like probably 200 or 300 times before it finally falls down the center.
00:13:06
Speaker
Well, that's very, very much what this analogy is trying to show. As Jonathan said, if you have something heavy like a bowling ball and it bends the trampoline, and then you have the golf ball, that can approximate what the earth does around the sun as it's going toward the divot, that dent.
00:13:23
Speaker
Now, if you use the same analogy and you try to explain a black hole, I think a black hole is where you have an object that is so massive that, in the case of the rubber sheet, it bends it down to an infinitely small point. It's almost like a tear in the object itself, which, if you're talking about space and time and tearing it, that's where things stop making a whole lot of intuitive sense. It's really a terrifying thing.
00:13:50
Speaker
I would say it's, I mean, me and Gabriel differ on this, but I don't think that black holes are terrifying just because I think that it's just for, I guess, maybe for my own sanity, maybe I'm being a little bit pedantic or even protective of my mind here. But I don't think that they're terrifying because they exist and we just have to think of them as existing. And since they're not conscious, there's no reason to be like necessarily terrified of them.
00:14:18
Speaker
Yeah. You remind me of like, um, who was, um, who was the, the Australian guy who had animals? Um, Steve Irwin. Yes. He was always trying to say, Oh, we've got a lovely black hole here. No need to be afraid. I certainly wish that I, I could not be afraid. What I'd like to do is I'd like to talk about, um, what does, what I find very disturbing about black holes. And the way I'd like to do that is I'd like to talk about the difference between a white dwarf, a neutron star and a black hole.
Star Evolution & White Dwarfs
00:14:48
Speaker
This might be a throwback to one of our earlier episodes of Breaking Math on Physics, and I believe that episode was called Language of the Universe, episode four, if I'm not mistaken, where I first introduced a brief discussion on the differences between black holes and neutron stars. Neutron stars are some of my favorite objects in the universe because they are terrifying like black holes, but they are also finite.
00:15:11
Speaker
So without further ado, I'd like to explain for audiences who may not be aware of this, what the difference is between a massive object like a white dwarf and a neutron star and a truly terrifying, well, I think so, black hole. So with a star, we know that we'll start the story off with a star that's vibrant and alive like our star, which is the sun.
00:15:37
Speaker
The star is very, very massive. It's the most massive object in our solar system and all of our planets, of course, orbit the star, which makes perfect sense with Einstein's relativity. That's what we would expect the path of the planets to be.
00:15:52
Speaker
Now something about the star is that it is burning with the burning gas that causes an outward pressure. So with our star you have both an outward pressure from the burning gas you also have the inward pressure of gravity and it's sort of at an equilibrium. Well it is at an equilibrium where the
00:16:10
Speaker
It's like if you exploded a bunch of iron filings and you had a magnet in the middle of them, but they kept exploding for millions of years. Yeah, something like that. The gravity harnesses the sun, so it doesn't spread out. Now, what's noteworthy here is that eventually the sun is going to run out of hydrogen to burn. Its current state is very temporary.
00:16:35
Speaker
When it runs out of hydrogen to burn, it will no longer be able to push outward. However, the mass of the sun is not going to change at all. You're still going to have the same amount of pressure pulling inward. So what's going to happen? Gravitational collapse. Exactly. I think in bigger stars, you have things like supernova and you have an explosion in some instances, which is a function of the gravitational collapse. The end result of something like our sun is, I believe our sun is going to become what's called a white dwarf.
00:17:03
Speaker
where all of the hydrogen atoms are suddenly pulled inward very very very fast and into a very dense thing called a white dwarf which essentially is a dead star. I believe the the pressure of a white dwarf even though the hydrogen is no longer burning the pressure of a white dwarf still causes it to emit light
00:17:21
Speaker
And what's interesting is that white is our sun is at almost limit for how big a white dwarf can be. And more on that after we discuss what neutron stars and black holes are. Yeah. So, you know, I sometimes think of a white dwarf as like a glowing ember in a campfire. That's after the fire has gone out, you still have a glowing piece of coal that still has some heat, but essentially the fire is dead.
00:17:45
Speaker
Now, this does not always happen. In fact, as Jonathan said, if you have a star that's, you know, not much bigger than our star, you're going to have more mass and therefore even more mass than what you have with a white dwarf. So with a more massive star, eventually when it reaches its death and it runs out of hydrogen and helium to burn, you will also have, there will also be gravitational collapse, but there will be a lot more gravity. And what happens then?
00:18:14
Speaker
Well, what happens then is that you essentially end up with a star made out of neutrons. Isn't that amazing? I mean, the gravity pulls in not only the atoms, but even the electrons in orbit are squashed right into the nucleus.
00:18:31
Speaker
Yeah, and of course the mechanism behind this is something complex and best left for a different episode, but the important thing about this is that it runs into something called the Chandrasekhar limit, and it was by the Indian mathematician, and I'll say his name as carefully as I can.
00:18:51
Speaker
Subrahmanyan Chandrasikhar
00:19:09
Speaker
or later or not at all, depending on, it's a mess, honestly. This is just fascinating for me. I mean, I've got all kinds of questions that I'd love to do some more reading up on, like what happens to those electrons? I mean, I don't think the electrons are destroyed, but I don't know.
00:19:25
Speaker
Yeah, I'm not sure if they're destroyed or if they're radiated outward because I know that white dwarfs resist gravitational collapse through electron degeneracy pressure, which is a type of fermion pressure, which is also
00:19:41
Speaker
present in neutrons. Now what affirming on is, it's something with, and don't worry about this next, it's fundamental as charge or mass, all particles have something called spin. And it basically says how you react to magnetic fields. But things with half integer spin, meaning spin that's like 1.5 or 2.5 or something like that, well it's not, or .5, those would all be examples of half integer spin.
00:20:10
Speaker
you have something called the poly-exclusion principles. This is one of my favorite topics and I think that this is key to the formation of a neutron star, right? Yeah, and the thing is that neutron stars don't collapse all the way because they have this exclusion that no two fermions can occupy the same, like, okay, you have all these variables such as mass, you have charge, you have position,
00:20:39
Speaker
and you have a spin and if you if you have two fermions cannot occupy the same one of those it's like their mailboxes and they all have a distinct address which means that if they try to go together the pressure that just is part of the universe's base form keeps them apart from one another.
00:20:58
Speaker
So you have this Fermi exclusion principle. And if I recall from my middle school days, that that's the same thing that keeps two electrons and only two in the outer shell of the first outer shell of an atom. Is that correct?
00:21:11
Speaker
I believe, yeah, I believe that's okay. Sure. So, so, uh, and what's also interesting is in that first orbital of an, of an atom, you have a spin up and a spin down. You can't have to spin up electrons. They have to be spin up and spin down. Obviously, as you get further out on an atom, uh, you know, the, the outer orbitals, uh, get a little bit more complex than that, but that's the essence of it.
00:21:31
Speaker
Yeah, they get more complex because you have more space, essentially, for electrons to exist together. So with neutron stars, it's just fascinating. It's a star of nothing but neutrons.
Neutron Stars & Black Hole Formation
00:21:44
Speaker
So they are, for all intents and purposes, invisible, although they do radiate radio waves from the poles, I believe, if they spin very fast. They're fascinating material.
00:21:56
Speaker
And I think that something the size of our star, or even a little bit larger, when it becomes a neutron star, it would get squashed down to something about the size of Manhattan, if I'm not mistaken. Now again, it all comes down to this thing called a Pauli exclusion principle. So one of the things that I like to ponder is, is the Pauli exclusion principle a fundamental constant of our universe and the way our universe currently exists? I would like to say yes, but I'm hesitant.
00:22:25
Speaker
What Gabriel's referring to is that neutron stars, if they're too massive, they seem to overcome the Pauli exclusion principle and turn into black holes. And black holes mean something from which light cannot escape, and they're smaller than neutron stars, so a black hole cannot be made out of anything that we know about that is outside of a black hole for the simple reason that the universe is based on observation.
00:22:55
Speaker
This episode is all about mathematics and how it relates to black holes and their more far-reaching consequences.
00:23:01
Speaker
To that end, our partner Brilliant.org has a course all about astronomy and how it relates to cosmology and the fate of the universe. I love how the course takes you through the pillars of astronomy, including the life cycles of stars and how that relates to star formation, stellar evolution, and even, yes, black holes. Astronomy is a fundamental way in which we observe the large universe, and this course provides a solid foundation in the essentials of this field. To support your education in math and physics, go to Brilliant.org slash breakingmath and sign up for free.
00:23:29
Speaker
The first 200 Breaking Math listeners can get 20% off the annual subscription, which we have been using. And now, back to the episode.
00:23:38
Speaker
The electrons are the way that they are because we see them. They have certain properties. Fermions and bosons have different properties. Atoms and cows have different properties. We observe different behaviors of different things. Sorry, did you say atoms and cows? Yeah, atoms and cows, like, you know, like, yeah. As we're describing, OK, OK, OK, I was going to say, well, cows are made of atoms, wouldn't that?
00:24:00
Speaker
Oh yeah, but I just mean a single atom versus one full cow. But just things that have different properties. However, because you cannot see beyond a black hole, you cannot observe beyond a black hole. And one thing that we've learned from quantum physics is that if you cannot observe something, it becomes in a sense meaningless.
00:24:18
Speaker
Yeah, fair enough. You know, I would like real quick to describe a little bit more about what I find disturbing about black holes. And again, with the rubber sheet analogy, if a neutron star was something so, so heavy that it would bend it down maybe a mile, like if you had a trampoline in your backyard and you had something bend it a mile down, well, you couldn't because trampolines aren't a mile above the ground.
00:24:42
Speaker
If you could, if you had a trampoline over the Graham Canyon or over some massive chasm and it would bend down a mile, that's tremendously deep and that's extremely powerful, but it's still finite. You know, it still has an end to it. The thing I don't like about black holes is I think using that same rubber sheet analogy, it would bend down forever. And that to me, that's an infinity that shouldn't exist in our universe.
00:25:04
Speaker
To me, though, that's a problem with the rubber sheet analogy more than it is a problem with black holes. Let me ask you this then. If something were to fall into a black hole, it would continuously accelerate essentially forever or until it touches the singularity. Is that correct? Not necessarily, because we can't see past the event horizon of a black hole. So what happens inside, quote unquote, of a black hole is the domain of the black hole itself.
00:25:34
Speaker
Okay. Yeah. So, so that could be beyond, you know, they say it breaks our standard model of physics or, you know, we need a new physics to describe it. I don't like the idea that it seems to destroy physics because that seems to destroy our most fundamental way of understanding things. You know, so it doesn't destroy all the physics. It just destroy, it destroys some of the ways that we've understood it so far. I mean, Einstein quote unquote destroyed Newton, but we still use Newton stuff all the time.
00:26:00
Speaker
Yeah, yeah, I see. You know, one other way I was thinking about it is outside of a black hole, just in normal empty space, if something is to accelerate, you need to add energy to it. If you're in a car, you can step on your gas in order to accelerate. But that costs gas. That costs energy.
00:26:18
Speaker
And I realize that you did say that we don't necessarily know what happens in a black hole, but it would seem that just due to gravity itself, when something falls into a black hole, it is continuously accelerated, but that's not the same as a free energy machine, right?
00:26:34
Speaker
Oh, no, in the same way that, I mean, when we sent the... What was that mission? Back in the 70s, we sent a spaceship out, Voyager. We sent Voyager out and it had a gravitational boost from Jupiter, where it slingshot around Jupiter and just went at an insane speed out of our solar system.
00:26:57
Speaker
And that didn't violate any laws of conservation of energy, because it's the same way that you might, as Randall Munroe of XKCD puts it, throw a tennis ball at a speeding freight train, or what did he say? He said a speeding... What are those trucks called? The big trucks with the boxes? Not Mac trucks, that's a brand, right?
00:27:20
Speaker
No, the general category. Semi-trucks? Semi-trucks. If you throw a tennis ball off at a semi-truck, it's going to accelerate it. So when it comes to gravity, there's no free energy. And it's the same way with magnets. The energy, in a certain way, this is very informal, so don't quote me on this, but it's found in the position, the distance, the potential. The potential is a more formal way of saying it. Interesting. OK.
00:27:49
Speaker
All right, I'm still trying to think of this from every angle, and I thought even though this is impossible because of the event horizon, I thought with a black hole, if at some point you could reverse it somehow, and again, don't ask me how. I'm just saying hypothetically, if you could, you could seemingly have an arbitrary amount of energy, right? I mean, your wind up, if you will, seems to be like,
00:28:13
Speaker
The longer you wait for an object in a black hole, the stronger your windup or the more potential energy you have. I don't know. It's the same way though that with like two objects that are infinitely far apart. I mean, you do have a finite amount of windup, but also if you accelerate something for an infinite amount of time, it doesn't get up to an infinite speed. The faster you go is the speed of light. Interesting. You do have a limit even with the limitless. Okay.
00:28:36
Speaker
So, Einstein found these objects very horrifying and terrifying, and I do too. And, you know, maybe I'll get more comfortable with these abysses. We explore them more, but so far I have not been.
00:28:51
Speaker
Following the contributions of Schwarzschild to our understanding of black holes, there was a great renewed interest in black holes. And there are many institutions that began intense study of the subject matter, especially Princeton.
00:29:06
Speaker
Now, there wasn't really a huge, I would say, revolution in thinking about black holes until the late 1960s. And it had to deal with certain physicists who were especially interested in things like quantum mechanics and how they would relate to black holes and also entropy and how it would possibly relate to black holes. Now, before we go further into how entropy affects black holes, let's talk a little bit more about what entropy is.
00:29:36
Speaker
So one way to describe entropy is it's a measure of how many possibilities there are for a system, any system. So that sounds a little nebulous, so let me give you a concrete example. Let's say that we have four different blocks and we put two of them in one stack and two of them in another stack.
00:30:00
Speaker
So we have four blocks and two stacks. For each stack, there's only two ways that you can configure the blocks, the first on top of the second or the second on top of the first. It doesn't matter which one you label as the first or second, which means that you have four times four equals 16 different possible configurations for all four blocks. And that's fine and dandy. But let's say you had one stack of blocks instead of two different stacks of blocks.
00:30:25
Speaker
It turns out that you would have 4 times 3 times 2 times 1, which is 24 different configurations. Now 24 is more than 16, and that's a property of entropy. Entropy has the property that it describes basically information. So the entropy of the second system is not 24, but it would be the log of 24, logarithm base 2 or 10 or whatever you want to do of 24.
00:30:49
Speaker
and logarithm base whatever of 16 and the logarithm just being like 2 to the what power equals 24 and 2 to the what power equals 16. That comes out to being about 4.58 bits for 24 different configurations and 4 for 16.
00:31:04
Speaker
So I think for our purposes and actually I think in actuality Information is essentially the inverse of entropy very high information. I'm sorry No, I'm sorry. It is not the inverse. It is exactly described. It is exactly proportional to entropy.
Thermodynamics & Black Hole Entropy
00:31:20
Speaker
Yeah, and another maybe a little more intuitive way to describe entropy is a lot of times it's associated with disorder because the more disordered something is, it takes a lot more time to describe it. If we take those blocks and we stack them neatly in alphabetical order from A to Z, that just, you just say there's a stack of blocks from A to Z. It's very simple. But if we throw one on the roof and then we burn one and then we mail one to China and we keep doing things like that until we get arrested,
00:31:49
Speaker
the blocks are going to be in such different positions that it can take you forever to describe where the blocks are. Yeah, exactly. A lot more information. Yeah. I was talking to my nephew about this, and I use the example of eggs in a carton versus eggs in a bag, especially eggs in a bag that's been dropped. Being able to describe where they are takes a lot more. It does take more energy to describe the location of all of the eggs or the constituent part of broken eggs in a bag than if they are safely organized in a box. So organization is another way of putting it. Organization is the inverse.
00:32:18
Speaker
Well, be careful about my definitions here. Organized eggs in a cart have less information than disorganized eggs or any other object in a bag or where they're not ordered. And you can see how this is a slightly weird property of the universe. But one thing that's fundamental about entropy
00:32:39
Speaker
is that time and entropy are closely related. Entropy goes from the more simple system to the more complex system always in a closed system. Think of a burning candle like the wick. The molecules are ordered in the wick and the wax wick and as they're burned, they are completely disarrayed in soot and ash and smoke.
00:33:02
Speaker
Yeah, there's thousands of different components to combustion. And that's why combustion is so often used as a model for entropy. So why are we hammering this point home for our listeners? The essential idea is think of disorder and think of, you know, high information, high disorder, low disorder, low information.
00:33:22
Speaker
And since entropy always increases, time goes forward as entropy increases. That's the definition really of time, meaning that if entropy started decreasing in the universe, time would seem to go backwards, which is a weird concept and something that we can't touch right now. Yeah, that's okay.
00:33:44
Speaker
So, and I think, as you touched on earlier, Entropy and Shannon's information theories, both topics that we've talked about in previous episodes, come together in a big way in this particular segment. So, back to black holes. Jonathan, would you like to introduce how Entropy influenced the thinking about black holes, especially for a physicist by the name of Jacob Beckenstein?
00:34:06
Speaker
Sure, Beckenstein was a physicist who had a lot to think about, black holes of course, and his thought experiment was a pretty simple but pretty revolutionary one. He said, let's suppose you had some black holes in a universe. The universe is a close, at least, you know, like an observable universe to be more precise. An observable universe is a closed system, so the entropy can only go up. Now let's say you had a bunch of hot gas.
00:34:35
Speaker
Hot gas has a very high entropy because it's very disordered. Molecules bouncing all over the place at different speeds. It's insane. And he said if you throw that into a black hole, and the black hole just has no entropy, then the black hole would eat the entropy and the number of the amount of entropy in the universe would go down. This is abhorrent. This does not happen.
00:34:56
Speaker
The possibility of the fact that your mother is not your own mother is much, much more possible than the universe violating the second law of thermodynamics. The fact that you think you're even alive is more probable than that fact. You cannot deny the second law of thermodynamics.
00:35:12
Speaker
It is the closest thing to being sacred that physics has. This is fascinating. Now, it's actually worth noting that before Bekenstein, actually, Bekenstein's own professor, I believe his name was John Wheeler, in fact. Is that right? It was John? His name was Wheeler. He had a statement about black holes that was before entropy was considered. And the statement was, black holes have no
00:35:36
Speaker
hair. Now what was interesting is it was thought that that phrase was scandalous at the time. We don't need to go into that, but eventually that phrase was published, but not before riling a whole bunch of feathers who thought that that was a very inappropriate phrase. But let's talk about that phrase real quick and exactly why they thought that. Sure.
00:35:54
Speaker
Black holes are a solution to the Einstein field equations, and because it's a math that we won't get into really until the next episode where it becomes a little bit where we talk more about the holographic principle. So a black hole can be described by five different components. It's mass energy, it's linear momentum, so how fast it's traveling,
00:36:17
Speaker
its angular momentum, or how fast it's rotating, its position relative to the observer and the electric charge. And that's all. And it is still true that at a large scale, that's how black holes can be considered by these few components. But it was thought that because it had so many few components to describe it, that it had low entropy. Because remember,
00:36:44
Speaker
Entropy has to do with complexity, it has to do with information, and if there's so little information describing it, then they must be low energy. But because of Bekenstein's observation that throwing hot gas into a black hole would reduce entropy in the universe, he basically showed that black holes have to have entropy, and they have quite a bit of entropy, actually.
00:37:06
Speaker
Wow, that's fascinating. When you talk about entropy and about information, I think one of the ways that I thought about it earlier was, I was trying to think of, if you had a bunch of iron filings that were randomly placed, those would have a high disorder and there'd be a lot of information. And then if you put a magnet above them, suddenly they'd move into a uniform formation, assuming they didn't all get stuck to the magnet, which they actually would. Yeah, and that's a local lowering of entropy for sure, but at the cost of potential.
00:37:35
Speaker
OK, right, right, right. So then, of course, again, I'm trying to think about the assumptions about black holes prior to Bekenstein. So they thought like a magnet, the black hole would probably make them all move kind of in a uniform direction toward the singularity, hence lowering information and entropy. But as you said earlier, that's not as intuitive as that might be. That's incorrect.
00:37:58
Speaker
Yeah, because entropy just simply does not decrease. It is such a steadfast rule. And it's a steadfast rule just because it's so mathematical. Because the only difference between time going forward and time going backward is entropy.
00:38:14
Speaker
And if we change that, then entropy would change within the entire universe, and time would not make sense in the global consideration of time within the observable universe. One of the things I was thinking about as we were driving to my house to record today was about this very topic. I actually thought about airplane propellers.
00:38:34
Speaker
You know how as a kid, sometimes you'll see either a helicopter or an airplane with propellers. You don't see the propeller spinning at all. It looks like there's nothing there. Where in fact there actually is something there. In fact, it's spinning ridiculously fast. You just can't see it.
00:38:50
Speaker
that's sort of like the entropy inside of a black hole where there's tons and tons and tons of entropy but rather than seeing something you know like just zeros or just ones it's so scrambled it's sort of like if you were to have you know a black sheet of paper and a white sheet of paper spinning together and you see gray it looks like it's uniform gray it's not it's it's it's discreetly black and white it's just
00:39:15
Speaker
so scrambled up that all we perceive is smooth gray. In fact, that's the example that's used in the book by Leonard Susskind, The Black Hole War, which we read in preparation for this episode, which I think that's a great example. Yeah, and one incredible result requires the understanding of how much one bit of information is.
Information & Black Hole Surface Area
00:39:35
Speaker
one bit of information. Actually, again, in fact, it was Bekenstein who calculated how adding one bit of information to a black hole would change it. This fascinates me because I'm like, wait a minute, wait a minute. What on earth is one bit? What in the physical universe is one bit of information? You're about to find out.
00:39:54
Speaker
Well, one bit of information, just when it comes to information itself, is a yes or no choice. So flip of a coin, interpreted, interpreted is a key word, is one bit of information. Two coins, two bits. 30 coins, 30 bits. So that's a yes or no choice. So how would you send one bit of information into a black hole? That sounds like a weird question.
00:40:17
Speaker
But it's not that word of a question when you consider photons if you sent a photo because photons have a thing called They're a wavelength if you've ever seen smoke come off a come off of a cigarette It's slightly blue, but when it's exhaled from someone's lungs, it's white
00:40:32
Speaker
And the reason why is because the particles are so small coming off of the tip of the cigarette that some of the light literally misses the particles of smoke. But when it combines with water vapor in the lungs, the particles become larger. And so when something has a wavelength that is larger than the thing it's interacting with, or it basically doesn't interact with it except probabilistically,
00:40:57
Speaker
And if it has the same wavelength and it's still probabilistic, it's always probabilistic, but if you have something with a very short wavelength and doing something with a very long wavelength like a door, because all objects have wavelengths which are very high in a discussion in their own right, you get a very accurate description of the door in terms of the door itself.
00:41:17
Speaker
Let's suppose we sent a photon with the wavelength of the Schwarzschild radius into a black hole with the same Schwarzschild radius as that. Let's say it's 1,000 feet or something like that in radius, you send a wavelength of 1,000 feet.
00:41:37
Speaker
that whether or not that interacted with the black hole will be a 50-50 shot. Now the way that that'll work is you send this into the black hole so it is one bit of information. Now the radius of the black hole is 2 times the mass of the black hole
00:41:56
Speaker
times the gravitational constant divided by the speed of light squared. It's a big radius, not big compared to how much mass it is. For example, the Earth would be the size of a cranberry if it were a black hole.
00:42:11
Speaker
So you'd only need a radio wave to increase the mass of the black hole. The energy of that photon would be equal to the Planck constant, which just tells you how much energy something has, depending on how much wavelength it has, times the speed of light divided by the Schwarzschild radius.
00:42:31
Speaker
And when you combine those two equations, you get that the change of mass is equal to the Planck constant divided by the speed of light times the Schwarzschild radius. And you might be thinking, why are you telling me all this? I have a headache, stop, stop, please stop telling me all this. And I'll tell you no, you have to listen to this because when you add one bit of information into any black hole, it doesn't matter if the black hole is as big as Jupiter or is as small as a P, it'll add the exact same amount of area to any black hole.
00:43:00
Speaker
is just a fundamental, and it turns out that that's a fundamental law of all black holes. And what does that say? It says that if you add entropy in the form of bits into a black hole, then it turns into area or that information and area are interchangeable in this universe.
00:43:19
Speaker
Wow, so I don't know if you use these words here, but I think the most profound part of this proof here is that one bit of information is one square plank unit, or a plank unit squared. And a plank unit is like the smallest distance that you could have.
00:43:40
Speaker
Yeah, and we have not yet done an episode on the Planck unit, and it's very, very hard to... It's hard to entirely describe the significance of it. It's revolutionary for quantum mechanics, but in this universe, it's not possible to get smaller than a Planck length, and for reasons we can't go into this podcast.
00:44:02
Speaker
But it just makes sense then that one bit of information is one plank length squared. It's like a little, like on an HDTV, it's like a single pixel, just much, much, much smaller. And another way of putting this is that everything that falls into a black hole, no matter what it is, is stored on the surface of the black hole. And we have this proof that you might at first think to be a little bit dubious, saying that area and information are the same thing.
00:44:30
Speaker
And so the amount of information on the surface of a black hole is proportional to the area of the surface of the black hole. And as Hawking showed, it's about a quarter of that. Actually, I think it's exactly a quarter.
00:44:45
Speaker
Oh, yeah. And if it's a quarter of that, then we have a linear relationship between information and the area of a black hole. But also, since there's a relationship between the area and the mass of a black hole, that means that for any black hole, we know how much information weighs, but not only how much information weighs, but how much weight has information with respect to that black hole. It's almost like a relative law of information itself, which is just fascinating to me.
00:45:15
Speaker
In fact, in the book that we're reading by Lynette Susskind, I think it actually works out, if you were to add one bit of information to a black hole, how much does it increase in mass? And what's that number? 10 to the negative 45th kilograms for a black hole made out of the sun. Wow. Wow. That's just, that's insane. So then that does vary by the starting mass of the black hole, right?
00:45:36
Speaker
Yeah, but the amount of area for each bit does not change. You might be thinking, how is that possible? And let me just give you a quick example. If you take a rope and tie it around the Earth and you add 30 feet to it, you could add fence posts around the entire rope and make the rope rise up five feet everywhere on the Earth. Yes, yes, exactly.
00:46:03
Speaker
But if you tied it around Jupiter, you'd still only need 30 feet or so. Yeah, that's right. If you tied it around the universe, you'd only need 30 feet or so to make it five feet more around in radius everywhere, which is unintuitive, but it's similar to this thing. Okay. Yes. I see what you're saying. That's actually a famous thought, you know, a fun math puzzle that my sister-in-law likes to say to my nephews around a dinner table. That's actually a great question.
00:46:28
Speaker
Yeah. And it just shows how things can be unintuitive even when we have all the information available.
Summary & Future Topics
00:46:35
Speaker
So we've come a long way here with black holes. We've started with a vague, massive object that's invisible. And then we've gotten to something that is literally a rip in space and time with Einstein. And now we're starting to actually talk about the degree of entropy in a black hole that's proven by Beckenstein.
00:46:55
Speaker
And also on top of that, the exact surface area of one bit of information. This is quite fascinating. You can see how this subject matter is really riveting for people who are interested in, well, the nature of this universe that we live in.
00:47:09
Speaker
Yeah, and one thing that we're going to be talking about next episode is frame dragging and how rotating black holes are different than normal black holes or I guess Schwarzschild black holes. We're going to be talking about how the edge of the universe relates to black holes because they're two sides of the exact same coin.
00:47:26
Speaker
We're also going to be talking about one of the most famous equations of the last century from Stephen Hawking, if not the most famous equation, and that is the information paradox equation, which relates to entropy, it relates to information, it also relates to temperature, and a really fascinating characteristic, the concept that the black holes over an obscene amount of time actually evaporate. That's a fascinating topic.
00:48:00
Speaker
In this episode, we discuss some of the nature and thermodynamics of black holes. On the next episode, we'll discuss some of the more seemingly creepy aspects of black holes and how they relate to things you may not have thought they related to.
00:48:13
Speaker
I'm Jonathan. And I'm Gabriel. And this has been Breaking Math. On this episode we had ourselves. That's right. Just you and me. Anything you want to plug? I want to plug this great show, Breaking Math. I'm so funny. Self-referential plug.
00:48:30
Speaker
Yeah, I would like to plug something. I would like to plug the book that I read about black holes. Now, I'll say something about this book. It wasn't all easy to read. I skipped around and I read some chapters, especially chapter eight, multiple times. The book is The Black Hole War by Leonard Susskind.
00:48:47
Speaker
And it came out a couple years ago. I don't quite remember how long ago it came out. But I believe that it's no longer up to date, which is the case with many things. There's been a lot of research into black holes and the information paradox since this book has been written. Yeah, which is insane because it's only eight years old. Yeah. Yeah. And there's a lot of stuff about black holes and wormholes and about even quantum entanglement just in the last
00:49:08
Speaker
six years that's really phenomenal. So that's one thing that's a great book. And I really want to say, so if you're reading it and you have a really tough time, I would say read chapter eight if nothing else. Wouldn't you say the same thing? Chapter eight is pretty great and they don't do build on one another, but I do want to add a caveat to what Gabriel said. He said it's difficult to read and he doesn't mean that it's difficult, I don't think, in the sense of
00:49:33
Speaker
mathematically, conceptually it could be difficult at sometimes, but it's also difficult to read in the sense that he says things like, and I love Leonard Susskind, I watch his lectures all the time, but after 15 miles I had reached exhaustion, but I was still a regrettable two miles from my warm office. Without my wallet, I didn't have even the necessary 20 cents to take the subway back. It's full of stuff like that. Yeah, it's like, why are you telling me that? It's like, get to the point, but it's still wonderful. He's kind of like a really adorable grandpa,
00:50:01
Speaker
who has some phenomenal stories and then just goes off about like politics or something like what yeah oh and it's funny when you watch this youtube lectures he's always eating something from starbucks but it's yeah oh yeah yeah totally so that and one more thing there's actually a british um a british documentary all about steven hawking's black hole information paradox i think it could be found at a couple of sources if you just do a you know if you just google it just
00:50:27
Speaker
BBC Information Paradox. I found it on a few websites, which I believe are legit. I believe they are. It was free. But that's another really great show about block holes and information. Also, there's a lot of stuff on YouTube that's great as well. There's some good PBS series as well. It's one of my favorite topics. I hope you enjoy it.
00:50:48
Speaker
Chandrasekhar, Sabrah Manyan Chandrasekhar, Sabrah Manyan Chandrasekhar. A guy by the name of Sabrah Manyan Chandrasekhar.