PAUL GINSPARG: It is my great pleasure to welcome you to the second of Lenny Susskind's free messenger lectures. Based on the number of people who stepped up to us and introduced themselves to Lenny, and how thrilled they are to meet him in person, ignoring me, I think I need an introduction more than he does. I'm Paul Ginsparg, I'm a professor of physics and information science here at Cornell.
And I'll say a few words about this particular lecture series and then I'll give a short introduction to Lenny. So Cornell's messenger lecture was established in 1924.
LEONARD SUSSKIND: Paul, can I ask you to call me by my actual given name? Leonardo.
PAUL GINSPARG: Leonardo da Susskind. By a gift from Dr. Hiramo da Messenger. Actually, Dr. Hiram Messenger, a Cornell graduate of 1880 and a longtime teacher of mathematics.
The terms of the original gift established-- pay attention-- a fund to provide a course of lectures on the evolution of civilization for the special purpose of raising a moral standard of our political, business, and social life. There was a period when anyone who was anyone in the pantheon of physics was a messenger lecturer here, going back to RA Millikan in 1925 remembered the oil drop.
Sir Arthur Eddington 1933 remembered the solar eclipse in 1919. J. Robert Oppenheimer in 1945-- well, you know. Fred Hoyle in 1960. Richard Feynman 50 years ago, 1964, up to including recent messenger lectures in the past decade. Sir Martin Rees, Steven Weinberg, Nima Arkani-Hamed.
I was told to ask, out of curiosity, how many of you were here for Feynman's lectures 50 years ago? So I count about 10 or 15 people. After the lecture on Monday, I was thrilled to-- somebody who was an undergraduate visiting his children here told me the same thing.
Now, tonight's speaker, Lenny Susskind, was born in the South Bronx of New York City in 1940. He went to City College of New York. Came to Cornell in 1962, both to escape taking over his father's plumbing business in the South Bronx and also to receive a PhD in theoretical physics in 1965.
He went to Berkeley. He was on the faculty of Yeshiva University until 1979. Since then, he's been a professor at Stanford, including most recently since 2009, the director of the Stanford Institute for Theoretical Physics.
He's one of the founders of string theory-- actually the one who introduced the notion that particles could be described by excitation states of a relativistic string. Has four decades of over 200 cutting-edge research articles, still growing rapidly, he-- oh. We lost it. He's one of the-- sorry.
He has various awards. National Academy of Sciences, American Academy of Arts and Sciences. I'll read only his J.J. Sakurai's prized citation for pioneering contributions to hadronic string models, lattice gauge theories, quantum chromodynamics, and dynamical symmetry breaking.
His expository gifts have carried over to the popular realm. He has authored a number of popular votes on subjects from cosmology, to black holes, to quantum mechanics. He's been on Nova shows discussing various aspects of physics. On this-- I have to move this over.
For this poster, I took a screen grab in the upper left from his popular book on black holes and information theory because he told me at lunch today, the audience would be disappointed not to see at least one picture of an actual black hole.
There's also-- I think that might be a galaxy [INAUDIBLE]. And he has a large number of online lectures. This was absolutely extraordinary to me. This is what people know. I searched for Leonard Susskind in Google Videos. I don't know if you can see it. This is not a hoax. 281,000 results for videos of Leonard Susskind-- sorry, Leonardo Susskind, on everything.
Popular lectures, the theoretical minimum, and all the rest. So everything from My Friend Richard Feynman, to My Battle with Steven Hawking. So what I'm going to do now, and it's with great trepidation as to whether or not this is going to work-- I'm going to set it up. In honor of this being the second lecture in the 50th anniversary, I'm going to show a two-minute clip, which you'll see is entirely appropriate, but also great fun, from Feynman's second messenger lecture.
I'm going to stop it at one point for an essential observation.
PAUL GINSPARG: Now, the observation I want to make here is that--
I have it on good authority that these are not only similar blackboards, but based on the very sensitive genomic screen, these are the identical blackboards used by Feynman. Now, when he comes on, he's going to say something about how he had been blinded by the lights in the first lecture, and he was happy to be able to see the audience, but I want it on record that, in none of his lectures did he ever complain about the blackboards.
Now, the words he uses will also be an incredibly prescient 50 years earlier second lecture in introduction to tonight's lecture.
-Hi. I can see the audience tonight, so I can see also from the size of it, which last time was a black blob in front of my eyes, that there must be many of you here who are not thoroughly familiar with physics. And also a number that are not too versed in mathematics, and I don't doubt that there are some who know neither physics nor mathematics very well.
That puts a considerable challenge on a speaker who is going to speak on the relation of physics and mathematics. A challenge which I, however, will not accept.
PAUL GINSPARG: And there, in the middle of the screen, is Lenny. Let us welcome Lenny Susskind.
LENNY SUSSKIND: All right. I have a couple of things to say before I start. The first thing I want you to know is that I actually did bring a jacket and tie to Ithaca with me. I'm not wearing it, but I did bring it. You know what? I'll prove it next time. I'll prove it next time. Dick was better dressed than I am.
The second thing I wanted to talk about was the way I begin lectures. I often write out a couple of paragraphs, and then I actually read them. Sometimes I don't, and somebody asked me, why do you sometimes do this and why do you sometimes not do it? And the answer is, it really does depend on what I had to drink before I lecture.
Tonight, I had a couple of beers, so I have this terrible fear of completely forgetting what I'm supposed to talk about, and so I write a few paragraphs to get me started. So if you don't mind, I will begin by reading my paragraphs. But even before that, I want to second the thing that Dick said. I actually don't know how to aim a lecture like this. I really don't. I have the feeling that my problem is similar to our President Obama's. Anything I say will leave almost everybody dissatisfied. But here goes.
Quantum mechanics is confusing. I think we will all agree. Oh, actually, I'm not sure. David Mermin probably doesn't agree. He understands it better than we do. But apart from David, yes. One of its peculiar features is the often different descriptions of the same thing that sometimes seem incompatible to our classical minds.
A couple of examples that come to mind are the fact that the electromagnetic radiation is sometimes described as waves and sometimes described as particles, but not both at the same time. Another one is that particles have positions, particles have velocities, but not both at the same time. This is confusing. If an electron can have a position, and that same electron can have a velocity, how can it not have a position and a velocity?
All of this has a name, at least I think it's the right name. It's called complementarity. Now, the reason I'm not sure is, I don't think anybody ever had the vaguest idea of what Bohr was talking about when we talked about complementarity, but that's the way I'll use it. That there are complementary descriptions of things that, at first sight, look truly incompatible.
Now, how does nature avoid contradictions? The answer is that she never allows us to measure incompatible properties at the same time. We cannot measure both the number of photons and the phase of an electromagnetic wave. We can at most measure one of the other. The same is true for position and velocity of a particle. We might say that an electron does not have a position and a velocity, it has a position or a velocity.
Black holes are very strange objects, to be sure, but black holes in a quantum mechanical world are worse than strange. They appear to contradict deeply held principles. However, a strange kind of large scale complementarity is at work, which again allows us to avoid contradictions, but just by the skin of our teeth. I will do my best to explain. I hope nobody has a gun.
The last time I explained something a little bit, called information conservation, before I actually start to talk on the blackboard, I just want to remind you about what information conservation means. The equations of physics never allow information to disappear. Let me give you an example of something that looks like information is disappearing.
Here's a box of some sort. I'm going to slide it across the table. No matter how I start it, no matter how fast I start it, no matter which way I give it a push, it comes to rest. It comes to rest someplace on the table. There are many different starting points that might lead it to come to rest at the same place. It seems like it's lost information.
Of course, we all know what happened. What happened is, it heated the table underneath. It excited the molecules, heat when he was formed by friction, and object slowed down. Why? because energy got transferred to the table. Not just energy that got transferred to the table. The information of the starting point also got transferred to the table.
If I could reconstruct every single molecule, every single degree of freedom in the table, I could, in principle-- not in fact, but in principle-- recover exactly what had happened to the box. I would, of course, have to include all the air in the room, and everything else.
There's hidden information in the world. Hidden often because it's hidden in degrees of freedom which are too small and too numerous to keep track of. This hidden information has a name. It's called entropy. All right. That was by way of reminding you of what I said last time. That's really all we need from what I said last time.
Now I want to start by discussing the equivalence principle. I told you last time that my interest in these lectures was going to be conflicts of principle, and it will be true. We will be talking about the conflict of the principle of information conservation and the equivalence principle.
So let me remind you what the equivalence principle is. The equivalence principle is, of course, about an equivalence. It's the equivalence between acceleration and gravity. If I was standing here, not in a room with a gravitational field, but I was standing in an elevator that was accelerating upward, Einstein's elevator, I would feel exactly the same thing as I feel sitting here in this room. Instead of gravity, we would have the acceleration.
If I took this-- I'd better not throw this. Here's something I can throw. If I did this, or if it was juggling or something, I would have exactly the same experience if I were in the upward accelerated elevator or if I were in the gravitational field. So a uniform gravitational field-- keep in mind, for the moment, uniform. A uniform gravitational field is exactly the same as a field of acceleration, let's call it.
Let's pursue that a minute. Let's think of an observer, a person who was in uniform acceleration along the x-axis. Normally, I think of the vertical axis as the z-axis. I don't know why, but I'm going to use for the vertical axis, the time axis. So for the moment, we'll just think of uniform acceleration along the x-axis.
And the equation for uniform acceleration along the x-axis-- I think probably most of you have seen it-- is that the position of an object is the starting position, call it x0, plus 1/2g. g is the acceleration of gravity. In this case, we're talking about a gravitational acceleration times t squared. There's another term that I could put in to represent the initial velocity, but I'm going to leave it out.
This is the formula for uniform acceleration. It would describe my trajectory if I were an x in an accelerated frame instead of standing still. All right. Now, one of the things about this formula, which tells you immediately that it can't really be the real truth, is that it also says that the velocity is equal to g times t.
What does that mean? That means, if you wait long enough, you'll achieve any velocity, and you know you can't achieve a velocity greater than the speed of light. So this has to break down. In fact, with the ordinary acceleration of gravity, it would take about a year, and after a year, this formula would say you're moving with the speed of light, a year and a half and you'd be moving 1 and 1/2 times the speed of light.
So this can't be right. Yeah. This can't be the right formula. Not in special relativity, in any case. So let me show you what a uniformly accelerated frame of reference looks like in special relativity. I'll draw it by beginning with a drawing of space time. Here's space time. The vertical axis, now, is time, but I won't draw it.
Whenever a physicist starts drawing space time, chances are, he will draw two diagonal lines. The two diagonal lines represent the light cone, they represent the trajectory of light rays, light rays going to the right, or light rays going to the left. There are two other directions of space. Time, x, two other directions, which I can't draw on the blackboard. The blackboard just doesn't have enough directions, and so I have to leave them to your imagination.
Now, a uniformly accelerated trajectory is one which looks like a hyperbola. The formula that I wrote before would be a parabola. x equals 1/2g t squared, or 1/2 plus an extra constant. The right formula for a uniformly accelerated observer would be a hyperbola, and that hyperbola would asymptote, would eventually become very, very close to the light rays.
The meaning of that is a uniformly accelerated observer in special relativity asymptotically moves with the speed of light. Apparently, stops accelerating, but of course, in his own frame of reference, he is accelerating. Now we can introduce a clock. Imagine that the observer has a clock, and let's tick off the observer's clock ticks.
This is t equals 0, this is t equals 1, and up here somewhere, this is t equals 2. And if we keep going up, and up, and up, asymptotically, we'll come to t equals infinity. We can draw this a little bit differently. We can imagine a family of observers. Imagine a family of observers, and we can draw a time axis here. Not a time axis. A surface that we'll call t equals 0.
Everywhere along here is t equals 0. Everywhere along here is t equals 1 for the accelerated reference frame. Every where along here is t equals 2. As we keep letting time go forward, eventually, t equals infinity becomes this lightlike line there. For those who are not familiar with this, this is new, you won't recognize it immediately, but this is what it looks like to have a uniformly accelerated reference frame.
OK. Now let's call this observer here. Let's give her a name. This is Alice. Why Alice? Well, for those who aren't physicists, physicists all know that the two physicists in the world are Alice and Bob. Alice and Bob were two names that were-- I don't know who first introduced them, but they represent two observers.
I think people just got tired of saying Observer A and Observer B. They became Alice and Bob. When you need a third one, it becomes who?
LENNY SUSSKIND: Charlie, of course. I don't know who the fourth one is.
SPEAKER 1: The eavesdropper.
LENNY SUSSKIND: That's Eve, the eavesdropper, but who's the fourth-- I don't know. David. OK. So this is Alice. So far, we haven't introduced Bob, and Alice is moving like that. What can Alice see? What's in Alice's experience? This is what I want to ask. What can Alice look out and see?
So what she can see has to do with the light rays that she can eventually detect. For example, if a light ray from here, from this point over here, comes up and hits Alice, she can see that point. So this is a point that Alice can look back and see by virtue of a light ray hitting her over here. Here's another place that she can see. She can see this place.
She can look back and see that point. This point of space and time over here. In fact, she can look back and see everything to the right, to the lower right of this 45 degree line here. But she cannot see anything up in here. Why? Because no light ray from here can reach Alice.
So from Alice's perspective, Alice, who has eternally accelerated, forever, and ever, and ever keeps accelerating, this portion of space time here is not part of her observable world. Well, let's not cross it out for the moment. Alice's observable world is to the right here, and Alice would call this line here a horizon. She can see nothing beyond it. Can that be read? Horizon. Yeah. She can see nothing beyond it. This is Alice's world.
Now let's introduce Bob. Let's make another picture for Bob. This one's getting too cluttered. Here's another picture for Bob. Bob is a fellow who doesn't mind falling past this line. He's not uniformly accelerated. He's just in free fall. Free fall means he moves on a straight line, and so he's just perfectly free. He is not dragged along by the accelerated reference frame. He just falls through over here.
So here's Bob. Here's Bob, here's Alice. Alice looks back and sees Bob. Bob can also look back and see Alice. In fact, Bob can continuously look back and see Alice. He never loses track of her. But Alice loses track of Bob. Once Bob crosses here, she cannot see him anymore.
In fact, the way Alice observes Bob is, she keeps looking back from later and later time. She looks back from later and later times, and gradually sees Bob approach this point. She can never see him cross that point, because she can never get light from behind this point. And in fact, it appears to her to take Bob an infinite amount of time to cross that horizon.
That's a peculiar discrepancy between the way they describe the world. Alice describes the world as only this half space here. She describes Bob as asymptotically approaching the horizon. And Bob says, nonsense. I simply fell through the horizon. I didn't get stuck here. Alice says, he got stuck there.
So there's a discrepancy in the way they think about the world. This doesn't seem much of a big deal. This is just a funniness about the way we draw coordinates. Let's go a little further. Let's introduce a bunch of Bobs. Here's a bunch of Bobs falling in with Bob. A whole bunch of Bobs. How does Alice see those Bobs?
She sees the Bobs pile up at the horizon. She sees a grand pile up at the horizon, where they all seem to slow down and get stuck at the horizon. The Bobs, they never heard of such a thing. They just said, we sailed nicely through the horizon. Felt nothing at the horizon.
This is what the equivalence principle says. The equivalence principle says that, if there is a gravitational field, it must somehow do the same thing. This is not a gravitational field, but if there is a gravitational field, it must somehow do the same thing. All right.
So let's move on, now, to non-uniform gravitational fields. The Earth's gravitational field is not uniform. If we only look at the world over a small area, we can approximate it by a uniform gravitational field, where the gravitational field points in the same direction, has the same value everywhere.
But if we want to think about the Earth, we have to imagine that this acceleration is radially symmetric, that we're sort of being accelerated outward away from the Earth in a radial way. You can't really draw that. Doesn't really quite make sense. But what we can do is, we can take the Earth, draw it as a sphere-- the sphere is the floor, if you like-- and imagine at every point, if we draw a small little patch near that point, we should be able to use this picture.
What this picture says is that Alice out here looking at this, what will she see? She will not be able to see-- excuse me. I'm not now talking about the Earth. Of course, I am talking about a black hole. I shouldn't have called this the Earth. I'm now talking about a black hole. I hope the Earth isn't the black hole.
All right. So we're talking about a black hole. The horizon over here simply becomes a surface over here. And what does Alice see? Alice sees Bob, and Bob slowly, asymptotically approaches the horizon. Gradually, gradually, gradually, he gets closer and closer. Incidentally, as he gets closer and closer, I might add that what happens to him is, he gets contracted, flattened, pancaked, and eventually just gets, roughly speaking, plastered against the horizon.
Plastered against the horizon. It takes forever, in her frame of reference. In her watching of this, it takes forever for Bob to reach the horizon. And if there are a lot of Bobs, they all pile up. In fact, it's worse than that. Or perhaps maybe not worse than that. Maybe it's better than that. Everything that ever fell toward the black hole, everything that made up the black hole, everything that fell inward, that composed the mass of the black hole, Alice sees as piled up at the horizon.
What does Bob see is Bob jumps in? Well, Bob sees exactly the same thing that he would see here. The equivalence principle says that Bob sees nothing special at this point here. He sees no pileup, he sees no dramatic event happening. This is a little bit odd. Physicists have known this for a long, long time, and they have simply said-- the words they use is, "This is an artifact of funny coordinates."
Is it an artifact of funny coordinates, or does it represent two different descriptions which have to be made compatible? I would submit that the answer is the latter. We have to have a description of the world in which both of these descriptions simultaneously make sense. So we're going to talk about that.
A little more now. Let's go a little further. From Alice's point of view, all of this pileup here that piles up at the horizon is hidden information. It's very hard to see, incidentally. The reason it's hard to see is because it all gets red shifted. Redshifted means the radiation from it becomes longer and longer wavelength, and it just gets very, very hard to see.
So when I say Alice sees it, I mean Alice's mathematical description of the world, or of the black hole, involves this sort of picture, but she really can't see it. It's too dim. Things are too close to the horizon. Radiation just takes too long to get out. And so this becomes hidden information.
What's stored near the horizon there in this very thin layer, this thin layer just above it, is all of the information, and that means all of the distinctions that might have to do with the different things that could have fallen to the black hole, they're all stored there. And they're stored there, essentially, forever. Well, not quite, as you'll see. But they're stored there, and they're hidden information.
That must mean that the black hole has entropy. Entropy and hidden information are the same thing. So we come to an interesting conclusion. The black hole hides information, must give rise to entropy. Entropy is a quantitative concept. It's the logarithm of the number of bits of information that can be hidden in this way. For our purposes, a bit of information could just be an elementary particle.
How much information can be stored at the horizon of a black hole? At first sight, you might think that because you can get closer, and closer, and closer, there's a kind of sedimentary glop, sedimentary sediment that settles closer and closer to the black hole, you might think you can store as much as you like.
But it's not true. There's a limit to how much information can be stored at the horizon of a black hole. It was discovered by Bekenstein around 1972, elaborated by Stephen Hawking, and it's probably one of the most important contributions to physics of the 20th century. I'm going to take you through the derivation. We're going to do it.
We're going to do the derivation, the mathematical derivation, with all the fancy calculus, and group theory, and integral equations, and category theory. We're going to use category theory. All right. So we're going to go through the calculation of how much entropy can be stored in a black hole.
If I'm successful at this, if I can get this across, even if I can get it on the blackboard, I will be very happy. First statement, which I didn't write down and I was supposed to write down before is, there's a connection between the radius of the black hole, and that means the radius of the horizon. The radius of the place where the pileup takes place, and the mass of a black hole.
Here's the formula. The radius of the horizon is equal to twice the mass of the black hole, Newton's constant, divided by the speed of light squared. In what follows, I am going to ignore all numbers like 2, and pi. That's OK, because I'm only interested in the moment for orders of magnitude. So let's ignore, irrelevant for my purposes, irrelevant things like 2, and just say the radius of a black hole is proportional to its mass.
We could use another symbol. We could use not equals, but sort of equals. Sort of equals means approximately equals. All right. Now, what we're going to do, we're going to start with a small black hole, and then we're going to build up a bigger black hole, and we're going to build up a bigger and bigger black hole by dropping stuff into it one bit at a time.
And each bit is going to be required to be only one bit of information. The bits that we throw in should only convey one bit of information. For example, we could throw in photons. You can have polarization. A photon can have a polarization. A photon moving that way can polarize this way or that way. We could take the bit of information to be what the polarization of the photon which fell in was.
OK. So we're going to throw in photons, and each photon brings in one bit of information. This photon here, this wave here, is supposed to be the wavelike character of the photon. But if we're not careful, if we throw in photons of short wavelength, they will actually have more than one bit of information.
For example, if we throw in-- just imagine we throw in a photon. Imagine a photon is an ordinary particle. If we throw in a photon, then there's not only the information about whether the particle was there. There's also the information about where it fell in. Where it fell in might have-- there may be many digits to how many digits we want to keep track of in the angular position of where the photon fell in.
That's a lot of information. I had not intended to throw in that information. The way to avoid that is to use the Heisenberg uncertainty principle and to say, let's take a photon whose wavelength is such that it's comparable to the radius of the horizon. A photon like that is delocalized, doesn't have a definite position. It has a fairly definite momentum, a fairly definite other thing, but it doesn't have a well-defined position. In fact, its position is spread out over its wavelength. Therefore, when that photon falls into the black hole, it conveys no information about where on the horizon it fell in.
So we're going to take photons of wavelengths lambda. Lambda is the wavelength, and that wavelength is going to be chosen to be the radius of a black hole. Now, you can say, why don't you use even bigger, longer wavelength photons? The answer happens to be that wavelengths longer than lambda will just bounce off the black hole and won't go in. That's something you can calculate. It has been calculated.
And so if you throw in photons of wavelengths R, there'll be no information about where the fell in, but they will go in. So that's our game. And now we're going to ask, when we throw in a photon of that wavelength, how much does the radius of the black hole change. We're going to do that calculation. All right.
So what we have to know is the following. We have to know the energy of a photon of wavelength lambda. This formula here is also an ancient and venerable formula. I think it's due to Einstein. And it's the energy of this photon is Planck's constant, times the speed of light, divided by the wave length.
Now, if you don't know that formula, I can't do anything for you. If you do know that formula, then we can proceed. In fact, even if you don't know that formula, now you do. So. The formula's not finished. We should divide by lambda. The smaller the wavelength, the bigger the energy, and the wavelength is R.
So the photon that we threw in has energy h bar c over R. That means that the energy of the black hole has changed. We've thrown in-- so let's write the change-- little delta stands for change. The change in the energy, now it's the energy of the black hole. That's just the energy we threw in, so it's also h bar c over R.
I was sure that Paul would tell me that I forgot a 2pi in this formula, but I did. OK. Delta E equals h bar c over R. Now, how much of a change in the mass does that correspond to? Well, for that, we use a formula that you do know. E equals mc cubed. No. Let me say squared. Sorry. E equals mc squared.
So that means that the change in the mass is the change in the energy divided by c squared. So we can now write-- let's put a comma here-- delta, the change in the mass of a black hole, is h bar c over R divided by c squared. That's h bar over R times c. That's the change in the mass of the black hole.
What's the change in the radius of the black hole? Well, here we know the connection between the radius and the mass. If the mass changes by a certain amount, then the radius changes by the same amount times g over c squared. So now I can write about how much the radius changes. Delta R is equal to h bar over R times g over c cubed. Now it is c cubed. Now it is c cubed. One c comes from here and two c's come from here.
And that's the form. Do I have it right? I think I have it right. Yeah. OK. This is a change in the radius. So what I found out is that when I drop one of these photons in, its radius changes by this much. But now I'm going to do a trick. I'm going to multiply by R. So let's multiply both sides by r, and we're going to get something that you may not find particularly interesting.
But if you think about it for a moment, you'll realize that what's on the left hand side here is not just R times the change in R. It's the change in the area. The change in the area. The area is proportional to R squared, and the change in the area is R times the change in R. It's a little calculus formula.
So what do we have here? We have the change in the area of the black hole. Let's erase what's in between. And in the right hand side, R has disappeared. We just have a universal constant. A constant which is made up out of Planck's constant, the Newton constant-- Did I tell you that g was the Newton constant? If I didn't, you should have asked me.
Please ask me. If I do that again and I forget to tell you what something is, just blurt out, what the hell is that. OK. The change in the area of a black hole. When you change the amount of information by one bit, is this universal constant here. h bar g over c cubed. What are the units of this quantity? The units are, of course, area.
This must be some area. It is an area. It's a number which is about 10 to the minus 70 square meters. That's also called the square of the Planck length. It's called the Planck area. So every time you increase the amount of hidden information-- we might as well now call it the entropy. Every time you increase the hidden information by one bit, you increase the area of the black hole by this amount.
Well, after you do this many times and you build up from, let's say we start with a tiny black hole and we built up a big black hole, the answer is that the area must be the number of bits that we threw in time this Planck area. We can write it a different way. I'm going to erase some things over here. I'm not ready for the next blackboard yet.
I'm going to erase over here, and what I'm going to write is that the area of the black hole is equal to the number of bits of information we threw in. That's called the entropy. It's called the hidden information. The symbol for entropy is S. I don't know why. I don't know where it came from.
Somebody told me that it represents Sadi, S-A-D-I. Sadi Carnot. I think it's "sah-dee." Is that the way you say it? "Say-dee," "sah-dee." Sadi Carnot, who was the first to introduce entropy into physics. And I asked, well, why didn't they call it c? Well, it turns out, I think the person who named it was named Clausius and didn't want to name it for himself, so I think this is true.
So it's called S. S is entropy. It's the area divided by the area of one bit of information. It's the total area, how many bits of information can you stick on it. And that means divided by h bar g, and a c cubed, which I'll take from the denominator and stick it up in the numerator, because that's where it goes.
Now, this formula is not exactly right. It's not right because I didn't keep track of the constants, and in fact, I didn't have enough control of the physics and the mathematics to keep track of the constants. But Stephen Hawking did keep track of the constants, and discovered that there was a 4 in the denominator here.
This is extremely interesting. I mean, it's more than interesting. It is, as I said, one of the great discoveries of 20th century physics, that a black hole has an entropy which is proportional to the area of a black hole. The area of the horizon of the black hole. Now, that's unusual.
The reason it's unusual is, most systems-- for example, the room here. The room is full of molecules. The molecules are too small to see. And so there's a lot of hidden information. How much information? Basically, the number of molecules. How many molecules? Well, it's proportional to the volume of the room.
In most cases, in fact, all cases that I know, besides this, as a rule, the entropy of a system is proportional to its volume. This is something different. This is something we never saw before, that the entropy of a system is proportional to its area. Now, it does occur elsewhere, but it's unusual. Any questions? Yes.
SPEAKER 2: Looks as if the constant's the wrong way up there.
LENNY SUSSKIND: I am a teacher. This is what I do.
SPEAKER 2: When you went from the delta A up there, looks as if the constant's the wrong way up.
LENNY SUSSKIND: Did I get them in the wrong place? It should be the area divided by this. Right? Yes?
SPEAKER 2: Yeah. S is the area divided by the--
LENNY SUSSKIND: Hm?
SPEAKER 2: S is the area divided by the--
LENNY SUSSKIND: Oh my god. Oh my lord. Thank you. S is the area divided by the Planck area. Entropy is the area divided by the Planck area. In other words, the black hole horizon, its area is equal to the number of bits of hidden information times the Planck area. Whoops.
OK. Let's go a little bit further with this. Black hole has entropy. It has energy. It has energy because E equals mc squared, and the black hole has a mass. It has energy and entropy. Thermodynamics tells you that when anything has entropy and energy, it has temperature. And one can calculate the temperature.
I'm not going to do the calculation here. I could show you how it works. It comes from a basic relationship between energy, entropy, and temperature. For those who know, it's equals DE equals TdS, but never mind. From these things, you can calculate the temperature, and the temperature of the black hole is h bar c cubed, I believe, over some number times the mass of the black hole times g.
That is the temperature of the black hole. Now, the reason we're interested in the fact that the black hole has temperatures is, I'm not so much interested in the exact formula. That's not the issue here. Just the mere fact that it has temperature means that it shines like a warm body. It shines like a black body. It shines like a heated object. Shines means it gives off radiation. It has a luminosity.
All right. Given that it has a luminosity, it means it gives off energy. Any warm body, if put in empty space, will radiate its energy away. So this black hole with a temperature will start radiating its energy away. That means the black hole evaporates. We could calculate if we wanted how long it takes for the black hole to evaporate. It's a very, very long time. But this was the discovery of Stephen Hawking sometime around 1974.
And as I said, it was really quite a monumental discovery. Black holes have entropy, they have energy, they have temperature. They behave like a bucket full of warm water, or something like that. Now, if there are no questions, I will go on.
Let's come over to this blackboard here. This is bad news for Bob. Well, maybe it is, maybe it isn't bad news for Bob. But let me tell you why it might be bad news for Bob. Bob, remember, is the fellow who jumps into the black hole. He's the one who likes to jump across the horizon. OK, why is it bad news? Well, let's think about this for a moment.
Here's the black hole, and if you go far away, you will discover photons coming out of it. This is the radiation. This is the black body radiation that comes out. And the black body radiation has a certain temperature. We wrote down what the temperature. It's h bar c, whatever it is. h bar c cubed over mg.
Associated with that is an energy for each photon. The photons have about that wavelength, and they just fly out. But a photon that appears over here with this temperature, it's also the energy of the photon. Temperature and energy, roughly related. This is also the energy of the photon.
When a photon materializes at the horizon and goes out, it loses energy. Why does it lose energy? For the same reason that if I threw this chalk up in the air, as it rises, it loses kinetic energy. The gravitational field is pulling it back, and the result is that it loses kinetic energy. The same thing happens to the photon as it goes out. It loses kinetic energy.
But by the time it gets out, it has this much kinetic energy. That must mean, when it started near the horizon, it had a lot more kinetic energy, and if it had more kinetic energy, it effectively means that things were much higher horizon. Bob falling in will not encounter a photon of this energy. This is very low because of the mass in the denominator here. He will encounter a photon of much shorter wavelength, much higher energy, just because, if it got to be of this energy by the time it got out, it must have been much higher energy when it left the horizon.
And in fact, you can calculate. The energy of a photon-- or let's just call it the effect of temperature. The effect of temperature at a distance d from the horizon-- little d-- that temperature is h bar c-- that's not the important thing-- divided by d. The implication is that, as you go closer and closer to the horizon, the temperature gets hotter, and hotter, and hotter.
That could be bad news for Bob. Bob, who's going to jump in, will apparently encounter a huge temperature as he gets near the horizon. On the other hand, something very peculiar, because the equivalence principle tells us that when Bob falls through the horizon, discovers nothing special. Just empty space near the horizon.
And so we have a kind of conflict. We have a real conflict, I think. Well, I don't think it's a real conflict. I think there's an answer to it. Let's just go back for a minute. What is Alice's description of this black hole? A pileup happens. The pileup carries lots of entropy. Apparently, it carries a lot of temperature. It's very hot near the horizon.
Where did this all come from? It came from the assumption that information is conserved. How did we use information was concerned? Well, we said anything that falls into the black hole is piled up near the horizon. It doesn't get lost. So information conservation on the one hand tells Alice that there's a pile up near the horizon.
Bob, on the other hand, uses the equivalence principle, and he says Einstein tells me that a gravitational field is nothing but a field of acceleration, and under those circumstances, I should discover nothing special when I pass the horizon. Now, I left out one concept here. I didn't intend to leave it out. I just went too fast.
The center of the black hole. What's happening at the center of the black hole? In the approximation where the gravitational field is uniform, the center is infinitely far down. In an approximation where the gravitational field is completely uniform, the center is infinitely far way. But if we take into account the curvature of the Earth or the curvature, in this case, of the horizon, the center is somewhere finitely down.
So somewhere in the center, things are finite. And according to the standard rules of general relativity, Bob should discover, when he falls through, a singularity at the center. The singularity is a very nasty point. Let's not worry about the mathematics of it. Let's just say that gravitation, as you get close enough to that point, becomes so strong that it tears you apart. Pulls you apart like toothpaste. Singularity's a bad place.
Where is the singularity on this diagram? Where is the singularity here? I'm going to show you where the singularity is, and I'll show you why Bob is in big trouble, not because of those high temperatures, but because of the singularity. The singularity is over here. That's the singularity of a black hole in this diagram.
Think about, for a moment, what happens if Bob finds himself back here. If Bob finds himself back here, that means inside the black hole, if he finds himself inside the black hole, he cannot escape. He cannot escape the singularity. To escape out here, he would have to go faster than the speed of light. He can't do that. He's inevitably going to crash into this. No matter how he moves, he can't get away from the singularity.
So that was one more point that I meant to stress, that there's something going on at the center of the black hole which is very deadly for Bob. But according to this story here, temperature, it's at the horizon of the black hole where Bob runs into trouble.
Well, which is it? Bob is not going to survive flowing into the black hole. That's for sure. But is he going to get himself in bad trouble at the horizon, because of this, or is he going to sail through the horizon freely and find himself destroyed at the singularity? That's the puzzle. That's the conflict of principle. that's the puzzle that this lecture is about.
OK. Which description is real? We could take a poll. Let's take a poll. Everybody has to vote. Everybody has to vote. Which is the truth? Is the truth that Bob runs into trouble and gets killed at the horizon, or that just a crazy artifact of silly coordinates? Or does he get killed when he gets to the singularity?
So I will ask you to raise your hand. For those who think that he gets killed at the horizon, raise your hand. How about those who think he gets killed at the singularity. I thought so. There's a very strong tendency, for one reason or another, to believe that Bob's description is better than Alice. I don't think it's sexism.
For some reason, people tend to prefer the Bob description, the description falling in. Why it is, I don't know. In my opinion, both of them have to be made consistent. They have to be consistent. And so let's talk for a moment about the possibility of making them consistent. And I will show you that they are consistent. OK.
So here's our horizon. And using the formula, which I erased, that the temperature is equal to-- what did I say it was-- h bar c cubed over d, over the distance from the horizon. That's the distance from the horizon. Let me take the place. The place here where the temperature is larger than a certain temperature. What temperature? Oh, the temperature at which Bob will be scalded. The temperature at which Bob--
Let's do something a little bit simpler. Instead of Bob, let's just drop an atom in. According to this picture, if the atom gets close enough to the horizon, it runs into a place where the temperature is hot enough to ionize it. So an atom falling in, according to this picture, will get ionized at a certain distance from the horizon. That distance is the temperature where the black hole or where this formula becomes what? 13.5 electron volts, or whatever it happens to be, the ionization energy.
On the other hand, the other picture says that the atom just sails through. Let's do an experiment. Let's do an experiment of the following kind. Charlie-- let's see. Alice is here, and she's watching all of this. Charlie is going to illuminate the atom with radiation that will bounce off so that Alice can see what's going on. This is how you see things. This is how you find things out. You illuminate the system with, let's say, photons, or with radiation, and take a look and see what's going on.
Question. Will Charlie's radiation come back to Alice and say the atom was ionized, or will it say that the ion was an unionized? OK. So let's see if we can analyze it. In order for these photons to see what's going on in this small distance, d, here, the photons have to have a small enough wavelength that they can resolve this distance.
You can't see something of a given size with photons of longer wavelength. So the photons have to have a wavelength, d or smaller. Lambda of Charlie's photons has to be d or smaller. From that, we can compute what the energy of the photons is. The energy of these photons will be-- perhaps no surprise, the energy of those photons will be h bar c cubed over d.
Did I get that right? h bar-- sorry. h bar c over d. The energy of the photons is h bar c-- is this right? This can't be right. c. It's c. It's c. Same thing.
All right. So here, then, is the energy of the photons that Charlie has to shine on this system. What's the problem? The problem is that these photons that Charlie uses to illuminate the atom are of a high enough energy to ionize the atom. In the attempt to try to find out whether the atom was ionized, quantum mechanics-- this is the complementarity. This is the uncertainty principle.
This goes back to what I originally started with. The experiment to try to see that the particle fell through without being lionized will itself ionize the atom. This is a very, very general pattern. Any attempt to try to discover whether there is a pile up over here by shining light or anything else on the system to see what's going on will produce exactly at the effect that the experiment was designed to show didn't happen.
For example, when Bob falls through, Alice may want to find out whether he gets killed, or roasted, or fried, or whatever it is at the horizon here. The only way she can find out is for Charlie to illuminate Bob with radiation and send it out to him. The answer will be that the radiation that Charlie has to use to see whether Bob was fried will have an energy hot enough to fry Bob.
So I'm going to stop around here and tell you that, as far as we can tell, all evidence is that both descriptions are consistent, but they're complementary. Complementary in the same sense that the wave particle theory of light is complementary, complementary in the same sense that you can describe an electron by a position or a velocity. Not both.
What we're learning is, you should not try to think of the description of this black hole simultaneously from the point of view of Alice and Bob. If you try that, you will run into trouble. One or the other. Either you do or you do not try to find out if Bob was roasted. If you don't try to find out, Bob will fall through with no trouble. But Alice will never know. If Alice does try to find out, she will roast Bob.
The evidence is that these two descriptions are not incompatible, but complementary. I will stop at that point. I think. I have lots more, but we're going to stop there.
PAUL GINSPARG: [INAUDIBLE].
Could I? Oh, I can? Want me to go on for another 5, 10 minutes? We started at 7:30.
PAUL GINSPARG: Everybody has to vote.
LENNY SUSSKIND: Yeah. OK.
PAUL GINSPARG: Yes? Should he go on or not?
LENNY SUSSKIND: OK, we can go on for a little while.
PAUL GINSPARG: And then what about no?
LENNY SUSSKIND: All right. OK. That's very nice. All right. What I promised to talk about in the title of the lecture was the holographic principle. We've not talked about the holographic principle. A holographic principle is another extremely odd, but apparently really-- what shall I say-- consequence. Logical consequence of what I've told you.
So let me tell you what the holographic principle is. It, again, has to do with where information is really stored, and how much information can be stored, but not, for the moment, in the vicinity of a black hole, but just in empty space. Let's suppose we have a region of space. There's a region of space. I chose it to be spherical for simplicity.
It could be a very big region of space. It doesn't have to be small. It could be big enough to contain galaxies, and planets, and all that kind of stuff, but of some finite radius. Some finite radius sphere. And the question is, what's the maximum amount of entropy that can be in there. The maximum amount of entropy that can be in a region is the maximum amount of information that can be hidden.
It's also a measure of how many degrees of freedom it takes to describe that region of space. The maximum entropy of a system is, roughly speaking, how many distinct coordinates does it take to describe that system. All right. So there's our question.
What's the maximum entropy? It can be here. It could be in the form of gas, it could be in the form of Swiss cheese, could be the form of people, whatever. How much hidden information, or how much information, period, can be inside, or how much entropy be inside there. So here's the thought experiment we're going to do. First of all, let's suppose that the entropy initially was S in. in stands for initial.
The initial entropy is called S in. Now let's imagine that we have a large shell of material. We create a large shell of material out of whatever material we have available. Interstellar, gas, whatever it happens to be, and we arrange it in a shell, and then we project it inward so that it starts to fall in toward this region.
Now let's choose the mass of the shell to be exactly right so that, when it's combined with a mass in here, it's exactly enough mass to make a black hole of precisely the same radius. You have to trust me. That's possible. There's no ambiguity about it, and there's no controversy about it. It is possible to create such a shell which is just right to create a black hole of exactly that size.
So the shell comes in, crosses the radius R, and leaves a black hole. What's the entropy of that black hole? The entropy of that black hole is the area of the region, which is the same formula, times c cubed over h bar g. Most important is that-- and it's the area in Planck units. 4 times that. It's the area in the units of this very, very small Planckian area.
What does this say? First of all, the second law of thermodynamics says that the entropy after you do this must be bigger than the entropy before. Entropy increases. Therefore, we learn the following. That the initial entropy must have been less than or equal to, could not have been bigger, than the area times this other stuff here.
That's a limit. That's a limit on how much entropy could have been in that region of space to begin with. It didn't really depend on whether there was a black hole there or whether there was a shell of matter there. Even if the shell of matter wasn't there, just the fact that we could have created it and sent it in tells us that the entropy in a certain region of space cannot be bigger, can never be bigger, than the area of that region in Planck units.
That's really interesting, because in all other physics, the maximum entropy of a region of space is proportional to the volume. It's also true in most other physics that the number of degrees of freedom that it takes to describe a region of space is proportional to its volume.
We have found out by the series of indirect arguments that any region of space can be described by a collection of degrees of freedom which are no more numerous than the area of the region in Planck units. That led to a conjecture. It led to a conjecture that any region of space can be described by some kind of theory describing degrees of freedom on the boundary of the region of space. That is what was called the holographic principle. Holographic, because it's like a hologram.
That was a conjecture. You might even call it a speculation, which is worse than a conjecture. But it was in 1994, so this conjecture was put forward. Most people at the time thought it was a screwball motion. But it took a couple of years, basically until 1998, until a young Argentinian physicist whose name you should know--
I ask people, do you know who Juan Maldacena I'm talking about physicists or physics groupies and so forth. Often, they don't. Juan Maldacena is, perhaps, the greatest physicist of his generation. I won't tell you how old he is. He's younger than me, so he's not necessarily greater than me, but he is certainly the greatest theoretical physicist of his generation.
I'm going to write his name. Maldacena. Mal. Mal stands for bad or something, but I don't know what the-- what does a bad of-- what is cena? It's Italian, I think. Bad of cena? Anyway, his first name is Juan. Juan Mald produced a perfectly rigorous description of this for a particular kind of space time called Anti-de Sitter space. But that's not what's important.
What's important is he dotted the I's, crossed the T's, and found an extremely precise mathematical description of the degrees of freedom on the boundary which would describe a region of space. And indeed, it turned out that there were no more of them than one per Planck area on the region of space.
So in some sense, the world is really described by a kind of holographic description on its own surface. Just to make it graphic, everything in this room, there is enough degrees of freedom on the walls of the room to describe everything in the room. It is as if the walls of the room could be thought of as a hologram describing what's happening in the interior.
OK. So we got through the holographic principle. Thank you.
PAUL GINSPARG: So while you're thinking about questions to ask the lecturer, I have three comments. The first is, Juan Maldecena will be giving the Bethe lectures this fall.
LENNY SUSSKIND: I wish I would be here.
PAUL GINSPARG: We'll see if Lenny's right.
LENNY SUSSKIND: You'll find out I'm right. I'm telling you.
PAUL GINSPARG: The second comment is, tomorrow's lecture will begin at 4:30 in this room, but there's a class that ends at 4:10, and the audio video people are going to be moving quickly to get everything ready. We won't be opening the doors until 4:15 so the guest will not interrupt the end of the class.
The third is that there will be a reception in the atrium between Clark and the new physical sciences building, there will be bunch of people here. There won't be enough food for all of you, but Lenny has promised he's stay until long after midnight to answer any further questions.
SPEAKER 3: Today or tomorrow?
PAUL GINSPARG: Right now. After.
LENNY SUSSKIND: Now?
SPEAKER 3: No, it's after this.
LENNY SUSSKIND: Oh. I thought you said tomorrow.
PAUL GINSPARG: Uh. Well, I had three comments. The second comment referred to the lecture tomorrow. The third comment referred to after the lecture today.
LENNY SUSSKIND: Ah. Got you.
PAUL GINSPARG: Do we have questions for Lenny? Yes.
SPEAKER 4: OK. So you said that if something is falling past the horizon, it can't cross back out, but simultaneously, you've said that the black holes are radiating temperature. So if things have fallen in, you've added energy to the system.
LENNY SUSSKIND: Mm. In one description, things fall in. In the other description, they never get in. In Alice's description, which is appropriate to somebody who stays outside-- remember, somebody who falls in doesn't see this radiation. Somebody who falls in has a short amount of time before they're squashed to annihilation at the singularity. Somebody who stays outside sees the radiation.
So we can ask, what is Alice's description of things. Alice says there's pile up, and all these degrees of freedom pile up at the horizon. They stick around there and gradually sort of one by one, they boil off. Bob, who fell into the black hole, says I got killed-- well, he doesn't say this, but he says, I was destroyed at the singularity, and simply wasn't around long enough to know whether the black hole evaporated.
You know, he's gone in a small fraction. The amount of time for a solar mass black hole, the numbers here are a little bit deceptive. A solar mass black hole is about a kilometer. You might think that a meter's worth of human being could fall through a horizon a kilometer big. No. It turns out that the tidal forces at the horizon would be so strong that they would tear you apart. But that's not the important thing.
A much smaller thing could fall through. A bacterium could fall in. How long would the bacterium have before it fell and hit the singularity? And the answer is, approximately the light transit time across the black hole. That would be about a millisecond, I think. I think it would be about a milli. For one kilometer, anybody can do that fast in their head? One kilometer? I think it's about a millisecond.
So that bacterium has about a millisecond. How long does it take the black hole to evaporate? Oh, just 1/10th of the 75 years or something like that. So the answer is that Bob has no experience of the evaporation. It's Alice whose story involves the evaporation of the black hole, and her story is that there's a pile up at the black hole, and the pileup gradually evaporates off the surface of the black hole. So it's quite confusing, but it's quantum mechanics. Quantum mechanics is supposed to be confusing.
PAUL GINSPARG: Yes.
SPEAKER 5: So how do we know that the black hole has a uniform radius?
LENNY SUSSKIND: Has a what?
PAUL GINSPARG: A uniform radius.
LENNY SUSSKIND: A uniform radius. OK. How do we know that a black hole is a sphere? Is that the question?
SPEAKER 5: Right.
LENNY SUSSKIND: Yeah. Of course, if it's a rotating black hole, which most black holes are, then it's a sort of oblate-- oblate. Do I have it right? Oblate is the broad one and prolate is the skinny one? Yeah. It's an oblate spheroid, which means a sort of ellipsoid. But if it's not rotating, it is spherical.
Now, one answer to it is a black hole is a thermodynamic system. A black hole is a thermodynamic system, and that means it's a system which will come to thermal equilibrium. When a system comes to thermal equilibrium, it becomes featureless. If a system is spherically-- no. Forget whether it's spherically symmetric.
If an idealized system, such as a water droplet in space or something like that, which is a thermodynamic system, is allowed to relax to thermal equilibrium, it will be perfectly round. One of the reasons is that the temperature has to be uniform everywhere across it. Temperature, pressure, and so forth, and so it will achieve a perfectly round behavior, and the black hole is also a thermodynamic system which comes to equilibrium.
The words that John Wheeler used was, a black hole has no hair, which means it's sort of bald, like my head, and perfectly round.
SPEAKER 5: And how do you describe the region between the singularity and the horizon?
LENNY SUSSKIND: Yeah. OK. But that's this region here. As long as you don't get-- how does who describe it? Alice or Bob? Charlie. How does who describe it?
SPEAKER 6: It has to be Bob?
LENNY SUSSKIND: Has to be Bob. Alice has no use for it. She never heard of it. She said, the world ended over here. Bob, yes. And what does Bob say? Bob says, it's just flat space. It's just a normal vacuum, normal empty space, except that it starts to get curved, and distorted, and deformed to the degree where it will become very harmful as he gets close to the singularity.
As long as he's not too close to the singularity, is just ordinary, empty space.
SPEAKER 5: So in some sense, the gravitation force is--
LENNY SUSSKIND: What's that?
SPEAKER 5: In some sense, the gravitation force is increasing?
LENNY SUSSKIND: Can you? I'm going deaf.
PAUL GINSPARG: He asked if, in some sense, is the gravitational force increase.
LENNY SUSSKIND: Yes. In some sense, the gravitational force is increasing, but it's increasing-- in particular, the tidal force is increasing. The tidal force is the thing that pulls you apart. The tidal force is increasing, but if we don't think about the tidal forces, if Bob is small enough that the tidal forces don't disrupt him, then he's just in free fall.
In free fall, there is no gravitational field. If you jump from the ceiling and fall down to the floor, with your eyes closed, of course, you don't realize that you're falling until you hit the floor. So Bob is in free fall. He says, everything is perfectly OK until he gets to close to where the geometry becomes highly deformed.
PAUL GINSPARG: And one can show, inevitably, in finite, proper time, he's sailing along and then, oh, bloop. So one more question and we should deconvene. Yes, [INAUDIBLE]?
SPEAKER 7: It's been a beautiful talk.
LENNY SUSSKIND: Thank you.
SPEAKER 7: Thank you very much. But what experiment can we do, what rational analysis can we use to prove you're wrong?
LENNY SUSSKIND: There's none.
SPEAKER 7: But, if matter's falling into a black hole, we know there's [? entropy in a ?] black hole. What can we look for?
LENNY SUSSKIND: I don't think this is going to say anything new, but what I would say is, this has implications for cosmology. Remember, in cosmology, I talked last time a little bit about cosmological horizons. OK? Everybody is falling through somebody's horizon in cosmology, in De Sitter space. Am I talking a language you are familiar with? No.
SPEAKER 7: I'll get it.
LENNY SUSSKIND: All right. In cosmology, everybody is falling through somebody else's horizon. And so just the fact that we don't get destroyed every fraction of a second means that horizons are smooth. You know, you're asking a question which I don't know the answer to. How do you make empirical science out of any of these questions?
This issue has been around since 1974. It has invaded physics to the point where I would say the majority of papers in theoretical physics, including condensed matter physics these days, is using the mathematics of horizons and so forth, but it really is troubling. All right, letme just put it bluntly. It's very, very troubling that we've entered such a range of parameters that it seems absolutely out of the question to do experiments.
Even if I told you that you could do the experiment of jumping in the black hole, I think you would tell me, well, get me the black hole and you try it. But, you know, it's a problem. I don't know the answer.
SPEAKER 8: Silly question. If space time is granular, and then you tried to use the holographic principle, doesn't it sound as if the surface area has to be smaller than the grain to get all that information in?
LENNY SUSSKIND: The whole surface area is smaller than the grain?
SPEAKER 8: No, no. I mean, if you look at the inside of the [INAUDIBLE] space and say, OK, how many grains do we have, and then we put--
LENNY SUSSKIND: In the inside of the space? Yeah. There are many, many more grains, apparently, than on the surface of the grains-- that's right. And what this is telling you is that those grains cannot be independently excited. There just aren't enough degrees of freedom so that every independent place can be independently excited.
SPEAKER 9: [INAUDIBLE].
LENNY SUSSKIND: Yes, yes, yes. But there's another way to think about it, which makes it less odd. If you try to excite, independently, the degrees of freedom in every region of space, you'd make a huge energy. You make an energy which would make a black hole bigger than the region that you're talking about. Right. So it's consistent, and right.
PAUL GINSPARG: OK. So there will be food for some of you. Lenny will provide food for thought for the rest of you.
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Theoretical physicist Leonard Susskind delivered the second of his three Messenger Lectures on "The Birth of the Universe and the Origin of Laws of Physics," April 30, 2014. Susskind is the Felix Bloch Professor of Theoretical Physics at Stanford University, and Director of the Stanford Institute for Theoretical Physics.