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MAXIM PERELSTEIN: Well, I want to welcome everybody to the third lecture in the Messenger Lecture series for this year, by Nima Arkani-Hamed. I'm Maxim Perelstein. I'm a professor in the physics department. And I want to introduce this lecture. So this is the third time you're here in this room this week. So by now, you must know where Nima got his PhD and all that kind of stuff. So I'll try to make it a little more personal.
So I first met Nima in '97 or '98. I was a graduate student at Stanford, working on a beautiful, but extremely esoteric subject called multi-loop scattering amplitudes in maximally supersymmetric theories. There was maybe about five people in the world working on that. It was fun. But you know how you're a graduate student, you work hard, but you get a little shifty. You get a little restless.
And so one day I ran into this new postdoc, sitting down the hall, Nima, who just came up with this amazing new hypothesis that space-time can have extra dimensions, which are a millimeter across, about this big, a drastic change in our ideas of space-time. So, of course, I drop everything I'm doing. I started working on large extra dimensions. It becomes my thesis. Nima goes on to invent a large number of amazingly inventive and amazing influential ideas, such as deconstruction, little Higgs theories, split supersymmetry, ghost condensation-- among other things, he was great at naming things-- theory of dark matter.
So 10 years later, he comes to Cornell to give Messenger Lectures. And in addition to Messenger Lectures, he gives us a few more technical lectures about the most exciting subject in theoretical physics right now. It's probably the most exciting due to Nima's work and his collaborators. And guess what that subject is? It's multi-loop scattering amplitudes in maximally supersymmetric theories.
[LAUGHTER]
It turns that this involves modifications to our ideas of space-time, compared to which large extra dimensions are just child's play. So space-time is doomed. I think it's not because there is a big black hole that LHC will create and eat us up.
[LAUGHTER]
I hope not, right?
But if you're sitting not on a chair, but on anything else, a big black hole will come and eat you. So there are some seats that are still left. So please take one of those. Nima.
[APPLAUSE]
NIMA ARKANI-HAMED: I must say that-- um, is this on?
MAXIM PERELSTEIN: Yup.
NIMA ARKANI-HAMED: I must say that from the moment I met Maxim, it was clear he was going to be a big-shot professor at a big-shot university one day. But I have to say it's a particular pleasure to be introduced by him. And thank you very much for your kind words. And I have really been enjoying reading your papers from 10 years ago, which were really beautiful papers. It's only to bad five people working on it at the time.
So I'm going to be talking about a dramatic subject today, which is the end of space-time and what we're going to do without it or how are we going to deal with the fact that we have to replace it with something else? This is the first of the set of lectures which are really explicitly devoted to questions we don't have complete answers to. That was the point. It's the future of fundamental physics.
So I will start by telling you what the major issues are and the really significant progress that has happened in theoretical physics in the last 15 years, taking a very major step on getting a good part of the answer or at least what it might look like. But I will really end by telling you what I think the central conceptual challenges in our field are today, that we just don't have good answers for. And I'll tell you what some of the strategies and attempts are for proceeding. So that'll happen towards the end of the lecture and we'll also dovetail in parts with tomorrow's lecture. But we're now starting to go into uncharted territory.
So let's go back to the picture of the scales that we think about in this part of physics a lot. And today is really all about-- it's at least going to begin to be all about what's happening near this Planck length of 10 to the minus 33 centimeters, where, as we've said a number of times, the effects of quantum mechanics and gravity become important and space-time is doomed.
So I told you already one understanding of what that 10 to the minus 33 scale is. Yesterday, we saw that in the appropriate natural units, the Newton constant that controls the strength of gravity is roughly 10 to the minus 33 centimeters squared. And the fact it was such a tiny number compared to any other length scale that we are used to talking about is simply a reflection of the extreme weakness of gravity relative to the other forces.
I want to open this up in a little more detail. And actually tell you what would happen if we took a pair of electrons and actually looked at the force between them as we brought them closer and closer and closer. So we get a more physical feeling for what's happening at the Planck length.
So imagine that we have this pair of electrons at very large distances apart from each other. And at those very large distances, there's a gravitational attraction between them. There's an electric repulsion between them. And as we said yesterday, the electric repulsion is 10 to the 42 times stronger than the gravitational attraction. And you've all learned in school that both the electric force law and the gravitational force law go, like, 1 over the distance squared. So it would look like, no matter what distance you put them from each other, that ratio would always be 10 to the minus 42. Gravity would always be way, way weaker than the electric force.
That's true until you get to distances that are comparable to around 10 to the minus 11 centimeters. 10 to the minus 11 centimeters is the length scale associated with the mass of the electron. What that means is when you start trying to put two electrons much closer to each other than 10 to the minus 11 centimeters, then the uncertainty principle forces the electrons to have some kinetic energy.
I'm putting them closer and closer together. So they have to have some uncertainty in their momentum. So they have to have more and more energy. But at around 10 to the minus 11 centimeters is the first time when that amount of kinetic energy they have starts becoming comparable to their mc squared energy. So it's starting to affect the amount of actual mass that's in the system at around 10 to minus 11 centimeters.
As you start pushing them closer and closer together, you have to supply more and more energy in that region of space to hold them closer and closer together, again by the uncertainty principle. And because gravity talks to mass, the gravitational force is proportional to the mass of the two bodies, you start seeing that the strength of the gravitational force starts going up. Simply the mere fact of holding them so close together is forcing so much mass into that little region that the gravitational force starts taking off, starts getting bigger.
The electric force, remember, its strength was this dimensionless number, around 1 over 137. It may vary gradually, varying with distance. But we're going to ignore that. But the gravitational force starts taking off and getting stronger and stronger.
And what happens is that it actually gets stronger, and stronger, and stronger. And at some point, actually around 10 to the minus 31 centimeters, if we kept extrapolating, it would cross the electromagnetic force. It would become as strong as electromagnetism around 10 to the minus 31.
At that point, they'd still be both relatively weak forces. That number, 1 over 137, is a kind of small number. The strength is kind of small. But at that point, it's caught up with electromagnetism. And if we keep on going a little further, at around 10 to the minus 33 centimeters we find it's just massively strong, absolutely massively strong.
So that's one way of talking about this Planck length. In the neighborhood of a Planck length is where gravity catches up in strength with all the other forces. That actually happens a little bit earlier than this Planck length, at around 10 to the minus 31 centimeters. At this 10 to minus 33, it's just the strongest thing around by far, dominating everything in sight.
Of course, that distance is vastly smaller than anything we've seen. Our highest energy accelerator is going to go to 10 to the minus 17 centimeters. That's 16 orders of magnitude larger in length than the Planck length. But still, we can in our mind's eye do this extrapolation. And that's what's happening around the Planck length.
Another very important thing about what's happening at the Planck length-- and this is really the little thought experiment that tells you the space and time just don't make sense-- is the following. You remember we learned because of the uncertainty principle that we can't know the position and the velocity of a particle at the same time. But that's OK. We can understand and probe physics at any distance we like-- perfectly compatible to the laws of quantum mechanics-- so long as we're willing to add more and more energy.
So if I want to probe things at shorter and shorter distances, I just have to pump more and more energy into the smaller and smaller region of space in order to see what's going on there. Again, as I've mentioned already, that's why we build these giant particle accelerators. They have to be so big to accelerate particles at such high energies ironically to probe very, very small distances. And that's just a direct reflection of the uncertainty principle.
So for a long time, that just works. You just keep going to shorter, shorter, shorter, shorter, shorter distances. But something new happens at some point. At some point, you put so much energy into such a tiny region of space that there is so much mass by E equals mc squared in that region that you collapse the region into a black hole.
You are trying to figure out what's going on inside this region of space by doing experiments at higher and higher energies. But the mere fact of doing the experiment, by the uncertainty principle, puts so much energy into the region you're trying to look at that you collapse it into a black hole. And you can't get any information out.
So you defeated your own purpose. You are trying to understand what's going on at very short scales and it stops working. It stops working when those distances you're probing at around 10 to the minus 33 centimeters, the times are 10 to the minus 43 seconds, and so on. What if you say damn, that's not what I wanted to happen? Let me build an even higher energy accelerator. Let me build an even more powerful microscope. What would happen?
You make an even bigger black hole. It gets even worse. If you wanted to probe what was going on at 10 to the minus 31 centimeters, OK, that would be OK. You could do it with a very, very high energy accelerator. But you go to even higher energies, you make a little black hole. You go to even higher energies than that, you can't even know what's going at 10 the minus 31 centimeters anymore. That's being collapsed into a black hole.
So this means that there's simply no operational way of measuring or probing physics in any way at distances in time smaller than 10 to the minus 33 centimeters, 10 to the minus 43 seconds. Now, at this point you could take two attitudes. One of them is that space and time actually do exist. They're there. It's just we can't measure it.
Everything we've learned in the history of physics says that that's a very reactionary attitude. And every quantity that we haven't managed to associate with something that can in principle be observed, that's telling us something very important. It's telling us that that quantity actually doesn't exist. There's no such thing as space-time at arbitrarily short scales. Something has to give.
Now at this point, let me dispense with one very naive idea that you might have. This naive idea is the basis of a number of other research directions. But it's really naive, nonetheless. So I don't want to spend a lot of time talking about it. But let's just mention it.
You might say, oh, I know what's going on. What's going on is there's some kind of atoms of space-time. It's just like in this table. If I look at the table at shorter and shorter distances, of course, it doesn't look like a table any more. I start seeing that it's made up out of other stuff. That's what's going on.
That will be just like before. It's so exciting. It's atoms of space-time. You get down to very short distances, maybe you see some kind of lattice. Maybe there's some sort of discrete structure. There's little cells, whose size is more to the Planck length, something like that.
Now, if you think about it one more second, there's a big problem with that idea. And that big problem is in whose reference frame is the size of the lattice 10 to the minus 33 centimeters? If it's in 10 to the minus 33 centimeters in my reference frame, someone else's reference frame it could be a lot longer. Someone else's reference frame, it could be a lot shorter.
It's very difficult to come up with this idea because of relativity, because of special relativity. It's very difficult to come up with an idea of discreteness in space-time in any way consistent with the laws of special relativity. And, indeed, all the people who pursue this idea are perfectly happy to abandon special relativity. And they do.
But another lesson I think we've learned over and over in the history of physics is that things don't progress. Things progress by making things more, and more, and more restrictive, in a tighter and tighter straitjacket. Not that all of a sudden, there's anarchy. And all rules, all bets are off. And everything changes.
So I'm not going to talk about atoms of space-time. And essentially, it's because it's in flagrant contradiction with the laws of special relativity. We will see instead what sort of thing can replace it. And when you see the sort of thing that can replace it completely consistent with the laws of special relativity, and you see how subtle and interesting it is, you will really appreciate even more deeply what's wrong with the idea of atoms of space-time.
OK. I want to say one other thing. Often this set of problems is associated with the words of making sense of quantum gravity, which makes sense, right? This issue arose precisely because of quantum mechanics and general relativity said in the same sentence, quantum mechanics forces, the uncertainty principle, more and more energy, smaller and smaller region, gravity turns it into a black hole. So it's quantum mechanics and gravity.
So it's really the problem of putting quantum mechanics and gravity together. But often, especially in popular expositions, you'll find the statements like, our difficulty is that we don't know how to put together general relativity, which is the theory of the very big, with quantum mechanics, which is a theory of the very small. And this is really overly simplistic. In fact, we have no difficulty talking about quantum gravity at large distances compared to the Planck length.
The real difficulty is we don't know what happens when gravity gets strong and all these fluctuations are gigantic at the Planck length. But at very large distances, there's no difficulty at all talking about quantum mechanics and gravity. In fact, people have calculated-- oops, oh, sorry. Well, let me get to that.
People have calculated what the first correction to Newton's laws is, due to quantum mechanics. There it is. It's Gmm over r squared. You've seen that.
It's one minus 27 over 2 pi squared, G. There's an hbar. There's a c. It's a quantum gravity correction to Newton's laws. There's a 27. So someone has actually done some work.
[LAUGHTER]
And I want to emphasize that you can really calculate this. It's a honest quantum gravity calculation. It's a tiny effect no one gives a damn about. OK. It's a minuscule, minuscule effect. But you can calculate it.
It's something in principle you could measure at very large distances. Our difficulties are not the words "quantum mechanics" and "gravity" in the same sentence. It would be absurd if that was our difficulty. I mean after all, we dropped atoms and gravitational fields. We can do quantum mechanics and gravity, no problem. OK
So our difficulty is that we don't know what's going on at very short distances when these quantum gravitational fluctuations become very large. We can calculate this so reliably because it's small. But we can calculate it. So the difficulty all has to do with what's going on at the Planck length.
Now, where else are these issues important? These issues hit you in the face in a variety of situations. For example, we talked about this idea of the Big Bang. So we go back, the universe gets smaller and smaller. At some point, the curvatures in the universe become comparable to the Planck length.
When that happens, we simply don't know what's going on. Often, people ask the question, what came before the Big Bang? Well, the real answer to the question is before we get to the Big Bang, we hit a situation where we don't know what time and space mean anymore. So the whole notion of "before" probably ceases to make sense.
It's one of the lessons we've learned again and again. Questions you can phrase in the English language are not necessarily good questions about the universe. And this is a good example of it. The whole notion of space and time is breaking down in the neighborhood of the Big Bang.
We don't even know if "before" makes sense. We don't know if "space" makes sense. Something is going on there. But to understand it, we have to come to grips with these issues. So that's one situation.
Another situation is what happens in the center of a black hole? In the center of a black hole, all mass is getting crunched down to arbitrarily high densities until it hits the Planck scale. And then we don't know what's going on again.
Incidentally, I'm sorry to draw this picture in this way. This is a very misleading representation of what happens. There is no center of a black hole in the sense that you fall in and then there's like a sinkhole somewhere. And you're like, oh, no. But you get sucked towards it.
Really, the singularity inside of a black hole is a moment in time. When you fall behind the horizon in the black hole, what happens is no matter what you do, no matter how you struggle, you hit the singularity and go kaput. But it's not some point in space.
But, anyway, I will sometimes abuse-- take-- it's not artistic license, I guess. It's scientific license, which is much worse-- but draw it as a point. But anyway, we don't know what's happening there. It's a situation where the laws we have-- our theories just break down.
They're just telling us we don't know. They throw up their hands. They beg for mercy. We don't know what's going on there. That's the sort of question we'd like to have an answer to.
When the theories you have, you push them to their limits and they beg for mercy, that's the time you know that you're asking a good question and that something has got to give. And so we have to figure out what's going on.
Now, there is another aspect to this, which also goes a little distance to showing you that there's something more subtle going on than some sort of discreteness in space-time. And it's a more refined and a more subtle point, but it's a very deep point about putting quantum mechanics and gravity together. Remember, something we learned about quantum mechanics is you've got to be very careful what actual observations makes sense. What is it that actually has sharp, sharp meaning?
And in quantum mechanics, we found out that the world isn't deterministic. It was probabilistic. And the things that have sharp meaning ultimately are doing an experiment infinitely many times, with an infinitely big apparatus. That was a conclusion that we came to yesterday.
So there's two infinities, two infinities in implicit in the discussion of any perfectly, perfectly sharply defined quantity in a quantum mechanical theory. Do the experiment infinitely often, with an infinitely big apparatus. When we add gravity, something bad happens. When we add gravity, one of those infinities is taken away from us.
Let's say I want to know what's going on? I want to do some observation of some system in this room, OK. Of course, we know we can do it. We can get the errors to be very small, smaller than anyone would care about.
But let's say you're a really hard-nosed person. You want to get that error down to zero, get a really sharp measurement. So you would start making the apparatus bigger, and bigger, and bigger. You have to do that.
But as you make the apparatus bigger and bigger, it becomes heavier and heavier. And at some point, the apparatus becomes so heavy that it collapses the room into a black hole. All of our problems are with these damn black holes, as we'll come back to in a second. But this is devastating because it means that if you want to do any sharp measurement in a fixed-size room, it's just some possible. But that's what we talk about when we're talking about doing measurements in space and time.
I do the measurement here. I do the measurement there. I do it in a fixed-size room. I write down the answers on my piece of paper. None of those things will ever converge to something perfectly sharply defined.
So we saw already that quantum mechanics implied a drastic reduction in what we could actually talk about relative to classical physics. And now combining quantum mechanics to gravity seems to be even further drastically reducing what we can talk about. We can't sharply talk about any measurement done in any fixed-size room.
What can we possibly do? What can possibly make sense? Well, the slogan is that the quantities that do make sense in a theory of gravity are observables that are not in the inside of the space-time. But are at the edges of the space-time, at the boundaries of the space-time, far away at infinity.
Let me do a simple example of what one of these observables can be. Let's say, I take a big measuring apparatus. And I just make it bigger, and bigger, and bigger, and bigger, as big as I like. But I start it moving up further, and further, and further away from the regime where something is going to happen. That's fine. If this apparatus is huge here, it's not going to make a black hole over there.
Let me have it shoot out a particle. So this apparatus shoots out a particle. Another one shoots out a particle there. Another big one shoots out a particle there.
We let the particles come in. The particles merrily collide with each other. They do whatever they do. They go back out. And they're measured by really, really big apparatuses, very, very far apart away from each other. They're not going to collapse anybody into a black hole. Everyone's fine.
So that's an experiment that I could do infinitely many times, with an infinitely big apparatus, and get an exact result. But it doesn't involve an experiment merrily in the inside of the space-time. You've got to get up to the edges of space-time, and throw things in from infinity, and see what happens when things go back out to infinity.
These things are called scattering experiments, for obvious reasons. So these sort of scattering experiments could yield perfectly defined observables. But they don't involve the inside of the space-- I mean the actual observation doesn't involve the inside of the space-time.
The way you might want to think about how it takes place might involve the words that that the particles come in. They hit each here and they go back out. But the ultimate thing you observe is not located inside of the space-time, but far away, at infinity, at the boundary of the space-time.
Now, since all of our difficulties here in one way or another stemmed from these black holes, let's just pause for a second to say a few words about them because some of their properties are going to be important for the rest of the discussion. So black holes, even in classical physics, caused people some consternation when they started thinking about them.
It's interesting that-- I think I told you that Einstein refused his prediction of the expanding universe. He refused to believe that his theory predicted an expanding universe. It did. He also refused to believe that his theory predicted black holes, which it does.
He tried as hard as he could to make these solutions go away because they were very confusing. But they don't go away. They're perfectly good solutions. In fact, they're one of the most dramatic solutions of general relativity, where it flexes its muscles and shows how different it is from Newtonian physics.
But one of the difficulties with them is, as we've discussed, there are regions of space-time where you can't ever get information back from them. You can throw something into a black hole and it's gone. Good bye. That was disturbing for a variety of reasons. One thing that bothered people is it seemed that, like, black holes might allow you to violate some very well-established laws of physics.
For example, there is something-- I won't have time to explain in detail, but I'm sure you've all heard about-- this idea of entropy. Entropy always increases. There's this thing called the second law of thermodynamics. Entropy increases. Entropy is a measure of the disorder in the system. And the total disorder in the universe is just increasing.
So black holes seem to give you a way to get around this. Because you could take some region of disorder, just thrown it into the black hole.
[LAUGHTER]
Oh, that's nice. Things are much more ordered now. It's just in the black hole. It's OK.
So that was a little bit strange. So that was one puzzling thing. But people also noticed other interesting things about black holes, also puzzling.
For example, people noticed that if you had one black hole and another black hole, if they merged, well, it makes sense that they make a bigger black hole. But they noticed that something more specific was true. That if you merge two black holes together, the new black hole, the area of the black hole, would be bigger than the sum of the areas of the two black holes that made it up. Again, this is just facts about classical black hole physics.
This fact, coupled with this strange feature that you seem to be able to lose entropy into a black hole, started this vague and then sharpened feeling in the early '70s that it actually makes sense to ascribe entropy to a black hole. And what should its entropy be? Whatever it is, it makes sense it should probably be proportional to its area. Because its area is the thing that always increases because of this fact. So there started to be this idea that black holes should have an entropy and that was somehow proportional to the area. People weren't quite sure what the constant of proportionality should be, but that black holes in some sense had an entropy.
Now, this was an idea that, amongst other people, Stephen Hawking found ridiculous. He found it ridiculous because he thought, well, normally things that have entropy, they have a lot of disorder. And things that have disorder are hot. So if the black hole is going to have an entropy, it had better be hot. In other words, it had better be emitting radiation constantly.
But every idiot knows that black holes are black. They don't emit radiation constantly. Except he found out that they do because of quantum mechanics.
So black holes are, in fact, hot. This picture of a classical black hole as just nothingness is just wrong, quantum mechanically. And roughly speaking, what's going on is this. It's again a consequence of these pairs of particles that can pop in and out of the vacuum all the time.
Remember, the horizon of a black hole is this a point of no return. If you're on one side of it, you always get sucked into the black hole. On the other side, you have a chance to escape. Now, imagine that you're a poor particle and antiparticle pair, that had the misfortune to pop out of the vacuum right in the neighborhood of the horizon. Your friend, who popped out on the other side, is just gone. You have a chance to escape out to infinity.
But this process of the particles and antiparticles popping in and out of the vacuum right in the neighborhood of the horizon gives rise to a phenomenon that particles come streaming out of the black hole. And, of course, total energy is conserved. So the black hole will gradually start losing mass.
So Hawking set up and did this calculation, tour-de-force a brilliant calculation. And he found that not only does it emit radiation, but it emits radiation exactly as if it had a finite temperature, like a perfect black body. Well, what we call in physics a perfect black body. But really, it's a beautiful, perfect-- it has exactly the properties that you need to ascribe in entropy.
And the entropy you should ascribe it is indeed the area of the black hole. But he also figured out precisely what the constant of proportionality is. The constant of proportionality is 1 over 4 times G newton. Remember, G newton in these natural units had units of length squared. So the area over G newton is indeed a number. It's a pure number, which is the entropy.
So quantum mechanically what happens is you make black holes and they evaporate. They evaporate by giving off this Hawking radiation and getting smaller and smaller. Now, after Hawing realized this, he realized something else. And he made this extremely interesting thought experiment, whose resolution had to wait 20 years, but which I'll now describe, which really drove a lot of thinking in this field.
So here's the thought experiment. Let's take one of these experiments, that we even said we're legally allowed to do. It's even a real observable. I'm going to send things in from infinity. I'm going to collide them with each other and see what comes out.
But now, let's send so much stuff in them. I'm just going to aim a whole bunch of laser beams in just the right spot. So I put so much energy into that region, I make a big black hole, a big fat black hole. Now, I make this big fat black hole-- by the way, so let's say I did that. And right after I did that, I decided to get rid of my laptop, some books, just for the heck of it, just-- my latest enemy-- just chuck them, chuck them in towards the black hole too.
So there's an awful lot of things there in the black hole now. There's lots of information about my laptop, my former friend.
[LAUGHTER]
OK, great. So now what happens?
OK, we made this big black hole. The black hole starts to radiate. At first, it's big. And so it's radiating relatively low energy radiation. I should have mentioned before that the temperature of the Hawking radiation is around 1 over the size of the black hole. Again, 1 over the size of the black hole in natural units. So it's pretty cold. But it's still radiating.
For a macroscopic black hole, it's so cold that we can't actually see it. For an astrophysical black hole that's out there in the center of the galaxy, this radiation is so cold that we can't see it. But it's there. And it's big for microscopic black holes.
So then it shrinks in size after its given off a lot of energy. And now, it's smaller. So it's giving our blue light, higher energy. Then it really, really gets really small. And then poof, it just disappears. This is what Hawing said. It's just gone now.
Now, we have all of this light streaming out. But at every step, this black hole was radiating as if it was just a perfect black body. In other words, the way the black hole radiates, it doesn't care at all where the black hole came from. If it's a black hole, the way it radiates is always the same.
So you make a black hole. And what comes out, apparently-- that's what Hawking's calculation said-- bears no memory whatsoever of what it took to make that black hole. You could make that black hole by aiming these laser beams. You could do it by aiming other laser beams.
You could throw in your friend, not throw your friend. It doesn't matter. If you make the same black hole, it'll radiate in exactly the same way. What comes out will look exactly the same.
So that was the puzzle. The puzzle was that it seems like you can lose information. That what happens at the end, knowing what happens at the end, we can't reconstruct in principle what happened in the beginning.
Now, of course, you might say, but isn't that true? You just told me in quantum mechanics, nothing is deterministic, et cetera, et cetera. That's true. So what I really mean is that we would have to do the experiment, not with just one black hole. We'd have to do this experiment infinitely often, make infinitely many black holes, do the experiment the same way every time. But ultimately, there are some quantities that we can extract. There are these sharply defined quantities that we can extract.
And, in principle, those quantities we can use to back-reconstruct. In the usual formulation of quantum mechanism, from that final information we can fully reconstruct what the initial information was, the initial information of the initial state factor, if you like. The initial state, the initial wave function, completely determines the final wave function, and vice versa. And yet this calculation seemed to say that that doesn't happen.
Now, this is a beautiful example of one of the things I told you about in the very first lecture. That we never know in physics ahead of time whether the thing to do is to be radical or conservative. And it's not even often obvious which is which.
So let me tell you what Hawking's attitude was. Hawking took what seemed like a very radical attitude. He said, this is great. This means that quantum mechanics is wrong and they I proved it. Other people found this very hard to swallow, partially because it was very hard to see how you could modify quantum mechanics in any way that was consistent, that was consistent with very basic principles.
So there was a real impasse. What do you do with this paradox? That's a real contradiction. Either quantum mechanics is wrong or somehow the information gets out. Either way, something is wrong.
So Hawking had what sounded like the radical attitude. But there was a smaller group of at first, apparently, curmudgeonly physicists, who were like, you know, we can't just throw quantum mechanics like that. It's the most spectacular, successful theory of all time. Somehow quantum mechanics has got to be right. And this calculation has got to be wrong.
And that's how it stood for many years, until the problem was finally resolved. Well, we'll see what the resolution looks like. But the answer is that, indeed, quantum mechanics is right. Hawking's calculation was very, very, very subtly wrong.
But, in fact, the nature of the answer that we're going to see in some ways is much more radical and involves really remarkable new ideas. We'll see what it takes for quantum mechanics to be right. For quantum mechanics to be right really needed the end of space-time. And we'll see in a very specific sense how that comes about and how normal theories manage to do it. And I think by the end, you'll get a sense for how remarkable the picture that's emerged has been.
Well, there are many of these qualitative puzzles. There are also other quantitative puzzles. And people started trying to make sense of putting quantum mechanics and gravity together. And the most ambitious and successful of these ideas has been related to string theory.
Now, one could give a whole series of lectures on string theory alone, which I'm not going to do. And I'm only going to focus in this lecture on its aspects that are really specifically relevant to this problem of quantum mechanics and gravity. There are other aspects to this whole program, that it could unify all the forces, and give us a consistent picture of everything, and all that stuff. Let's put that aside. I'll talk about that to some extent tomorrow.
But let's just talk about its initial raison d'etre, which was to come up with a consistent way of putting quantum mechanics and gravity together. So here's very roughly the way the story works. I want to tell you the story the way string theorists thought about string theory before 1995 and the way string theory is described in popular books in 2010. Which is that the whole big idea here is that we're replacing particles with strings.
Instead of having point particles, we had little loops of string. Those little loops of string are around 10 to the minus 31 centimeters big, not 10 to the minus 33. They're around 10 to the minus 31 centimeters big. Remember, that 10 to the minus 31 centimeters was around the distance at which electromagnetism and gravity became comparable? So the strings are actually a little bigger than the Planck length. This is very important. It's an important part of what makes the theory tractable.
And the idea is that, OK, so we don't have points. We have these little loops of string. And an advantage of having them be big like this is that those little microscopic black holes can't eat them up. They're bigger than the black hole is. So you can start imagining scattering these particles together-- these strings together, sorry-- seeing what happens. You could hit these Planckian energies. And now, things aren't necessarily going haywire.
I should say that's that you say why stop here? Why not have it be something else, not loops of strings, little loops of membranes or something else? We'll come to that in a second. But there was a very good reason people were excited about strings. This is the sort of thing that's harder to-- I mean it's something you'll just have to take my word for or their word for.
But it took a number of miracles for this picture to be mathematically consistent. Anything else is just impossible to make mathematically consistent. It was really remarkable that this even had a prayer. People had talked vaguely about this going all the way back to the 1940s and 1950s. And no one could make any sense of it. It seemed very hard to imagine that it would make mathematical sense. The fact that it did make mathematical sense in rather surprising and intricate ways was what convinced people that it was interesting and on the right track.
Anyway, string theory has a variety of consequences. I'm not going to spend a lot of time talking about it because they're not that relevant for our discussion. They'll be relevant later.
One of the things that comes out of it is a really remarkable extension of our notion of space-time associated with the word supersymmetry. There's something that's much less interesting than supersymmetry. But it's still there. And that's these ideas of extra dimensions of space. Supersymmetry is really a much more-- it's a much deeper and more interesting idea. But extra dimensions come along as well anyway.
And there was this picture that we don't see these extra dimensions because they were curled up on these tiny sizes, compared to our big, macroscopic four-dimensional world. I'm not going to talk about it a lot, again because it's not very relevant to our discussion.
If we imagine how this picture that you have strings, instead of particles, impacts our picture of these stick figure interactions we were drawing before, we see that instead of having a stick figure interaction like that, all these particles are replaced by little loops of string. And it looks like something much smoother. And that had a number of very nice consequences. And people could start doing concrete calculations and life was good.
But there was something hanging over the head of the string theorists, even in the 1980s, where this picture first developed, through to the early '90s, which is that eventually they would also have a problem with black holes, not when the black holes are 10 to the minus 33 centimeters big. But if you collided these particles at high enough energies, the black holes would get bigger and bigger. And they would eventually become 10 to the minus 31 centimeters big and swallow up even the strings.
So there was a problem. But it still wasn't completely solving the problem. It was helping you get going. It was a step in the right direction. But there was still something missing.
That was string theory before 1995. After 1995-- well, something happened in 1995 evidently. And after 1995, people realized that amongst other things, string theory is not a theory of strings. In fact, we're going to see in a bit that string theory is actually particle physics. But we'll come to that in a moment.
It's not a theory of strings. There's all sorts of interesting objects in it. There are strings. There are even point-like particles in the more correct formulation of the theory. And there's even higher dimensional objects, membranes. And these things can all talk to each other. So strings can end on membranes. They could interact with each other in a very interesting way.
And another interesting thing that happened is that the actual size that these things ended up needing to have could actually be quite a lot more variable now. The strings could be bigger. The extra dimensions could be bigger. There was just many, many more options that became available. Again, that's a topic for maybe tomorrow or Friday.
But the theoretically most important realization early in this period was that, in fact, this theory that people were talking about and trying to describe in various approximations turned out to be seen to be a single theory. There was one theory. And it wasn't of a variety that we had encountered before in physics. See, normally, we think about classical theories. And we take classical theories and we imagine quantitizing classical theories.
But we're used to the idea that we have some quantum theory and it has some classical limit. But there's one classical theory and one quantum theory and they just go hand in hand. This was the first example of something really rather remarkable and new, which is that there's a single quantum theory. But the single quantum theory turned out to have both many, many different kinds of solutions and also many different kinds of classical limits.
So that's very strange. You would take two classical theories that looked utterly completely different. And think that surely these theories have nothing to do with each other. OK, I can make them quantum mechanical. They had nothing to do with each other before. They'll will have nothing to do with each other after I make them quantum mechanical.
But it's the other way around. There was one quantum theory, which turned out to be possible to have several classical limits. This was a very early surprise, that said that there was even more unification going on in this set of ideas than people had realized.
An immediate consequence of all of these ingredients that I told you is that people managed to understand where this entropy of the black holes came from. Remember, I told you that Hawking figured out that the black holes have a temperature and they have an entropy. But remember, we also know that entropy is a measure of the amount of disorder in the system. And if you wanted to really understand where that entropy came from microscopically, you would have to understand what the microscopic constituents of whatever you're looking at are. The entropy just counts how many different possible ways could you put those microscopic constituents together so they look the same macroscopically.
When you have a bigger entropy, it's made out of more stuff. You can put it together and arrange it in different ways so that it looks the same microscopically. But if you want to come up with a microscopic, a real deep understanding for where that entropy is coming from, you should be able to actually see what's making it up and just count how many ways could you put it together so that it looks the same macroscopically.
Well, people managed to put all these ingredients, strings and branes together, and understand, not for the sorts of black holes we were talking about a second ago, but cousins of them, very special sorts of black holes that carry both mass and a variety of kinds of electric charge. But for these very special sorts of black holes, that also have entropy, people were able to figure out how to count precisely all of those microstates in terms of these strings and the branes.
And roughly speaking, they managed to set up a situation where you could gradually turn a collection of strings and branes into a black hole, in such a way that you weren't changing the number of microstates at any point in the process. So you could keep track of every single thing. And so, therefore, you could count the number of states to begin with and then match it to the entropy of the black hole at the end.
They did this calculation. And it came out exactly Area over 4G newton, with every pi, every factor of two, everything perfectly correct. It wasn't some vague squiggle. This is another tour-de-force calculation. And big qualitative ideas are very, very important in physics. Concrete solid calculations at work are also very, very important for telling you that you're on the right track. When something works to every factor of two and every factor of pi in this surprising way, you feel like there's something correct about it.
So that was a huge accomplishment of this period. But still, even this calculation didn't address this question of what happened to the information? I wasn't talking about the sort of black hole that can be made and evaporate. These are very special sorts of black holes that turned out not to evaporate, for reasons I won't explain-- one of the reasons that the calculation could be performed.
So another qualitative insight emerged, which started heading in a more radical direction. And this has to do with another remarkable fact about black holes. They have an entropy. In fact, not only do they have an entropy, but if you have some region of space and if you want to imagine what's the biggest amount of entropy that I can have in that region of space, the biggest amount entropy you can have in that region of space is if you make it a black hole.
Any smaller amount of entropy, it will not be black. For example, just as a simple example, let's say you take a big hot ball of gas, a big hot of gas. And you put so much of it in there that it's just about to collapse into a black hole. Just a second before it's about to collapse into a black hole, you can open up your undergraduate textbooks in thermal physics and calculate the entropy inside that amount of radiation.
Then a second later, it collapses into a black hole. But you can ask what was the entropy just as it was about to collapse? The answer is it's minuscule compared to the final entropy of the black hole. It's actually proportional to the area over G newton to the power of 3/4. Which if the number is huge, if it's written in the power 3/4, it's vastly, vastly tinier then the final entropy.
So the entropy goes up dramatically when you make a black hole. But the entropy is telling you-- at least, it's telling you what the maximum number of possible-- well, it's telling you something about the number of sort of microstates that you can have inside the region. Again, the entropy is a count of all the possible ways you could put something together so that it will look the same from the outside.
But if you can't possibly make the entropy bigger than the entropy of the black hole, then that suggests that there's a sort of maximum number of possible different configurations you can have inside. But what is that maximum number of possible configurations? It's the area of the region in units of the newton constant, in units of the Planck scale.
That's very perplexing. Because normally in physics, if we're talking about the number of possible different states of air molecules in this room, you would think that that number of states would scale like the volume of the room. If you made the room twice as big so it was eight times the volume, well, in every little cell you have exactly the same amount. Everything looks the same. So you would think that everything is-- you're increasing by a factor of eight the total number of possible states you could be talking about, the total number of possible-- more technically, the total number of possible degrees of freedom you could be talking about.
But that seems to be wrong. It seems like that the maximum number of degrees of freedom that it makes sense to talk about goes like the area of the room. This is another very important qualitative observation and suggests the following startling picture.
That actually to describe what's going on inside this room, the actual correct physics resides on the boundary of the room. If you want to think there's little pixels. And the pixels have an area which are little Planckian cells, that in order to describe anything going on in this room, you just cover the boundary of the room finely with these pixels and little lights go off here and there.
As things move around the room, they're encoded somehow in lights going on and off in the pixels that are on the boundary of the room. If you put more and more in stuff in the room so that it starts getting close to being a black hole, more and more of these pixels are lighting up. If you put so much stuff that it's just about to collapse into the black hole, all the pixels are lit up.
But all the information is at the boundary. It's not inside the volume. It's at the boundary. You see, this is yet another indication of what we saw from our more general arguments about quantum observables.
It's not inside the space. It's at infinity. It's far away or it's at the boundary. It's somewhere else. It's not what you would think. The inside of the space doesn't make sense. What's controlling the physics lives on the boundary of the space.
As I said, this was another important qualitative idea. And when I said that the subject made spectacular progress in the last 13 years or so, it's because a completely quantitative and working version of this idea was finally discovered in 1997. So I want to tell you what it is.
But by the way, this whole thing-- the slogan is that the world is a hologram. That just like a hologram is really two-dimensional, it looks three-dimensional. It encodes three-dimensional information in a very nontrivial way. Holograms were invented by extremely smart people. And they encode the information. They make things look three-dimensional because of lots of interesting correlations between what's going on in the hologram.
But still, it's fundamentally two-dimensional. And it mimics and it fakes a third dimension. But it's really fundamentally two-dimensional. So similarly, that's the idea. Quantum mechanics and gravity force the world to really be like a hologram.
So now I want to tell you how this works in an example. And the statement is the following. Let's talk about what quantum mechanics and gravity might look like in a world that's not exactly like our world, but could be made so close to it for so many purposes that it certainly suffices for many of the questions we're asking.
The technical name for this world as an anti-de Sitter space. But what it really is is some curved space-time. It is a curved space-time. Our space-time is also a little curved. There's the Sun there. There's the Earth there, and so on. But on average, it's pretty flat.
But this is really a space-time that's sort of uniformly curved everywhere. It's uniformly curved everywhere. Of course, even if it's uniformly curved everywhere, if the scale of the curvature is 10 million light years, then I wouldn't notice it in this room. So what I'm telling you could well work to describe a universe that has a curvature that's 10 million light years. But it is important that it's curved.
Now, in this space, once again the kinds of observables that we want to talk about is you might-- ah, so the important feature of this curvature turns out to be-- and we'll talk about what this geometry looks like a little bit more. But the important point turns out to be that there's really naturally a boundary in this space. There's naturally a boundary in this space-time.
But the boundary of that space-time is just like a normal 3 plus 1 dimensional universe. The boundary is just flat 3 plus 1 dimensional space. In that case, the interior would be a 5-dimensional space. So this would be a five-dimensional theory of gravity on the inside. And it would have a boundary. Just like this room is three-dimensional and the boundary is two-dimensional, it's just like that.
It really looks like a tin can. It's a universe that looks like a tin can. It has a boundary. It has an inside. And we want to describe how to do physics.
If you don't want to imagine a higher dimensional world, we could just do everything else, imagining that it's four-dimensional inside and the boundary is 2 plus 1 dimensional. Literally like that wall, if the wall is at infinity and the space has this uniform curvature.
Now in this situation, we can ask again, we sit at this boundary. We send particles in. They bang into each other. They go back out. We make all the measurements. And that's what we should try to hope to predict and understand. That's the problem of figuring out what quantum gravity looks like in this space-time.
So there's an answer to the question, what is, what describes quantum gravity in this D plus one-dimensional space? And if there's four-dimensional quantum gravity on the inside of this tin can, what is the theory? What is the theory that tells you what happens when you scatter all these particles together?
The answer is that it's a good old-fashioned quantum field theory of ordinary particles interacting according to the laws of special relativity, just the standard old particle physics sorts of theories that we were talking about yesterday and the day before, which lives on the boundary of that space. Every single question you might want to ask about what's going on inside is secretly answered by a theory that doesn't have an inside. There in no inside.
The theory doesn't even have gravity. That theory doesn't have strings. It's just a relatively boring particle physics theory. It's not all that boring. It's in a range of parameters of such theories that we didn't know how to consider before or didn't know how to study before, when these interaction strengths between the particles become very, very large. But still, it's a theory we could put on the computer. I mean we know exactly what to do with it. There's no deep mysteries about space-time, or fluctuations, or anything. It doesn't have gravity.
This is the shocking, startling thing. That we knew that somehow on general grounds, the only observables were associated with things going on at the boundary. But the really remarkable thing is that the theory that computes what those observables are doesn't even have an inside. It's just a good old-fashioned quantum field theory that lives on the boundary.
So there's many slogans. So this deep, mysterious thing is just the same as something that is just an extreme limit of a theory or set of theories written down by people and studied since the 1930s. String theory equals particle physics.
Gravity emerges. String emerges. This whole space emerges. But it emerges from the dynamics of something that we understand much, much better.
We'll see how it works in a little more detail. But it's another example of something of the phenomenon we've seen over and over again. Remember, I told you that even when we got to a quantum mechanics, wasn't it strange we lose so much predictivity about the world? We can predict so many fewer things than we could in classical physics. And yet, strangely, we find that what was left that we could predict was somehow much more unified that we would have suspected before.
The same thing just happened to us. We had these thought experiments about gravity. We're like, oh, that is just really terrible. The only things that we can predict are out at the boundaries of the space. What about the inside of the space? That's really too bad.
And it's true. But we can predict even fewer things. But it allows two things that seem to have nothing to do with each other to be identified. Good old-fashioned particle physics is the same as quantum gravity in one higher dimension.
Now, I really want to tell you this will not go over a long way. But I want to tell you really what this looks like in some detail because it's pure magic. So first of all, let me tell you a little bit more about this space.
As I said, this is a curved space. It's a uniformly curved space. And there's a variety of ways of representing what this curvature looks like. But one of them is like a trumpet.
So this is really a picture of what distances and times look like as you head in towards the middle of the space, away from the boundary. This is out towards the boundary. That's going in towards the middle of the space. As you see, the areas around you are contracting.
That has a very interesting consequence that if you talk about some big region here, which is large compared to the scale of this curvature, the areas and the volumes are proportional to each other. So that's good because it's easier for this to satisfy this holographic principle because there isn't such a big difference between the area and volume after off in this highly curved space. That's one interesting thing.
Another fact about it is if I go back to the tin can picture, there is a sort of gravitational-- this curvature or this warping of the space means that if I put a massive particle off here, somewhere not quite at the boundary, off here somewhere, what it wants to do is just bounce back and forth around the middle. So it just sloshes back and forth. It's getting gravitationally attracted towards the center of the space.
Light, of course, can make it out all the way. And actually can make it all the way out to the boundary and bounce back in a finite time. So it's an infinite amount of distance off to the boundary. But things can get there and come back in a finite time. I'm just trying to get you to visualize with this space looks like a little bit.
Now, how can it possibly be, how can it possibly be, that a theory that doesn't have the interior, a theory that just lives on the boundary and only knows of the boundary, how can it possibly be that it describes physics in the middle of the space? For example, how are you going to tell it if a particle is over here, or over there, or over there, or somewhere else, just something as basic as where is it in this extra direction? What information can encode where it is in the extra dimension, in this extra direction?
The answer is lovely. If you put a little point mass here, what corresponds to it in this hologram? What corresponds to it in this boundary? It's just putting a lump of energy that has some size here. So a particle sitting here in the middle of the space corresponds, in this other description, to a little lump of energy. A particle that's sitting further away towards the middle is a bigger lump of energy. Even further, would be even bigger.
So the position in this extra space is the overall scale size in what's going on at the boundary. Remember, what I'm telling you is that there's a description for everything that's going on inside this tin can, where you never have to say there's an inside of the tin can. You only ever talk about things that are going on the outside.
So I have to tell you how to translate everything that makes you think there is an inside of a tin can. I have to tell you how to translate every one of those things into a statement that you can just say "on the boundary." And this is one piece of the dictionary. Being in different positions in the middle just corresponds to having different sizes of lumps of energy.
Now, here's another remarkable thing. So if I take two particles now, two particles out in the middle of the space, they're sitting in the middle of the space. And Newton tells us that they attract each other gravitationally, just classical gravitational attraction out in the middle of the space here.
How can I possibly include that? That's gravity, right. That's gravity. How am I going to get that if I don't have the middle of the space?
The answer is that, OK, you have these two particles. So you have this lump of energy here and you have a lump of energy there. Do you remember this basic vacuum polarization effect that we talked about a number of times, that when you have some particle sitting here, something else sitting there, that you can affect what's going on in this cloud of virtual particles in its vicinity? And we've seen the effects of that a number of times already.
Well, in this theory it turns out that this guy over here affects the virtual cloud around it. That guy affects the virtual cloud around it. And the net effect of this very quantum mechanical process is that there is an attraction between these two lumps of energy. And that attraction is exactly what you would compute from Newton's laws in the interior. Of course, it's Newton's laws in this curved space. But Newton's laws in this curved space are nothing other than this quantum attraction between these lumps of energy in this boundary description.
So this is really remarkable. A classical force, what you call a classical force on one side, is due to a very highly quantum mechanical process on the other side. There's no inside of the space. There's no gravity. What you have is a theory of nice particles and forces. As I said, in a particular regime of its parameters, but which mimics everything that you would like to call gravity in the space and everything on the inside.
So I'm just telling you more and more about how to decode this hologram. More and more about what properties of what's going on at the boundary you should associate with things that are going on the middle of the space.
OK, something else. What about a black hole sitting in the middle of the space? A black hole sitting in the middle of the space corresponds to just heating up the entire boundary, just making it a hot fluid, a hot soup. That's great. Hawking radiation, all of those things are just completely obvious. You're just heating everything up.
So if you take what's going on in this boundary, you just heat everything up uniformly, then all of that physics is identical to as if you had a black hole sitting in the middle of the space. So this immediately tells you that there can be no information loss because let's say I wanted to now model the process. That I take these particles, I send them into the middle of the space. I make a big black hole. I wait that they evaporate. They come back out.
In the middle here, everything is riddled about confusions about quantum mechanics, space, time, all this stuff. But we have a completely equivalent description of that process in the boundary. What does the hologram tell us? The hologram tells us that we have two lumps of energy. There's no inside, right. We have lumps of energy.
Their going inside means that these two lumps of energy grow. They hit each other. They make a hot soup of messy interacting stuff. That cools down. And things go back out.
There's no gravity here. There's no reason for any violation of the laws of quantum mechanics. In fact, the theory is just quantum mechanics and relativity of the sort that I told you about two lectures ago. There's, in particular, no information loss.
So this really just conclusively proves that, in fact, there was no information loss. Quantum mechanics is valid. But what gave is that this picture of the inside of the space was just not there.
There is an equivalent description of what's going on. We did get rid of some of the space-time. We at least got rid of the space on the inside. We got rid of gravity on the inside. And there was a completely equivalent description of what's going on in terms of this much more standard physics, but encoded in these very subtle holographic ways.
I hope you see now how different this is than atoms of space-time. So this is something which is much, much more interesting. And it doesn't violate the laws of special relativity. And it doesn't violate the laws of quantum mechanics.
So why are we not happy? Why aren't we done? If I can just take a few more minutes. That was the success. So why aren't we just done?
For many questions, we are sort of in principle done. But, in fact, there's a very, very important difference between this universe, which we in principle understand very well, and our own universe. This universe was like a static tin can. Our universe came from a Big Bang and is expanding. It's not static. It's time-dependent.
For a long time, it was expanding, expanding, expanding. And the rate of that expansion was slowing down. People thought for a while that that would probably go on forever. And we would eventually just keep decelerating, decelerating, decelerating. The universe would get bigger, and bigger, and bigger. And eventually, it would become infinitely big.
So if waited long enough in this world, you would see everything that there is. The light from everything that was ever emitted you would be able to collect and see what it was, in principle.
Even this picture of cosmology, by the way, is still restricting us a lot more for the sorts of observables we can talk about because now there's a Big Bang in our past. So we're not free to set up any initial states we want, scatter them any way we want, and so on. There's even fewer things that we can talk it still, even fewer things.
But there are some things we could talk about. If we waited for a long, long time, we could count to see how many red stars there were, how many blue stars there were. We could do that with very, very big apparatuses without collapsing everyone to a black hole and so on. So there's still something to talk about.
But 13 years ago, something very dramatic happened. Which is that there was evidence that the universe is actually not decelerating, but just started picking up again. And the rate of expansion, instead of decelerating and slowing down, has started speeding up. And it's speeding up at a constant rate. It's accelerating.
And people first noticed this by seeing that when you look at the velocity of these more, and more, and more distant objects-- they first did it with supernovae-- that instead of lying on the straight line, it actually went up, and up, and up, and up a little more. So it was getting faster and faster, the further and further back you went.
So that was the first evidence. In fact, the experiments that measured it were designed in their design reports. They were designed to measure the deceleration of the universe. And they have to report the negative value for the deceleration because they actually saw that it was accelerating. No one unexpected that.
Now, the fact that it's accelerating is often said to be due to something called dark energy. But, in fact, we have a very good idea for what it could be. There is a completely straightforward thing that it could be, which is just a non-zero energy density of just the vacuum. So if the vacuum had a non-zero energy per unit volume, then exactly this phenomenon pops straight out of our standard understanding of general relativity and cosmology.
The formula I wrote down yesterday for the rate at which a universe expands-- the formula that looks like this, where rho is the energy density of the universe, if you just may rho a constant, which is what you would get if it was just the energy of the vacuum-- there's always vacuum there. You can't get rid of it. As the universe gets bigger and bigger, there's always vacuum there. So if that vacuum energy is there, then what would happen is the universe would start doubling in size at a uniform rate. It would start accelerating in its expansion.
So we actually live in an interesting time. Earlier in the history of the universe, it was completely dominated by radiation, photons. Matter took over from radiation a while ago, 100,000 years after the Big Bang. Back then, the value of the vacuum energy was totally minuscule compared to everything else. But just in the neighborhood of where we are today, all of them are sort of comparable to each other. We're living in a very interesting cosmological time, so that it was possible to discover these things.
Now, this means that the deep future of our universe, it will just start doubling in size in a uniform way. Every 10 billion years or so, it'll double in size. And it won't slow down. It will just keep doubling in size, double in size, double in size.
That means something. That something that's right now 20 billion light years away from us, let's say it emits a beam of light now. And we'll never, ever see that light.
There is a poor piece of light trying to get to us. This is the sky far away. 10 billion years later, we've doubled in size. The light is moving as fast as it can. It's moving over one cell. But it can't beat the fact is that the next moves over, we've put-- it moves one cell over. We put two cells in the middle of it. It moves one cell over. The next doubling size, we put two cells in the middle of it.
We're just inventing space all the time at such a fast rate than even light can't overtake it. Nothing is moving faster than light. It's just the rate at which space is being invented everywhere is so fast that even light, streaming along, just can't make it.
So we can draw a sort of picture like this. That there was a Big Bang. We talked about this period of inflation. We talked about a hot universe.
But in a sense, we're starting to inflate again now. We're accelerating. We're doubling in size at a uniform rate. We're accelerating again. And light from this guy will just never reach us, just never, ever reach us.
So that means that this is what the universe will look like in a long time. In a hundred billion years or so, there will just be me. Well, not just me. But me and-- well, me and all of us in our galaxy. Me and all of us in our galaxy, which are being held together gravitationally. We're not being ripped apart by this expansion of the universe. Were being held together.
But we'd be sitting there. Everything else around us is getting sucked away. Everything else around us is getting pulled apart from us away by this accelerated expansion. So we would be lonely sitting there in our galaxy. Looking out, we wouldn't see any other galaxies anymore.
In fact, this thermal microwave background radiation that we talked about before, even most of that would be stretched out and gone. This would be a very bad time to be a cosmologist. It'll be very hard to convince people that there was a Big Bang. They'd say, well, sure looks like we rule the universe. Here we are. We're all alone. I don't see any cosmic microwave background.
I'd say, no, no, you don't understand. It was there. It started accelerating and inflating away. They'd say, yeah, right, sure. So this is a particularly good time to be doing cosmology because it'll start getting harder pretty soon.
[LAUGHTER]
But it means something very startling. There are regions of space that we'll never see again. There's something out there. The light from out there we'll never see again. What we see out there basically today is what we're ever going to see. If this picture of the accelerated expansion is correct, we're never going to see more.
In particular, will never see an infinite amount of anything. We don't have an infinite amount of material out of which to build detectors. We can't separate things from each other arbitrarily in order to minimize the interaction they have with each other.
On top of all of that, the situation is really quite analogous to being-- roughly analogous to being inside a black hole. There isn't a singularity that's going to kill us-- well, we'll talk about that tomorrow-- but not obviously, not very obviously.
But there is this sort of horizon out there. And just like black holes radiate, that horizon also radiates. Now, this temperature is very, very, very low. But there would be radiation coming from-- it's really like we're on the inside of a hot tin can.
So the situation is that our future is finite. We don't get access to any more things. And we're, furthermore, being slowly cooked all the time, a very low temperature, but slowly cooked all the time.
OK. This is a really severe paradox. Well, I don't know if it's a paradox. But we don't know what we're doing.
Because of quantum mechanics, if I want to make any predictions about what this universe-- any sharp predictions in this world, I have to do an experiment. We've heard the mantra now many times, with an infinitely large detector, an infinite number of times. I can't do either one of those things in this universe.
I can't do it with an infinitely large detector because I don't even have an amount of material. I'm stuck in this finite-sized tin can. And actually, it turns out I can't do it infinitely many times either because I'm constantly being bombarded all the time by these pesky radiation coming from the horizon, which has got a really low temperature. It's true. But if you're cooked at a uniform rate, even at a very low temperature, for 10 to the 120 years, you'll start feeling it. It'll start interfering with your ability to do the experiment and not have some random fluctuation in your apparatus because of the fact that it's constantly being bombarded. Actually, it turns out to be around 10 to the 10th to the 120 years, but anyway.
But it's a real question. We don't even know what to talk about in such a universe. In that other tin can universe that we're talking about, where quantum gravity was the same as quantum field theory, everything was great. We could talk about these observables on the boundary. We found a theory for them. Everything was fine. But we don't even know what to talk about.
But this is what our universe is, not this other imaginary universe. Our universe forces us to come to grips with these issues.
All of this is pointing to something, which is that that previous picture that we talked about that was so wonderfully successful, space emerged, gravity emerged, strings emerged. All of those things happened in a quite magical way. There was one thing that did not emerge. You did not hear me say that time emerged in that picture. That's because time didn't emerge.
The time that happened inside the space, things moving along in time, was exactly the same time that happened in this boundary. Things moved in time in the boundary. Things also happened in time on the interior. But the two times were the same.
We didn't have to abandon the notion of time. We got rid of basically everything else. But we didn't have to abandon the notion of time.
And that's clearly the next frontier. We have to figure out how to get time out of nothing? We figured out to get gravity, strings, space out of nothing. We have out how to get time out of nothing. And that's really the hardest thing. Because physics, if absolutely nothing else, is about talking about things-- the relationship between things at one time versus another. So we don't know how to do that.
All right. I don't have time to tell you about these other things. But let me just end by saying that I hope you've seen space-time is doomed. We have some ideas for what replaces it. There are some beautiful ideas about how space can emerge and gravity can emerge. We have real issues about where time comes from.
And tomorrow, we're going to pick up the story once again with this accelerating universe, but with a different feature of it, that I didn't emphasize at all. Which is that if we go back to this realization that our universe is accelerating, I told you there's no problem. We have this energy density of the vacuum. It's that number. Everything is great. Well, tomorrow I will tell you that number is absolutely insane.
No one in a million years would have predicted that that should have been the energy of the vacuum. It's a completely insanely tiny number. And understanding where that number could come from is another very major challenge that we'll talk about. And will end up being related to yet other potential extensions of our notion of space-time, including the idea that we're just a small part of a very, very, very vast multiverse. And that we have to figure out what that means, on top of everything else. So that's the plan for tomorrow. Thanks a lot.
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Renowned theoretical physicist Nima Arkani-Hamed delivered the third in his series of five Messenger lectures on "The Future of Fundamental Physics" Oct. 6.
Formerly a professor at Harvard, Arkani-Hamed currently sits on the faculty at the prestigious Institute for Advanced Study in Princeton, New Jersey, where Einstein served from 1933 until his death in 1955.
The Messenger lectures are sponsored by the University Lectures Committee. The lectures were established in 1924 by a gift from Hiram Messenger, who graduated from Cornell in 1880.
