 Good morning and welcome everybody to this valedictory function of the orientation come selection camp of the physics Olympia. This is the first session of this function where we traditionally have a lecture and we are very pleased today to have with us Professor Shiraz Minwala, senior professor at the Tartine Institute of Fundamental Research and it would not be too much to say that one of the most well-known theoretical phases of the country and internationally also. So I will give a more formal introduction of Shiraz later. Right now I would invite him to give his talk on black holes. Thanks a lot and thanks for the invitation to come and speak here. Okay, let's start. So it does this work? Can you hear me from here? Excellent. Okay, so my talk today is titled black holes and is meant to introduce you to fascinating objects that we now know almost for sure exist in our universe and are fascinating for various reasons. I want to tell you about them and why they're interesting. Okay, so the story starts about a hundred years ago, a little more than a hundred years ago with Einstein who put forward a new theory called the general theory of relativity. Einstein about a hundred years ago was trying to reconcile two known theories of physics, the special theory of relativity, a framework in which physics was supposed to be based, and Newton's theory of gravity. And the theory he finally came up with to achieve this reconciliation was a marvelous theory. Okay, what he tried, what he postulated and since been experimentally verified in a number of ways, is that gravity is a consequence of a remarkable fact of the following fact that the geometry of space and time in our universe is not static but dynamic. Okay, well, let me say a little more than I'll try to explain. The idea is that space and time are not some unmoving stage on which dynamic happens. Space and times of participants in the dynamics of the universe. And the way they participate is that the geometry of space and time warp, bend, react to dynamics, and in turn affect dynamics. Okay, this allow, you know, in the process of dynamics all kinds of things can happen. For instance, all of space and time can expand and is expanding in the universe as we speak. The ripples of space and time exist. They're called gravitons. Wonderful things happen. I tried to explain to you, I tried, well, I mentioned that Einstein's theory of general relativity was an attempt to make a relativistic version of Newton's laws of gravity. And so a minimal thing that it does is inappropriate regimes, that is when masses are not too large and when velocities are not too high, it reduces to Newton's laws of gravity. You know, that's what it set out to do. It did that. However, you know, it's a hallmark of really great, of really fundamental advances in physics that they achieve more than they set out to do. Einstein set out to try to reconcile special relativity and gravity. And then he found a theory that had in it much more than a souped up version of Newton's laws of gravity. He had, he found a theory that predicted entirely new phenomena, entirely new phenomena that Newton's laws of gravity were just, I mean, could not be addressed in the theory of Newton's theory of gravity. One of these phenomena is the existence of gravitational waves. These ripples of space-time that give rise to wave-like excitation, very much like ripples of the electromagnetic field give rise to light is a completely new prediction of Einstein's theory of gravity. There's no analog of these gravitons or gravitational waves in Newton's theory of gravity. Another thing that his theory took from, you know, allowed us to take, another thing that his theory did is allow us to study the dynamics of the universe as a whole. How the universe as a whole could expand the contract and took the question of cosmology, the study of the beginning of the universe and its evolution history since it began from the realm of religion and speculation into science. Okay? But I'm, my talk is neither about gravitational waves nor about cosmology, both of which could be fascinating subjects for it, talks. My talk is about a new, a third new phenomenon that Einstein's theory of general relativity predicted, which again has no analog in Newton's theory. And this, this new phenomenon was the, was the existence of a new class of, new class of objects called black holes. This is what I'm going to tell you about. Only a few months after, you know, Einstein famously is reported to have said soon after publishing his equations of general relativity that his equations are so complicated that human beings will never find exact solutions to these equations. You know, people who invent theories are often not the best at understanding them and are predicting how they will be used and Einstein was wrong about this. A few months after he published his equations, the Schabka-Schwarzschild discovered the first, the first exact solution to Einstein's equations, apart from the trivial solutions in flat space. Now, the, this solution is very interesting. So I told you that Einstein's theory describes how the curvature of space time is affected by matter and in turn affects matter. Okay? That's the structure of these equations. But the simplest context to study these equations in which there's no matter other than space time itself. Okay? Already, his equations are non-trivial even in this context. A, a context where there's no matter just space time itself. The dynamics of space time itself, for instance, these rippling geometries of space time, are already non-trivial things to study within his theory. And Schwarzschild found a solution to the equations of motion describing just pure space time with no matter itself. The solution he found was very interesting. It, far away, there was an origin in the solution. Okay? So the solution, roughly speaking, could be thought of as some object. Far away from this origin, space time just became flat. But when you went near to this, this origin, space time became more and more warped in a way I will describe to you in more detail. In a way, in a way that was sort of strange. This was an exact solution to Einstein's equations. Okay? Some, a lump of warped space time sitting somewhere in space. This solution is now, now called a black hole. Now, okay, I'm gonna tell you a little more about this solution. This is meant to be some sort of strange, highly inactive, okay, some sort of cartoon of a black hole. Every, far out here will be a flat space. And as you go inside, you go to the origin of the, of the space time towards where the black hole is sitting, space becomes more and more. Inside a spherical black hole, spherically symmetrical black hole, which is all I talk about in this talk for simplicity. There are two surfaces of particular interest. Is a surface of, a sphere of a finite radius which has a name. It's called the event horizon. I'll tell you what that is. And then there is what in this diagram shows up as a point called the singularity. It's like the center of the black hole. Okay? As I mentioned before, the solution describes warped space with nothing in it. There's no star or fluid or anything like that. It's just space itself warped by itself. At least outside the singularity, where the equations of our theory break down and we don't really know what's going on there. There could be stuff hidden there that we don't see. But everywhere outside that singularity, the solution is just pure space in this warped way. Okay? Now, the really funny thing about black hole is that there is a region of space time which is from which nothing can ever emerge out of the black hole. Okay? So suppose you're in a spaceship, you're going like this. If you were here, you wanted to, you could reverse your spaceship and accelerate out past now if you had a powerful enough motor. But if you went like this and you happen to go inside this region, you go past the lentor horizon, no matter how fast you turn around, no matter how fast you reverse, how fast you shoot your motor, you will be dragged into the singularity. This is true of rockets, but it's also true of light. If you were here and you tried to send a signal to your friend by sending a, oops, by sending a, oops, by sending a laser burst. Fortunately in this room, I can do it. You tried this in the black hole, you sent your laser light, the light, you thought you were sending it outwards, but it's, you know, even if you pushed it outwards, the light would go back in and bang into the singular. Nothing, not even light from this region, can emerge outside, outside the event, the horizon of the black hole. The event horizon is this point of no return. Once you go in, nothing ever comes out, not even information. Classically, okay? Within classical Einstein's equations. We'll come back to this as we've got. Okay. Now, I want to explain, I've said a lot of words and all of this sounds very weird to most of you, so I want to give you a rough analogy which has many inaccuracies, but also, but, but, but, but, but, but capture some features of what the black hole is. Okay. So this analogy goes as follows. Imagine a lake. Imagine a lake which is far away from a central point, still and placid, nothing's happening, the water is just still. Okay. But there is a central point in the lake in which, from, from which in the floor of the lake, water is somehow being sucked out. I don't know how, maybe there's an alien who's got a huge pump, whatever, whatever. Okay. Water is being sucked out from the central point of the lake. So now, because of the law of conservation of mass of water in this problem, okay, water is being sucked out, so it must be flowing from the outside of the lake into that place where it's being sucked out. Now, you can imagine that because as you go nearer and nearer to this point, the surface area is becoming smaller and smaller. A certain amount of volume flow has to happen because a certain volume is being sucked out for you in that time. That means the water has to be moving faster and faster into the, into the, into this region. So the flow of this water, let's say it's radial and it's going like this and it's moving, the water is moving faster and faster and faster as you go towards the same, the, the central point. Now, imagine your, you are a tourist motoring on this lake and you have a boat that, that has a maximum speed with respect to still water. That maximum speed is 100 kilometers per hour. Okay. This blue line here is the line at which the speed of the water as it moves towards the central point exceeds 100 kilometers per hour. Here it's less than 100 kilometers per hour. Here it's more than that. Is this clear? You're motoring around in this lake and as you motored around, you want, you're curious about the singularity. You come here, you take a look at it and then decide, okay, I don't want to go further because I'll be sucked in. So you turn around and you motored out. You can do that because the speed of your boat faster than the speed of the water with respect to the ground. But if you make the mistake of getting in here, you turn around and you try to motor back, it doesn't matter. You're going to be sucked in because the speed of the water moving towards the singularity is faster than the, that, that your boat can go with respect to the water. Now, of course, in this particular problem, this blue line here, so this blue line is the analog of the event horizon. This place where the water is being sucked out is the analog of the singularity. And in this particular problem, this blue line here, he's sort of observer dependent because somebody else came with a faster boat. He would have another blue line maybe here. Somebody came with a slower boat, he would have another blue line maybe here. In the universe, there is a speed limit. Nothing moves faster than the speed of light. So we should imagine that there is a speed limit for boats in this analogy. Boats can be slower, but they can't be faster than a particular speed. And we've drawn that blue line here as blue line for the fastest boat. Once you do that, nothing that ever goes in can ever come out. It's sort of like a black hole, okay? There are many things about this analogy that are inaccurate. And yet it's not a worthless analogy because it gives you a way of thinking of what a word would be. Okay? Excellent. So this is what the black hole solution looks like. Any questions or comments? Any brief questions? Okay, any time through my talk, you want to interrupt me with questions, comments or objections? Fine. I reserve the right to say I'll answer later if it's going to take me too far off. And anything brief, clarificatory that you want to ask? Yes? Loudly? Yes? Pollution then. That's coming. It's a very good question I'm going to tell you about. Okay, excellent question. The question, okay, let's go on. That's what we're going to talk about. Now, so far what I told you about was just about a solution, a mathematical solution of Einstein's equations of general relativity. Somebody wrote this down on a pen and paper. The question of whether these black holes are real and if so, how are they formed, which is basically your question, was not addressed by what I told you. Okay? And in fact, Einstein, you know, when he, when he saw the solution, was very appreciative of its mathematical beauty and elegance, but felt that it was a physically unreasonable solution. It described then solution to these equations of motion that could not reasonably be created in a reasonable physical situation, because it was just too weird. He published several papers objecting to the physical nature of the solution. Yeah, not an example of a person who creates a theory, not understanding it very well. Einstein had many wrong papers. And that's great. Somebody who doesn't have the courage to be wrong will never say anything important, that's right. Okay, in my opinion. Okay, so, so, so, so, but anyway, so Einstein and all, all his friends, and you know, all the people around the 1959, then he, by and large, regarded the solution as a mathematical artifact without corresponding physical reality, at least in the world that we, we see. And this story changed with this gentleman here, Subramaniam Chandrasekhar, who began to ask where at first sight seemed to be a completely different question. That's the question you were alluding to. The question that Chandrasekhar asked was, what happens to stars after they finish burning all their nuclear fuel? So now what does this question mean? You see, a star is a huge collection of matter. And if you've got this huge collection of matter, it's self, you know, matters gravitates. So the matter of the star self-gravitates. And if gravity was the only force in the game, all the matter in the star would, would want to scrunch down to the central singular, the center of the sphere. What stops that? Okay, we know the sun is stable, it's not scrunching down into the center of anything. Why not? No, what, what opposed to gravity? Well, when the sun is burning, it's burning its hydrogen and helium through fusion, it's producing a lot of heat. So the sun is very hot, it comes with an associated pressure. And ultimately, it's this, the banging of these molecules, the pressure of the sun that opposes the force of gravity. Okay, pressure differential is more, more accurate than you guys know all about this. Oppose the gravitational attraction and allow for a steady equivalent. Okay, this pressure differential can only come with a temperature differential. The temperature differential can only come because of constant source of heat through the fusion. Great, that's how a star survives in equilibrium when it's alive and it's burning heat. But all stars eventually run out of nuclear fuel, our sun will probably in about five billion years run out of hydrogen, it'll all be helium. Maybe some additional fusion processes will happen, helium into something else and so on. Eventually that will all stop. So once all the nuclear fuel is burnt out, what happens to this, the star? Now this pressure differential, this temperature induced pressure differential cannot sustain it anymore. So the star will stop, the scrunch onto itself. And unless some other opposing force opposes this scrunch, the scrunch will keep going. So the question that Chandrasekhar asked was what could oppose this scrunch? And what he did was to apply the, what was then a quite, quite recently developed theory, the theory of quantum mechanics to this question. You see, within the theory of quantum mechanics, all of you guys, everyone in this room ready, knows that within quantum mechanics there are two kinds of particles. They're fermions and bosons. The constituents of matter, the elementary constituents of matter at a simple level, let's say electrons and protons, are both fermions. Now all of you know that fermions have this famous, obey this, famous Fermi exclusion principle, which tells you that if you've got two identical fermions, let's say, two electrons, they cannot be in exactly the same state. Fermions dislike each other, dislike their, like, tigers. They dislike their, their identical other fermions. And the mathematical version of that statement is that you've got a fermion in one state, you cannot put an electron, once you cannot put another electron in exactly the same state. Now, suppose I've got a little box, a box of some, some radius, and I put a bunch of electrons. I put one electron, it goes into the lowest energy state it can find. I put the next electron, it cannot go back, it go into that lowest energy state. It has to go into slightly higher energy state. It does this by having slightly higher momentum. Okay? I put the next electron and it does this by having even more momentum and so on. So you see that when you now, as, as the momentum of the electron keeps increasing, its energy keeps increasing. As you add it up, what you get is a certain energy of the substance, which is a function of the volume in which it, it lives. Now since the momenta of this electron are quantized in units of one over r, if you do the calculation what you will find is that the energy of this, the energy that comes just the ground state energies coming just from quantum mechanics of this substance becomes larger and larger as r becomes smaller and smaller. What does this mean? This means that there is an effective quantum pressure opposing a bunch of electrons to scrunch to a smaller volume. Is this clear? Because they can only do that by going to higher and higher energies. Is this clear? It's the derivative of the energy with respect to the radius gives you an effective opposing force that opposes scrunching. He can realize this and he, he, he postulated that this, this, this force has a name called the degeneracy pressure. That this degeneracy pressure of electrons perhaps is the opposing force that allows matter to equilibrate from a star once it's run out of nuclear fuel. And indeed we believe that, that this is correct for stars that are small enough, roughly less than twice the mass of the sun. Now there's a slightly complicated story involving electrons and protons squeezing into neutrons and slightly complicated details which I wouldn't, wouldn't get, get into. But what Chandrasekhar also found was that when the mass of the remaining star was large enough, let's say significantly larger than twice the mass of the sun, this opposing quantum force was not strong enough. Not even this degeneracy pressure could resist the pull of gravity. And indeed Chandrasekhar could think of nothing, nothing that could resist this crunching pull of gravity. So he suggested that well since there's no option what the matter must be doing is just continually from this big enough star, then you're scrunching down, down, down, down, down, down, down, until they reach a point. Now what happens at this point? God knows. It's where the equations of physics as we know them break down. But everywhere else there's no matter left, so it's crunched away. So what's left is just a lump of space and time. And then Chandrasekhar looked around and said, has anyone described this lump of space and time? Uh-huh. Schwarzschild had. He proposed that what was left behind was a black hole. When it was stars that were massive enough. Is this clear? Did that answer your question? Okay, now if Chandrasekhar, please, once they shrink into this point, once they go right down to this point, you see what happens there technically at this point there's matter with infinite density and our, the equations of physics as we know them break down. You know, to answer your question we would need to have let's say a quantum theory of gravity answer what was on it. So I don't know the answer. I, well they don't lose their identity, okay. I mean that would violate the basic rules of quantum mechanics but exactly the details of what happens. Uh, uh, nobody knows it. Good question. Let's see. So now if Chandrasekhar is right, this is exactly the opposite of what Einstein said. You know Einstein said, well Schwarzschild solution is a nice solution. It sounds physically unreasonable to them. Must be, there's probably some rule preventing these things from ever forming. Chandrasekhar not only, I'm basically identified a very plausible physical mechanism for the formation of these black holes and argued that massive enough stars always end up as black holes. So roughly speaking, there must be as many black holes in the universe as stars. Okay, people give, this is too rough. Astrophysicists have ranges of estimates. One, one estimate is roughly 1,000 as many black holes as stars. Now how many stars are there in the visible universe? That's something like 100 billion stars in our galaxy, 10 to the 11 stars in our galaxy and something like 10 to the 11 galaxies in the visible universe. So something like 10 to the power 22 stars in the visible universe. Okay, let's say 1,000 degrees of black holes, then there are 10 to the power 19 black holes in the visible universe and that's a lot of black holes. So not only can they form, they do form and copious numbers, our universe is full of them. And I think the 10 to the 19 maybe an underestimate. I don't, I think it's unlikely to be an over. We don't really know. LIGO will find it. We don't really know. Okay, so now suddenly if you believe Chandrasekhar, not only are these black holes physical, the universe is full of them. Okay, so there's a long story which I'm not going, I'm not describing all of this. There's the work involved in the singularity theorems and so on. I will not describe that here. There's a long story but increasingly since the 1960s, increasingly physicists took these ideas of Chandrasekhar which we initially, largely ignored. There's this beautiful paper by Oppenheimer and Schneider, but largely ignored. But increasingly various developments in the theory of gravity forced people to take this idea more and more seriously. And from the 60s and 17s onwards astrophysicists started looking out into the sky to see, could they identify, could they actually see black holes? This might seem like a strange question because I told you that black holes are those things that cannot be seen. You know, once you go inside the black hole, even if you shine this laser, it doesn't come out. Nothing comes out of black holes. How can you see a black hole? Well, the way you see a black hole is not by seeing the black hole itself, but by seeing the effect it has on nearby things. You see, a black hole is a very compact, massive object. And so, if there's, let's say, a star near it, it can start ripping away the gas from that star which will come and sort of circulate around the black hole, come very near to it and then swish it. While it goes, it emits a great deal of energy. So you can see that. Okay? So astrophysicists started looking for such things. And as soon as they started looking, they found many, many candidates for such things. Many objects that looked like they were caused by an object so compact, like by the gravitational force of an object so compact that nothing in known physics, it could be nothing in known physics other than that. Wonder whether there were things that you don't know about that could be causing this swirl. That's not a black hole. This was answered at least, like 80% answered. In a beautiful experiment performed over the last decade and whose results were first announced maybe two years ago, this experiment called LIGO. LIGO was an experiment in which physicists and earth looked for these gravitational waves, these ripples in space-time that I told you about. These waves had never been seen before. And about two years ago, LIGO announced that they'd seen gravitational waves. That was very exciting. But even more exciting than the fact that they seen gravitational waves was the fact that their analysis revealed that the gravitational wave that they saw had a remarkable source. The source was this. About 1.2 billion light-years away. There were two black holes, each about 30 times the mass of the sun, orbiting around each other. As they orbited, they lost energy into gravitational radiation. So they came nearer and nearer and nearer to each other. At some point they came so close together that the two event horizons banged into each other. And the black holes merged. In a dramatic event, they went from two black holes to one black hole. LIGO claims to have seen the gravitational waves coming out out of this collision. Now, take two minutes to convey the drama of what the event that LIGO claims to have seen You see, I told you that these two black holes each have energy at roughly 30 times that of the sun. This collision, according to LIGO, took place, well, you can estimate it, took place in about a hundredth of a second. In the process of the collision, the net amount of energy released was energy equal to three solar masses. Now, just to understand how dramatic that is, let's remember the following. The sun will live for 10 billion years. In those 10 billion years, it will lose roughly one percent of its rest mass as solar radiation. The sun is giving out a lot of energy. What it does is turn one percent of rest mass into energy over a period of 10 billion years. Now, a year is, you guys probably know better than me, something like 10 to the past seven seconds. So, let's see, 10 billion years is about 10 to the 17 seconds. In that process, in 10 to the 17 seconds, this object loses one hundredth of a solar mass. In this black hole, this black hole collision lost three solar masses, so 300 times as much energy in one hundredth of a second. So, the intensity of energy emission was 10 to the 17 times 300 times 100. So, that's, what is it? 3 into 10 to the 21 times the intensity of emission of energy from the sun in this event. We've already estimated that there are about 10 to the 21, 10 to the 22 stars in the whole universe. So, if you believe my estimate, the intensity of energy emission from this bank was equal to approximately the sum of the intensities of emissions of all the stars in the visible universe for that one hundredth of a second. It was quite an event. LIGO has done a more serious estimate than we did just now. They came up with the estimate of 50 times the intensity of all the stars in the universe in that one bank, only for one hundredth of a second. But the intensity was amazing, blinded. The reason that we were not all blinded by it was all the eminent and gravitational radiation, which started this. It was eminent and light, ooh, it would have been an event. Okay, the great thing about this is that this, this, this, the LIGO experiment has caught, you know, has caught the black holes behaving extremely black-holy and actually seen it, because the two event horizons of the black holes are black. So, if these black holes were some other thing that was twice the radius of a black hole surrounded by some other mysterious thing, they were, their collision would behave very differently from the collision of two actual black holes. LIGO already claims to have done some analysis to say that what they see is in good agreement with what you'd expect from General Reynolds between the collision of two black holes. In another 10 years they will have seen thousands of events. There'd be no doubt that what we've seen here is an actual black hole, the black holes of General Reynolds. Okay, so let's, let's, let's shed skepticism. Let's say that there's already very good reason to believe, and there will be even better reason to believe in 10 years. We'll be clinching that what we've, what LIGO has seen is actual black holes. So now, I think by now the observational evidence that black holes exist is more or less convincing. Okay, so now that we, we've gone from Einstein's skepticism to almost charity of the existence of these black holes, let's try to get them, get to know them a little better. Okay, any questions or comments about this? Okay, excellent. This is what I've already said. Okay, now within Einstein's, within the, I, I want to tell you a little bit more about black holes. Okay, so people studying classical, Einstein's classical theory of general relativity have proved many theorems about the behaviors of black holes. I want to quote two of them to you. These two theorems were actually both proved by Hawking. Both of which you made crucial use of work done by this, this man, this professor, Professor A. K. Reichaudry, a great Indian general relativist who taught in Presidency College for many, many years before he passed away, I'm not sure exactly when, but maybe 10 or 15 years ago. And whose work has had deep and lasting impact on the study of the theory of black holes. Okay, the two theorems I want to quote to you are the following. The first theorem is this, the area of a black hole, the area of the eventualization of a black hole can stay the same in time or can increase in time but can never do. This is theorem number one. The second theorem is that technically it says that horizons are bifurcated, which colloquially means that one black hole can never split up in an early time as time progresses forward. One black hole can never, never split up into two. Now, what does this mean for, you know, for, for a black hole? Well, one thing it means is that a black hole within classical general relativity can never be destroyed. Okay, because you see one, suppose I've got an object, let's say, I don't know, a cricket ball. What do you mean by destroying it? Well, you could break it up into many little pieces that would destroy the ball, but black holes cannot be broken up into pieces. Or if you could somehow, I don't know, let it evaporate away or something like that, that would destroy it, but, but black hole areas cannot decrease. So black hole cannot become smaller than it was, cannot break up into many pieces, can't be destroyed. Let's, just, just to give you a feeling for what, what this means, let's imagine the following thought experiment. Yes. Sir, can a black hole be stressed? What does stress mean? Are they entitled forces? Can it be squished? Yes, it can. But no matter what you do, the net size of the event horizon, the net area of the event horizon cannot be made to decrease. You can stretch it out, for instance. So even, it can be stressed out indefinitely. You see, this stretching indefinitely will require putting some force. Yeah, the black hole likes to be a sphere. Just like a soap bubble likes to be a sphere. Okay? Now, by applying appropriate forces, you can stretch it out. But as you stretch more and more and more, you require more and more force. It's true that by doing extreme things, you can stretch it out indefinitely. That's true. Why can it never break? Okay, why can it never break? Yeah, yeah, yeah, I see where you're coming from. You're saying, well, if I can stretch it out like this, I can, it can do this and break up. That soap bubbles can break. It's true. Let me be more precise. The theorem actually states that any solution of general relativity that describes the breaking of two black holes necessarily encounters a singularity. That is a place where something becomes infinite. So the precise theorem is that within the classic, when that happens, classical general relativity breaks down. Okay? So the precise theorem is that classical general relativity cannot describe the breaking of two black holes. Okay? It is possible that in reality, usually such possibilities that have been studied involves some quantum stuff. Okay? That allows some sort of breaking, but not within classical general relativity. In fact, we will, I will tell you about in the last part of the talk, another way in which black holes are destroyed by quantum, by Hawking radiation. Okay? I feel I've not answered your question very well. Why can the black holes ever break? It's just a fact. You know, I mean, I don't know what else to say. I'll try to see if I can think. Yeah. But you understand, right? It's just a mathematical theorem you can prove. That there's no solution of general relativity that can describe such a thing without a singularity happening somewhere. If singularity has never happened, this will never happen. Okay? Okay. Excellent. I think I've even said it here. A black hole can never be destroyed without doing something clever in quantum. That's the, the escape clause. Okay? Okay? But just, just, just to give you a feeling for how, what this means. You know, suppose tomorrow ISRO issues us and alert says, okay, we've got bad news. There's a black hole and collision caused with a, it's going to come and a month from now we are all going to be swallowed up in the black hole. What can we do? Well, you know, maybe once we communicate to the US, President Trump will say, unleash the nuclear arsenal at the black hole. Break that guy up into the smithereens. It won't work. This, this thing will absorb the radiation of nuclear energy and become bigger. Okay? Okay. Actually, the only thing we could do in such a situation is to divert the black hole. Black holes can't be destroyed, but they can be pushed away. We could try to do that. That would be the sensible thing. Okay? So let's, let's, let's, let's move on. Black holes are also, well, I want to tell you more about black holes. Black holes, it turns out that black holes are the most compact objects in the universe. What does this mean? It, this means that if I give you a, suppose I take a black hole of a certain mass, I give you something else of the same mass. Let's assume for my statement to make it precise for the moment that it's both are spherically symmetric. Then that something else will have to have a radius that is larger than the radius of the black hole. This is the theorem you can put. Okay? There is no object of a given mass that is smaller than the black hole of that mass. So the black holes, black holes are the most compact, compact objects that exist. Okay? But when I say radius, I'm talking the radius of the event horizon. Okay? One more thing that I want to tell you about, and that's very important, is this. That black holes are like objects. They can move around, do all kinds of things. But take them in flat spacetime, the spacetime where nothing is happening, and you let them settle down. Maybe you take a black hole, you throw an asteroid into it, switches around, does all kinds of things. Then you wait for a while, it settles down. You wait long enough, black holes settle down into solutions, all of which we know, which are labeled by very few parameters. They're labeled by the mass of the black hole and the angular momentum of the black hole. Also if they charge black holes by the charge of the black hole. Okay? But there's a few, a small finite number of numbers, depending on the assumptions like three for angular momentum plus one for mass, let's say there's no charge, four numbers. By four numbers, completely label the black hole solution. Okay? The full set of equilibrium black hole solutions label just by four conserved charts. This statement is something sometimes called the no hair theorem. John Wheeler I believe said, worded the statement in a way I can't quite understand. I mean I don't know why he used the words, but he worded the statement with these words, black holes have no hair. Forget those words. It's the statement that equilibrium solutions of black holes are characterized by four conserved charges in the actions of mass. Okay? Now this is quite dramatic. It's dramatic for the following reason. We've talked about how black holes form. Black holes form let's say by the collective stars. Now let's say I've got star number one. You've got star number two. Both stars have no zero angular momentum in my thought. My star was lithium rich. Your star was I don't know beryllium rich. I don't know where all of those are possible. That's okay. Okay? Both of them collapse into black holes of the same mass. The final black hole solution doesn't care about whether the black hole was formed by the lithium rich star or the beryllium rich star. It's just the same solution. It's strange about this that we're going to come to in a little while. Yes? What about a black hole made of anti-matter? Made of anti-matter. You see anti-matter, if it's neutral anti-matter, well you're probably thinking of something like positrons. Okay? That will be distinguished by black holes made of just electrons by charge. But suppose I had a black hole made of anti-electrons and anti protons. Okay? Neutral charge, the final black hole would be exactly the same solution. A black hole made of electrons and protons. So the black hole remembers nothing about what makes it up other than conservator. But wasn't there something like amount of anti-matter plus matter is conserved or something? Amount of anti-matter plus matter is conserved. Uh, no, in fact it's not true. There's this CpT theorem and in fact within uh, in fact within uh, within the theory of gravity. There is a theorem you can do. It says that no global symmetries are absolutely conserved in a theory of gravity. And one of the reasons for that is precisely what you're saying. Ordinary processes might conserve a global symmetry, something like what you're saying, some version of what you're saying. But black hole process is violent. Okay? Quantum mechanics together with gravity violates all global symmetry. Only things that are absolutely conserved are gate charges. Gate charge. Okay? So it's something like angular momentum, angular momentum, those are absolutely conserved. Charge is absolutely conserved because it's coupled to a gauge field, electromagnet. No global symmetry is absolutely conserved in the presence of black hole. Okay? So it's exactly the same solution. Yeah, it's a very good question. Okay. Now, change in many ways, right? You've got completely different matter giving you the same eventual solution. Think of this violates some basic rule of physics, more or less like the intuition you have. Okay? And this feature of black holes leads us to an almost paradox. Okay? That I will now spend the rest of the talk trying to explain to you. It's called Hawking's Information Paradox. I'm going to try to explain the paradox to you and what we can say about it. Okay? But in order to do that, I'm going to have to introduce a completely new notion, one that has not entered our talk before. This is the notion of energy. Okay? So hang on with me for five minutes before we take a detour into something that has no black holes in it. We'll come back to the black hole five minutes. Okay? All of you know that physical systems are characterized by energy. Yeah, energy is actually quite a strange concept. When energy was first, as a concept was first invented, it had a limited, there was a limited idea of what energy was. But as people realized that physics had more and more and more than they first thought, as for instance the electromagnetic field came into physics, the old notions of energy would no longer be conserved. But people kept realizing that you could enlarge the notion of energy so that this enlarged object was a conserved one. Okay? So physical systems are governed by energy. Formulas for energy keep changing as you keep, as your understanding of physics becomes more and more sophisticated. But the fact that there exists an energy which is always conserved has never changed because it follows from a cement. Now, in the 1800s, physics is studying improbably enough steam engines hit upon another fundamental law of nature that reminds you in some ways of the conservation of energy and in some ways it's different. And what they explained to us was the following. That if you look at complicated enough systems, I will say a little bit about what that means, then in addition to being characterized by the energy these complicated enough systems are characterized by another number called the entropy. And the claim of these physicists in the 1800s was that the entropy always also obeyed a sort of law. Now the law of entropy wasn't that entropy was always conserved. That's how it works with energy. That's not how it works with entropy. The law of entropy is more interesting. The law is that entropy can remain constant or can increase in time but can never decrease. Once you account for all the entropy of all parts of the universe, just to plus in time. This law is sometimes called the second law of thermodynamics. The law that energy is always conserved when you heat energy is sometimes called the first law of thermodynamics. That's a simple law. This is the most sophisticated in many ways more interesting law. Entropy can remain the same or can increase but can never decrease. Now I want to explain this to you in the next three or four minutes to give you a rough intuitive notion of what's going on and why this is so. Imagine the following experiment. I've got two chambers. One of which I've got hydrogen molecules. The other which I've got oxygen molecules and I've got a stopper between these two chambers. So the right has only oxygen and the left has only hydrogen. Okay now if I open the stopper what's going to happen? Well some of the oxygen from the right will move to the left, some of the hydrogen from the left will move to the right and we'll end up in a situation something like this. Now I were to do the following experiment in my head. You know I started with a situation like this with the stopper closed and then I released the stopper. Would the reverse happen? It's very likely to happen. This way would require a miracle especially if the 20 10 to the power 23 molecules there are 10 to the power 23 molecules there. Okay one way of formalizing this idea goes as follows. Let's suppose that there was a way of measuring how many possible states the oxygen molecules could be. For instance suppose you divided up this flask into little cubicles and the oxygen being in any one of these cubicles was one a molecule being in any one of these cubicles was one state. It's very crude. That's not how the actual counting is done but just roughly. Okay now if you count how many states are there for the hydrogen and oxygen to be in when all the hydrogen is here and all the oxygen is here. That's clearly less than the number of states for the hydrogen and oxygen to be in if all the hydrogen is allowed to go everywhere all the oxygen is allowed to go everywhere. So just probabilistically a motion that starts with fewer allowed states will tend to the place where there's more allowed room. Just probabilistically something this is a very unprobable configuration if you've created it and you uncock it it's likely to move here the reverse is very unlikely to happen. Okay entropy is a measure of how many different states a system has can be in subject to some overall macroscopic constraints overall macroscopic constraints in this situation would be the volume of the two beakers the number of hydrogen molecules the number of oxygen. Subject to these overall macroscopic constraints usually conserved points like the number of molecules. Okay how many states can the system be in and the idea is simply that the law of increase of entropy is simply a statement of probability that systems tend to the place where there's more face space more kinds of ways in which they can be rather than the other way around. Okay so I've already said this this motion seems likely this motion seems very unlike now although what I'm going to say here between no no further role in the talk there's a caveat I thought I should mention and then we can forget about it but let me mention it the law of increase of entropy is a probabilistic statement and not an absolute statement unlike the law of conservation of energy okay at the microscopic level these molecules moving to this has a reverse motion that can happen it's just that when you start with molecules in this system the chance that your initial conditions will be so tuned so that you will undertake that reverse motion is very very small so the law of increase of entropy is a statement about probabilities and a statement about probabilities that becomes overwhelmingly like certainties when the number of constituents becomes very very large so it's not an absolute law like conservation of energy but a probabilistic statement however probabilistic statement that is certain limits namely the limits in which the number of constituents goes to infinity becomes essentially okay so with that caveat out of the way let's move on now as I said though the second law of thermodynamics applies only probabilistically it works almost with almost unfailing accuracy in large enough systems third sense in which the law of entropy increase is as a robust as a law of energy increase is as we learn more and more physics we have to change our way of counting states the states the system can deal change our way of giving formulas for entropy but as we learn more and more physics what changes is our formulas for entropy not the fact that there always exists an entropy and that their entropy always increases so it's a universal law sort of meta law that goes beyond the particular theories of physics we happen to be working with at the moment like the law of conservation of energy so now with this oh one last thing I have to say about entropy is that it's entropy is closely tied to the notion of temperature I won't try to explain this to you unless someone presses me but in the study of statistical mechanics of thermodynamics there's a formula for the temperature of a system once you know its energy if you know the entropy of a system as a function of its energy then one over del S by del del E gives you the temperature okay this is a fact which I won't try to explain to you unless you press me but now let's go back okay so enough of our for our interlude on entropy now let's come back to black hole now you see I told you that the law of conservation of entropy oh sorry increase of entropy was a universal law it survived for instance the people working with steam engines knew only about classical and they formulated the law on the classical arena quantum mechanics happened about a hundred years ago quantum mechanics revolutionized physics changed our conceptual framework for many things physics it did not among the other things it did not change the law of conservation of energy another thing that did not change was the laws of thermodynamics laws of thermodynamics sailed through unscathed by the change from classical quantum mechanics thermodynamics continues on formulas for entropy state but not the laws okay so in the 1970s a young israeli physicist jake beckonstein who was a phd student at Princeton working under john wheeler was called by his advisor was asked to consider the following thing his advisor wheeler suggested to him that it seemed like in the presence of black hole the theory of gravity violates the second law of thermodynamics and his pictures way of saying it was well suppose i'm sitting outside of black hole i'm bringing a cup of tea and i throw the cup of tea into the black hole the tea carries some entropy once the cup of tea vanishes behind the event horizon that the cup all knowledge of what what made up the black hole in particular knowledge of the entropy so it would appear that from a practical point of view from the point of view of somebody sitting outside the event horizon the entropy of a black hole the entropy of the universe has decreased violating the law of conservation of the law of increase of entropy well all our previous experience with physics before is that when you have a conflict between a theory and thermodynamics thermodynamics wins so what's going on here now beckonstein was just around the time wheeler suggested this problem to beckonstein Hawking had come up with this area increased theorem the theorem that event horizon areas of black holes always increase and never decrease and these two things remind or beckonstein remind felt that this this sounded interesting and what he did was the then suggest the following he suggested that thermodynamics arrives in the presence of black holes we have to change the formula for it and there is a new term in the equation for entropy one term does entropy of everything else in the universe and there's another term which is proportional to the area of the event horizon of all black holes with you and when you add this extra term in the formula for entropy then entropy will always increase this is something he proposed and kept through many thought experiments that if you modify the law of area in entropy increase in this fashion if you modify the law of entropy increase in this fashion then the second law of thermodynamics is never violated beckonstein performed relatively crude thought experiments by now we understand this very well and this seems correct there are many sort of semi proofs of this okay now so beckonstein went to her to propose that black holes actually carried their entropy and the formula for the entropy where I written that down is here the entropy is the he didn't know the proportionality the formula for the entropy was a proportion some number times the area of the event horizon now you know Hawking in written text admits that this paper of beckonstein irritated him very much he felt that it was a classic example of misuse of wonderful physics his beautiful area increased here which was a precise mathematical statement had been misused into this thermodynamic mesh okay and he didn't like it at all so he said about trying to find an argument to disapprove to to demonstrate that this was not true and one of the things he did was to use this basic thermodynamic fact if a black hole carries an entropy proportionality area according to this formula it must carry a temperature and Hawking thought that the first self manifestly ridiculous because things that carry a temperature radiate but it's you know an intrinsic feature of black holes that things go into them and nothing comes out how could it be just shows that this beckonstein stuff was not or was it beckonstein was unable to compute beckonstein was unable to you know seriously compute to compute the proportionality constant behind the area behind the area but rough argument suggested that the proportionality constant was proportional to one over h bar h bar is new is the flank constant now if the entropy is proportional to one over h bar then deli by del s is proportional to h bar so what you would get is a temperature that in the classical limit goes to zero and everything we've said about black holes for our classical state so Hawking began to worry about where that might actually be that black holes once you included quantum mechanics into the game actually did leave and then he did what is one of the one of the in my opinion one of the classic calculations that last 45 years and it's classic for the fact that it's extremely simple anyone you know anyone of you once you took a quantum field theory course could have done it just that you wouldn't think of it he um um he he uh he he did the following calculation he asked if you took free quantum fields let's say free electrons in the presence of a black hole what happens to them okay so the quantum behavior of these electrons is modified in a certain way and then he did the calculation I won't try to try to review the calculation when he did the calculation he found that what happens is that the electrons start quantum mechanically getting emitted out of the black hole okay in precisely the way that electrons wouldn't be if a black hole was at a particular temperature okay um I need to finish in the next three four minutes so I will not try to say more about this but he found that quantum mechanically black holes radiate at a given temperature and he found exactly what the temperature was his calculation gave him an exact number and then he found that that temperature was precise that that that radiation was black body radiation it looked exactly thermal if you and matched exactly with beckonstein's idea if you identified the entropy of the black hole with c cube a divided by 4 times Newton's constant uh oh and I've missed the h bar sorry there's an h bar here h bar c cube a divided by 4 times okay now this was confirmation of the idea that he set out to kill looks like black holes do have a uh an entropy do carry an entropy because they radiate they obey the laws of thermal dynamics if you associate the entropy this entropy to the black hole okay great so at first side it seems now we're all happy the second law of thermodynamics has been rescued black holes carry and everyone's happy however Hawking realized that there was something about this picture that doesn't seem quite right okay let me remind you what an entropy is I've already told you that the entropy is a measure of the number of microstates the system can be in consistent with a certain number of macroscopic constraints in this case that the conservation laws conserve quantities mass angular momentum of the black hole by the way if you compute um if you compute the entropy of a solar mass black hole using this formula you will find that the number of states associated with that black hole is e to the power s where s is 10 to the power 78 so e to the power 10 to the power 78 the number of states associated with the black hole it's an enormous number of states now there seems something wrong about this is it okay five moments okay there seems something wrong about this and you know there's something dichotomous about some part of my talk I told you about a feature about the classical black holes they forget lose forget all information about what made them what collapsed to make them classical black holes in classical general reality they carry almost no information they carry information only about conservation on the other hand we now associate an entropy with black holes which tells you that the fundamental description of the black hole is associated with an enormous number of states classically the black all black holes with the same charges identical quantum mechanically it must be that there is an enormous number of states associated with the black hole how doesn't this sound sort of problematic okay now let's see how can it be that a black hole is classically so simple and get quantum mechanically so complex and analogy may help us consider a glass of water okay um consider a glass of water water can switch around to do all kinds of interesting things we should just leave it at rest once it's finished switching and it comes to equilibrium equilibrium is very simple characterized by just a couple of quantities the amount of water and let's say the temperature once you know those two things in equilibrium you know everything there is to know about that water okay yet a glass of water at a particular temperature carries an entropy an entropy that all of you have studied about when you study third one I know so there is a description namely the description of hydrogen in which the glass of water in equilibrium is characterized by just the conserved charges yet we know that as a fundamental there's an enormous number of micro states that make up the that that equilibrium configuration of water there's these micro states are the details of how the molecules of the water are moving around to produce that water so before the atomic theory of what of matter you might have been puzzled how can water which is a unique state carries so much entropy we now know that that unique state is just a statement of ignorance we're describing some the systems of coarse-grained way that are basic microscopic degrees of freedom we're not keeping track of that gives rise to the entropy so it must be that something like this is true of gravity the equations of gravity must be something like the equations of hydrodynamics that are giving you a coarse-grained description of reality throwing away detailed information of some more fundamental degrees of freedom like the atoms of the water okay so this is roughly the picture that we we believe we have about black holes and yet we're not yet home dry Hawking pointed out that even if we believe everything I've just told you there's something strange about about this whole setup and that goes as follows Hawking did the following thought experiments the thought experiment the thought experiment he did was this was a version of the experiment we did by 20 minutes ago consider two stars of exactly equal mass one lithium rich the other very both of them collapse to give you black holes of exactly the same of exactly the same of exactly the same mass now at the quantum level you could say well there was a lot of quantum information in the in the in the detailed construction of the matter that made up that went to make up the black hole the black hole that detailed information has not been lost precisely because black holes carry entropy this individual microstates the black hole can be in that can record what meant went up to make up the matter quantum again we don't know how that could happen now Hawking said well let's take this one step further now what happens with black hole he discovered that the black holes read it and as far as he could tell the radiation that he could get he got was determined just by the classical black holes and nothing else so both these different black holes radiate and lose energy and eventually just radiate away to nothing in as far as Hawking could tell exactly the same way and there's if if this picture is correct needs to fundamental contradiction with the laws of all of physics really but in particular the laws of quantum mechanics because you see what you would have is two different initial states namely the beryllian rich star and the lithium rich star evolving into identical final states which never happened technically in quantum mechanics evolution is unitary evolution okay you can never have if two final states of the same the initial states are okay so Hawking posed this as a paradox and actually proposed that eventually when we understand the resolution of this we will find that we have to modify the laws of quantum mechanics to make them consistent with this with black hole we believe that's not the case okay quantum mechanics research is string theory of the last 40 years and in particular the adseft correspondence show no evidence that any anything needs to be changed the laws of quantum mechanics we we believe that this is Hawking's paradox is a false paradox but let's let's let's let's talk about it a little more in order to understand the paradox let's first contrast it with a non-paradox okay let's consider two different glasses of water okay two different glasses of water that were formed in some of the different states both of these glasses of water evaporate away and lead to nothing at the thermodynamic level the evaporation that the gas the thermodynamic properties of the gas that's evaporated away would would be identical for these two glasses of water and you might think we have the same paradox because your different initial state is going to identical final state but of course it's a false paradox the detailed state of this evaporated matter depends on the detailed states of the of the water molecules in the initial glass you kept track of the detailed state of the final matter matter will be influenced by what the actual water molecules were doing at the beginning and sort of you know the state of the vapor would be in detail in one-to-one correspondence with the state of the original gas water molecules in the glass there's no paradox here we understand everything really well so you might think well suppose that perhaps that's that's that's also what's going on with black holes we don't know what black holes are somehow composed of elementary constituents except in toy models in the ADS CFP correspondence of string theory we understand what these constituents are okay but except in special toy models we don't know what these constituents are but you might say well perhaps the detailed state of the Hawking radiation is being influenced by what these constituents are but Hawking you know Hawking asked us to take this a little further suppose these constituents live near the singularity of the black hole then by causality we've already seen that there's general features of physics that tell us that nothing inside the black hole can affect what's going on outside the black hole in particular Hawking's calculation of Hawking radiation happened entirely from the eventurized if the if the atoms of the black hole live near the singularity it does not seem that their details could by causal considerations could be imprinted on this outgoing radiation so it must be that somehow the atoms of the stuff live near the eventurized and if that's the case and maybe that's the picture you might have but a group of physicists including Sameer Mathur from Ohio State University and various other people are Almeri, Marolf, Pulsinski and Sully over three four years ago made precise a statement that somehow in some way we always knew that that if it were true that there were atom atom likes configurations near near the event horizon and they imprinted their information into Hawking radiation then that would cause the black holes after radiating a bit to become quantum mechanically extremely different from what they were classically in particular if a black hole started with some mass and radiated half its mass away at the end of the radiation process if we believe semi classical laws of physics the black hole horizon which was a smooth place which is rich spaceship could just sail through in classical theory would it appears have turned into a terrible quantum firewall of a place which if you banged into you would be destroyed I know I'm going very fast through this but okay let me just say all of these theorems make assumptions they make assumptions about how physics works and it could be that one of these assumptions breaks down okay and indeed these two gentlemen Kiryakos Papadodimas and Suvrat Raju both of whom were my PhD students at some point came up with a beautiful suggestion a solution to how Hawking's paradox might be evaded this pattern I won't want to try to describe I won't try to describe this in any detail but just to say that you know as I said all these all of these paradoxes make assumptions one of the assumptions is that the laws the equations you work with have a certain locality and Kiryakos and Suvrat observed that within the ADSAFP correspondence of string theory there's a certain fundamental breakdown in this locality which occurs in precisely the way that would be needed in in order to evade evade Hawking's paradox so they haven't understood in detail how this happened they haven't understood in detail how the information about what state went into make up the black hole is recorded in the Hawking radiation but that's the eventual picture the eventual picture is that we believe string theory is telling that the Hawking radiation carries the information of the black hole and signals a breakdown in the locality in certain fundamental in the fundamental equations of gravity in a very interesting and yet to be completely understood now let's believe that this is correct if this is correct then Hawking's information paradox holds within it deep lessons about the fundamental quantum description of gravity or space of time and it may be that the fact that black holes preserve information is proof of interesting stuff that space time is emergent yeah you won't try to describe in detail that space time emerges outside out of a non-space time description okay unless there are questions I would try to what I want to say is that Hawking's information paradox is an interesting paradox which probably has interesting implications for the fundamental formulation of quantum gravity 40 years after his formulation of this paradox is the subject of active research and holds potential promise I mean the person who completely cleans this up in a completely satisfactory quantitative way will become famous that's become the next Einstein and may may may understand deep lessons about the nature of quantum gravity which we still don't understand okay sorry sorry for this vagueness at the end of this you know the last part of the talk was not understandable I understand okay so try to forget that ask questions about the first part which is probably more there's no such thing as a bad question unless it's a terrible you say you said that when two black holes collide they radiate gravitationally yes when do they radiate in the gravitational waves or as electromagnetic how do you know where how would it how would you distinguish how would we distinguish yes you see collisions of black holes something I should emphasize is this you see collisions of black holes are incredibly simple events because black hole solutions initial black hole solutions are just simple solutions of general relativity and in order to understand what this collision is doing all you have to do is to solve the well understood equations of general relativity with well understood initial conditions such in principle completely algorithmic now if your black holes initially are uncharged then they are solutions of just general relativity with no matter and the final evolution will give you rise to some solution of general relativity with no matter with with no matter at all in particular no charge okay and therefore can only be in vibrations of the myth there's just no room for anything else on the other hand were the black holes to be charged black holes okay then these are solutions of general relativity coupled to Maxwell's and then the collision could give rise to roughly equal amounts of gravity and photo however it is believed that astrophysically there are basically no charge roughly speaking because if there was charge you know it would ionize the medium around it and neutralize it for the same reason that there are no charge planets roughly so realistically in an astrophysical situation the only way in which you could really emit a significant radiation through the electromagnetic radiation of black hole collision is by secondary effects let's say there's a charged plasma around the black hole the this could excite that plasma which could then secondarily primarily almost all the energy becomes gravity let's distinguish this from collisions of two neutrons if you have two neutron stars colliding okay then there will be significant amount of both gravitational as well as magnetic radiation and this by the way is in topical because about six months ago LIGO announced that they saw an event which they thought was a collision of two neutron stars quite remarkably they found it fast enough so that optical optical instruments could go and and zero in on that so now we have this so-called multi messenger event which has been seen in LIGO and also been seen optically confirming everything that they they feel this cannot be done with black holes as far as we are but with neutrons as well please okay you start gravitational waves of the same forms as that of a lepromagnetic identical Doppler shifts are basically kinematic you know you've got a wave how it gets stretched out or not depending on what frame of that what makes up that wave as long as it moves at the speed of light no difference just identical okay this is about the second part of your talk you mentioned that when you introduce entropy yes you said this is introduced for a system which is complicated enough yes and that of course we understand in terms of chaos and ergodicity in all that yes so when you introduce this concept of entropy for black holes in the picture of string theory whatever yes so is there any indication that black hole system is also ergodic and chaotic okay it's an excellent question so the first thing i want to say is that the complicated enough yeah has to do with chaos and ergodicity in fact the answer is yes let me say many one aspect of the complicated enough is that there should be many moving parts okay there should be a parameter in the system such that when taken to infinity the probability of entropy decreasing goes to zero now in familiar systems that parameter could be taken to be avagadro if you've made you've got a stuff made up of a certain number of molecules as that number of molecules goes to infinity the probability of entropy decreasing goes to zero okay now you could ask is there a parameter in the black hole situation now even though even without understanding the detailed structure of what makes up the black hole entropy we can answer this question by just looking at the formula for the entropy something divided by h bar so the limit h bar goes to zero takes the number of moving parts to infinity that's the limit in which the area increase the entropy increase theorem should okay so that's part of the point of being complicated enough in the sense of black holes this is the h bar however there is a deeper answer to your question there is a good reason to believe that black hole motion is essentially ergodic and in fact essentially Maximillian let me explain this let's not look at the actual reaching of chaos let's look at the approach to chaos as you know very well the approach to chaos happens through exponential dependence on the initial conditions through the so-called Lyapunovian right you take two initial conditions that are separated by a distance delta x and after time t they're separated by delta x times e to the power a times t within classical physics there is no known lower upper bound on this quantity a it can be anything however there has been a recently proved theorem about you can define a similar thing in the quantum theory i will have to tell you how you define a similar thing and there's been a theorem proved that in any quantum system this a quantity cannot exceed two pi times the temperature divided by h bar let me write or maybe i've got the two pi wrong the two pi could be up or down i may not be remembering that right but so e to the power times time this is the maximal possible divergence speed of divergence of anything the evidence first came from ads cft but the proof is a proof in quantum mechanics okay this was proved by Maldesena Schenker and Stanton okay so simple proof uses causality the input is causality okay now you can show that within ads cft ads cft describes the correlation function of a dual quantum field and you can compute the correlation functions that define this quantity in ads cft at a given temperature when you compute at a given temperature that that computation is dominated by the behavior of gravity in the presence of a black hole and so you can do that calculation gravitationally and you find that this value there is saturated that there's maximum possible values in fact that's of course how it you saw by doing many different calculations that you always got this number and then you came up with the proof that this was enough okay so there is a sense in which motions determine the microscopic motions this is a classical motion of black holes it's just dissipate it's like hydrogen it's the water molecules that are moving chaotic you need ads cft to prove this microscopic motion and when you do that you find not only is it chaotic it's maximum note that as h bar goes to zero this upper bound goes to infinity consistent with the statement that there's no no bound in classical space the initial part of your talk you said i mean the black hole as a pure spacetime geometry yes what is the sense of time there does it have an arrow at that point that's a very good question time in the structure of general relativity goes as far as every spacetime is a lorenzian manifold what does this mean let's remind ourselves what it means for something to be a euclidean a euclidean manifold in mathematics is a space such that any little bit of it technically a neighborhood is isomorphic to a little bit technically a neighborhood of rd the d-dimensional manifold is something such that it could be curving around let's say two-dimensional let's say this object here is some two-dimensional manifold that could be curving around in some strange way let's take a little region of it it's identical to a little region of a plane that's a little region of r2 okay the notion of the in general relativity spacetimes are spacetimes in which you have a four-dimensional spacetime such that any little region of it is identical to a small region of minkowski space okay now minkowski space comes with minkowski space as anyone who studied special relativity and i know all of you are experts in special relativity uh you know that minkowski space comes with a light code okay this is time and this is space the space is three-dimensional there's a light code there's a light code structure that is all important for the study of causality an observer can move like this but he cannot move like this okay so in any little neighborhood of our spacetime manifold there is a little light code and these light cones get sewn together to make the global light now what happens in a black hole is this suppose we've got the event horizon here the light cones which are like this start as we go nearer and nearer the event horizon start tipping over and at some point the light cone on the event horizon becomes like this when you go into the event horizon becomes like this now what is the rule of physics the rule of physics is that observers can only move towards the future in your light so here the rule of physics is that observers can only move into the in the light cone and towards the future this is something we're all doing locally the rules of physics always stay the same everything interesting in general relativity is not happening locally locally everything is like a little little bit of minkauci so it's the sewing up because of this tipping up the local rule remains that observers can only move like this and you see that this has translated into the most the statement that things can only move towards the center of the black hole and not away so the same rule that outside tells you you can't move backwards in time because the light concept if you ever tell you you can't move outside you know so it's the same local rule that is brought that is constructed out of the sewing together of these local light at least one question from a student this is you guys are the guys for whom this talk is I know there was some from the talk or you can ask me anybody from the students okay if not let's thank Shiraz once more and we'll start the we're running a little late so we'll start the program at 1145 yeah and there's a tea outside so please have tea outside and come back at 1145 for the data part of