Solving the Puzzle of Black Holes Hawking Entropy and a Theory of Everything

In talking about, in talking about black holes, of course. Stephen Hawking is the great scientist who comes to mind, so I'd like to spend little bit of time thinking about what it is that Hawking told us about black holes. What were the insights that he gave us, and what puzzles emerged from his work that we are still struggling to resolve and probably will continue to struggle with for some time? And toward that end, as little bit of background, we need to have some understanding of concept that everybody is familiar with it at some level or another, but let's use this opportunity to get us all in the same place. And that's concept of entropy. So entropy is word that's free, they used, and the culture, we often use it to describe certain amount of going from order to disorder, that's going from low entropy to high entropy. That's sort of the intuition that the culture has about this word. So let's start with that very basic way of thinking about entropy. And good example would be, let's say take my book, throw it up in the air, it starts very ordered. All the pages were in complete, numerical order, but as they come down, it's overwhelmingly likely that they will not land in numerical order. They'll land in highly disordered state and that gives you some sense of what it means to go from an ordered arrangement to highly disordered arrangement, going from low entropy to higher entropy. Now, of course, that's just sort of fanciful, everyday example. If we head more toward physics example, we could imagine having box of gas, and imagine it has small number of atoms in there, and imagine that they're all neatly arranged in this nice cubic lattice. This would be very ordered or low entropy configuration of the gas, like the pages all in numerical order. Now, contrast that with box of gas that has whole lot of atoms, whole lot of particles are all bouncing around in completely haphazard way. This would be very disordered state, high entropy state. And another way of thinking about the distinction between low entropy and high entropy, which we'll be making use of in just moment is the amount of information that would be required to describe these two configurations. Because look on the left, you don't need lot of data. It's cubic lattice. You've got three by three by three atom filling out that lattice. And if gave you the separation of them, say 10 centimeters, that basically describes the configuration on the left. You don't need lot of information to describe the system that has low entropy. But let's say want to describe the configuration on the right. Well, basically have to tell you where each and every particle is located. That's the only way could describe it. Look at all the information need to describe that configuration. So the idea is low entropy, not lot of information. High entropy takes lot of information that's hidden in this configuration of particles and that's kind of the distinction between order, disorder, low entropy, high entropy, low information, high information. Now of course I'm talking in loose language so that we can all get feel for the critical ideas, but you can make this all quite precise. And this gentleman over here, you know who this is? Who's this? Ludwig Boltzmann. That's right. And this is his, his famous tombstone, and you see that formula on the top of his tombstone, and that is the formula that makes these statements precise. This is formula that you learn if you take course in statistical mechanics, thermodynamics. stands for entropy. is Boltzmann's constant, and the rest of it has to do with the number of configuration of the ingredients of the particles that make up the system. You. You got job at Harvard as faculty, but that's okay. We're still. We're still good- You are doing very well too. Thank you. And should say my most exciting intellectual adventures in string theory have been the papers that we have done together. lot of fun with you Brian. Thank you. So it's been an honor to work with you. So thank you. So, so let's, let's now get into this subject here. We heard little bit about entropy and about the second law that entropy increases over time, but in the 1970s, people like Bekenstein and, and Hawking, and they began to worry about how the second law of thermodynamics would behave in the environment of black hole. And give us sense of what they were puzzling about. You. You got job at Harvard as faculty, but that's okay. We're still. We're still good- You are doing very well too. Thank you. And should say my most exciting intellectual adventures in string theory have been the papers that we have done together. lot of fun with you Brian. Thank you. So it's been an honor to work with you. So thank you. So, so let's, let's now get into this subject here. We heard little bit about entropy and about the second law that entropy increases over time, but in the 1970s, people like Bekenstein and, and Hawking, and they began to worry about how the second law of thermodynamics would behave in the environment of black hole. And give us sense of what they were puzzling about. So part of the puzzle was the objects they knew falls into the black holes, but they cannot come out. So objects that are falling in have entropy, just like what Brian already described, you have some information about where they are, how they are, but once you fall into black hole then you don't have anymore access to that information. So what happened to that information? So that seems like you lost entropy. Entropy went down and that's against what Boltzmann said. So can you give concrete example of that? So think we have little silly example. If took my wallet, it has information, it has entropy. throw it into the black hole and you're asking what happened to that entropy? Did it disappear from the universe? Exactly. So that was the kind of question that they were struggling to understand. How could this be consistent? Right. So, they began thinking about this and then there were few ideas that they were thinking about, and in particular the properties of the horizon of black hole- The edge. The edge, the event horizon that we thought- The edge in which- beyond which you can- no light can escape. They found it had some interesting properties, and in particular they began studying what is the area, and what is the area of this edge, and how does it change. And in particular, what happens if you throw objects into this black hole, what happens to this area? And one of the surprising things they found was that when you throw objects in, this area cannot stay put, and in fact increases. Yep. So they began to think about, okay, what could this mean about this- What, what this could be related in some way compensated by the fact that you're losing something, the entropy. Could it be somehow compensated by the fact that this area is growing? So they begin to ask the question, could the area of this event horizon be in any way related to the lost entropy? So we actually can unpack that. So let's just go through couple of little examples here. So this is the standards. Second Law of thermodynamics. Entropy grows over time. Let's just take the two boxes that started with to illustrate what you're describing. If we take those two boxes and they put them together, what happened to the entropy? It goes up. The Boltzmann's thing. Right. So we have the combined entropy is greater than the individuals entropy. Great. Now you're describing something that doesn't have to do with gas in box. You're talking about the geometry of black hole. So for example, they studied what happens if you study one black hole with one area of event horizon and another black hole with different area, and you bring them together, you merge them and you get new black hole with new area. And they found that the area of the new formed black hole is bigger than the sum of the two areas, just like the entropy had that property. The entropy resulting in the box when you combined them was bigger than the individual entropies. So the entropy sounded very similar to the area, or areas sounded very similar to the entropy. So they began asking, could it be that the area and entropy are proportional to one another for black holes? And indeed they found formula that we can bring up here. So entropy is the area, and the proportionality constant is little bit complicated but you don't really have to worry too much about that, this number on the right hand side, but the basic lesson is that they're guessing, they're suggesting that the area of the event horizon is somehow holding on to the entropy, holding onto the information in some sense that's inside the black hole. Now you gave an interesting thought experiment about how to test that. You're talking about throwing something in and watching the area, and it just so happens that we have little segment on that here. So why don't we go through that. So here's a-. So take us. So what are we seeing here? And then we'll do your little thought experiment. So here guess are seeing black hole. The red black hole. Yes, exactly. With the event horizon of it, you can see that the idea that the information is proportional to the area with some proportionality constant. If each individual element of the information is like one square there, we just divided this sphere into these areas. So these represent bits of information on that surface of that black hole. So those little squares can roughly speaking, be thought of as Planck areas. These in the Planck length that described sort of the smallest length that makes sense in in quantum theory of gravity. Squares of the size 10 to the minus 33 centimeters or so if you want to. Yeah. Each side of the square is so- So you basically count the number of those little squares that are on the event horizon, and that's the amount of entropy it has. So if we now took- If this is true, and we were to take particle that sort of carried one unit of entropy, what would we expect to happen if this is true? We expect this area to grow exactly by one square. So let's, let's sort of do that. We have- This is actually real real film of, of space here. So let's get our particle going in to this guy over here. Good. So the one on the left was the original black hole. The one on the right is the larger one after particles fall in. So if we take all those little squares, and we smear them out on the larger black hole, what we'd expect to happen, something like that. Exactly. So one one more square. So we throw sort of one particle in that carries one unit of disorder, and the black hole eats it. It grows little bit, and in fact you can calculate that the amount by which it will expand can accommodate one more of these little tiles. And that makes us consistent way of thinking about black balls. So. So this was like an important- How, mean, how important was, was this kind of an insight? Well, this was the amazing inside because it was the first time where Einstein's theory of relativity was combined with quantum effects. And that is what Hawking used to actually argue that the entropy of black hole has something to do with the area. And so this combination of quantum theory and Einstein's theory was quite novel, and this was in early seventies, mid sevens. So is this the end of the story when it comes to entropy and black holes? Like what, what, what issues does this leave over? Well actually the problem was that Hawking's argument showed that there must be information, but on the other hand, the horizon of black hole is featureless, so you don't see these points, squares or anything. So there seems to be no room for information to be stored on it. So in fact, if you use Einstein's equations and solve it, you find there is exactly one solution for spherical symmetric black hole. One solution means it has zero entropy. It's exactly ordered state. There's no choice. One piece of information. One piece of information. And that is inconsistent with what Hawking found when he combined Einstein's theory of quantum mechanics, where he found huge amount of information. And the question is where is this huge amount of information stored? Is it the shape of the horizon, or is it- What aspect of black hole encodes the microscopic structure of the black hole that was open when Hawking discovered this fact. So, so for Hawking's idea to work, let's just go back to my little example of sort of throwing the wallet in. So, so in that little example threw the wallet and he said where did the information go? So, so the new idea, if it's true, would be take my wallet and throw it in and rather than it disappearing, the idea roughly speaking would be the information that is in my wallet, the configuration of all the atoms and molecules that goes toward the black hole. And we want it to be the case that that information kind of like smears itself out on the outside and is encoded in sort of bits, ones and zeros. That's the picture. That's the picture. And the principal were this true, could actually extract that information in principle and use it to rebuild anything that fell in. In this case it'd be my wallet. So that's the idea. That's if the information is not lost, that information is there. That's what we would expect. Yes. That's the hole. And the puzzle is, is that actually- Is that actually true? And how do you actually see this information and where is it stored? Or how do you account for the microstates of the black hole or the entropy of the black hole? And that's where you come in. Well me and my colleagues and string theory began studying these questions more seriously with the advent of duality, some interests in string theory. So firstly- Can you say few words about string theory? should say few words about string theory. So string theory is, is framework where we believe combines Einstein's theory of relativity with quantum theory in consistent framework. And as part of it, it demands that the entities making up the matter is not, are not just point particles, but entities like strings or membranes or extended objects have to play key role. And so that's one aspect of string theory. It is little, if you want to take us through- This is just little example, what string theory might look like. So you've got piece of matter, and we're going to dive into it. And then you're saying that if it goes sufficiently deep- So if you go to the beginning, inside, you'll see the atoms first. And you know, inside the atoms you see the nucleus. And the nucleus you see here, like it's the protons and neutrons, I'm assuming. Then you go inside them, you see the quarks, and once you see the quark you can go inside them, and if you zoom in further you might see actually that these featureless quarks that you think are point particle actually are tiny strings, vibrating strings of some sort, and that's the kind of picture that we expect to be true in the context of string theory. That point particles are point-like because we are not zoomed in sufficiently close to them, and if you zoom in enough, they will have features like strings. How big would those- We don't know exactly, but it could be like 10 to the minus 30 centimeter- It's somewhat close to the Planck scale that we discussed already in the context of the areas of the horizon of the black hole. So that's the basic feature of string theory, but as one of the aspects of string theory that was originally one of the negative features, was that string theory predicted that the spatial dimension of the universe is not three, and that was manifested in contradiction with the fact that we think we live in three spatial dimensions. We have three spatial dimensions. String theory seem to demand that we have nine spatial dimensions, and so this was contradiction originally, and physicists thought how could this possibly work? And one resolution was, these extra- six extra dimensions that we have are so tiny, like little circles at every point in space. Let's imagine six dimensions, six circles at each point, six types of circles which are so tiny that you cannot see. So the idea would be the macroscopic dimensions are only three. The big ones are three, but the tiny ones are six. And the tiny ones are hard to see unless you really zoom in and that's difficult to do with the energies that are available to us in our experiments today. But the idea would be there. You've got these, these big dimensions that we know about. These extra dimensions, and then how do you relate this to this idea of black hole? this was actually negative feature for string theory. So string theorists were kind of struggling. Yeah, we have these six extra dimension. It's little embarrassing. It's tiny. You don't see them. And then other people said, sure, sure, yes, yes. So this was the state of the art to mid-nineties where we didn't know what to do with these extra dimensions. They were there just to make the theory consistent and we were hiding it away in tiny little things. And then came the question about the black holes and the black hole had the opposite problem. The problem with the black hole was we were missing information of the black hole. So where were they? What were that information? It turned out that these two problems canceled each other out. And so this- the problem that was where are they in-? Where are the ingredients or the degrees of freedom that constitutes black hole ended up in the internal spatial, tiny spacial things that was talking about. So as you can see here, the red, donut shape object, there is to represent these tiny internal dimensions. And the blue sheet that you see represents the macroscopic three macroscopic dimensions. And so the idea there is that if you take string or membrane or one of these extended objects and wrap it around the cycles on these external tiny spaces like torus or like circle or donut or whatnot shape you have, it will create mass at the point where you're wrapping around that circle, and the more it wraps around, the more mass is going to be constituting there, and the more it's gonna shape and distort the geometry of space around it. And if that is enough of it, it's going to create this warped space, which is what we call the black hole. So the black hole itself can be viewed as these strings or membranes wrapped around these extra cycles of this tiny space. And so then if the question is what accounts for the degrees of freedom of the black hole it are, these are precisely the strings or membranes wrapped around these extra dimensions. How many are there? How many degrees of freedom are there? What are the entropy? Gets translated to how many ways these extra strings or membranes can wrap around these extra dimensions. And there are many ways and that accounted for the entropy of the black hole. So with Andy- Andy Strominger and did actually computation of how much entropy is there in these wrappings of these string around the external, around these internal dimensions of string theory. And we found exact match with prediction that Beckenstein and Hawking had made about the entropy of these black holes. So just to quickly unpack little bit. So, so basically you found new way of describing black holes in string theory, which makes use of the extra curled up dimensions. The red part of this image that we find here. And you're saying that by wrapping these strings or ingredients of string theory around the extra dimensions, you can create warping in space that looks just like the black hole that we've been talking about for many years before string theory even was on anybody's mind. But in this description you can do direct count of the amount of information necessary to describe the situation and bang on, it matches what Hawking said. Yes. So, so this gives us insight into one possible way of describing the internal structure of black hole, which in many ways think they, the community views as one of the most important pieces of data that string theory may be going in the right direction. We don't have experimental support for these ideas, but here's mathematical piece of experimental support, if you will, that it matches ideas that had been on the table for 25, 30 years. So. So this is key step forward, but now let's turn to some of the puzzles that still remain and part of the puzzles surround something that's tightly related to what we've been talking about, which is something called Hawking radiation. Can you just tell us bit about what Hawking radiation is? So Hawking after discovering the properties of having information in the black and the entropy, he also noticed that the black hose actually are not quite black. And what happens if you go near the event horizon of the black hole. What happens is that you get pairs of particles created out of vacuum. These are quantum fluctuations, the quantum fluctuations always creating particles and antiparticles in pairs, and typically they go in and out without doing anything, but if they are near the event horizon, one of them per chance could go towards the inside of the black hole and then one of them could go outside. And the moment she goes inside has no way of getting out, so stuck there, but the one going out so can leave and go far away from it all the way to infinity, and that looks like radiation from the viewpoint of somebody outside. So in other words, from the viewpoint of somebody outside, the black hole is actually radiating energy. As it radiates energy, it loses mass, and it shrinks and shrinks and shrinks, until after while the black hole totally disappears where there's no more energy left to, to emit. So once the, once the black hole disappears, the question is what happens to all that information that went into the black hole? What happened to that area was talking about? Where is that information gone? So this was question that Hawking basically posed, perhaps you recall the discussion between Einstein and Bohr that Einstein did not like the probabilistic aspect of quantum mechanics. And so he said, God does not play dice. And Niels Bohr responded by saying to Einstein, stopped telling God what to do, and Hawking added his own wrinkle on this. He said, well, what happens if you throw dice inside the black hole, and the black hole after while disappears and with it disappears the dice? So, and then he said, not only does God throw dice, he sometimes throws it, throws them where they cannot be seen. And then you made little little cake for him. mWe had the lucky occasion of having Stephen Hawking as our guests at his home couple of years ago when he was visiting Harvard, and then we made the cake in the form of black hole with dice. And the book, his famous book, which is actually made of- It's cake and it's actually gluten free because he was allergic. So, so, so Stephen had his cake and he could eat it too. So yeah. So in that, in that picture, you see that there are two, two dice hanging there floating towards the black hole, and it's going in and that's the, the issue of the information puzzle. You lose the information of what, what is the dice roll, and that's what's called the information puzzle because if the dice rolls some number and the black hole evaporates, it disappears. With it disappears the information about what the role of the dice was, and that's called the information posture, which we still are struggling to understand even today. Now, most people in the field for long time have anticipated that somehow the information does get out with that radiation that we saw coming off of the black hole. Now for while, Stephen Hawking said that the information would, would not command- And that quote was quite serious. He was saying that we have to rethink quantum mechanics because the information does not come back out. He then changed his mind on that. Yes. So part of the reason for this is the discovery of more detailed version of what is called holography. In other words, this, this began with the work already back in the, in the long time ago, but became more precise with the work of one mother center where it was discovered how that you can describe gravitational system precisely using objects which are much simpler to understand. And in particular in that system you could prove mathematically rigorously, there's no way you can lose information. So therefore, indirectly there was proof that whatever happens to black holes, the information should get out. So that part was kind of beginning to be established using these kinds of arguments and string theory. But the wrinkle is that even though there's this mathematical proof that this should happen, we still did not know exactly how, and some people thought maybe the information gets out with this radiation, as you were saying, Brian, very gradually and very tiny bit at time, but gradually builds up to give you information out. And this is what, in the early 2000s, believe, Stephen agreed that this is the resolution. But still think, and think the community still believes we haven't really understood how this happens exactly. So the information puzzle about how this information comes out is still one of the big mysteries of black holes, and we don't really understand it. So do you think it's possible that we will one day conclude that the original view of Hawking was right, that the information does not get at? Is that still on the table in your view? Well, think it sounds much less likely given that we think we understand dual description and other description of these black holes. So it seems like that seems much less likely now, but you know, we can never say we know everything about the subject. That's our current understanding. It could of course evolve, but even even with that understanding, it's not- Our understanding is not sufficiently detailed to convince us of this, of how it happens. Now lot of our discussion here has been focusing on what happens at the edge of black hole, at the event horizon to black hole that it may be where the information is stored, that may be where the entropy is stored. We haven't said much about what happens at the center. Can we just spend little time talking about that? Well, one of the things that happens in the center of the black hole is that there's this singularity, this infinity that Brian discussed already, this division by zero analogy, and we don't believe that's physical. We believe that's just the phantom of our equation. It's just the mathematical thing that comes out and that means that our description breaks down. Not that there is an infinity. Now quickly, the infinity could be physically that you've got all this mass crushed to zero size. Zero size is not physical. Yes for us, there nothing smaller than Planck's length, which is about 10 to the minus 33 centimeters. The notion of space breaks down anything smaller than that, so there cannot be an infinity in physics, we don't believe so, so we don't understand what is happening at that, at that in that way. And the understanding of this singularity, we are not at all close to understanding what's happening there. We know that there should be good description, but we don't have it. And why that's important is that, well, we want to know what's happening inside the black holes. Of course we can send somebody and find out what happens to that infinity, but that would not be fair thing. And even if we did that, even if we said, well, we send some creature not us, the robot or something, the robot cannot come back to tell us what happened or what they saw. So because nothing can escape from black holes. So it won't be useful. So we have to figure it out without going inside the black hole, what's going on there, and that's still, we have no, not only you don't have observational data, we don't have enough theoretical understanding. And we think it's important not just for the black holes, but it turns out that those infinities that we see, if you try to pass through those infinities to the other side, so to speak, it turns out it looks very much like the infinities we see at the beginning of our own universe. It is as if our universe is emerging from inside of one of these black holes. Along those lines. And don't lose your train of thought there. Oftentimes people think of the center of the black hole as location in space. The center. Is that the right way of- No, actually it's like in given time, it happens at given time and that's because the nature of space and time changes as you enter the black hole. When you cross the event horizon of the black hole. So one of the reasons why you can't get out is because you can't stop time from ticking forward. Exactly. So going forward and forward in time is like going closer and closer to the singularity- And then at certain time you hit the singularity. That's the time. It's at the given time is that- So the singularity could be the end of time in some sense. It could be, but we don't believe that. And then what you would say what's the after that, and that looks very much like what could be new universe or something. So we are interested in understanding the resolution to the issue of singularity of black holes, and think it's very exciting. We know that black holes, as we discussed already in this program, there are huge amount of evidence that they exist or they are there. We cannot say. We're just imagining. So what is that? What is singularity? What are we learning from it? This one of the most beautiful, mysterious objects think in the universe, and hopefully, by some experiments and maybe by some theoretical understanding, we will have some progress in that. So what's your guess? mean if we come back and we invite you back for the, dunno, the 2025 World Science Festival, can you give us the answer to what happens at the center of black hole? Maybe invite in 2050 if I'm alive. 2050. All right. Good. good prediction. safe prediction. Please join me in thanking Cumrun Vafa. Thank you. Thank you very much. Fantastic. Goodnight everybody. Thank you.