Become a Creator today!Start creating today - Share your story with the world!
Start for free
00:00:00
00:00:01
Black Holes: The Abyss Part 2
3.4k Plays4 months ago
Happy Summer! We have another week of our summer break after this episode from 2018. We will be back shortly with more content and surprises!
Black holes are objects that seem exotic to us because they have properties that boggle our comparatively mild-mannered minds. These are objects that light cannot escape from, yet glow with the energy they have captured until they evaporate out all of their mass. They thus have temperature, but Einstein's general theory of relativity predicts a paradoxically smooth form. And perhaps most mind-boggling of all, it seems at first glance that they have the ability to erase information. So what is black hole thermodynamics? How does it interact with the fabric of space? And what are virtual particles?
Keywords: Black holes, gravity, universe, physics, ai, machine learning, education, statistics, engineering, humanity
Subscribe to Breaking Math wherever you get your podcasts.
Become a patron of Breaking Math for as little as a buck a month
Follow Breaking Math on Twitter, Instagram, LinkedIn, Website, YouTube, TikTok
Follow Autumn on Twitter and Instagram
Follow Gabe on Twitter.
Become a guest here
email: [email protected]
Recommended
Transcript
Introduction and Recap
00:00:00
Speaker
Hello listeners, this is Jonathan Baca and I'm Gabrielle Hesh and you're listening to breaking math This episode is a part in a series about black holes You can go back and listen to the episode before this one But we will recap it before we get into the meat of this episode So you can listen to this one as a standalone
Understanding Schwarzschild Radius
00:00:18
Speaker
episode. We hope you enjoy Previously on breaking math So in our last episode, we learned quite a bit about the history of black holes. First first of all, we learned that the idea of black holes goes all the way back to the 18th century with physicist Pierre Simon de la Place and the English cleric, John Michel. Back then, they were called dark stars because they hypothesized that light could not escape them. It was pretty fascinating that they thought that an object could be so massive that light itself can't escape it.
00:00:52
Speaker
Using Einstein's field equations, which describe but how curved space makes matter move and how matter curved space, the German astronomer Carl Schwarzschild was able to calculate what became known as the Schwarzschild radius. The
Historical Theories and Entropy
00:01:05
Speaker
Schwarzschild radius describes a sphere. The Schwarzschild radius describes a boundary surrounding a black hole singularity. If you pass this boundary, there is no way of escaping. Light cannot escape this boundary, nothing can. And we talked a lot about escape velocity too. Right, escape velocity is the velocity that an object needs to be traveling in order to escape a gravitational well. For instance, on Earth, the escape velocity is, what is it? It's pretty fast, isn't it? It's like seven miles a second. Yeah, it's ridiculously fast. um Now, in a black hole, the escape velocity is greater than c, the speed of light. And as we know, nothing can go faster than the speed of light. Therefore, nothing can escape a black hole once it has entered the Schwarzschild radius.
00:01:46
Speaker
There were a lot of ideas in the 20th century that were wrong about black holes. yeah before the nineteen sixty s john wheeler who was a physicist at princeton express a belief that black holes are nearly perfect spheres with very little injurropy ah he said that black holes have no hair meaning no defining features Then in the 1960s, a physicist by the name of Jacob Beckenstein, who is also at Princeton, proved that black holes do, in fact, have very high entropy. Otherwise, they would violate the second law of thermodynamics. He
Black Holes and Information Theory
00:02:18
Speaker
proved this by a thought experiment imagining a jar of very hot gas, where there's very high entropy, being thrown into a black hole.
00:02:26
Speaker
And then it was proven that um the total entropy of the universe wouldn't decrease if the jar was thrown into a black hole. Therefore, there must be entropy accounted for. And this revolutionized the views of physics. We also talked a little bit about information theory. One bit of information is a yes or no choice, and we talked about how information theory is related to entropy. The same formula used for entropy in physics is used to describe information. Bekenstein knew that information theory and entropy were related, of course, and he did a thought experiment to that effect to deduce how much mass adding one bit of information to a black hole would add, and mass and the surface area of a black hole are related.
00:03:07
Speaker
And
The Information Paradox and Black Hole Thermodynamics
00:03:08
Speaker
we found out that using a photon with a wavelength equal to the radius of a black hole, the Schwarzschild radius, adding one bit of information would add one square plank length of surface area to the black hole, which is a very tiny amount. But it does not matter the size of the black hole, the same amount of area would be added every time. That's amazing. So essentially he he showed a model in which one bit of information is sort of like one pixel on an HD television, an ultra, ultra, ultra HD television, but still it's a one it's one pixel. I think that's just fascinating.
00:03:41
Speaker
Yeah, and so the statement that black holes have hair is at an extremely microscopic sale at the plank length wrong even though at any other scale basically it's correct. And we also talked about how information that cannot be retrieved using macroscopic processes. That's to say a process that does not know the exact location of every particle in the system is called entropy. It's a measure of disorder of a system. So when you mix dye into a glass of water, that increases the entropy because there's more disorder.
00:04:12
Speaker
So as you can see, the thought processes and the models for black holes changed a lot in the 20th century. And in this particular episode, we're going to talk about how else it has changed, especially with something called Stephen Hawking's information paradox. That's going to involve quite a bit. One of the things that we're going to talk about before we get into his paradox is a concept of virtual particles. That's right. So without further ado, here's the episode into the abyss.
00:04:42
Speaker
But black holes are objects that seem exotic to us because they have properties that boggle our comparatively mild-mannered minds. These are objects that light cannot escape from, yet glow with the energy they have captured until they evaporate out all their mass. They thus have temperature, but Einstein's general theory of relativity it predicts a paradoxically smooth form, and perhaps most mind-bogglingly of all, it seems at first glance that they have the ability to erase information. So what is black hole thermodynamics? How does it interact with the fabric of space? And what are virtual particles? All this and more on this, part two on a series of black holes, episode 31, Into the Abyss.
00:05:23
Speaker
I'm Jonathan. And I'm Gabriel. And you're listening to Breaking Math. Just a quick correction from last episode. When I was talking about the blocks, I said that one of them had four bits and the other one had 4.58 bits. I meant two versus 4.58 bits because there'd be four different ways to arrange two sets of blocks in ah different in different orders, as long as you're changing the order of each set of blocks. You know, I'm genuinely curious how many of our listeners actually caught that mistake. Just curious. We haven't gotten any email from it, ah but if you want to email us, you can email us at breakingmathpodcast at gmail dot.com. And of course you get the tensor poster on Patreon for a donation of $35 or more. The poster is going to ship probably next week or so. So what are we going to be talking about on this episode? follow up to last week of course we already talked about what we did last week this week we're gonna get more into gosh I would say the way I heard about this this is almost the pinnacle of physics in the last century and that is Stephen Hawking's information paradox that's where Stephen Hawking took what Beckenstein had done with entropy of a black hole and and he got into proving that black holes also have temperature and he also proved that black holes, we'll get into into the details here, but essentially he proved that over a long, long time black holes slowly ah evaporate, which is crazy. I never would have thought that would have happened.
00:06:47
Speaker
And by a long time, we don't even mean billions of years, which the universe has existed for a few billion. We don't even mean trillions of years. For the largest black holes, they're predicted to evaporate after 10 to 120 years, which is a one with 120 zeros after it. That's just unimaginably long. Unimaginably long. Now, this is an interesting thing because this is not just some insignificant fact about nature. Black holes evaporating have some drastic consequences, especially for things like basic causality in the universe. Now, there's a show that I was watching a while back. It was on the BBC and it was called um the Information Paradox. Yeah, I think we might have mentioned that on the last episode.
00:07:25
Speaker
This, anyways, the BBC documentary on the information paradox is what really got me interested in black holes and in this this entire idea. ah Essentially, this documentary opens up with this idea where the narrator says something to the effect of what if truly understanding the universe was impossible and science was in fact wasting its time trying to do so. This is one of the consequences of the black hole information paradox. And of course, it's an information paradox because our observations say the opposite, that the universe is very understandable at a macroscopic level. yeah And so that's that's that's what we mean by paradox. We don't mean that they are ah inherently paradoxical. We just mean that the way that we understand them is paradoxical, yeah or that we used to understand them. yeah
00:08:15
Speaker
and And I think real quick, so when we talk about when we talk about black holes evaporating, this is different from destroying information by burning it. Because obviously, when you destroy information, you you can certainly scramble it so that you have no idea which particle went where. And it would take you, you know, an impossibly long time to put, you know, all the ashes back together in order to, you know, restore paper. This is actually ah information matter going into a black hole. And then when the black hole evaporates, there's no trace of it again, this represents a ah problem. Yeah, because there's not a direct cause and effect, which we'll be talking about. We'll also be talking about ah virtual particles. And without further ado, here's the episode.
00:08:54
Speaker
Now we're going to be talking about black hole thermodynamics. Thermodynamics is essentially the study of heat. um I don't know why it's not called thermology, but there you go. I like that word, thermology. Yeah, and and actually I've also seen it as as ah described as thermodynamics as information destruction. Now, are we talking destruction or information scrambling? Well, we're going to be talking right now about what would happen if you did destroy information. ah like What would be the consequences of having information being destroyed? Because without getting too much into the next episode, ah there is the concept of virtual black holes popping in and out of existence all the time. And with there's also white holes, things like that, which are the opposite of black holes. But what we're going to be talking about is what would happen if black holes really did destroy information in the universe, even if we don't talk about the virtual ones.
00:09:44
Speaker
Yeah. And and in in in order to get and into those real consequences, you know, like we talk about things like the fact that information is reversible, or at least theoretically reversible. Yeah, even because if you burn a piece of paper, every every interaction that is done at the microscopic level, this that's key microscopic and on macroscopic level is a direct cause and effect. You know, like the ah plasma that is generated in the form of fire comes from the excitation of electrons and the emission of heat, things like that. um I don't know very much about burning. So part of that might be a little wrong, but there you go.
00:10:18
Speaker
Yeah, yeah. and and And again, you know, everything still exists, even though it's just big time messy, it still exists. You still have the ashes, the carbon, and the the atoms are still real. and It's not even a relativistic thing. You haven't transferred the matter into energy, which can happen, but you know, it so it still exists. And at the event horizon of a black hole, which is, you remember, you cannot get information back from, ah if you think about throwing a book at a black hole, there's nothing, at once a but book passes the event horizon, that could retrieve the book. You can't, even if you tie it to a string, you'll probably get pulled in too. So that's what that's why it seems like it destroys information. You could
Virtual Particles and Quantum Fields
00:10:55
Speaker
throw, I mean, you have a radio station, you beam energy at it, it's destroyed. You throw a brain that remembers things, it's destroyed, yeah seemingly.
00:11:02
Speaker
so So, and again, now now, this is just talking about just when you throw something into a black hole. Later, we're going to talk about the eventual consequence of when the black hole no longer exists, which is another information paradox as it were. But what but what you're mentioning right now certainly is relevant. Yeah, absolutely. And ah so destroying information means generating entropy. And why is this? or it It's a little bit ah involved and we're going to go into that. And before we do that, we're going to remind you that entropy is information that can't be retrieved using a microscopic process. So if you if you basically come across a book that's burned, you can't know how to like put it back together because remember you have um ah things like
00:11:46
Speaker
Heisenberg's uncertainty principle, which states that you can't no but you can't know enough to put it back together, basically. yeah And also, one of the things ah that weve we've we've talked about before are are you know states, where if you know the order of something, like let's just say you've got blocks that are numbered or or another example is if you've got time stamps on something when they're produced that information exists, but there are tons of examples in this world where where you know and an event happens in a or an order of events happen and there is no such time stamp and it becomes impossible to to know what order things happen just because that information is not available. Let's say you're destroying information. Why does that generate entropy?
00:12:33
Speaker
We're gonna answer the opposite of that first. Why does having information mean lowering entropy? And remember, what lowering entropy means is making something ordered. And that can be done as long as you increase entropy somewhere else. An example of that is learning, what you're doing right now, hopefully. When you learn something, you're putting order into your brain. But in doing so, you're burning a lot of stuff in your body. The body is basically a very slow burning reaction. It's oxidization. And I think it's very consistent with the law of thermodynamics where the amount of energy that it takes to learn is greater than or the the amount of disorder in your brain metabolizing your food and in your radio or your iPods or your podcast players playing this. That amount of energy is greater than the amount of energy stored or you know, I will say it's greater than or equal to.
00:13:24
Speaker
but Okay, yeah, greater than or or equal to the- But probably greater than. Yeah, the the learning that's happening. So so although you know learning is certainly ordering information in your brain, that there' there's greater than or equal to disorder happening elsewhere. Yeah, and an example of this is ah called Sizzlar's engine. And what Sizzlar's engine is, is imagine you have a tube with a divider in the middle of it and two pistons on either end. A piston is kind of like a disk on a pole, kind of like if you took an umbrella and opened it too much. So it seals off each so it like it seals off each side of the cylinder.
00:14:05
Speaker
Now in this setup, we have exactly one gas molecule. The gas molecule is either on the very left or the very right hand side of the divider. And there's nothing else in between. betweenen Wow. So yeah, there's no air. So like there wouldn't be air pressure, right? When the pistons go in. There wouldn't be air pressure, but the molecule itself would push the, would bounce around creating a little pressure of its own. And that's the, and that's the thing is that if we know which side the particles, let's say the particles on the left hand side, what we could do is we can move the right hand piston close to the divider and then open the divider up and then the piston would be pushed back creating useful work.
00:14:43
Speaker
Now, how do we know this? That's not important. But the fact that we did know this means that we created work out of something that you're not supposed to be able to create work from, which is a microscopic process. So that means that we're lowering entropy. So in the example of Sizzler's engine, you create useful work, you lower entropy, and create usable work. And because it did that, it expended a little bit of its energy. And remember, temperature is a measure of how much energy is in like a gas or something like that. It's a measure of like vibrational energy. it's it's a It's a property of a macroscopic system. even And we're kind of treating a one-particle system as a macroscopic system in this instance because the processes we have are macroscopic. But the temperature of the cylinder is lowered. So we're lowering in temperature by knowing something. And we possessed exactly one bit of information, left or right. So one bit of information lowers temperature. So we know that entropy and information are very similar. So why does destroying information not mean reducing entropy?
00:15:44
Speaker
And the reason why is, of course, because entropy is not usable, usable information in a macroscopic way. It's like if you translated English to, I don't know, Martian, like you wouldn't be able to get that information back, but it would still exist. So even though it's a hidden information, it still exists. So in a way, because when you're destroying information, you're also taking away usable information, it looks the same to the universe. I can see why this might be confusing for the listeners because earlier we had said that information is proportional to entropy. The more entropy in a system, the more information. However, that includes that includes hidden information as well.
00:16:18
Speaker
Yeah, structured information ah it does not really contain entropy. And it's a very it's in it's kind of a rough rough model treating information exactly as you ah entropy, because they they are slightly different. But they're close enough when you're talking about a macroscopic process, especially when you mean that this is not usable information. So so you say usable, this basically, if I'm not mistaken, correct me if I'm wrong, we're talking about meaningful information. So so so yes, when you you know you have disorder, there's more information, but meaningful information is you know like when you get ah you know a word that's in our dictionary, it's a fixed word. But if you add a bunch of more letters to it, there's more information, but it's not meaningful.
00:17:02
Speaker
right Yeah. And, um, really what it means in the physical sense too, it has to do with the second law of thermodynamics. A quick recap of that. The second law means that you can't extract work from a system as unless there's a heat difference. There's a lot of energy in the air around you, but if you had a heat engine, you would have to either have a ice cube or a, or a lighter to make the heat engine generate work. So entropy is that cannot be extracted as work. So the crux of this is, if black holes really could destroy information, the universe would get very hot very quickly. um It would heat up to ah a ridiculous temperature in a very short amount of time.
00:17:45
Speaker
What's interesting is is that for so long, Stephen Hawking still held that his information paradox was true. And I'm wondering if the way you're thinking about it is sort of the way Leonard Susskind was thinking about it in some sense. Leonard Susskind had an intuition that the information paradox was wrong, but he couldn't quite formulate why it was, at least for a period of about 30 years or so. And one of the reasons why is because we understand that black holes exist, but the fact that they have entropy means that there are processes that are going on ah there because there's a right above the event horizon of the black hole right outside it. We'll go into this next episode a little more, but a lot of stuff is going on right outside of a black hole. And it's my view that this is what happens to the information that the edge of a black hole is the only thing that exists and inside of it,
00:18:31
Speaker
is something that would be useless to talk about because it doesn't exist. But we'll talk about that more in the next episode. Before
Hawking Radiation and Black Hole Evaporation
00:18:39
Speaker
we get more into Stephen Hawking's information paradox, we need to first talk about something called virtual particles. Which aren't even real particles. Gosh. So yes, virtual particles are a thing. They're they're they're not real, but they are a thing. that This is one of the stranger things that I've read about in the last 10 years, and and I've really struggled to understand them. I think it wasn't until I was listening to a podcast, I'll go ahead and plug it here, um I was listening to a podcast called Ask Science Mike, which is a great podcast actually, and he he actually explained ah virtual particles, and I think it was fantastic. He explained them essentially by saying that there are fields everywhere in our universe, and ah virtual particles exist from perturbations in the fields.
00:19:24
Speaker
um Which is interesting, um you know, I haven't thought of it from that aspect what would you say Well, I think we've got to talk about what fields mean first what virtual particles are and like who how they're discovered Yeah, like how many virtual particles are there? You know what I mean? Like I I know there's quite a few subatomic particles. I mean, there's there's bosons, there's W bosons and Z bosons, and, you know, the force mitigators and gauge bosons. I mean, I forgotten what each each of those are. It's been a while, but you know, and yeah I think of virtual particles, yeah like, you know, how, like, they're measurable. um And so how many are there? And I guess we also, we also discussed, why are they called virtual particles if they're not even real?
00:20:06
Speaker
Well, they were invented by Feynman, and I say invented because virtual particles, because they don't exist until ah we measure them really, they have to be measured with an increase of energy. Now, let's talk about what a quantum field is, what a field is in like in physics. Imagine the surface of a very still pond of water, and let's say you throw something in it, it ripples outward. That is kind of like, that that is a wave. And matter exists in waves, but kind of a weird wave that stays around um and where the frequency of the wave is actually the the momentum of that particle.
00:20:43
Speaker
there Did you know I recently read about a physicist, de Broglie, I believe is his name, the French guy. And I think he actually said that that if if photons can be both waves and particles, then he also said that means that matter or or particles can also be waves. is that Is that related to virtual particles? That is a little bit related to virtual particles. And how we're going to talk about how it's related, we have to talk about, again, Fourier sequences, which I think we did on an episode called frequency. Yeah, yeah so so real quick here, just to underscore the the point that I just said about de Broglie, that would mean that everything, you know, quarks or or neutrons and and protons also have a wave property, just like electrons have a wave and a particle property. it's It's strange. Yeah, and matter is made out of waves, and um we're going to play a sound really quick.
00:21:39
Speaker
That sound is called a triangle wave. Now a triangle wave can be actually made out of sine waves, which are the purest wave in a sense, which sound like this.
00:21:51
Speaker
Of course, for for our our listeners who may be, you know, eighth grade and younger, when you think about a sine wave, it's just a very simple oscillation. Like if you're riding your bicycle, it's it's the pattern that your foot does, the up and down and up and down. That's a sine wave. Yeah, and the left to right is not is also a sine wave, but it's in a different axis. And um so the volume, ah we could measure that as the volume at any point. For a triangle wave, the alt volume would go up steadily, then down steadily, then up steadily, then down steadily. And for a sine wave, it it would go up slowly, up quickly, slow out as it got to the top, then go down slowly, speed up, and then it would go do that over and over again, like a bicycle.
00:22:32
Speaker
And but we could measure it like that, or we can measure the frequency. Now the frequency of a sine wave is something very simple. it's It's one number. But what's the frequency of a triangle wave? We could answer this by building a triangle wave out of sine waves. And that sounds like this.
00:22:55
Speaker
You could actually play around with something like this on our website, breakingmathpodcast.com, and go to the Fourier sequence app. That's right. That's right. There's a Fourier sequence app app that that actually shows how, gosh, how how how do you describe it essentially? Essentially, you start off with a very, very basic sine wave, almost like you know riding a bicycle and pedaling, but then you you sort of add sine waves on top of sine waves. It would be like if you added a smaller pedal that spirals around on top of the other pedal. yeah i suppose I don't know if that's a good example. Oh, that's a great example Yeah, and then a smaller one in and a smaller one and a smaller one things just get crazy And the radius of the first pedal would be let's say one foot ah That's a very big pedal But still the second pedal would be half a foot the third one would be a third of a foot and so on If you look at the very smallest pedal on something for a triangle wave It would go up steadily and then sharply go down and then sharply go up
00:23:44
Speaker
Man, I just I love thinking about sine waves and then you can talk about things like Constructive and destructive interference and how you can change the shape of them and modulate them That's it's it's quite amazing. And when we're talking about volume, that's a one-dimensional thing with one dimension It means technically kind of two-dimensional but it's one-dimensional at its base because it only goes up and down as time goes on whereas matter is a we exist as four-dimensional waves, x, y, z, and t, which is forward, backward, left, right, up and down, and past and future.
00:24:17
Speaker
So matter is also made out of waves, where the height of the waves is kind of like the probability that matter exists at that spot, and the frequency is the momentum. So just like volume and frequency are what's called a Fourier pair, meaning that a Fourier transform can turn one into the other, go from time into frequency space, you could do the same thing with matter using position and momentum. Position is a Fourier transform of momentum and vice versa. Wow, that's that's amazing. that That's mind blowing right there. That just blows my mind. And that's actually why um the uncertainty principle exists. Because let's say you had a wave that existed at one point and nowhere else. We would know exactly where that was. But if we knew exactly where that was, we wouldn't know how fast it was going. And the reason why is because you could imagine on this pond, if you tried to make a and ah splash that was just one point, you would have to splash basically everywhere in the pond, if this infinitely large pond.
00:25:16
Speaker
And so you would not know anything about the momentum because you'd be splashing everywhere, where splashing is kind of like adding, defining momentum. It's it's a difficult analogy, so you might need to listen to this twice, but it's ah essential for understanding virtual particles. And the reason why it's essential for understanding virtual particles is because the vacuum of free space does not have zero energy. Yeah, that's that's amazing. Just the fact of what you just said there, that a vacuum does not have zero energy. And there's actually physical evidence of virtual particles in a vacuum. And it's something that I recently saw on a very old rerun of the Big Bang Theory, where they talk about the cashmere effect. This is a really trippy thing. Jonathan, do you want to hit on the cashmere effect?
00:26:03
Speaker
Oh yeah, the Casimir effect is, let's say you have two plates that are very, very close together. The waves and that are generated in free space be ah will of like matter and everything will either push apart or bring together these plates, kind of like magnetic ah energy. And and yeah, so so basically you have these two plates in a vacuum and there will be a positive pressure that either pushes them together. And again, isn't it dependent on ah the space between them? It's a fraction of the wavelength, isn't that right? Yeah, it has to do with ah how if the if it's a full wave or a half wave and destructive and constructive interference, but it is measurable. And you can there's actually a video on the Wikipedia article about it that shows how to do it with sound waves. Yeah. So wow. So OK. And again, I believe that virtual particles in a vacuum average out to zero energy, but that's only the average. Well, zero measurable energy um using like using zero useful energy, basically. ah And the reason why is because you could imagine the pond again, is the pond of existence. Because there is an energy, there is a certain probability that a particle will exist everywhere and everywhere. But the particle is not defined so that one of the particles can have negative energy. And what this means is that these particles, you could think of them as popping in and out of space, a pair of them, one going forward in time and one going backward in time.
00:27:24
Speaker
and just popping in and out of existence faster than you can measure it. And the way we currently understand it, if I'm not mistaken, is it's always pairs. Is that right? Oh, yeah. I mean, yeah, because it's never triplets. I mean, I know it's a dumb question, but I need it. I need to clarify for my own edification. Oh, yeah. No, they're definitely ah pairs. And the reason why they are pairs is because in the pond of existence, there is a construction operator and an annihilation operator. And what what is an operator? An operator is just a type of wave. It operates. Yeah, so there's one wave that creates new particles and one wave that destroys old particles and the reason the way that this kind of works is like ah Let's say you send a photon you let's say let's say you go to a black light ah party you and there's black lights everywhere and people are glowing because they are drawing on each other with highlighters. That sounds like a lot of fun actually. That is fun. Can we actually add some background tech now to this section? Absolutely. you know So when this happens you have this photon of that's almost and ultraviolet that hits the highlighter. It hits one of the molecules in the highlighter and it hits the electron and in doing so it beams out another photon. Now, what happens is that the old photon is destroyed we're using the destruction operator, but the construction operator ah creates a new photon. And this is of course done because when you beam it to the electron, the electron jumps up a shell, it it gets farther away from the nucleus, and then it does it, it but it's not stable there, so it goes back down, and in doing so it generates a different energy photon, a different wavelength photon that you can see. So I'm imagining, the way you're describing that, you want to know what's happening in my head right now? At first it was a pool table where, you know, you had billiard balls disappear when they hit other billiard balls, but then a whole Rube Goldberg machine where parts of it disappear when other parts start happening.
00:29:14
Speaker
And that's kind of what happens in the universe. The universe is not how we see it. It it can only be understood through analogy, but it can and be understood very well through analogy. And so what happens is that you have these construction operators, these destruction operators, everything is in harmony. And you can only measure a virtual particle is if you interfere with it using energy. For example, if you bring two plates very close together, you have the Casimir effect. Or if you destroy part of space using a black hole, you can also detect these particles. Wow. So I think one of the main takeaways before we go into the next section is that virtual particles do come in pairs and they pop into existence and then they and and they annihilate extremely quickly. And they have to have its pair to annihilate. Like they crash back into each other. And this is different than real particle annihilation really quickly, because the terms are exactly the same, but they mean different things. When two real particles annihilate, they just change into two different types of particles. They or yeah they they do something that's that's conserved. Annihilation, in this sense, means complete destruction with nothing ah to show for it. yeah That's crazy. So that's almost like, as you said earlier, destructive interference in the wave. it just It's not anymore. It was, and now it's not. And that's just, ah, it's weird. It's just weird.
00:30:27
Speaker
Yeah, you can think of the universe as an accountant that is very, very picky about things. And you can think of a virtual particle as a somebody returning something that they bought before anybody notices. I like that. I like that. Yeah, some sneaky accounting there. So why are we talking about virtual particles and black holes? How are they related? Okay, so Stephen Hawking knew about virtual particles, of course, and he began to think about, well, how do these how does this phenomena of virtual particles relate to black holes? As we said earlier, um in the vacuum of space, virtual particles are always popping in and out of existence in pairs. So that means that even at the edge of a black hole, at the event horizon and near the event horizon, virtual particles are popping in and out out of existence there too. So now imagine that you have one of these virtual particles popping in and out of existence near the edge of a black hole. One of them falls in and the other one doesn't. Now that means that when we're talking about the pond of existence that the annihilation operator basically kind of fell into the black hole but the construction operator
00:31:33
Speaker
did not, which means that the black hole gets smaller and as the particle goes away. And the reason why the black hole gets smaller is because, like we said, the universe is an accountant. What happens to the particle is it falls into the black holes in none of our business because inside a black hole is not measurable. Yeah, and that that's a really interesting, um I would say crush ah crux of quantum mechanics is what's measurable and what's not measurable. And that's just something we kind of have to give up, wouldn't you say, in terms of understanding understanding the universe is just what happens inside. We we can't do that with with modern physics and it may not be possible at all.
00:32:09
Speaker
So then, you know, when we're talking about virtual particles and black holes, ah Jonathan said when the annihilate, how did you say it? Annihilator operator. Annihilation operator. the Annihilation operator goes into the black hole. The black hole gets smaller. Something else that I want to say. ah When I was first reading about it, it highlighted the fact that outside of the black hole, you now have a real particle that exists in the universe that didn't exist before. And this is very, very important because again, As Jonathan said, this is the universe being an accountant, as it were. So you don't have you don't have any more net mass or any more net energy. So although although you have a real particle that's that's now radiating away from the event horizon, and that is actually called Hawking radiation. and and It's a very, very low energy radiation, but it definitely exists.
00:32:59
Speaker
Well, for normal um black holes, it's very low, but the sum of all the energy over the black hole's lifetime is very great. It converts all of the energy and into radiation. And when we're talking about the pond of existence, ah you might be wondering how black hole looks on that. So just imagine you have this infinitely large pond, and every and the thing is, there are splashes everywhere, but the splashes cancel out. That's ah zero point energy, and that's those are virtual particles, the things canceling out. um So now you can imagine you build a an island, ah a round island in the pond. That's kind of like a black hole, because a black hole reflects the operators. The annihilation and the construction operators but ah are essentially reflected.
00:33:42
Speaker
And I say essentially because they have to be redesigned around a black hole. and But the annihilation operator the new annihilation operator is a combination of the old annihilation and construction operators, and so is the new construction operator. And the crux of this is is that because there's not a lot of because the splashes that would be happening on the island are not happening, and because the splashes that are happening outside of it are reflected, that essentially creates radiation. It's the same thing with the macroscopic view versus the microscopic view. yeah And as we said earlier, with that annihilation operator, the black hole very, very slowly gets whittled down and after a very, very long time, in fact evaporates. ah Now that is not an insignificant event. As we said earlier, this has huge consequences. As we
Consequences of Information Destruction
00:34:30
Speaker
had mentioned, we know that whatever goes into a black hole will never go out.
00:34:36
Speaker
So what about when a black hole evaporates? Is information destroyed? And I'd like to say, so if you were to add up all of the Hawking radiation that that that comes off of a black hole from the virtual particles, if you were to add that up, would that add up to less than the sum of all the mass of other things that went into the black hole? ah the mass Well, the the energy is the same. When it's a large black hole, it evaporates mostly photons. Very small black holes e evaporate things like electrons and other things. They're extremely high energy, as we'll talk about in a bit. But energy is conserved with a black hole. um like And remember, mass is energy, E equals mc squared.
00:35:14
Speaker
So in other words, like let's say that you did have a black hole of any arbitrary size and and it and throughout its lifetime it sucked in a bunch of things, you know, a nebula, some planets, all kinds of things. the The effect of virtual particles over the life of the black hole will eventually cause it to evaporate. What I'm saying is, will any information, according to Stephen Hawking, any information about what what went into it, when that black hole evaporates, it's gone. Is that right? Essentially, but I think an analogy will clear up why there's confusion around this, because the black holes ah the black hole evaporating actually solves the violation of the second law of thermodynamics. Otherwise, black holes would reduce entropy, because if you threw a very hot
00:35:54
Speaker
think of gas into the black hole like we said in the previous episode the universe would get a little bit less entropy and that is impossible because any closed system can't reduce its entropy. but So the analogy I'm thinking of is imagine a couple of kids in an overcoat trying to get into the movies. So they get into the movies, and then they get out of the overcoat, and then they're walking around, they get popcorn, whatever, one of them, and then one of them goes to the bathroom and changes their shirt because they, I don't know, they wanted a new shirt. You can't see them going into the bathroom, and and then they and then say they go through an air duct or something, we don't know what happens, and they come out of a different bathroom. Everything is conserved, but we weren't able to follow them. If we had cameras, we would have a ah gap. And that's what is really disturbing to a lot of people,
00:36:37
Speaker
is that there doesn't seem to be a direct line of accounting. However, there's been research, which we'll talk about in the next episode, which says what might happen to the information and how the information is actually conserved on the surface of the black hole. Because as we said, one bit of information increases the surface of a black hole no matter what size it is by the same amount. Now, when this first came out, there were a lot of physicists who were very bothered by this, including Leonard Susskind, as well as a gentleman who I'm having a very hard time pronouncing. it's I feel bad about this. Gerald T. Hoofed.
00:37:09
Speaker
yeah i don't know to of i don't know If we can edit that out or not. I'm sorry, I'm not doing great. Dutch names are hard to pronounce. Yes, yes they are. He's a very, very brilliant physicist, but they were very, very bothered by it. Actually, watching the Hawking Paradox documentary on the BBC, Leonard Susskind recounts after he was first introduced to the idea of the information paradox. He was so shaken by it that on his drive home he said that it was raining and there was a lot of fog on his windshield. At every single stop he would sit and with his finger do physics equations on the window because he just didn't want to believe it. It was just that disturbing for him. So um maybe we can talk about a few reasons why somebody like Leonard Susskind might find the idea very disturbing.
00:37:56
Speaker
Yeah, like um for example, like if Hawking radiation didn't exist, it would violate the second law of thermodynamics and information destruction itself does violate the second law of thermodynamics. So even if that violation is like instantaneous or temporary, it is an apparent violation. Maybe it's a virtual violation. There are other scary consequences of of the idea of information being destroyed. This essentially means that causality doesn't exist, which would mean that the way we know things about the universe would essentially not work.
00:38:28
Speaker
Because remember, causality and the conservation of information means that everything has a everything has a unique previous state. So let's imagine you have a book and you throw it into a black hole and it's destroyed. If a black hole can destroy information, that means you could have thrown anything into that, which means there's a whole bunch of different past circumstances which could exist. You destroy causality because you create basically a web of causality, which isn't the way that we currently understand causality in the in the quantum field theory.
00:39:01
Speaker
and And for physicists who are really you know far deep into this area, there's other consequences as well. For instance, there's no reason why virtual particles are sorry virtual black holes couldn't exist in your own mind that would destroy your own memories. You couldn't even trust your own memories. this is ah the the The idea of information and being destroyed is a very, very big deal. Yeah, because it means that yeah if causality doesn't exist, then everything that Francis Bacon said about science is wrong, and essentially.
00:39:32
Speaker
So, the destruction of information is not good. It's no bueno. However, we know that in the universe we don't have the destruction of of information even with all the mysteries surrounding black holes. We know this because... The universe is not boiling hot. The
Black Hole Temperatures and Energetics
00:39:50
Speaker
know universe the universe would be extremely hot if black holes destroyed information. Because at the beginning of the universe, it created a lot of very small black holes. um And some of them still exist. And yes, black holes are or at the center of galaxies. They're part of our universe, so they would heat it up very quickly because of the destruction of information. Yeah. While researching this episode, actually, we we did some calculations of some some black holes. Jonathan, what is what's the cap what what's the temperature of a Schwarzschild black hole that is one kilogram?
00:40:22
Speaker
And just real quick, a Schwarzschild black hole is a black hole that is not rotating and that has no charge. Because remember a black hole from last episode must, it has like three properties. It has mass, it has charge, and it has rotational energy. Yeah. So again, just, uh, we, Jonathan did some calculations, you know, uh, including the temperature of a one kilogram black hole. And the temperature is 1.22 times 10 to the 26th. That's one to two with 24 zeros after it degrees Kelvin. And that's just from all the, the, the, the, the crazy virtual particles popping out, right?
00:41:00
Speaker
Yeah. And just, and just to give you an idea of a hot that it's, um, because it's so high, um, because Kelvin is just 273.15 degrees more than Celsius at any point. So you can't have negative degrees Kelvin, but 1.22 times 10 to the 26 degrees Celsius is very hot. It's, it's so many, it's, it's almost incompetent. There's no real, there's nothing we could talk about that would make that make sense. It's just so hot. Yeah, I think it's hotter than like a but then an atomic bomb, actually, even. It's hotter than the set center of the sun. It's very hot. And this black hole over its lifetime will turn all of its mass into radiation. And that radiation over its time lifetime is equal to 21,000 kilotons of TNT. Wow, that's a lot. That is an excellent power source.
00:41:51
Speaker
Yeah, it's a little dangerous. Anything that it touches gets sucked in and destroyed forever. Or does it doest get destroyed? Fantastic source of sci-fi. And I'm sure I have not read all the sci-fi that utilizes black holes. Now, how do we get to 1.22 times 10 to the 26 degrees Kelvin? That's the temperature of a black hole formula that was discovered by ah Hawking and friends. Now we're going to talk about the formula and it's pretty mathy, but it involves so many constants that are used in other formulas and they're used in physics that it's kind of mind boggling. It kind of combines them all. Yeah. I don't, I don't keep a lot of math details in my head, but for the sake of this formula that we're going to talk about, if you can at least, you know, pay attention to it to each of the constants that that we talk about yeah and you can, you know, process it later. That's, that's kind of the way that, you know, that, that was my take on it. Uh, at least it's, it's, it's certainly helpful.
00:42:43
Speaker
Yeah, so it's equal to the reduced Planck constant divided by all of the following. The speed of light times 8 pi g over c to the fourth. That's 8 pi times the gravitational constant divided by the speed of light to the fourth power. And we'll show why we're using that twice, the speed of light twice, because some of you might be wondering why didn't I put the speed of light cubed on top. Times the Boltzmann constant times the mass of the black hole. No, it's a good, it's like a, you know, eight triple quadruple Decker sandwich made out of all the physics constants altogether.
Mathematics of Black Hole Entropy
00:43:15
Speaker
Oh, yeah, it's amazing. And the reason and we're going to go through each one of these. So first, the reduced plank constant. ah we're We need to talk about that. We're going to talk about the plank constant real quick.
00:43:25
Speaker
Let's say you have a photon with a certain frequency. The energy of that photon is equal to the Planck constant times the frequency, but it's used in so many other things too. One of which being the uncertainty principle, the modern Heisenberg uncertainty principle, which says that the standard deviation, meaning how fuzzy basically, The momentum is times how fuzzy the position is, is greater than or equal to the Planck constant divided by four pi. And the reduced Planck constant is just the Planck constant divided by two pi. And it's sometimes called h bar. So that's the first constant and everything else is, it divides by. Oh goodness, man, I am, I'm i'm out of breath. I feel like I'm in like a really hard spin class and I'm, I'm, I'm just, you know, it's my first day and that was the first rep. You know what I mean? So, so.
00:44:13
Speaker
Stay with us. Stay with us. But we're gonna up the resistance. Maybe we'll put some more techno music. Throw the techno on there and get through this equation. Whoo! Feels good. Feels good. Okay. So next constant is 8 pi g divided by c to the fourth. And that is related to how matter and the curvature of space-time relate to one another. Because if you remember, the fact of gravity exists because of the curvature of space-time. So space tells matter where to go, and matter tells space how to curve. So it's the Einstein tensor, which is a whole thing. You can actually check it out on our tensor poster. is equal to 8 pi g over c to the fourth times the stress energy, which in the stress energy is things like the energy of the particle itself, like E equals mc squared, the momentum, the shear stress, and the pressure. All of those create gravity waves, and that is the constant that relates to those. Oh man, so again, if we're using the analogy of the spin class at a gym, I feel like I'm just out. My legs are still pumping, but I'm just kind of in a zombie middle. And remember, real quick about that too, G is the gravitational constant, and that is the force that is between two massive objects. So you take the mass of the two objects, multiply them together, divide by the square of the distance between them,
00:45:29
Speaker
And then you have your proportionality constant, which is G, and it's very, very small, because gravity is incredibly weak. To prove that to yourself, rub the balloon on your head, and then put it on something else that sticks to it. That is doing more than the pull of the entire earth on that balloon. And after that, we got some that that are... Okay, this is the downhill part of the skin class. We're going to reduce the spin. The resistance. Yeah. So now we're we're now shifting to low gear here. The next part of the equation we'll talk about is C, which is just the speed of light. Whee! And the speed of light is how fast information can go. Nothing can exceed the speed of light because if it did, I mean, it just doesn't. You know, like if you look at the formulas for even special relativity, going faster than the speed of light just doesn't make sense as a thing. It's like saying, I want bread that's extra bready.
00:46:20
Speaker
Sorry, man. Far out, man. Far out. All right. So what do we have up next? Boltzmann's Constant, which is K with a subscript B? Is that right? Yeah. And Boltzmann's Constant shows how much energy of particle has in the gas. So let's say you have a balloon with a certain temperature. You multiply that temperature by the Boltzmann constant, and you get the energy of one particle in that gas, an ideal gas. And it doesn't matter what the gas is, it'll have the same thing. And so that's an energy of each particle given the temperature of many. And that's different than E equals hf, which is the plant constant, which is the energy of one particle given the frequency of one.
00:47:01
Speaker
So we have microscopic laws, macroscopic laws, and all these are multiplied together on the bottom. them oh i'm sorry interrupt the flow Okay, I was gonna say, we are now down to home stretch. We've got one more owner's equation, and that is with capital M. Which is just the mass of the black hole. How much it weighs. And you might say, how can we put a black hole on a scale? It'll just suck in the scale. We can measure the mass of an object in space just by seeing how it pulls on other things, because remember gravity exists. And black holes do not violate gravity. If we made the earth the size of a cranberry, it would change into a black hole. And the moon would just keep orbiting, and the black hole would keep orbiting the sun until it evaporated, and it'd be fine, basically. Yeah, so you don't measure it directly. Kind of like, uh...
00:47:48
Speaker
Yeah, yeah, exactly. Sometimes you don't want to measure how heavy something is, but you can infer. Yeah. And the weird thing about this is because the mass is on the bottom. That means that bigger black holes are colder. So the bigger black hole is the colder it is. So finally, when you put all these constants together, we've got s, which is the entropy of the black hole, equals k subscript b times the area of the black hole times the speed of light cubed, all divided by 4 times the gravitational constant times h-bar. That's all of it.
00:48:24
Speaker
And that means that the entropy of a black hole is dependent only on the area of a black hole, which means that the boundary of a black hole, the area that is the black hole, is the only thing that creates entropy. And that means that entropy must be stored there. So the information paradox really at this point is, does the Hawking radiation conserve the information or does it create new information or what happens there? does cause Is causality violated? What happens to the network of causality? We have to answer these all in the next episode, probably. That's right. And that is the equation that I get a tattoo
Listener Interaction and Closing
00:48:56
Speaker
of if I, if I were to ever get a physics tattoo. How many, how many emails would it take for you to get that tattoo? Do I have to commit to this? So Gabe has always talked about the information paradox formula and how he might want to get it as a tattoo. And I've been so on the fence about it, but, but here we go on, on on the record. If, if, if we get a hundred emails in a period of two months, then I will get this tattoo.
00:49:23
Speaker
Yeah. also So just email us with anything, questions, whatever. And we're not going to count the emails we you do for business. ah And if we get 100 of them, Gabe will get a tattoo and we'll post the video of him getting the tattoo on the Facebook page. That's right. You heard it here first.
00:49:40
Speaker
Information seems to be destroyed when it goes into a black hole, but we observe a universe in which that does not occur. However, we do not technically know if the information that goes into a black hole is the same that comes out, because we lose the train of accounting as soon as we involve the event horizon. And studying black holes involves much of the physics that we have discovered in the last century and a half. So what can we do with such interesting objects, and what is the current state and the future of the study of black holes? Stay tuned for the next episode. I'm Jonathan. And I'm Gabriel. And this has been Breaking Math. ah Gabriel, do we have anything to plug? Yes, yes. We've got a substantial amount of things to plug. Oh, I made a list. I have a list of reading that I've done. The first thing I want to plug is not related to block is not related to to to black holes. The first thing that I want to plug is a work of fan fiction that I've read that is so fantastic that I just wanted to share it with all the Breaking Math listeners.
00:50:32
Speaker
I read a book called Harry Potter and the methods of rationality. It's written by—oh, I'm not prepared here. as written I will get you the author's name. Essentially, you know a devoted fan of Harry Potter rewrote the story. Not not the entire story, but parts of it. um as though his parents are not the Dursleys, but but professors at a college. I think maybe chemistry, and mathematics, or something like that, I don't quite recall, but every chapter has Harry intensely analyzing phenomena around him, you know, in the magical world of Harry Potter, and he looks at it from all angles, and he uses the the Francis Bacon method, you know, the the Francis Bacon scientific
00:51:14
Speaker
method where you test for positives and for negatives. And, you know, he he talks about the mathematics of Richard Feynman and the book Godel Escher Bach. In fact, the chapter I just read was a chapter where he has the sorting hat on and he he is thinking about the consciousness and the awareness of the hat. And the hat reads his mind thinking about the hat's own consciousness and it becomes a strange loop. And the the hat becomes conscious all of a sudden. and it It references Godel Escherbach. it's Which we did an episode on. Yeah, yeah yeah we did. so So if you guys are looking for something really great to read, I would i would Google search Harry Potter and the methods of rationality. It's so enjoyable.
00:51:56
Speaker
We also have the poster to plug. The prints have been ordered and they're shipping soon. So if you want to get a poster, they're $35 on our Patreon. ah The price will go down at some point. ah But right now that's basically a breaking even for us. We just want the poster to be, you know, we want you, we don't want you to have this poster. We worked hard on it. Yes, it's it's been a very, very long process to get the prints correct. Earlier this past year, we had prints from so many shops and they't with they were never of a quality that we were happy with. They always had strange lines on them where the paper was too thin. We finally found the shop that that has the equipment to get us what we need. So those posters are are being printed and they are being shipped very, very soon. So yeah, make sure you check out the Patreon. which What's the Patreon website?
00:52:44
Speaker
patreon dot.com slash breakingmathpodcast we're also on facebook at facebook dot.com slash breakingmathpodcast twitter at breakingmathpod we have a website breakingmathpodcast.com which will be updated soon and it really needs to be updated and ah one thing i also want to plug is pbs spacetime it helped a lot with this episode um at least you know getting some of the stuff primed and it goes into a lot of detail. it's if If you like ah but another plug, three blue, one brown, it's very much like that. Yeah. Was there a specific episode of Space Time or did you just kind of flip around on it? Oh, i I lost count of how many episodes I've watched. Okay. Yeah. And of course, as you guys know, we've mentioned so many times, I think that the book, The Black Hole War by Leonard Susskind is absolutely required reading. I did say that parts of it are, how do I explain this?
00:53:35
Speaker
There's so much brilliance in it that it's a must read, but I'll also say that there's other segments where he's not talking about black holes, and instead he's talking about, you know... Sandwiches and stuff. Yeah, yeah, yeah. Sandwiches or shopping or something. But it is charming. You get to love him throughout the book. You get to love him, yeah. So I would say it's a must read for sure. ah That as well as the the the do the BBC documentary on the information paradox as well. All good stuff. And I think lastly, there's and a podcast that we really, really admire. A very good friend of the show. Yes, a very good friend of the show. um There's a podcast that we want to talk about called the Mad Scientist Podcast.
00:54:15
Speaker
Now, this is neat. It's a slightly different style than breaking math. They talk about science, skepticism, and culture, which means that a lot of the episodes are about ah the occult. They've got episodes that are are about medical marijuana and the science behind it. They've got a recent series, a four-part series, all about UFOs. Now, let me tell you about the one of the founders, Chris Cogswell. He actually has his PhD in chemical engineering. We actually had an episode, ah it was a joint episode for um all the podcasts, called, ah if you look it up, it's BFNB1 and BFNB2. The first one is food for thought, and the second one is thought for food. Yeah, yeah, yeah. That acronym stands for blank, for non-blank. It's part of
00:55:05
Speaker
which is now called CNCA Podcast, which is a podcast collective that we're associated with. Yeah, yeah. There was a lot of plugs there. I realize that. That's quite all right. You know, there's a lot of plugs in this episode.
00:55:22
Speaker
In a world where black holes run rampant, a hero is needed, and one comes just in time. This is something, an adventure in space and time. Uh-huh. Oh no, I've divided by zero. You divided by what? It's become sentient. I know you're virtual, but I have a gun! Black hole highest. Coming to a theater. Somewhere. Probably.