Topics in String Theory | Lecture 8
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Susskind, Leonard
(March 7, 2011) Leonard Susskind gives a lecture on string theory and particle physics that dives into the idea of cosmic horizons as well as the relationship between ultraviolet and infrared light. In the last of course of this series, Leonard Susskind continues his exploration of string theory that attempts to reconcile quantum mechanics and general relativity. In particular, the course focuses on string theory with regard to important issues in contemporary physics.
 Stanford University. One thing that I'm confused about is vacuum energy. I know what you said when you get the plank scale that explodes. My question in particular, does electron positon, positron hair production in annihilation actually occur, or is that excluded by? Actually occur means what? That you can actually see out of the vacuum electrons and positrons coming out? Let's say the measured vacuum density, if that would be a fair measure. Electron, positron, virtual pair production. Virtual pairs do contribute to the vacuum energy. That doesn't mean that you should see electron, positron pairs popping out of nowhere. It does not mean that. It means they pop out and go back in a very short time, subject to the uncertainty principle. Quantum fluctuation and not real, genuine particles being produced. It's much like the zero point energy of a harmonic oscillator. Classically, the harmonic oscillator just sits still at the bottom of its potential well. Quantum mechanically, it cannot both sit still and be at the bottom of the potential. There's competition and it's resolved by a little bit of quantum fluctuation. That quantum fluctuation has energy. How much is it? A half h bar omega. It's always there. Can't get rid of it. It's the least energy that the oscillator can have. And of course, that doesn't mean looking at the oscillator. Most quantum oscillators are too small to see. It is the ground state. And as the ground state, it means it's the state in which nothing is happening to the extent that nothing can happen. Same is true with electron, positron pairs. Same is true with virtual photons. And in fact, it's a closely related phenomenon. The electromagnetic field, for example, in a cavity, electromagnetic field in a cavity is just a collection of harmonic oscillators. Each oscillator has a little bit of zero point energy. Add them all up. You call that the vacuum energy. I was saving that for a little bit later because it plays an important role in cosmology. Are you also going to talk again about the 120 orders of magnitude? No, I think I won't spend a lot of time on that. I'm not going to mention it, but no, I thought tonight, in any case, we mainly concentrate on cosmic horizons. What a horizon is, why there are horizons, and a little bit about the cosmology, the equations of cosmology, how they lead to an accelerated expanding universe and why the accelerated expanding universe has event horizons. I thought that would be a good thing to do tonight. Since we spent a lot of time learning about what a horizon is, we're not quite finished learning what a horizon is. No four minutes. You mentioned last week, I guess, was the ultraviolet and red relationship, and you said you were going to be getting to that tonight? Well, we can talk about it. Maybe we'll start with that. Maybe. It's not really on my list of things to do tonight. Maybe we'll get to it, though, if I run out of things. Okay. Wait three minutes and sit and think. That's a question. Let's lay it to the question back there about vacuum energy. If there's a vacuum, there is a positive energy density. Why would that have some kind of refractive effect on it? Refractive. Optically refractive effect. What is refractive? Refractive, remind me which is refraction, reflection, and respect the speed of light. Speed of light, for the speed of light, for example, would be ever so slightly less than the quantum vacuum at different free space. Yeah. The vacuum energy, it's a good question. It's a good question. Most energy that you think about, if you have that energy present, it creates a situation where the world is not Lorenzen variant. I don't mean that the theory is not Lorenzen variant. I mean that the configuration of the world is not Lorenzen variant. What does it mean? What exactly does that mean? Let's first talk about something simpler. Translation invariance. Every place is like every other place, right? No, it's not true. Stanford University is not like Berkeley. The Earth is not like the Sun, and it's certainly not like interstellar space. So what do we mean when we say the world has translation invariance? We don't mean that the configuration of the world, every place looks the same. We mean if we took the whole thing and moved it, it would look the same. But moving it means moving us, moving Berkeley, moving everything else. Now, in the same way, you talk about Lorenzen variance, is the world Lorenzen variant? No, I'll tell you why in a minute. Are the equations of physics Lorenzen variant? Yes. So what does it mean for the world not to be Lorenzen variant? Well, it means that for whatever reason, the stuff in the universe does not look the same from every frame of reference. If I were to whiz by you at a thousand miles a second, you would look different to me than if I was standing still. So stuff in the world, stuff in the world, breaks the symmetries that we talk about. They create special locations. They create special directions. The equations of physics are supposed to be rotationally invariant. That means the same, no matter how we rotate. True enough, the equations of physics are rotationally invariant, but is this room rotationally invariant? Certainly not. If I look that way, I see you. If I look that way, I see something else. So the presence of stuff breaks symmetries, and presence of stuff breaks Lorenzen variance. The world is not Lorenzen variant by virtue of the fact that there's stuff in it. Now, most of the times the stuff, energy and so forth, as I said, is not Lorenzen variant, and the result of that is in particular a result on the motion of light waves. Light waves moving through materials have different velocity than they have moving through empty space. The rule about the speed of light is that light always travels with the same velocity in empty space, but that's another way of saying that empty space is Lorenzen variant. Empty space is Lorenzen variant. Empty space is translation invariant, it's rotation invariant. Empty space has no preferred frames of reference. And the statement that light always travels with the same velocity is a statement about a Lorenzen variant world where there is no preferred frame of reference. When there's a preferred frame of reference due to the presence of material, light can travel at any velocity lower than the speed of light, than the usual speed of light. Now, vacuum energy is very special. Vacuum energy is the one exceptional situation where the presence of that energy is Lorenzen variant. It does not pick out a reference frame. If you weren't here and the worm wasn't here and I was in absolutely empty space, nevertheless, there would be vacuum energy in this room. Vacuum energy has a certain value, whatever its numerical value is, a certain number of joules per cubic meter, very, very tiny, but nevertheless, finite. And so you might think, well, that's like a material being here, it's like a stuff being here, and a stuff being here would define a frame of reference where the stuff would be at rest. Vacuum energy is very special that way. It does not define a frame of reference. I could detect the vacuum energy with a vacuum energy detector. I won't tell you how to build one. I'll tell you how to build one in a little while. We'll talk about how you build a vacuum energy detector. I could detect the vacuum energy and see how much is here. Or we get some number. I would then go whizzing by as fast as I can go and do the same experiment. If it was ordinary material, I would discover that the energy changes, the energy that material changes with my velocity. In particular, if I whizz past a gas, then of course the gas looks higher energy. Why? Because all the molecules appear to be moving past me. Vacuum energy is the one special case where my vacuum energy detector will give the same answer no matter how I'm moving. So vacuum energy is the special case where it's present, does not pick out a special frame of reference, does not violate Lorentz invariance, and it's a special case where light will move with the speed of light independently of the observer. Now moving with the speed of light means something. It means when the light ray passes you, passes right in front of your nose, and you have your clocks and your detectors locally, right near you, you measure the same velocity of light. Because of the expansion of the universe and so forth, it becomes a more problematic question of whether light moves relative to us far, far away when the light is far away from us. Does it move with the speed of light? And we're going to talk about that. The answer is no, not in general. It moves relative to local observers where that light is passing by them with the same speed of light, but it doesn't necessarily move relative to us with the speed of light. Faster or slower? In this discussion, is vacuum energy synonymous with cosmological? Yes, absolutely. All right, so let's begin there. Why don't we begin there? If we take the universe and fill it with full of particles, then let's say ordinary particles, protons, fill it up. I don't mean fill it up so that every proton is against every other proton. I just mean create some uniform density of protons. And of course, the universe is something like that. It's not so different. There's a more or less uniform energy density throughout all of space. It tends to be clustered into stars and stuff like that, but it's still bigger than superclusters of galaxies. It's homogeneously distributed, rather diffuse, something if I remember, something like about 50 protons per cubic meter on the average. And there's some energy density. Let's give that energy density a name. Standard terminology for a density is rho. And now that could stand for the mass density, but it's not being the same thing. All right, so that's the mass density. That's consistent with a bunch of particles. Now what happens, and it's the number of particles per unit volume, or the number of particles times the mass of each particle divided by a volume of space. Take a volume of space. The density of energy is the mass of each particle, let's suppose there's only one kind of particle. Particles are standing still, and not photons. Let's not get involved with photons right now. They're just protons standing still, protons, neutrons, electrons, atoms. The mass of an individual particle times the number of particles in a volume divided by the volume, that's the density. Now what happens to that density as the universe begins to expand? Well, the number of particles doesn't change, at least not if they're ordinary particles. The number of particles doesn't change, their mass doesn't change. What does change is the volume of a region of space occupied by those particles. So we take some particles that started out in a region of space, a square on a side, a hundred light years on a side, a hundred light years, a hundred light years the other way too. It has a volume of a million cubic light years, which is some number in cubic meters. That's the volume that goes in here. The number of particles stays the same, and now the universe expands, everything expands with it on sufficiently big scales. The volume of space, let's say over some period of time, doubles. When the volume of space doubles, what? I'm sorry, I meant the radius of, I didn't mean the volume. When the radius of this cube, the linear dimension of the cube, doubles, what happens to the density? Well, when the radius doubles, the volume gets multiplied by eight, and the density becomes one-eighth of what it started. OK? So ordinary material has the behavior, and this is true for photons, for anything, not the exact formula is a little bit different for photons, but the general rule is that as the universe expands, the energy density dilutes. It decreases. The density of energy decreases just because everything separates. Again, vacuum energy is the exception. Vacuum energy is the exception. Vacuum energy has the property, vacuum energy density. Vacuum energy density has the property that as the universe expands, as the volume of space expands, the amount of, the density of energy does not change. The density, not the amount in the box, but the amount per cubic meter, always stays the same. If the universe were to expand by a factor of a hundred in every direction, the volume of the region of space would go up by a million. Nevertheless, the density of vacuum energy in this room would stay exactly the same. So it's a property of empty space, which just doesn't change as time goes on. It doesn't change as the observer moves past it. It doesn't change when the observer rotates. It's not clustered. Uniformly filling space picks out no direction, picks out no Lorentz frame, and picks out no special time in the sense that as the universe expands, it doesn't change. That's, energy of that type is called vacuum energy. If it exists at all, and it does seem to exist. What do we care? Why should we care if the universe were filled with something that we call vacuum energy? We do experiments, and the only thing we're interested in this room, if we're doing experiments in this room, is differences of energy. The zero point of energy really usually doesn't matter for anything. The thing which usually counts is differences of energy. What's the energy difference between an excited atom and an unexcited atom? What's the difference of having of the energy with or without the atom? The energy of an atom counts, but that's because we can compare it to the energy of a configuration without the atom. But just empty space, who cares? We could just put as much energy into an empty space as we like as long as that energy cannot turn into any other kind of energy. As long as it doesn't come into the questions of differences of energy, why should we care if there's something called vacuum energy in the room? Well, the answer is what I said is we don't care about the zero of energy until gravity becomes important. Once gravity becomes important, what's the source of the gravitational field? The source of the gravitational field is mass. Mass is energy. Vacuum energy gravitates. It has a gravitational effect. It has an effect on spacetime. And it has an effect on spacetime, which is different than if it weren't there. So let's talk about, well, the first question is, what's the origin of vacuum energy? And as I said, or somebody asked me earlier, vacuum energy is, it wasn't quite asked to me in these terms, but vacuum energy is a feature of quantum field theory. Quantum field theory, quantum electrodynamics, quantum, the standard model of particle physics, all quantum field theories have vacuum energy. It's just a zero point energy of the oscillating fields, if you like. There's a contribution to it from the electromagnetic field. There's a contribution to it from the electron-positron field. There's a contribution to it from quark fields, even if there are no quarks present. The point is, you don't need to have anything really present to have that energy. What does it do to? It's due to virtual creation and annihilation. Virtual, remember, means that it happens so fast that a little bit of energy that is created and annihilated satisfies the uncertainty principle. If you create some particles out of nothing, then they can only last for a very short time because of the energy time uncertainty principle. Let's leave it at that. But the net result is that there is energy density called rho. And it fills up space. It's everywhere. It's Lorentz invariant. It doesn't matter how you're moving and so forth. Vacuum energy. Let's call it rho vacuum. Now, what does it do? It has an effect on the right-hand side of Einstein's equations. Let me remind you what Einstein's equations are. The details of them we don't need to know. I will write down as much detail as you need to know, which is very little. But let me just remind you what Einstein's equations say. On the left-hand side of Einstein's equations is a tensor called g mu nu. g stands for gravity, I think. It's a tensor. And it's made up out of the metric components of space, the metric of space. It's made up out of the metric tensor, the metric tensor is little g mu nu of x. It's on the left-hand side. It's got to do with curvature. It's got to do with derivatives of the metric, Christoffel symbols, all sorts of awful stuff. And by the time you write it down, it takes up what g mu nu is. It takes up a fair piece of space. But it's got to do with, oh, I said it has to do with gravity. I'm not sure it has to do with gravity. I think maybe it has to do with geometry, the g. It's the geometric side of Einstein's equations. It says some property of the geometry of space-time, this is space-time geometry, not the geometry of space alone. And on the right-hand side, in other words, it's the gravitational field. Gravitational field, geometric field, however you want to think about it. Einstein's field equations are that this on the left-hand side is equal to something, a numerical constant, but I'm not interested in numerical constant. It's 8 pi over 3, but not important. On the right-hand side is the source of the gravitational field. And the source of the gravitational field is energy and momentum. Energy, also known as mass, and the motion of masses, which is also known as momentum. So on the right-hand side, there's another tensor. It's called T mu nu. I don't know why it's called T. But it's a thing which is made up out of particles, out of ordinary fields, not the gravitational field, the electric field, the magnetic field. And if you know what this is, you know what the ordinary energy density is, the density of particles and so forth. That's Einstein's field equation. And on the right-hand side is all of various kinds of energy, both those which dilute particles, ordinary stuff, light, the stuff which dilutes when the universe expands, but also the stuff which doesn't dilute, the vacuum energy. So that's the reason we have to know about vacuum energy, to know how the geometry of spacetime behaves. If there's only vacuum energy here, I'll tell you what, I'm going to write these equations, I'm going to write them out in full, but for one very, very special case. I told you that the left-hand side is made up out of the metric tensor. And let's take a very, very special case for what geometry, for what spacetime geometry is like. In some sense, it's the simplest geometry that's not just static space just sitting there. I think we actually wrote it out last time. It was the expanding, uniformly expanding space that I told you to think of as an infinite rubber sheet, where the infinite rubber sheet is being stretched so that the coordinate marks on it, we take a rubber sheet, we mark it with coordinate marks, x equals one, x equals two, x equals three, x equals four, and then we start to stretch it. The actual distance between neighboring points increases with time, and that's represented by writing a metric, the metric ds squared, that's the spacetime distance between pairs of points, neighboring pairs of points, just as it would be in ordinary Minkowski space, we start with a minus, this is pen, this pen seems weak, minus dT squared, that's all we would have in ordinary Minkowski space, meaning flat spacetime, and then what we would have in Minkowski space would be plus dx squared. Now, dx squared here stands for dx squared plus dy squared plus dz squared, which I won't bother writing, it stands for the ordinary Euclidean metric of ordinary space. There would be some speeds of light in here, if I kept the speed of light around but I'm not, but that's not the expanding spacetime, the expanding spacetime you introduce one more thing, a thing called a scale factor, A of t. A of t is a function of time, is not a function of space, and it tells you, oops, I'm sorry, this should be A squared of t, A squared of t. What it says is that if you take two points, let's say x equals zero and x equals one, what's the distance between them? Well, it's one unit. No, it's not one unit because x doesn't actually measure physical distance, it's this which measures physical distance, the actual distance between them at any instant of time is A. And as A changes with time, the distance between these marked points on the Rubber sheet world, the Rubber sheet world, x equals zero, x equals one, the marked points separate, so A grows with time. It doesn't matter if they're separated by one unit, two units, if we want to separate them by, let's say this is x equals seven, x equals 13, then the distance between them is delta x, that's 13 minus seven, which is four, thank you. 13 minus seven times A, A of t, four. 13 minus seven is A, is it four? No, 13 minus seven is six, thank you. Right, okay, that's the meaning of this metric. They don't have to be on the x-axis, they can be off the x-axis, but you get the idea. Okay, this is a non-trivial geometry. It has space-time curvature, unless A is constant, then it's just flat space, good old flat space. If A is varying with time doing things, maybe just expanding, that's enough to make the space-time not flat. Space may be flat, but space-time is a different story. So space-time, expanding space-time is a new kind of geometry that's good. Okay, now let's write down Einstein's equations. What do Einstein's equations have to do with? They have to do with how A varies with time. A is the only thing in this geometry that's not known. It's the only degree of freedom for this kind of geometry. For a geometry of this type, all of the geometry is coded in the properties of A of t, which means A is a function of time. So Einstein's field equations must be simply equations for how A varies with time, and they are, and here's what they say. They say that the A by dt squared is equal. I'm going to put in all the factors now. 8 pi over 3 times Newton's constant times the energy density of matter. The assumption here is that the energy density of matter is completely uniform in space. You could have such a thing if space was filled up with a uniform radiation bath all at the same temperature. You could have such a thing if the universe were filled up with atoms at a uniform density. So we don't have to write rho as a function of x because rho is not a function of x, but of course we do have to write rho as a function of time. So rho is some function of time, but what is rho as a function of time? This, if we, oh, I'm sorry, there's a 1 over A squared here. It's 1 over A squared. It's A dot over A squared. Do you remember from last time what A dot over A is? It's the Hubble constant, but let's leave it there. A dot over A squared, that's it. OK, now this equation also has another significance in general relativity. It's one which confuses people. I get zillions of emails. I get so many emails about this particular point that I have got, can't keep up with them, and I don't even try anymore. This equation is also an equation for energy conservation. It expresses energy conservation in general relativity. The right-hand side is all of the energy of ordinary stuff, but it does not include the energy of the gravitational field. The gravitational field, because it's varying with time, A of t, also has energy, and that energy is a kind of kinetic energy. It's a kinetic energy of motion or a kinetic energy of the time dependence of the metric of spacetime. What is that energy? That energy is simply the energy of the varying gravitation. It's called the varying scale factor. Energy of the varying scale factor, A is called the scale factor. One odd thing about it is that it's negative in general relativity. Minus A dot over A squared is probably a factor of 3 over 8 pi g. 3 over 8 pi g, I think. It's not the number that's important. It's important that in the formal structure, in the mathematical structure of general relativity, gravity has energy, and some piece of the gravitational energy is negative, and it corresponds to a negative kind of kinetic energy of the gravitational field. So with this equation, it can be written as minus this plus this is equal to zero. Minus this plus this is equal to zero. Well, zero is conserved. Zero doesn't change with time. Zero is zero. It doesn't change with time. So among other things, among the many things that this says, that energy doesn't change with time. It also says the total energy of the world is zero. But it includes this piece coming from expanding space time, which you're not used to counting. When we count the energy of this room, we don't normally think about the fact that the universe is expanding. We just ignore it, and we say the energy in the room is positive. Okay. Good. Let's put it back the way that it was. This equals this. Let's go a little further now. Let's ask what row of T is like. What the time dependence of the energy density in the world is like. So let's come back to this box. There's not a real box. I've taken a region of space and just drawn a mathematical box. The mathematical box has a side delta x equals one on each side. How much energy is in it? Well, it's the number of particles inside it, number of particles inside it, which is not going to change. As the box expands, the particles move with it, so that if this room was part of the expanding universe, there would be a lot of particles or a bunch of protons. The protons wouldn't expand, but the distance between them would expand so as to keep the proton density uniform. So there's n protons in there. Each one has mass m. That's how much total mass is in there. What's the volume of space? The volume of space is delta x times delta x times delta x. That's one. Times a cubed. Why a cubed? Because the actual length here is delta x times a. So the side of the box here, I've chosen a box whose side at any given time is a of t. And so to find the density, I have to divide it by a of t cubed. The only important thing here is that at any time, if I know the energy, this is the energy density, if I know the energy at any given time, then I know it at all times, not because it stays the same, but because it always scales like 1 over a of t cubed. So we can say that rho of t then is some number, we could call it rho zero, it's some particular instant of time divided by a of t cubed. So now we can come back to this formula over here. We can say, uh-huh, I now have an equation for a. It's Einstein's equation for an expanding universe. And what goes over here is some constant rho naught over a cubed. When we can solve this equation, maybe we should solve it. Maybe we should solve it. It's not a difficult equation. It's much easier to solve if we take all the constants, 8 pi g over 3 times rho naught, and absorb them into another constant. Let's just call that other constant, let's just call it rho naught. And thereby erase this. OK, how do we solve this equation here? Let's write out what it is. I bet you can solve it. I'll have you solve it. But first I'll manipulate it a little bit. This is the adt squared. There's an a squared in the denominator, but I'm going to multiply through and put it on the other side. So we multiply by a squared and we get rho naught divided by a, is that right? Right. And now I take the square root of both sides, rho naught, divided by a, I take the square root of both sides, square root of rho naught, a to the one-half, and I multiply so that it reads a to the one-half times dA equals square root of rho naught times dt. a to the one-half on this side times dA, or just we could write it this way. Can you solve that? Yep, say it again. Right, what we're doing is integrating both sides of this equation. What Michael suggested is to integrate both sides of this equation on the right-hand side we get square root of rho naught times time. The integral of dt is just t. And this one is what? Three-halves, a to the one-half, so let's integrate it. That's three-halves. No, two-thirds, two-thirds, a to the three-halves. Is that correct? And now we know a is a function of time. Forgetting all the numerical constants, two, three, rho naught, what it says is that a grows like t to the two-thirds power. The scale factor grows like t to the two-thirds power because the energy density has a particular behavior. That's cosmology. That's standard, absolutely standard cosmological expansion. A growing like t to the two-thirds. Let's just draw it. A growing, let's first draw a growing like t. There's a growing like t. A t. t to the two-thirds is smaller. It grows less rapidly than t itself. This large t is smaller, and it looks like this. Not quite a parabola, but it looks like that. It slows down. A uniform line like this would be an a, which grows linearly with time. It would be like a coordinate of a particle with linear motion. That's not what happens. What happens is it turns over and it appears to go slower and slower. The universe decelerates in this kind of cosmology. This is a decelerating cosmology. What is it that causes the universe to decelerate? Gravity is pulling everything. Yeah. Another way to think about it is stuff started out moving away from each other, and gravity just grabbed the hole of it and slowed it down. And eventually it will get slower and slower. OK, now let's ask what happens if we have a different kind of energy on the right-hand side. We could put in, just for fun, I'm not going to do it tonight, we could put in radiation energy, energy solely due to radiation as if there were no particles, only photons in the universe. That would correspond, you can work it out yourself, that would correspond to putting A to the fourth here. We don't have to go into that now, it's just a fact. You can solve the equations and find out how the universe would expand if it were radiation dominated. The particular version that I gave you is particle dominated. It's usually called matter dominated, matter as opposed to radiation where the energy density decreases like 1 over A cubed. OK, but what happens now if there is vacuum energy, if there really is vacuum energy? Vacuum energy has the property that it never changes under any circumstances, the density itself. The density itself does not change under any circumstances. We need to give it a name. We'll put back, for the moment, we'll put back the 8 pi G's and so forth. Theory tells us that it doesn't change. Quantum field theory tells us it doesn't change. It tells us that it's a sort of universal constant that can be calculated. Now, does that mean that the universe that we know with certainty that the universe is filled up with a kind of energy which doesn't dilute as the universe expands? Theory doesn't tell us that. What tells us that is experiment. And we'll talk about the experiment in a minute. We'll talk about the observation in a minute. Theory tells us only what to put on the right-hand side for various different hypotheses about what the energy density is. If the energy density is vacuum energy in the sense of quantum field theory, then it really is just constant. So that means what goes on on the right-hand side here does not vary with A. Does not vary as the universe expands. And let's just call it row vacuum. Let's call it the vacuum energy density. Row vacuum is 8 pi g over 3, sorry, A dot over A squared is 8 pi g over 3 times this constant value of the energy density. Now, first of all, if this is correct, then one thing it says is that the Hubble constant, A dot over A is the Hubble constant. This is also H squared. It tells us that H squared, or H also, 8 pi g over 3 times the vacuum energy, that this doesn't change with time. Remember what the Hubble constant is. The Hubble constant is the relationship between distance and velocity. Velocity is Hubble constant times distance. So it says for all time in this kind of world, and this kind of world for all time, the relationship between velocity and distance is unchanged and always the same numerical value. That incidentally would not be true in the other solution we wrote down. Normally, for other kinds of cosmology, the Hubble constant decreases with time. Here, it just stays constant. An implication is that the distance out to which you have to go in order to get to the speed of light, where stuff is moving away from you with the speed of light, that distance is always the same. It's given by C equals H times D, or equivalently C over H is equal to D. C is a constant. H is a constant. The distance out to the place where things are receding with the speed of light from us is fixed. It doesn't change with time. So that's one implication of what this means to say that there's a constant value here. But let's see now if we can now solve Einstein's equations for an expanding universe. Under the assumption that the only energy density now, the only energy density is vacuum energy density. So what do we have? We have A dot over A squared is equal to 8 pi g over 3 times rho, but that's just the number H squared. Let's just call it H squared. Now, I'm using H now not to mean definition. It's not definition now. Previously, A dot over A was the definition of the Hubble constant. Now, it's the numerical value of 8 pi g over 3 times rho v. It's a number. It's a number that we don't know how to calculate it. We don't know the theory of it. But it's a number. H squared is now a number. Everybody understand the difference between saying H squared is A dot over A as a definition and H squared is A dot over A squared as a numerical relationship. Here, H is not defined this way. It just is a number. It's measured? It is measured. It is measured. It's a measured number. And it's not expected. And there's evidence that it is not changing with time. Right? So it's a number. How do we solve this equation? Well, first of all, we take its square root A dot. We multiply it by A. And it reads A dot is equal to H times A. I think some fraction of you can solve this equation. Nobody can solve it? Right. A dot A is equal to E to the Ht. This is the equation that tells us that the growth rate of A is proportional to A itself, exponential growth with time. A thing which changes by an amount proportional to its own value exponentially grows. And the growth factor, the time factor, the time constant for this is H. E to the Ht is the way this universe is growing. Exponentially. You can put a constant in front of it. The constant doesn't matter much. So this is an exponentially growing universe. That's the result of vacuum energy. Vacuum energy really is important. And the way to detect it, well, I'll give you two ways to detect it. One is astronomical. Measure the Hubble constant. Measure the Hubble constant over a period of time and see whether it's staying constant. Or you can measure the scale factor A over a period of time and see whether it's expanding exponentially. There's another way that you can measure it in the laboratory. There's no way you're actually going to do this. But if you have, the laboratory now means way out and out of space, take two objects which are very light and therefore their gravitational between them is not very important. Start them out at rest. And if they're light enough so the gravity between them is unimportant, they will just expand with the expansion. And you can measure the rate at which the distance between them expands with time. Make sure there's nothing else present. Complete vacuum. So the only possibility is vacuum energy distorting space. And you would discover that these begin to accelerate and begin to expand, separate from each other exponentially. So that's how you would build a vacuum energy detector. Now of course the effect in the real world is so minuscule that you couldn't even begin to hope to detect this in the laboratory, but you can detect it cosmologically. How do you detect it cosmologically? A number of ways, but you say, well we have to check that H is constant with time. That's going to take a long, long time. But no, not really. When we look out at the distant universe, we see it at different times. And we can actually measure the Hubble constant at different times by looking at it at different length scales. Or we can look, what comes to the same thing is we can measure the expansion rate or we can measure how A varies with time. All right, so I'll tell you now how it varies with time. Oh, wait. Let's point something else out. The real world doesn't just have an H squared. It also has some other energy density. Let's add then the other energy density. It was called rho naught divided by A cubed, for example. When A is small, when the universe is small, when the scale factor is small, this term is going to be much bigger than this. Why is that? Because A is in the denominator. That's why. So early on in the real world, this is not very important. And this may be very important when A is small. In that case, we worked it out a few minutes ago, and we found out that A goes like T to the two thirds. It goes like this. But now as A grows, eventually this term will get small, whereas this one is not getting small. This one's not diluting. This one is diluting. So eventually the diluting term gets smaller than the non-diluting term. And then after that, you can basically forget it. Once this term gets smaller than this term, you can forget it. And how does A behave? A behaves like E to the HT. So what you expect in a world that has both kinds of energy is that early on the scale factor, or the radius of the universe, or the distance between distant galaxies, grows this way. But then at some time, when the vacuum energy becomes more important than the other forms of energy, it takes off exponentially, exponentially something like that. So when you saw that differential equation, you end up with a C in front of the integration constant in front of the indian. Yeah, there's an integration constant here. How do you know that that integration constant isn't really small? Well, you measure it. You measure it. You measure it. You measure it. And here's T. Here's A. And of course, when you measure A, you don't just measure A now, as I said. You measure A by looking out deeper and deeper into distant galaxies. You can measure the distance between different distant galaxies at earlier times. And so at one time, you can measure this whole curve up to some point. You can't measure what it's going to be in the future, but you can measure what it was in the past. So if this is now, we get to measure all of this curve. And it looks something like that. It looks something like that, approximately something like this. With this branch of it here being fit rather accurately to an exponential, I won't try to tell you how accurately. It's about a 5%. The proper figure of merit is about a 5%. It's not a deviation. It's exponential plus or minus 5%. OK, so that looks to most cosmologists, that looks fairly convincing that we live in a world with vacuum energy. Certainly, we don't know anything else that would have much chance of doing precisely this. And it's not a fact in the same sense that we know that the electron exists so that the fine structure constant is 1 over 137 point blah, blah, blah, blah. But by now, it's pretty much considered a fact of cosmology that we live in a world with vacuum energy and that we live in an exponentially expanding world. That's pretty remarkable in a way. A question? A good question. In some books, I've seen if you take our position right down and draw the slope down, now we can, if that slope intersects to the left of the origin, you can see back to the big bang as it goes up, there will be a time where you can't see past the big bang. Is that, in other words, we'll expand it away so far that I don't quite understand that pointer is it important? Well, I'm not sure it's an issue of expanding back to, tracing back to the big bang. Or to the cosmic background radiation. In the sense that the Hubble distance is right into the age of the universe. In the sense that the Hubble, yeah. Right, the Hubble, right, in this exponentially increasing world here, the Hubble scale is not related to the age of the universe. It's just a constant, right? But it's someone you, I don't know, there's a horizon. There's a horizon. We're going to talk about horizons. We're going to talk about event horizons. There's a lot of different kinds of horizons on the market. But the only ones which are deep and fundamental interest are event horizons. So we're going to talk about event horizons. So let's now just put in our back pocket this piece of knowledge that the world is exponentially expanding. That we can summarize everything that was on the blackboard. Oh, there is an interesting fact. If the world continues to exponentially expand like this, then all of the dilutable matter, the kind, the energy that dilutes with expansion, is just going to get more and more and more diluted. So in some number of billions of years, the material in the universe will stretch out so much that it will become of negligible density. And the only energy left over will be the vacuum energy. It will become a very dull world. But so be it. Does that mean that the galaxies which aren't expanding today because gravity doesn't? No, they won't expand. They won't expand. What will happen is the few galaxies which are close enough to us and which happen just by accident more or less to be falling toward us. There's one or two galaxies which are actually falling toward us. That's a sort of, when we say that everything uniformly expands, of course it uniformly expands plus or minus little bits of fluctuation. It happens that there's a little bit of fluctuation in our neighborhood which happened to cause the Andromeda galaxy to be somewhat moving toward us. So us and the Andromeda are not going to separate from each other. But if you go out a few more galaxies, they're certainly moving with the flow. Anything which is moving with the flow will eventually depart. It will just grow and separate and separate and separate. And after a few hundred billion years, all that will be left for us to see is us, our galaxy, and that's it. We will be living in a world where we'll be able to see out to enormously large distances but you won't see anything. Astronomers who were born at that time and don't have a good record of history will come to the amazing conclusion that they are alone in the universe. Yeah. Doesn't that say that you eventually exceed the gravitational attraction because our galaxy is gravitationally bound to a group? Yeah. And if the group expands then... No, no, no. If the group expands then we're not gravitationally bound. Right. So when you said all we could see is our galaxy. And the Andromeda. Which by that time probably will have crashed into us and formed some sort of single structure. Are each of the clusters gravitationally bound to each other also? Correct. Yeah. So... No, no, no, no, no. It depends on what you mean by a cluster. If you look at a few galaxies out, not very many, just a few, one or two more, they're moving away from us. Even though they form what's called a cluster. No, they're not gravitationally bound. No, they're not gravitationally bound. Gravitationally bound means that they won't separate. So the measured gravitationally bound depends on this expansion rate? Yeah, but not very much. But not hardly at all. Hardly at all. For nearby things, if they're gravitationally bound, can just mean whether... Just forget this expansion and just ask whether they're moving relative to each other with less than or greater than the escape velocity. That's what gravity... Now, it happens, there's a more or less accidental fact that the Andromeda happens to be moving toward us. A few galaxies out, they're moving away from us. And they will continue to move away from us. They're moving away from us with greater than the escape velocity. And they'll continue. Yeah, it's a big galaxy. The gravitational constant is a constant that never changes in time. But the outward force due to the vacuum energy is a function of time. It grows... It grows with distance. It grows linearly with distance. It grows with distance, but not with time. No, not with time. The acceptance of far as things separate with time. Two objects will separate with time. The distance will become larger. And for that reason, the repulsive force will increase. But the repulsive force does not increase the... OK, you have a small repulsive... Where is the repulsive force now? And over a period of billions of years, the repulsive force... It seems to me that equation is sort of cumulative. There's a bigger amount of repulsive force. That's only because things get further away from each other. But a galaxy won't get farther away from each other. If the galaxy... If two galaxies are at the same distance at a later time as they are now, the component of the little repulsion from the vacuum energy will be the same. What I was asking is that the net gravitational force is G minus the expansion force. And if the force, the tendency towards the band increases, it wouldn't be that effect be that G very gradually decreases. That doesn't make sense. What are you saying is that the force is a function of distance, not a function of... It's a function of distance, not a function of time. If you have two objects, if you have two objects at distance r between them at any time, doesn't matter what time, there are two components to the force. One of them is G times M1, M2 divided by r squared. And then there's another force on each of them, there's another force which is proportional to... It has a G in it, but mainly it's proportional to and repulsive. This is minus meaning attractive, plus a repulsive force proportional to r itself to the distance between them. Coefficient doesn't change. Coefficient don't change. What does change is the distance between things. So as the distance grows, this will become less important, this will become more important. But you're going to ask, where does this become important relative to this? The answer is when the distance scales are pretty cosmological. You know, five billion light years or something like that. The crossover, the coefficient of this piece is so small that the crossover between the normal gravitational attraction and the bit of repulsion is on the billions of light years scale. So that's why you can't measure it by an laboratory. So it's not just a function of expanding space that is represented as a repulsive force. It's actually... It is the expanding space, but you can fake it by a force which is pushing everything else apart from each other. It's a slight fake, but it sort of works the right way. Yeah. Okay. We've now... Now, does that kind of... I think satisfies my curiosity here that in the early universe these galaxies were gravitational bound and the expansion has now taken them to where the vacuum expansion, this other factor, are. In the early universe they were not gravitation. Yes, it's crossed over. So the gravitational attraction field is small and below. When they get far enough apart, then the repulsion becomes more important than the attraction. Right. So in the early universe they were gravitational. Yeah, this was not important in the early universe. Right. That's another statement. It's two different statements to say that they're gravitationally bound and that the important force was the ordinary gravitational force. If we shoot a rocket off the surface of the Earth at one and a half times the escape velocity, it is not gravitationally bound. But still the most important force on it is the ordinary gravitational force. These are two different statements. One statement, gravitational bound, means it's going to come back. The other statement is this may be more important than this in the early universe than it was, but it doesn't mean that the galaxies were gravitationally bound. They were still moving apart from each other with greater than the escape velocity. For the most part, I mean, pretty much. The galaxies were shot out pretty much greater than the escape velocity between them. Roughly, very close to the escape velocity, very, very close to the escape velocity, extremely close to the escape velocity. And then they either turn around and fall back in or they accelerate away from each other, depending on whether there's this kind of new energy. Okay, let's take a little break. And when we come back, I will tell you more about the geometry of this kind of space and why it has horizons, what it means for it to have horizons. So now that we know that in a couple of hundred billion years, the universe is going to be described. And it's getting that way pretty quickly, actually. On the scale of tens of billions of years, we're getting there pretty fast. We're getting to a condition where the geometry of space-time is described by a metric. And the metric is e to the 2HT, that's a squared, times dx squared. And again, dx squared stands for dx squared plus dy squared plus dz squared, and this is the scale factor. This space-time, it's a space-time, it has a geometric property, it has a name. Anybody know what the name is? The sitter space, the sitter space, dE. It's the sitter space. It should be called the sitter space-time, obviously, but it's called the sitter space. And we'd like to examine its properties a little bit. Not because we're going to be sending rocket ships out there to explore it. That's going to take a while, but just for the intellectual value of understanding the space-time that we live in. That's pretty weird. It's pretty weird. Of course, it looks weird, but it's weirder than it looks. It's weird in that it has horizons. Before we get into horizons of the sitter space, let's just talk for a moment about what horizons are in general. And I'll give, as the illustration, the horizon of a black hole. Remember, we worked out what a black hole formation, a black hole which is created by an infalling collection of material, what it looks like, and we drew a pen. Remember the Penrose diagram? Penrose diagram looked like that. Before we do that, we're going to use Penrose diagrams, because it's helpful. But let's first draw the Penrose diagram of ordinary flat space. There's ordinary flat space. There's points far away. It's spatial infinity. Here's time infinity. Light rays come in from here, and they go out from here, and all that sort of stuff. And that's just empty space. A black hole space-time looks like this. Oh, one other point. Imagine an object, a massive object, not a photon, a massive object in this space. How does a massive object move through space-time? Well, all massive objects wind up at the same place. They were massless objects. Photons go out the sides here, moving with a speed of light. Massive objects, they all go to this point up here. Now, that's not really a point. It's not really a point. It's a whole big space-time that's been smooshed onto the diagram to look small. Obviously, if we wanted to take all of space-time and squeeze it onto the blackboard, we're going to have to distort it pretty badly. And it's very distorted up here, in that there's an infinite amount of space and time up in this corner. Likewise here, likewise here, in fact, on all of the edges. And all of the massive objects eventually wind up at this corner. That's not to say that they get close together. It's just to say that this is a very big place up here. Okay. Now, let's think of the massive objects in the black hole space. In the black hole space, you can have one of two things happening. The massive object can fall into the horizon and hit the singularity. Let's avoid that. Let's avoid that. Let's talk about the objects which don't fall into the singularity. Where do they go? They go to here. They all do. Again, that doesn't mean that they literally get close together. This just means there's a lot of space time out there. All of these people wind up. Let's draw the horizon. The horizon's over here. Okay. Now, let's ask what these people here can see as they look back into the past. They can see only those things which are inside their backward light cone. The backward light cone means take all the light rays which can get to this point over here. All the light rays come in from here and take that region. That's the region that he over here can be aware of. When he looks back, he can only see the things inside his light cone. Same for this guy over here. Same for this observer. They can see the things in their backward light cone. Now, how much can they see given arbitrary amounts of time? Given arbitrary amounts of time, they can look back from later and later times. And eventually, they can see all of this blue region here. What they cannot see is what's in here. A horizon is by definition the place which separates the region that these observers can see from where they can't see. Given enough time, you draw the backward light cone of a very future point along that observer's trajectory, and that is the region he can see. Everything on the other side is behind the horizon, and the horizon is the separation between them. Here, there's no horizon. An observer way up here gets back, looks back, and sees everything. Sees the entire space-time. No horizon. Here, there's a horizon. So, I just remind you of that to tell you what a horizon is. We're going to take this metric, and we're going to do a little bit of mathematics on it. It's not hard mathematics. It's pretty easy mathematics. It's not particularly abstract, but it helps us visualize the space-time. What we're going to do with it is we're going, I'm not going to try to take all of space and squeeze it onto the blackboard. I'm going to leave space infinite. But what I'm going to take do is take the time axis, and I'm going to smush it. I'm going to make a coordinate transformation of time so that I can put the entire time from minus infinity, well, not from minus infinity, minus infinity will be deep down in the floor, but plus infinity will be in some finite place here, because we're going to want to study what this world is like at late times, and I'd like to get the whole infinite future onto the blackboard. Not the whole infinite future of space, but the whole infinite future of time. So, we're going to make a coordinate transformation, and we're going to do this in a way that makes light rays move on 45-degree angles in this space. It's always good to do that. Always get the light rays moving at 45-degree angles, and it's easy to see what can communicate with what. All right, so here's the trick. We're going to change coordinates. We're going to introduce a new coordinate. I'm going to call it capital T, and it's a function of the old T, and it's going to have the property. Here's the property it's going to have. It's going to say such that dT squared, what appears here, is going to be equal to e to the 2HT, exactly what appears here, times d capital T squared. This is the definition of capital T. It's a new time coordinate, and it's related to the old time coordinate by a pretty simple equation. We're going to solve it. And then we're going to rewrite the metric with the new time in place of the old time, and the metric will have a nice appearance that will allow us to explore it very nicely. Okay, how do I solve for big T as a function of little t? The first thing I do is take the square root of both sides of the equation. dT is equal to e to the hT d capital T. And now divide by e to the hT. What do I get on the left-hand side if I divide by e to the hT? e to the minus hT, right? So I'll put an e to the minus hT here, and now we have dT. So all we have to do to find the relationship between big T and little t is to integrate. The integral on the right-hand side just gives us T. What about the left-hand side? Anybody can do that integral? Dime for anybody? One over h. One over h. Minus sign? Minus sign. e to the minus hT, right? Plus a constant, but we can fix the constant by just shifting T here. Right, there's a constant, but I can shift the constant. Suppose I put a constant. There's a constant. Now I just redefine T so that it's T plus C. It doesn't make any difference, right? Okay, now notice first of all capital T is negative. Let's just think about what happens to it. This one over h here is not so important. Let's just think about what happens to it. What is it like at very remote past times? Let's say when little t here is extremely negative, what happens to e to the minus hT? It becomes very big. If t is negative, and we have e to the minus, then this becomes e to a positive number, this gets very big, but there's a negative sign out here. So that means the remote past where little t is very, very negative is also the remote past from the point of view of big t. When little t gets very, very negative, so does big t. All right? So in both variables, the remote past is way down through the floor and the basement and even deeper. What about the remote future? What happens in the remote future to capital T? The remote future is when little t is very, very big. What happens to this when little t is very big? It goes to zero. So big t goes to zero in the remote future. Now we can draw it. The time axis, let's draw the time, the big t axis, the big t axis goes into the very remote past in the negative region, deep past. But where is the infinite future? The infinite future is right at capital T equals zero. Capital T equals zero. That also is little t equals infinity. So big t runs from minus infinity up to zero. That's one way of getting not the whole geometry on the blackboard, but getting the remote future onto the blackboard. That was a goal. Okay, let's see if we can rewrite the metric. Let's see if we can rewrite the metric in terms of big t instead of little t. So here we have, let's, little t squared. What's the relation between little dt here? Little dt, here it is. Little dt is equal to e to the big ht times big t, right? Good. I just multiplied by e to the ht to get it over on the side over here. And now we come to this metric over here. We have minus dt squared. So this is equal then to minus dt squared, but that's e to the 2ht, the big t squared. And then we have plus e to the 2ht dx squared. In other words, I've engineered it so that the e to the 2ht would come on the outside. It would be equal to e to the 2ht times the common Minkowski space metric. This would be the metric of ordinary flat spacetime minus dt squared plus dx squared plus dy squared plus dz squared, but it has this factor in front of it. What can we do with it? I want to re-express it in terms of capital T. So let's do that. Let's first multiply by h. e to the minus ht equals h times big t. All I did was multiply by h. And now let's square it. If I square it, the minus sign goes away and I get e to the minus 2ht. And on this side, I get h squared t squared. And finally, I take one over it so that e to the 2ht is equal to 1 over h squared t squared. Go through it, check the arithmetic. You'll find it works that way. And now we can write this metric just by substituting for e to the 2ht, 1 over h squared t squared. So let's put that there. In fact, let's write it over here. The s squared is equal to 1 over h squared t squared times minus dt squared plus dx squared. Now this is a long story, nice simple result, but the main point of it, the main point of it is that we've written the metric as an ordinary Minkowski space metric times a function with the same function appearing multiplying all of this. How do light rays move in this geometry? What's the rule for a light ray? Ds is zero. Right, light rays move on paths of zero proper time. What does that say? That says that you can forget this factor altogether when you're thinking about a light ray. If you want to know how a light ray moves, you don't need to know about this factor. All you need is this factor. Right? Everybody see that? If you want to know how a light ray moves, it moves exactly as it would if this were just ordinary flat spacetime. How do light rays move in flat spacetime? On straight lines tilted at a 45 degree axis to the vertical. We now know in this geometry how light rays move. Let's draw the geometry. Here it goes. It starts out deep in the past. We can't go down that deep into the basement, but it ends. This is space. Space now goes this way and this way. It ends at capital T equals zero. How can it end at capital T equals zero? Little t went on and on forever. Well, it's just that the metric, when big t gets small, the metric here blows up. It gets big. So that means there's a lot of time in here. The metrical distance between neighboring points gets very large when t goes to zero. So it's highly distorted. I mean, it's a highly distorted geometry. The one advantage of it is that light rays move on 45 degree axes. So we now know how light rays move. They move like that. If they're moving out of the blackboard, they also move on 45 degrees, but I'm not going to try to draw it. I can't draw it. All right, good. Now let's ask what an observer can see. Imagine an observer in this world. I'll make them blue. Here she is. She's standing still in her own reference frame. Time is going on. More and more and more time is unfolding. And eventually, she gets up to the top. Now how long does that take according to her watch? Forever. Forever, right? It takes forever for that to happen. But on this diagram, on this formal mathematical abstract description of this interspace, the terminal point is over there. That's the end of her existence. It doesn't bother her at all because her clock has gone off to infinity. Let's ask what she can see. What she can see is she looks back. All right, let's draw that in red. Here she is right over here. That's Alice. And Alice looks back and sees everything that she can see within her light cone. Remember, the whole purpose of this coordinate transformation was to make it easy to follow light rays, 45 degrees. If we want more dimensions, we'll have to extend this out to make a cone out of it. But you can see all of the main features from here. She can see everything behind her here. But if she waits a little while, she sees more. And if she waits a little longer, she sees more. But no matter how long she waits, the most that she can ever see is this. What about things out here? Nothing. If there's somebody out here, she can never know about it in principle. She has a horizon. The red line out here is her event horizon. It is her private event horizon. Other people have different event horizons. But this is her private event horizon. She cannot see anything out beyond it. In principle, as a matter of real principle, she cannot, at least by classical relativistic reasoning, by classical light reasoning and so forth, she cannot see anything. All kinds of things could be going on there. Her friend Bob might have originated over here, and she might have been able to see Bob in the past. From here, she could even, Bob is moving, is Bob. Alice, Bob. In the past, Alice was able to send messages to Bob. Bob was able to send messages to Alice. And now Bob passes Alice's horizon. From there on in, Bob cannot send the message to Alice. Bob may be able to see Alice, but Alice cannot see Bob. So from this point on, Bob has passed out of her horizon. He's gone. That's what's going to happen. Unless, of course, if she's holding hands, what does it look like if she's holding hands with Bob and she doesn't want to let him go? Yeah. Then it looks like this. Now, does this mean they really get closer and closer together? No, it doesn't. It doesn't because the metric is getting bigger and bigger out here. So the actual metrical distance between them, the actual proper distance between them can stay constant. And if it does stay constant, they would appear to join up here, but they're not really getting any closer together. So if they're bound together, if they're holding hands, this is what it looks like. And they can continuously forever and ever send messages back and forth. No limits. But if Bob lets go of Alice and goes with the flow, if he's, let's go implies, among other things, that he gets far enough away that the gravitational pull between them doesn't hold them together. If he lets go and passes outside her horizon, they simply fall out of causal contact. That's the expression for it. They've fallen out of causal contact. In this particular instance, Bob can send a message, sorry, can receive a message from Alice, but he can never send a message to Alice. Alice up here cannot send a message to Bob either, but Bob also has his own private event horizon. Let's make Bob's event horizon in green. Here's Bob's event horizon. Bob cannot see anything outside of his event horizon, so he can see Alice early on, but he can't see Alice after she passes out of her horizon. So he won't be able to see Alice at his time. Bob won't be able to see Alice at his time. That's right. Well, you can never see anything at the same time, period. Right. You can never see. The only thing you can never see is those things which are back inside your past light cone. So you never can see anybody at the same time. But once Bob has passed through here, Alice can never see him. If this really was flat space, if it really wasn't this peculiar, the sitterspace with this factor here, and then Alice would have to separate it, then Alice could keep going and look back at Bob. But Alice comes to the end of her world line over here, and simply cannot look back. So this is the character of the sitterspace. It has horizons. In fact, every single observer has his own private event horizon. In this case, Alice and Bob constitute one observer. And maybe we'll talk a little bit more about these event horizons next time. I'm getting a little tired, so I think I'll quit soon. But we apparently live in such a world. And in such a world, it's not Alice and Bob, but it's galaxies. Lots of galaxies. Here we are now. We can see about 10 to the 22 galaxies around us. As time goes on, they will pass through the horizon and pass out of, let's say we're Alice. They will pass out of Alice's horizon, and Alice will not be able to see them anymore. They will be gone. So as I said, Alice will be alone in the world for all practical purposes. Excuse me. Is there a theory that in the far distance, we would not be able to see any galaxies at all? That's right. That's right. They will have a... Galaxy, Alice, and Bob will... You might... Okay, let's ask whether Alice at a very, very late time here can see Bob back here. In a sense, no. It looks like he can, but in a sense, no, and I'll explain why. This is Alice here. Alice's time steps look like this. Many, many time steps here. Now, imagine that Bob sends her a signal, and that signal is going to come up to here. Alice is going to look back and look for Bob back here. Well, you say she can see him. But in fact, because her time clocks are so bunched up over here, it means the radiation that she sees from Bob is incredibly redshifted. That how many beats of Bob's clock gets squeezed into Alice's time scale here, one beat of her clock says stretched out over an enormous time scale from Alice's world. So when Alice looks back and sees Bob from way, way up here, he's enormously redshifted. You can't see very, very redshifted things. His wavelength, the wavelength of his light and the energy of his light will be so stretched out. So in fact, really, Alice has no chance of really seeing Bob over here. He's just been redshifted to hell and... Yeah, like a black hole. Like on the horizon. That's right. It's exactly the same as in the horizon of a black hole. So in that sense, at very, very late time, Alice will not be able to see any of the galaxies, except in this exceedingly redshifted sense, which is useless. She can't detect such low energy radiation, and she will just find herself alone. And so will Bob, all other galaxies having passed out of their horizon. Is it the same with the black hole, which it takes forever to reach the horizon? Yeah, yeah, but okay. It is the same. It is the same. Well, it is the same. Alice looks back. Has Bob crossed her... Where is Bob? Here's Bob over here. Has Bob crossed the horizon? No. Alice looks back. Has Bob crossed the horizon? No. Alice looks back. Has Bob crossed the horizon? No. She never sees him cross the horizon. But do objects falling into black hole become infinitely redshifted as well? Yes. Yes. It's exactly the same thing. It's exactly the same thing. Here's the black hole. Here's who stays outside. Let's put Alice outside. Yeah, let's put Alice outside. Let's keep the color coding. Blue? Blue for Alice? Alice is red. Bob is blue. No, here's Alice. Alice is blue. Bob is green. I don't know. Here's Alice. I usually send Alice into the black hole, but this time Alice gets to stay outside. And here's her clock. Her clock is going faster and faster and faster and faster and faster. It's not going faster and faster, but on this diagram, all her time intervals are bunched up, same as they are over here. And Bob, who is now green, is falling through the horizon. Alice looks back. Does she ever get to see Bob fall through? No. Bob's signals, his light waves or his waves of his hand that he sends out, appear to get slower and slower to her because they're drawn out over longer and longer time scales, which is another way of saying Alice becomes infinitely redshifted. The light from Alice becomes infinitely redshifted from Bob's perspective. So it's exactly the same thing. Yeah. So in Alice's real time, in her proper time, she's slowing down. She's going at her proper time, but on the scale, it's getting compressed, right? Well, on the scale, not on her scale. Well, she's just ticking along happily. No, that's Bob. That's Bob. You're right. That's Bob, yes. So Alice's scale time, right? Her proper time is still her proper time. So she looks back and she sees Bob slowing down. But she not only sees him slowing down, she sees his atom slowing down, she sees the radiation that's coming out. Yeah, she sees not only Bob slowing down, but she sees the atom slowing down, she sees the emission of photons getting slower and slower, the photons getting longer and longer wavelength, smaller and smaller frequency, and eventually his photons become so monumentally long wavelength that they can't be seen at all. The energy of them gets diminished to negligible. Yeah. How would she measure the temperature of the surface right above the horizon? Well, to do that, she has to send down a thermometer and make sure the thermometer doesn't fall into the black hole. So to keep it from falling through the black hole, she'd send down a thermometer, measure the temperature, pull it back up and see what it recorded. But that's not what she's doing with Bob. And the same incidentally, same thing here. If Alice, so here would be the experiment. Alice has a very long rope, a long cable. On the end of the cable is a thermometer. She throws it out hard enough that it's beyond the escape velocity from a local cluster of galaxies and waits a long time for it to migrate out toward the horizon, her horizon. But she doesn't let it go through the horizon. She then pulls it back in. And after she's pulled it back in, she sees what the temperature that it recorded was. She will also see a high temperature, but we'll come to that. Horizons are always hot in a sense, but we'll come to that. This is so far the classical description of the sitter space. Full of horizons. There's horizons everywhere. No matter where Alice is, there's somebody, she's passing through somebody's horizon. So wherever you are, you're passing through somebody's horizon. Say goodbye because you won't see them again. I'm trying to understand what it looks like in the untransformed, in the little thing. They're at the same distance. I'm going to leave that for you to do an exercise. It's an exercise to see what it looks like. No, it's a good exercise. Just undo what I did. I'm going to just say that my assumption is they're not moving apart. The x is not expanding there. No, no, no, no, no. The whole metric is blowing up, including the x parts of it. So the distance between points, if you keep the points at fixed coordinate separation, here Bob and Alice are at fixed x's. The distance between them is getting bigger and bigger. It goes to e to the infinity. That's right. The distance between them blows up when they get up to, right, that's another fact of the sitter space. Now, something you can work out is you can compare the distances between these two points, or the distance between these two points, between the distance comparing. How far apart are these two? How far apart are these two? What do you think the answer is? Are they growing? Are they shrinking? No, it's staying the same. Staying the same. If you draw a horizontal line, work it out. Here's the metric. Draw a horizontal line here and ask for the distance, let's say, from the center to the horizon over here. And compare it with the distance from the center to the horizon over here. Compare it with the distance from the center to the horizon over here. You'll find they're all the same. The coordinate distance is shrinking, but the metric is increasing, and the distance to the horizon is always the same. That is connected with the fact that the Hubble constant is constant, and so the distance to where the speed of light is, where the things are moving away with the speed of light is constant. So Alice doesn't see things changing with time. Everything stays the same. She just sees a horizon around her, and that horizon stays at a fixed distance from her. So would she be seeing the projection of all things in the past be projected onto that horizon space? Because they'll never go away, right? They'll never go away, but she sees it harder and harder to see them because they get redshifted. Now, that doesn't take into account the Hawking radiation coming from the horizon, but what you say is right. Okay, so assuming that we don't pass through each other's horizons, we'll meet here next week.
 Stanford University.  One thing that I'm confused about is vacuum energy.  I know what you said when you get the plank scale that explodes.  My question in particular, does electron positon,  positron hair production in annihilation actually occur,  or is that excluded by?  Actually occur means what?  That you can actually see out of the vacuum electrons and positrons coming out?  Let's say the measured vacuum density, if that would be a fair measure.  Electron, positron, virtual pair production.  Virtual pairs do contribute to the vacuum energy.  That doesn't mean that you should see electron, positron pairs popping out of nowhere.  It does not mean that.  It means they pop out and go back in a very short time,  subject to the uncertainty principle.  Quantum fluctuation and not real, genuine particles being produced.  It's much like the zero point energy of a harmonic oscillator.  Classically, the harmonic oscillator just sits still at the bottom of its potential well.  Quantum mechanically, it cannot both sit still and be at the bottom of the potential.  There's competition and it's resolved by a little bit of quantum fluctuation.  That quantum fluctuation has energy.  How much is it?  A half h bar omega.  It's always there.  Can't get rid of it.  It's the least energy that the oscillator can have.  And of course, that doesn't mean looking at the oscillator.  Most quantum oscillators are too small to see.  It is the ground state.  And as the ground state, it means it's the state in which  nothing is happening to the extent that nothing can happen.  Same is true with electron, positron pairs.  Same is true with virtual photons.  And in fact, it's a closely related phenomenon.  The electromagnetic field, for example, in a cavity,  electromagnetic field in a cavity is just a collection of harmonic oscillators.  Each oscillator has a little bit of zero point energy.  Add them all up.  You call that the vacuum energy.  I was saving that for a little bit later because it plays an important role in cosmology.  Are you also going to talk again about the 120 orders of magnitude?  No, I think I won't spend a lot of time on that.  I'm not going to mention it, but no, I thought tonight, in any case,  we mainly concentrate on cosmic horizons.  What a horizon is, why there are horizons, and a little bit about the cosmology,  the equations of cosmology, how they lead to an accelerated expanding universe  and why the accelerated expanding universe has event horizons.  I thought that would be a good thing to do tonight.  Since we spent a lot of time learning about what a horizon is,  we're not quite finished learning what a horizon is.  No four minutes.  You mentioned last week, I guess, was the ultraviolet and red relationship,  and you said you were going to be getting to that tonight?  Well, we can talk about it.  Maybe we'll start with that.  Maybe. It's not really on my list of things to do tonight.  Maybe we'll get to it, though, if I run out of things.  Okay.  Wait three minutes and sit and think.  That's a question.  Let's lay it to the question back there about vacuum energy.  If there's a vacuum, there is a positive energy density.  Why would that have some kind of refractive effect on it?  Refractive.  Optically refractive effect.  What is refractive?  Refractive, remind me which is refraction, reflection,  and respect the speed of light.  Speed of light, for the speed of light, for example,  would be ever so slightly less than the quantum vacuum  at different free space.  Yeah.  The vacuum energy, it's a good question.  It's a good question.  Most energy that you think about,  if you have that energy present,  it creates a situation where the world is not Lorenzen variant.  I don't mean that the theory is not Lorenzen variant.  I mean that the configuration of the world is not Lorenzen variant.  What does it mean?  What exactly does that mean?  Let's first talk about something simpler.  Translation invariance.  Every place is like every other place, right?  No, it's not true.  Stanford University is not like Berkeley.  The Earth is not like the Sun,  and it's certainly not like interstellar space.  So what do we mean when we say the world has translation invariance?  We don't mean that the configuration of the world,  every place looks the same.  We mean if we took the whole thing and moved it,  it would look the same.  But moving it means moving us,  moving Berkeley, moving everything else.  Now, in the same way, you talk about Lorenzen variance,  is the world Lorenzen variant?  No, I'll tell you why in a minute.  Are the equations of physics Lorenzen variant?  Yes.  So what does it mean for the world not to be Lorenzen variant?  Well, it means that for whatever reason,  the stuff in the universe does not look the same  from every frame of reference.  If I were to whiz by you at a thousand miles a second,  you would look different to me than if I was standing still.  So stuff in the world, stuff in the world,  breaks the symmetries that we talk about.  They create special locations.  They create special directions.  The equations of physics are supposed to be rotationally invariant.  That means the same, no matter how we rotate.  True enough, the equations of physics are rotationally invariant,  but is this room rotationally invariant?  Certainly not.  If I look that way, I see you.  If I look that way, I see something else.  So the presence of stuff breaks symmetries,  and presence of stuff breaks Lorenzen variance.  The world is not Lorenzen variant by virtue of the fact that there's stuff in it.  Now, most of the times the stuff, energy and so forth,  as I said, is not Lorenzen variant,  and the result of that is in particular a result on the motion of light waves.  Light waves moving through materials have different velocity  than they have moving through empty space.  The rule about the speed of light is that light always travels  with the same velocity in empty space,  but that's another way of saying that empty space is Lorenzen variant.  Empty space is Lorenzen variant.  Empty space is translation invariant, it's rotation invariant.  Empty space has no preferred frames of reference.  And the statement that light always travels with the same velocity  is a statement about a Lorenzen variant world  where there is no preferred frame of reference.  When there's a preferred frame of reference due to the presence of material,  light can travel at any velocity lower than the speed of light,  than the usual speed of light.  Now, vacuum energy is very special.  Vacuum energy is the one exceptional situation  where the presence of that energy is Lorenzen variant.  It does not pick out a reference frame.  If you weren't here and the worm wasn't here  and I was in absolutely empty space,  nevertheless, there would be vacuum energy in this room.  Vacuum energy has a certain value, whatever its numerical value is,  a certain number of joules per cubic meter, very, very tiny,  but nevertheless, finite.  And so you might think, well, that's like a material being here,  it's like a stuff being here,  and a stuff being here would define a frame of reference  where the stuff would be at rest.  Vacuum energy is very special that way.  It does not define a frame of reference.  I could detect the vacuum energy with a vacuum energy detector.  I won't tell you how to build one.  I'll tell you how to build one in a little while.  We'll talk about how you build a vacuum energy detector.  I could detect the vacuum energy and see how much is here.  Or we get some number.  I would then go whizzing by as fast as I can go  and do the same experiment.  If it was ordinary material,  I would discover that the energy changes,  the energy that material changes with my velocity.  In particular, if I whizz past a gas,  then of course the gas looks higher energy.  Why?  Because all the molecules appear to be moving past me.  Vacuum energy is the one special case  where my vacuum energy detector will give the same answer  no matter how I'm moving.  So vacuum energy is the special case where it's present,  does not pick out a special frame of reference,  does not violate Lorentz invariance,  and it's a special case where light will move  with the speed of light independently of the observer.  Now moving with the speed of light means something.  It means when the light ray passes you,  passes right in front of your nose,  and you have your clocks and your detectors locally,  right near you, you measure the same velocity of light.  Because of the expansion of the universe and so forth,  it becomes a more problematic question of whether light  moves relative to us far, far away when the light is far away from us.  Does it move with the speed of light?  And we're going to talk about that.  The answer is no, not in general.  It moves relative to local observers where that light is passing by them  with the same speed of light,  but it doesn't necessarily move relative to us with the speed of light.  Faster or slower?  In this discussion, is vacuum energy synonymous with cosmological?  Yes, absolutely.  All right, so let's begin there.  Why don't we begin there?  If we take the universe and fill it with full of particles,  then let's say ordinary particles, protons, fill it up.  I don't mean fill it up so that every proton is against every other proton.  I just mean create some uniform density of protons.  And of course, the universe is something like that.  It's not so different.  There's a more or less uniform energy density throughout all of space.  It tends to be clustered into stars and stuff like that,  but it's still bigger than superclusters of galaxies.  It's homogeneously distributed, rather diffuse,  something if I remember, something like about 50 protons per cubic meter on the average.  And there's some energy density.  Let's give that energy density a name.  Standard terminology for a density is rho.  And now that could stand for the mass density,  but it's not being the same thing.  All right, so that's the mass density.  That's consistent with a bunch of particles.  Now what happens, and it's the number of particles per unit volume,  or the number of particles times the mass of each particle  divided by a volume of space.  Take a volume of space.  The density of energy is the mass of each particle,  let's suppose there's only one kind of particle.  Particles are standing still, and not photons.  Let's not get involved with photons right now.  They're just protons standing still, protons, neutrons, electrons, atoms.  The mass of an individual particle times the number of particles in a volume  divided by the volume, that's the density.  Now what happens to that density as the universe begins to expand?  Well, the number of particles doesn't change,  at least not if they're ordinary particles.  The number of particles doesn't change,  their mass doesn't change.  What does change is the volume of a region of space  occupied by those particles.  So we take some particles that started out in a region of space,  a square on a side, a hundred light years on a side,  a hundred light years, a hundred light years the other way too.  It has a volume of a million cubic light years,  which is some number in cubic meters.  That's the volume that goes in here.  The number of particles stays the same,  and now the universe expands, everything expands with it on  sufficiently big scales.  The volume of space, let's say over some period of time,  doubles.  When the volume of space doubles, what?  I'm sorry, I meant the radius of, I didn't mean the volume.  When the radius of this cube, the linear dimension of the cube,  doubles, what happens to the density?  Well, when the radius doubles, the volume gets multiplied by  eight, and the density becomes one-eighth of what it started.  OK?  So ordinary material has the behavior,  and this is true for photons, for anything,  not the exact formula is a little bit different for photons,  but the general rule is that as the universe expands,  the energy density dilutes.  It decreases.  The density of energy decreases just because everything separates.  Again, vacuum energy is the exception.  Vacuum energy is the exception.  Vacuum energy has the property, vacuum energy density.  Vacuum energy density has the property that as the universe  expands, as the volume of space expands,  the amount of, the density of energy does not change.  The density, not the amount in the box,  but the amount per cubic meter, always stays the same.  If the universe were to expand by a factor of a hundred  in every direction, the volume of the region of space  would go up by a million.  Nevertheless, the density of vacuum energy in this room  would stay exactly the same.  So it's a property of empty space, which just doesn't change  as time goes on.  It doesn't change as the observer moves past it.  It doesn't change when the observer rotates.  It's not clustered.  Uniformly filling space picks out no direction,  picks out no Lorentz frame, and picks out no special time  in the sense that as the universe expands, it doesn't change.  That's, energy of that type is called vacuum energy.  If it exists at all, and it does seem to exist.  What do we care?  Why should we care if the universe were filled with something  that we call vacuum energy?  We do experiments, and the only thing we're interested  in this room, if we're doing experiments in this room,  is differences of energy.  The zero point of energy really usually doesn't matter  for anything.  The thing which usually counts is differences of energy.  What's the energy difference between an excited atom  and an unexcited atom?  What's the difference of having of the energy with  or without the atom?  The energy of an atom counts, but that's because we can compare  it to the energy of a configuration without the atom.  But just empty space, who cares?  We could just put as much energy into an empty space  as we like as long as that energy cannot turn  into any other kind of energy.  As long as it doesn't come into the questions of differences  of energy, why should we care if there's something called  vacuum energy in the room?  Well, the answer is what I said is we don't care about the  zero of energy until gravity becomes important.  Once gravity becomes important, what's the source  of the gravitational field?  The source of the gravitational field is mass.  Mass is energy.  Vacuum energy gravitates.  It has a gravitational effect.  It has an effect on spacetime.  And it has an effect on spacetime, which is different  than if it weren't there.  So let's talk about, well, the first question is,  what's the origin of vacuum energy?  And as I said, or somebody asked me earlier, vacuum energy  is, it wasn't quite asked to me in these terms,  but vacuum energy is a feature of quantum field theory.  Quantum field theory, quantum electrodynamics,  quantum, the standard model of particle physics,  all quantum field theories have vacuum energy.  It's just a zero point energy of the oscillating fields,  if you like.  There's a contribution to it from the electromagnetic field.  There's a contribution to it from the electron-positron field.  There's a contribution to it from quark fields,  even if there are no quarks present.  The point is, you don't need to have anything really present  to have that energy.  What does it do to?  It's due to virtual creation and annihilation.  Virtual, remember, means that it happens so fast  that a little bit of energy that is created and annihilated  satisfies the uncertainty principle.  If you create some particles out of nothing,  then they can only last for a very short time because of the  energy time uncertainty principle.  Let's leave it at that.  But the net result is that there is energy density called rho.  And it fills up space.  It's everywhere.  It's Lorentz invariant.  It doesn't matter how you're moving and so forth.  Vacuum energy.  Let's call it rho vacuum.  Now, what does it do?  It has an effect on the right-hand side of Einstein's equations.  Let me remind you what Einstein's equations are.  The details of them we don't need to know.  I will write down as much detail as you need to know,  which is very little.  But let me just remind you what Einstein's equations say.  On the left-hand side of Einstein's equations is a tensor  called g mu nu.  g stands for gravity, I think.  It's a tensor.  And it's made up out of the metric components of space,  the metric of space.  It's made up out of the metric tensor,  the metric tensor is little g mu nu of x.  It's on the left-hand side.  It's got to do with curvature.  It's got to do with derivatives of the metric,  Christoffel symbols, all sorts of awful stuff.  And by the time you write it down, it takes up what g mu nu is.  It takes up a fair piece of space.  But it's got to do with, oh, I said it has to do with gravity.  I'm not sure it has to do with gravity.  I think maybe it has to do with geometry, the g.  It's the geometric side of Einstein's equations.  It says some property of the geometry of space-time,  this is space-time geometry, not the geometry of space alone.  And on the right-hand side, in other words,  it's the gravitational field.  Gravitational field, geometric field,  however you want to think about it.  Einstein's field equations are that this on the left-hand side  is equal to something, a numerical constant,  but I'm not interested in numerical constant.  It's 8 pi over 3, but not important.  On the right-hand side is the source of the gravitational field.  And the source of the gravitational field is energy and momentum.  Energy, also known as mass, and the motion of masses,  which is also known as momentum.  So on the right-hand side, there's another tensor.  It's called T mu nu.  I don't know why it's called T.  But it's a thing which is made up out of particles,  out of ordinary fields, not the gravitational field,  the electric field, the magnetic field.  And if you know what this is, you know what the ordinary  energy density is, the density of particles and so forth.  That's Einstein's field equation.  And on the right-hand side is all of various kinds of energy,  both those which dilute particles, ordinary stuff, light,  the stuff which dilutes when the universe expands,  but also the stuff which doesn't dilute, the vacuum energy.  So that's the reason we have to know about vacuum energy,  to know how the geometry of spacetime behaves.  If there's only vacuum energy here, I'll tell you what,  I'm going to write these equations, I'm going to write them out in full,  but for one very, very special case.  I told you that the left-hand side is made up out of the metric tensor.  And let's take a very, very special case for what geometry,  for what spacetime geometry is like.  In some sense, it's the simplest geometry that's not just static space  just sitting there.  I think we actually wrote it out last time.  It was the expanding, uniformly expanding space that I told you to think of  as an infinite rubber sheet, where the infinite rubber sheet is being stretched  so that the coordinate marks on it, we take a rubber sheet,  we mark it with coordinate marks, x equals one, x equals two,  x equals three, x equals four, and then we start to stretch it.  The actual distance between neighboring points increases with time,  and that's represented by writing a metric, the metric ds squared,  that's the spacetime distance between pairs of points, neighboring pairs of points,  just as it would be in ordinary Minkowski space,  we start with a minus, this is pen, this pen seems weak,  minus dT squared, that's all we would have in ordinary Minkowski space,  meaning flat spacetime,  and then what we would have in Minkowski space would be plus dx squared.  Now, dx squared here stands for dx squared plus dy squared plus dz squared,  which I won't bother writing,  it stands for the ordinary Euclidean metric of ordinary space.  There would be some speeds of light in here,  if I kept the speed of light around but I'm not,  but that's not the expanding spacetime,  the expanding spacetime you introduce one more thing,  a thing called a scale factor, A of t.  A of t is a function of time, is not a function of space,  and it tells you, oops, I'm sorry, this should be A squared of t,  A squared of t.  What it says is that if you take two points,  let's say x equals zero and x equals one,  what's the distance between them?  Well, it's one unit.  No, it's not one unit because x doesn't actually measure physical distance,  it's this which measures physical distance,  the actual distance between them at any instant of time is A.  And as A changes with time,  the distance between these marked points on the Rubber sheet world,  the Rubber sheet world, x equals zero,  x equals one, the marked points separate, so A grows with time.  It doesn't matter if they're separated by one unit, two units,  if we want to separate them by, let's say this is x equals seven,  x equals 13,  then the distance between them is delta x,  that's 13 minus seven, which is four, thank you.  13 minus seven times A, A of t, four.  13 minus seven is A, is it four?  No, 13 minus seven is six, thank you.  Right, okay, that's the meaning of this metric.  They don't have to be on the x-axis,  they can be off the x-axis, but you get the idea.  Okay, this is a non-trivial geometry.  It has space-time curvature, unless A is constant,  then it's just flat space, good old flat space.  If A is varying with time doing things,  maybe just expanding, that's enough to make the space-time not flat.  Space may be flat, but space-time is a different story.  So space-time, expanding space-time is a new kind of geometry that's good.  Okay, now let's write down Einstein's equations.  What do Einstein's equations have to do with?  They have to do with how A varies with time.  A is the only thing in this geometry that's not known.  It's the only degree of freedom for this kind of geometry.  For a geometry of this type, all of the geometry is coded  in the properties of A of t, which means A is a function of time.  So Einstein's field equations must be simply equations  for how A varies with time, and they are, and here's what they say.  They say that the A by dt squared is equal.  I'm going to put in all the factors now.  8 pi over 3 times Newton's constant  times the energy density of matter.  The assumption here is that the energy density of matter  is completely uniform in space.  You could have such a thing if space was filled up  with a uniform radiation bath all at the same temperature.  You could have such a thing if the universe were filled up  with atoms at a uniform density.  So we don't have to write rho as a function of x  because rho is not a function of x, but of course we do have to  write rho as a function of time.  So rho is some function of time, but what is rho as a function  of time?  This, if we, oh, I'm sorry, there's a 1 over A squared here.  It's 1 over A squared. It's A dot over A squared.  Do you remember from last time what A dot over A is?  It's the Hubble constant, but let's leave it there.  A dot over A squared, that's it.  OK, now this equation also has another significance in general  relativity.  It's one which confuses people.  I get zillions of emails.  I get so many emails about this particular point that I have got,  can't keep up with them, and I don't even try anymore.  This equation is also an equation for energy conservation.  It expresses energy conservation in general relativity.  The right-hand side is all of the energy of ordinary stuff,  but it does not include the energy of the gravitational field.  The gravitational field, because it's varying with time, A of t,  also has energy, and that energy is a kind of kinetic energy.  It's a kinetic energy of motion or a kinetic energy of the time  dependence of the metric of spacetime.  What is that energy?  That energy is simply the energy of the varying gravitation.  It's called the varying scale factor.  Energy of the varying scale factor, A is called the scale factor.  One odd thing about it is that it's negative in general relativity.  Minus A dot over A squared is probably a factor of 3 over 8 pi g.  3 over 8 pi g, I think.  It's not the number that's important.  It's important that in the formal structure,  in the mathematical structure of general relativity,  gravity has energy, and some piece of the gravitational energy  is negative, and it corresponds to a negative kind of kinetic  energy of the gravitational field.  So with this equation, it can be written as minus this plus this is equal to zero.  Minus this plus this is equal to zero.  Well, zero is conserved.  Zero doesn't change with time.  Zero is zero.  It doesn't change with time.  So among other things, among the many things that this says,  that energy doesn't change with time.  It also says the total energy of the world is zero.  But it includes this piece coming from expanding space time,  which you're not used to counting.  When we count the energy of this room,  we don't normally think about the fact that the universe is expanding.  We just ignore it, and we say the energy in the room is positive.  Okay.  Good.  Let's put it back the way that it was.  This equals this.  Let's go a little further now.  Let's ask what row of T is like.  What the time dependence of the energy density in the world is like.  So let's come back to this box.  There's not a real box.  I've taken a region of space and just drawn a mathematical box.  The mathematical box has a side delta x equals one on each side.  How much energy is in it?  Well, it's the number of particles inside it,  number of particles inside it, which is not going to change.  As the box expands, the particles move with it,  so that if this room was part of the expanding universe,  there would be a lot of particles or a bunch of protons.  The protons wouldn't expand,  but the distance between them would expand so as to keep the proton density uniform.  So there's n protons in there.  Each one has mass m.  That's how much total mass is in there.  What's the volume of space?  The volume of space is delta x times delta x times delta x.  That's one.  Times a cubed.  Why a cubed?  Because the actual length here is delta x times a.  So the side of the box here,  I've chosen a box whose side at any given time is a of t.  And so to find the density,  I have to divide it by a of t cubed.  The only important thing here  is that at any time,  if I know the energy,  this is the energy density,  if I know the energy at any given time,  then I know it at all times,  not because it stays the same,  but because it always scales like 1 over a of t cubed.  So we can say that rho of t then is some number,  we could call it rho zero,  it's some particular instant of time  divided by a of t cubed.  So now we can come back to this formula over here.  We can say, uh-huh, I now have an equation for a.  It's Einstein's equation for an expanding universe.  And what goes over here is some constant  rho naught over a cubed.  When we can solve this equation,  maybe we should solve it.  Maybe we should solve it.  It's not a difficult equation.  It's much easier to solve if we take all the constants,  8 pi g over 3 times rho naught,  and absorb them into another constant.  Let's just call that other constant,  let's just call it rho naught.  And thereby erase this.  OK, how do we solve this equation here?  Let's write out what it is.  I bet you can solve it.  I'll have you solve it.  But first I'll manipulate it a little bit.  This is the adt squared.  There's an a squared in the denominator,  but I'm going to multiply through  and put it on the other side.  So we multiply by a squared  and we get rho naught divided by a, is that right?  Right.  And now I take the square root of both sides,  rho naught,  divided by a, I take the square root of both sides,  square root of rho naught,  a to the one-half,  and I multiply so that it reads  a to the one-half times dA  equals square root of rho naught times dt.  a to the one-half on this side times dA,  or just we could write it this way.  Can you solve that?  Yep, say it again.  Right, what we're doing is integrating  both sides of this equation.  What Michael suggested is to integrate  both sides of this equation  on the right-hand side we get  square root of rho naught times time.  The integral of dt is just t.  And this one is what?  Three-halves, a to the one-half,  so let's integrate it.  That's three-halves.  No, two-thirds, two-thirds,  a to the three-halves.  Is that correct?  And now we know a is a function of time.  Forgetting all the numerical constants,  two, three, rho naught,  what it says is that a grows  like t to the two-thirds power.  The scale factor grows like t to the two-thirds power  because the energy density has a particular behavior.  That's cosmology. That's standard,  absolutely standard cosmological expansion.  A growing like t to the two-thirds.  Let's just draw it.  A growing, let's first draw a growing like t.  There's a growing like t.  A t.  t to the two-thirds is smaller.  It grows less rapidly than t itself.  This large t is smaller, and it looks like this.  Not quite a parabola, but it looks like that.  It slows down.  A uniform line like this would be an a,  which grows linearly with time.  It would be like a coordinate of a particle  with linear motion.  That's not what happens.  What happens is it turns over  and it appears to go slower and slower.  The universe decelerates in this kind of cosmology.  This is a decelerating cosmology.  What is it that causes the universe to decelerate?  Gravity is pulling everything.  Yeah.  Another way to think about it is stuff started out  moving away from each other, and gravity just grabbed  the hole of it and slowed it down.  And eventually it will get slower and slower.  OK, now let's ask what happens if we have a different kind  of energy on the right-hand side.  We could put in, just for fun, I'm not going to do it tonight,  we could put in radiation energy,  energy solely due to radiation as if there were no particles,  only photons in the universe.  That would correspond, you can work it out yourself,  that would correspond to putting A to the fourth here.  We don't have to go into that now, it's just a fact.  You can solve the equations and find out how the universe  would expand if it were radiation dominated.  The particular version that I gave you is particle dominated.  It's usually called matter dominated, matter as opposed  to radiation where the energy density decreases  like 1 over A cubed.  OK, but what happens now if there is vacuum energy,  if there really is vacuum energy?  Vacuum energy has the property that it never changes  under any circumstances, the density itself.  The density itself does not change under any circumstances.  We need to give it a name.  We'll put back, for the moment, we'll put back the 8 pi  G's and so forth.  Theory tells us that it doesn't change.  Quantum field theory tells us it doesn't change.  It tells us that it's a sort of universal constant that can be  calculated.  Now, does that mean that the universe that we know with  certainty that the universe is filled up with a kind of energy  which doesn't dilute as the universe expands?  Theory doesn't tell us that.  What tells us that is experiment.  And we'll talk about the experiment in a minute.  We'll talk about the observation in a minute.  Theory tells us only what to put on the right-hand side  for various different hypotheses about what the energy density  is.  If the energy density is vacuum energy in the sense of quantum  field theory, then it really is just constant.  So that means what goes on on the right-hand side here does not  vary with A. Does not vary as the universe expands.  And let's just call it row vacuum.  Let's call it the vacuum energy density.  Row vacuum is 8 pi g over 3, sorry,  A dot over A squared is 8 pi g over 3 times this constant value  of the energy density.  Now, first of all, if this is correct,  then one thing it says is that the Hubble constant,  A dot over A is the Hubble constant.  This is also H squared.  It tells us that H squared, or H also,  8 pi g over 3 times the vacuum energy,  that this doesn't change with time.  Remember what the Hubble constant is.  The Hubble constant is the relationship between distance  and velocity.  Velocity is Hubble constant times distance.  So it says for all time in this kind of world,  and this kind of world for all time,  the relationship between velocity and distance is unchanged  and always the same numerical value.  That incidentally would not be true in the other solution  we wrote down.  Normally, for other kinds of cosmology,  the Hubble constant decreases with time.  Here, it just stays constant.  An implication is that the distance out to which you have  to go in order to get to the speed of light,  where stuff is moving away from you with the speed of light,  that distance is always the same.  It's given by C equals H times D,  or equivalently C over H is equal to D.  C is a constant.  H is a constant.  The distance out to the place where things are receding  with the speed of light from us is fixed.  It doesn't change with time.  So that's one implication of what this means to say  that there's a constant value here.  But let's see now if we can now solve Einstein's equations  for an expanding universe.  Under the assumption that the only energy density now,  the only energy density is vacuum energy density.  So what do we have?  We have A dot over A squared is equal to 8 pi g over 3 times  rho, but that's just the number H squared.  Let's just call it H squared.  Now, I'm using H now not to mean definition.  It's not definition now.  Previously, A dot over A was the definition of the Hubble  constant.  Now, it's the numerical value of 8 pi g over 3 times rho v.  It's a number.  It's a number that we don't know how to calculate it.  We don't know the theory of it.  But it's a number.  H squared is now a number.  Everybody understand the difference between saying H squared  is A dot over A as a definition and H squared is A dot over A  squared as a numerical relationship.  Here, H is not defined this way.  It just is a number.  It's measured?  It is measured.  It is measured.  It's a measured number.  And it's not expected.  And there's evidence that it is not changing with time.  Right?  So it's a number.  How do we solve this equation?  Well, first of all, we take its square root A dot.  We multiply it by A. And it reads A dot is equal to H  times A. I think some fraction of you can solve this  equation.  Nobody can solve it?  Right.  A dot A is equal to E to the Ht.  This is the equation that tells us that the growth rate of A  is proportional to A itself, exponential growth with time.  A thing which changes by an amount proportional to its own  value exponentially grows.  And the growth factor, the time factor, the time constant  for this is H.  E to the Ht is the way this universe is growing.  Exponentially.  You can put a constant in front of it.  The constant doesn't matter much.  So this is an exponentially growing universe.  That's the result of vacuum energy.  Vacuum energy really is important.  And the way to detect it, well, I'll give you two ways to  detect it.  One is astronomical.  Measure the Hubble constant.  Measure the Hubble constant over a period of time and see  whether it's staying constant.  Or you can measure the scale factor A over a period of time  and see whether it's expanding exponentially.  There's another way that you can measure it in the laboratory.  There's no way you're actually going to do this.  But if you have, the laboratory now means way out and out of  space, take two objects which are very light and therefore  their gravitational between them is not very important.  Start them out at rest.  And if they're light enough so the gravity between them is  unimportant, they will just expand with the expansion.  And you can measure the rate at which the distance between them  expands with time.  Make sure there's nothing else present.  Complete vacuum.  So the only possibility is vacuum energy distorting space.  And you would discover that these begin to accelerate and begin  to expand, separate from each other exponentially.  So that's how you would build a vacuum energy detector.  Now of course the effect in the real world is so minuscule  that you couldn't even begin to hope to detect this in the  laboratory, but you can detect it cosmologically.  How do you detect it cosmologically?  A number of ways, but you say, well we have to check that H  is constant with time.  That's going to take a long, long time.  But no, not really.  When we look out at the distant universe,  we see it at different times.  And we can actually measure the Hubble constant at different  times by looking at it at different length scales.  Or we can look, what comes to the same thing is we can measure  the expansion rate or we can measure how A varies with time.  All right, so I'll tell you now how it varies with time.  Oh, wait.  Let's point something else out.  The real world doesn't just have an H squared.  It also has some other energy density.  Let's add then the other energy density.  It was called rho naught divided by A cubed, for example.  When A is small, when the universe is small,  when the scale factor is small, this term is going to be much  bigger than this.  Why is that?  Because A is in the denominator.  That's why.  So early on in the real world, this is not very important.  And this may be very important when A is small.  In that case, we worked it out a few minutes ago,  and we found out that A goes like T to the two thirds.  It goes like this.  But now as A grows, eventually this term will get small,  whereas this one is not getting small.  This one's not diluting.  This one is diluting.  So eventually the diluting term gets smaller than the  non-diluting term.  And then after that, you can basically forget it.  Once this term gets smaller than this term, you can forget it.  And how does A behave?  A behaves like E to the HT.  So what you expect in a world that has both kinds of energy  is that early on the scale factor, or the radius of the  universe, or the distance between distant galaxies,  grows this way.  But then at some time, when the vacuum energy becomes more  important than the other forms of energy, it takes off  exponentially, exponentially something like that.  So when you saw that differential equation, you end up with  a C in front of the integration constant in front of the  indian.  Yeah, there's an integration constant here.  How do you know that that integration constant isn't  really small?  Well, you measure it.  You measure it.  You measure it.  You measure it.  And here's T. Here's A. And of course, when you measure A,  you don't just measure A now, as I said.  You measure A by looking out deeper and deeper into distant  galaxies.  You can measure the distance between different distant  galaxies at earlier times.  And so at one time, you can measure this whole curve up to  some point.  You can't measure what it's going to be in the future,  but you can measure what it was in the past.  So if this is now, we get to measure all of this curve.  And it looks something like that.  It looks something like that, approximately something like  this.  With this branch of it here being fit rather accurately to  an exponential, I won't try to tell you how accurately.  It's about a 5%.  The proper figure of merit is about a 5%.  It's not a deviation.  It's exponential plus or minus 5%.  OK, so that looks to most cosmologists,  that looks fairly convincing that we live in a world with  vacuum energy.  Certainly, we don't know anything else that would have much  chance of doing precisely this.  And it's not a fact in the same sense that we know that the  electron exists so that the fine structure constant is 1 over  137 point blah, blah, blah, blah.  But by now, it's pretty much considered a fact of cosmology  that we live in a world with vacuum energy and that we live  in an exponentially expanding world.  That's pretty remarkable in a way.  A question?  A good question.  In some books, I've seen if you take our position right down  and draw the slope down, now we can, if that slope intersects  to the left of the origin, you can see back to the big bang  as it goes up, there will be a time where you can't see past  the big bang.  Is that, in other words, we'll expand it away so far that I  don't quite understand that pointer is it important?  Well, I'm not sure it's an issue of expanding back to,  tracing back to the big bang.  Or to the cosmic background radiation.  In the sense that the Hubble distance is right into the age  of the universe.  In the sense that the Hubble, yeah.  Right, the Hubble, right, in this exponentially increasing  world here, the Hubble scale is not related to the age of the  universe.  It's just a constant, right?  But it's someone you, I don't know, there's a horizon.  There's a horizon.  We're going to talk about horizons.  We're going to talk about event horizons.  There's a lot of different kinds of horizons on the market.  But the only ones which are deep and fundamental interest are  event horizons.  So we're going to talk about event horizons.  So let's now just put in our back pocket this piece of  knowledge that the world is exponentially expanding.  That we can summarize everything that was on the blackboard.  Oh, there is an interesting fact.  If the world continues to exponentially expand like this,  then all of the dilutable matter, the kind, the energy that  dilutes with expansion, is just going to get more and more  and more diluted.  So in some number of billions of years, the material in the  universe will stretch out so much that it will become of  negligible density.  And the only energy left over will be the vacuum energy.  It will become a very dull world.  But so be it.  Does that mean that the galaxies which aren't expanding today  because gravity doesn't?  No, they won't expand.  They won't expand.  What will happen is the few galaxies which are close enough  to us and which happen just by accident more or less to be  falling toward us.  There's one or two galaxies which are actually falling toward  us.  That's a sort of, when we say that everything uniformly  expands, of course it uniformly expands plus or minus little  bits of fluctuation.  It happens that there's a little bit of fluctuation in our  neighborhood which happened to cause the Andromeda galaxy to  be somewhat moving toward us.  So us and the Andromeda are not going to separate from each  other.  But if you go out a few more galaxies, they're certainly  moving with the flow.  Anything which is moving with the flow will eventually  depart.  It will just grow and separate and separate and separate.  And after a few hundred billion years, all that will be left  for us to see is us, our galaxy, and that's it.  We will be living in a world where we'll be able to see out to  enormously large distances but you won't see anything.  Astronomers who were born at that time and don't have a good  record of history will come to the amazing conclusion that they  are alone in the universe.  Yeah.  Doesn't that say that you eventually exceed the gravitational  attraction because our galaxy is gravitationally bound to a  group?  Yeah.  And if the group expands then...  No, no, no.  If the group expands then we're not gravitationally bound.  Right.  So when you said all we could see is our galaxy.  And the Andromeda.  Which by that time probably will have crashed into us and formed  some sort of single structure.  Are each of the clusters gravitationally bound to each other  also?  Correct.  Yeah.  So...  No, no, no, no, no.  It depends on what you mean by a cluster.  If you look at a few galaxies out, not very many, just a few,  one or two more, they're moving away from us.  Even though they form what's called a cluster.  No, they're not gravitationally bound.  No, they're not gravitationally bound.  Gravitationally bound means that they won't separate.  So the measured gravitationally bound depends on this  expansion rate?  Yeah, but not very much.  But not hardly at all.  Hardly at all.  For nearby things, if they're gravitationally bound,  can just mean whether...  Just forget this expansion and just ask whether they're moving  relative to each other with less than or greater than the  escape velocity.  That's what gravity...  Now, it happens, there's a more or less accidental fact that  the Andromeda happens to be moving toward us.  A few galaxies out, they're moving away from us.  And they will continue to move away from us.  They're moving away from us with greater than the escape  velocity.  And they'll continue.  Yeah, it's a big galaxy.  The gravitational constant is a constant that never changes in  time.  But the outward force due to the vacuum energy is a function  of time.  It grows...  It grows with distance.  It grows linearly with distance.  It grows with distance, but not with time.  No, not with time.  The acceptance of far as things separate with time.  Two objects will separate with time.  The distance will become larger.  And for that reason, the repulsive force will increase.  But the repulsive force does not increase the...  OK, you have a small repulsive...  Where is the repulsive force now?  And over a period of billions of years, the repulsive force...  It seems to me that equation is sort of cumulative.  There's a bigger amount of repulsive force.  That's only because things get further away from each other.  But a galaxy won't get farther away from each other.  If the galaxy...  If two galaxies are at the same distance at a later time as they  are now, the component of the little repulsion from the vacuum  energy will be the same.  What I was asking is that the net gravitational force is  G minus the expansion force.  And if the force, the tendency towards the band increases,  it wouldn't be that effect be that G very gradually decreases.  That doesn't make sense.  What are you saying is that the force is a function of distance,  not a function of...  It's a function of distance, not a function of time.  If you have two objects, if you have two objects at distance r  between them at any time, doesn't matter what time,  there are two components to the force.  One of them is G times M1, M2 divided by r squared.  And then there's another force on each of them,  there's another force which is proportional to...  It has a G in it, but mainly it's proportional to and repulsive.  This is minus meaning attractive, plus a repulsive force  proportional to r itself to the distance between them.  Coefficient doesn't change.  Coefficient don't change.  What does change is the distance between things.  So as the distance grows, this will become less important,  this will become more important.  But you're going to ask, where does this become important relative to this?  The answer is when the distance scales are pretty cosmological.  You know, five billion light years or something like that.  The crossover, the coefficient of this piece is so small  that the crossover between the normal gravitational attraction  and the bit of repulsion is on the billions of light years scale.  So that's why you can't measure it by an laboratory.  So it's not just a function of expanding space that is represented as a repulsive force.  It's actually...  It is the expanding space, but you can fake it by a force  which is pushing everything else apart from each other.  It's a slight fake, but it sort of works the right way.  Yeah.  Okay.  We've now...  Now, does that kind of...  I think satisfies my curiosity here that in the early universe  these galaxies were gravitational bound  and the expansion has now taken them to where the vacuum expansion, this other factor, are.  In the early universe they were not gravitation.  Yes, it's crossed over.  So the gravitational attraction field is small and below.  When they get far enough apart, then the repulsion becomes more important than the attraction.  Right. So in the early universe they were gravitational.  Yeah, this was not important in the early universe.  Right.  That's another statement.  It's two different statements to say that they're gravitationally bound  and that the important force was the ordinary gravitational force.  If we shoot a rocket off the surface of the Earth at one and a half times the escape velocity,  it is not gravitationally bound.  But still the most important force on it is the ordinary gravitational force.  These are two different statements.  One statement, gravitational bound, means it's going to come back.  The other statement is this may be more important than this in the early universe than it was,  but it doesn't mean that the galaxies were gravitationally bound.  They were still moving apart from each other with greater than the escape velocity.  For the most part, I mean, pretty much.  The galaxies were shot out pretty much greater than the escape velocity between them.  Roughly, very close to the escape velocity, very, very close to the escape velocity,  extremely close to the escape velocity.  And then they either turn around and fall back in or they accelerate away from each other,  depending on whether there's this kind of new energy.  Okay, let's take a little break.  And when we come back, I will tell you more about the geometry of this kind of space  and why it has horizons, what it means for it to have horizons.  So now that we know that in a couple of hundred billion years, the universe is going to be described.  And it's getting that way pretty quickly, actually. On the scale of tens of billions of years,  we're getting there pretty fast.  We're getting to a condition where the geometry of space-time is described by a metric.  And the metric is e to the 2HT, that's a squared, times dx squared.  And again, dx squared stands for dx squared plus dy squared plus dz squared,  and this is the scale factor.  This space-time, it's a space-time, it has a geometric property, it has a name.  Anybody know what the name is?  The sitter space, the sitter space, dE.  It's the sitter space.  It should be called the sitter space-time, obviously, but it's called the sitter space.  And we'd like to examine its properties a little bit.  Not because we're going to be sending rocket ships out there to explore it.  That's going to take a while, but just for the intellectual value of understanding the space-time that we live in.  That's pretty weird. It's pretty weird.  Of course, it looks weird, but it's weirder than it looks.  It's weird in that it has horizons.  Before we get into horizons of the sitter space, let's just talk for a moment about what horizons are in general.  And I'll give, as the illustration, the horizon of a black hole.  Remember, we worked out what a black hole formation, a black hole which is created by an infalling collection of material,  what it looks like, and we drew a pen.  Remember the Penrose diagram?  Penrose diagram looked like that.  Before we do that, we're going to use Penrose diagrams, because it's helpful.  But let's first draw the Penrose diagram of ordinary flat space.  There's ordinary flat space.  There's points far away.  It's spatial infinity.  Here's time infinity.  Light rays come in from here, and they go out from here, and all that sort of stuff.  And that's just empty space.  A black hole space-time looks like this.  Oh, one other point.  Imagine an object, a massive object, not a photon, a massive object in this space.  How does a massive object move through space-time?  Well, all massive objects wind up at the same place.  They were massless objects.  Photons go out the sides here, moving with a speed of light.  Massive objects, they all go to this point up here.  Now, that's not really a point.  It's not really a point.  It's a whole big space-time that's been smooshed onto the diagram to look small.  Obviously, if we wanted to take all of space-time and squeeze it onto the blackboard,  we're going to have to distort it pretty badly.  And it's very distorted up here, in that there's an infinite amount of space and time up in this corner.  Likewise here, likewise here, in fact, on all of the edges.  And all of the massive objects eventually wind up at this corner.  That's not to say that they get close together.  It's just to say that this is a very big place up here.  Okay.  Now, let's think of the massive objects in the black hole space.  In the black hole space, you can have one of two things happening.  The massive object can fall into the horizon and hit the singularity.  Let's avoid that.  Let's avoid that.  Let's talk about the objects which don't fall into the singularity.  Where do they go?  They go to here.  They all do.  Again, that doesn't mean that they literally get close together.  This just means there's a lot of space time out there.  All of these people wind up.  Let's draw the horizon.  The horizon's over here.  Okay.  Now, let's ask what these people here can see as they look back into the past.  They can see only those things which are inside their backward light cone.  The backward light cone means take all the light rays which can get to this point over here.  All the light rays come in from here and take that region.  That's the region that he over here can be aware of.  When he looks back, he can only see the things inside his light cone.  Same for this guy over here.  Same for this observer.  They can see the things in their backward light cone.  Now, how much can they see given arbitrary amounts of time?  Given arbitrary amounts of time, they can look back from later and later times.  And eventually, they can see all of this blue region here.  What they cannot see is what's in here.  A horizon is by definition the place which separates the region that these observers can see from where they can't see.  Given enough time, you draw the backward light cone of a very future point along that observer's trajectory,  and that is the region he can see.  Everything on the other side is behind the horizon, and the horizon is the separation between them.  Here, there's no horizon.  An observer way up here gets back, looks back, and sees everything.  Sees the entire space-time.  No horizon.  Here, there's a horizon.  So, I just remind you of that to tell you what a horizon is.  We're going to take this metric, and we're going to do a little bit of mathematics on it.  It's not hard mathematics.  It's pretty easy mathematics.  It's not particularly abstract, but it helps us visualize the space-time.  What we're going to do with it is we're going, I'm not going to try to take all of space and squeeze it onto the blackboard.  I'm going to leave space infinite.  But what I'm going to take do is take the time axis, and I'm going to smush it.  I'm going to make a coordinate transformation of time so that I can put the entire time from minus infinity,  well, not from minus infinity, minus infinity will be deep down in the floor,  but plus infinity will be in some finite place here,  because we're going to want to study what this world is like at late times,  and I'd like to get the whole infinite future onto the blackboard.  Not the whole infinite future of space, but the whole infinite future of time.  So, we're going to make a coordinate transformation,  and we're going to do this in a way that makes light rays move on 45-degree angles in this space.  It's always good to do that.  Always get the light rays moving at 45-degree angles,  and it's easy to see what can communicate with what.  All right, so here's the trick.  We're going to change coordinates.  We're going to introduce a new coordinate.  I'm going to call it capital T, and it's a function of the old T,  and it's going to have the property.  Here's the property it's going to have.  It's going to say such that dT squared, what appears here,  is going to be equal to e to the 2HT,  exactly what appears here, times d capital T squared.  This is the definition of capital T.  It's a new time coordinate,  and it's related to the old time coordinate by a pretty simple equation.  We're going to solve it.  And then we're going to rewrite the metric with the new time in place of the old time,  and the metric will have a nice appearance that will allow us to explore it very nicely.  Okay, how do I solve for big T as a function of little t?  The first thing I do is take the square root of both sides of the equation.  dT is equal to e to the hT d capital T.  And now divide by e to the hT.  What do I get on the left-hand side if I divide by e to the hT?  e to the minus hT, right?  So I'll put an e to the minus hT here, and now we have dT.  So all we have to do to find the relationship between big T and little t is to integrate.  The integral on the right-hand side just gives us T.  What about the left-hand side?  Anybody can do that integral?  Dime for anybody?  One over h.  One over h.  Minus sign?  Minus sign.  e to the minus hT, right?  Plus a constant, but we can fix the constant by just shifting T here.  Right, there's a constant, but I can shift the constant.  Suppose I put a constant.  There's a constant.  Now I just redefine T so that it's T plus C.  It doesn't make any difference, right?  Okay, now notice first of all capital T is negative.  Let's just think about what happens to it.  This one over h here is not so important.  Let's just think about what happens to it.  What is it like at very remote past times?  Let's say when little t here is extremely negative, what happens to e to the minus hT?  It becomes very big.  If t is negative, and we have e to the minus, then this becomes e to a positive number,  this gets very big, but there's a negative sign out here.  So that means the remote past where little t is very, very negative is also the remote past from the point of view of big t.  When little t gets very, very negative, so does big t.  All right?  So in both variables, the remote past is way down through the floor and the basement and even deeper.  What about the remote future?  What happens in the remote future to capital T?  The remote future is when little t is very, very big.  What happens to this when little t is very big?  It goes to zero.  So big t goes to zero in the remote future.  Now we can draw it.  The time axis, let's draw the time, the big t axis, the big t axis goes into the very remote past in the negative region, deep past.  But where is the infinite future?  The infinite future is right at capital T equals zero.  Capital T equals zero.  That also is little t equals infinity.  So big t runs from minus infinity up to zero.  That's one way of getting not the whole geometry on the blackboard, but getting the remote future onto the blackboard.  That was a goal.  Okay, let's see if we can rewrite the metric.  Let's see if we can rewrite the metric in terms of big t instead of little t.  So here we have, let's, little t squared.  What's the relation between little dt here?  Little dt, here it is.  Little dt is equal to e to the big ht times big t, right?  Good.  I just multiplied by e to the ht to get it over on the side over here.  And now we come to this metric over here.  We have minus dt squared.  So this is equal then to minus dt squared, but that's e to the 2ht, the big t squared.  And then we have plus e to the 2ht dx squared.  In other words, I've engineered it so that the e to the 2ht would come on the outside.  It would be equal to e to the 2ht times the common Minkowski space metric.  This would be the metric of ordinary flat spacetime minus dt squared plus dx squared plus dy squared plus dz squared,  but it has this factor in front of it.  What can we do with it?  I want to re-express it in terms of capital T.  So let's do that.  Let's first multiply by h.  e to the minus ht equals h times big t.  All I did was multiply by h.  And now let's square it.  If I square it, the minus sign goes away and I get e to the minus 2ht.  And on this side, I get h squared t squared.  And finally, I take one over it so that e to the 2ht is equal to 1 over h squared t squared.  Go through it, check the arithmetic.  You'll find it works that way.  And now we can write this metric just by substituting for e to the 2ht, 1 over h squared t squared.  So let's put that there.  In fact, let's write it over here.  The s squared is equal to 1 over h squared t squared times minus dt squared plus dx squared.  Now this is a long story, nice simple result, but the main point of it,  the main point of it is that we've written the metric as an ordinary Minkowski space metric times a function  with the same function appearing multiplying all of this.  How do light rays move in this geometry?  What's the rule for a light ray?  Ds is zero.  Right, light rays move on paths of zero proper time.  What does that say?  That says that you can forget this factor altogether when you're thinking about a light ray.  If you want to know how a light ray moves, you don't need to know about this factor.  All you need is this factor.  Right?  Everybody see that?  If you want to know how a light ray moves, it moves exactly as it would if this were just ordinary flat spacetime.  How do light rays move in flat spacetime?  On straight lines tilted at a 45 degree axis to the vertical.  We now know in this geometry how light rays move.  Let's draw the geometry.  Here it goes.  It starts out deep in the past.  We can't go down that deep into the basement, but it ends.  This is space.  Space now goes this way and this way.  It ends at capital T equals zero.  How can it end at capital T equals zero?  Little t went on and on forever.  Well, it's just that the metric, when big t gets small, the metric here blows up.  It gets big.  So that means there's a lot of time in here.  The metrical distance between neighboring points gets very large when t goes to zero.  So it's highly distorted.  I mean, it's a highly distorted geometry.  The one advantage of it is that light rays move on 45 degree axes.  So we now know how light rays move.  They move like that.  If they're moving out of the blackboard, they also move on 45 degrees, but I'm not going to try to draw it.  I can't draw it.  All right, good.  Now let's ask what an observer can see.  Imagine an observer in this world.  I'll make them blue.  Here she is.  She's standing still in her own reference frame.  Time is going on.  More and more and more time is unfolding.  And eventually, she gets up to the top.  Now how long does that take according to her watch?  Forever.  Forever, right?  It takes forever for that to happen.  But on this diagram, on this formal mathematical abstract description of this interspace,  the terminal point is over there.  That's the end of her existence.  It doesn't bother her at all because her clock has gone off to infinity.  Let's ask what she can see.  What she can see is she looks back.  All right, let's draw that in red.  Here she is right over here.  That's Alice.  And Alice looks back and sees everything that she can see within her light cone.  Remember, the whole purpose of this coordinate transformation was to make it easy to follow light rays, 45 degrees.  If we want more dimensions, we'll have to extend this out to make a cone out of it.  But you can see all of the main features from here.  She can see everything behind her here.  But if she waits a little while, she sees more.  And if she waits a little longer, she sees more.  But no matter how long she waits, the most that she can ever see is this.  What about things out here?  Nothing.  If there's somebody out here, she can never know about it in principle.  She has a horizon.  The red line out here is her event horizon.  It is her private event horizon.  Other people have different event horizons.  But this is her private event horizon.  She cannot see anything out beyond it.  In principle, as a matter of real principle, she cannot, at least by classical relativistic reasoning, by classical light reasoning and so forth,  she cannot see anything.  All kinds of things could be going on there.  Her friend Bob might have originated over here, and she might have been able to see Bob in the past.  From here, she could even, Bob is moving, is Bob.  Alice, Bob.  In the past, Alice was able to send messages to Bob.  Bob was able to send messages to Alice.  And now Bob passes Alice's horizon.  From there on in, Bob cannot send the message to Alice.  Bob may be able to see Alice, but Alice cannot see Bob.  So from this point on, Bob has passed out of her horizon.  He's gone.  That's what's going to happen.  Unless, of course, if she's holding hands, what does it look like if she's holding hands with Bob and she doesn't want to let him go?  Yeah.  Then it looks like this.  Now, does this mean they really get closer and closer together?  No, it doesn't.  It doesn't because the metric is getting bigger and bigger out here.  So the actual metrical distance between them, the actual proper distance between them can stay constant.  And if it does stay constant, they would appear to join up here, but they're not really getting any closer together.  So if they're bound together, if they're holding hands, this is what it looks like.  And they can continuously forever and ever send messages back and forth.  No limits.  But if Bob lets go of Alice and goes with the flow, if he's, let's go implies, among other things,  that he gets far enough away that the gravitational pull between them doesn't hold them together.  If he lets go and passes outside her horizon, they simply fall out of causal contact.  That's the expression for it.  They've fallen out of causal contact.  In this particular instance, Bob can send a message, sorry, can receive a message from Alice,  but he can never send a message to Alice.  Alice up here cannot send a message to Bob either,  but Bob also has his own private event horizon.  Let's make Bob's event horizon in green.  Here's Bob's event horizon.  Bob cannot see anything outside of his event horizon, so he can see Alice early on,  but he can't see Alice after she passes out of her horizon.  So he won't be able to see Alice at his time.  Bob won't be able to see Alice at his time.  That's right.  Well, you can never see anything at the same time, period.  Right.  You can never see.  The only thing you can never see is those things which are back inside your past light cone.  So you never can see anybody at the same time.  But once Bob has passed through here, Alice can never see him.  If this really was flat space, if it really wasn't this peculiar, the sitterspace with this factor here,  and then Alice would have to separate it, then Alice could keep going and look back at Bob.  But Alice comes to the end of her world line over here, and simply cannot look back.  So this is the character of the sitterspace.  It has horizons.  In fact, every single observer has his own private event horizon.  In this case, Alice and Bob constitute one observer.  And maybe we'll talk a little bit more about these event horizons next time.  I'm getting a little tired, so I think I'll quit soon.  But we apparently live in such a world.  And in such a world, it's not Alice and Bob, but it's galaxies.  Lots of galaxies.  Here we are now.  We can see about 10 to the 22 galaxies around us.  As time goes on, they will pass through the horizon and pass out of, let's say we're Alice.  They will pass out of Alice's horizon, and Alice will not be able to see them anymore.  They will be gone.  So as I said, Alice will be alone in the world for all practical purposes.  Excuse me.  Is there a theory that in the far distance, we would not be able to see any galaxies at all?  That's right.  That's right.  They will have a...  Galaxy, Alice, and Bob will...  You might...  Okay, let's ask whether Alice at a very, very late time here can see Bob back here.  In a sense, no.  It looks like he can, but in a sense, no, and I'll explain why.  This is Alice here.  Alice's time steps look like this.  Many, many time steps here.  Now, imagine that Bob sends her a signal, and that signal is going to come up to here.  Alice is going to look back and look for Bob back here.  Well, you say she can see him.  But in fact, because her time clocks are so bunched up over here,  it means the radiation that she sees from Bob is incredibly redshifted.  That how many beats of Bob's clock gets squeezed into Alice's time scale here,  one beat of her clock says stretched out over an enormous time scale from Alice's world.  So when Alice looks back and sees Bob from way, way up here, he's enormously redshifted.  You can't see very, very redshifted things.  His wavelength, the wavelength of his light and the energy of his light will be so stretched out.  So in fact, really, Alice has no chance of really seeing Bob over here.  He's just been redshifted to hell and...  Yeah, like a black hole.  Like on the horizon.  That's right. It's exactly the same as in the horizon of a black hole.  So in that sense, at very, very late time, Alice will not be able to see any of the galaxies,  except in this exceedingly redshifted sense, which is useless.  She can't detect such low energy radiation, and she will just find herself alone.  And so will Bob, all other galaxies having passed out of their horizon.  Is it the same with the black hole, which it takes forever to reach the horizon?  Yeah, yeah, but okay. It is the same. It is the same.  Well, it is the same. Alice looks back. Has Bob crossed her... Where is Bob?  Here's Bob over here. Has Bob crossed the horizon? No. Alice looks back. Has Bob crossed the horizon?  No. Alice looks back. Has Bob crossed the horizon? No. She never sees him cross the horizon.  But do objects falling into black hole become infinitely redshifted as well?  Yes. Yes. It's exactly the same thing. It's exactly the same thing.  Here's the black hole.  Here's who stays outside. Let's put Alice outside. Yeah, let's put Alice outside.  Let's keep the color coding. Blue? Blue for Alice?  Alice is red. Bob is blue. No, here's Alice. Alice is blue.  Bob is green.  I don't know. Here's Alice.  I usually send Alice into the black hole, but this time Alice gets to stay outside.  And here's her clock. Her clock is going faster and faster and faster and faster and faster.  It's not going faster and faster, but on this diagram, all her time intervals are bunched up, same as they are over here.  And Bob, who is now green, is falling through the horizon. Alice looks back.  Does she ever get to see Bob fall through? No. Bob's signals, his light waves or his waves of his hand that he sends out,  appear to get slower and slower to her because they're drawn out over longer and longer time scales,  which is another way of saying Alice becomes infinitely redshifted.  The light from Alice becomes infinitely redshifted from Bob's perspective.  So it's exactly the same thing. Yeah.  So in Alice's real time, in her proper time, she's slowing down.  She's going at her proper time, but on the scale, it's getting compressed, right?  Well, on the scale, not on her scale. Well, she's just ticking along happily.  No, that's Bob. That's Bob. You're right. That's Bob, yes.  So Alice's scale time, right? Her proper time is still her proper time.  So she looks back and she sees Bob slowing down.  But she not only sees him slowing down, she sees his atom slowing down, she sees the radiation that's coming out.  Yeah, she sees not only Bob slowing down, but she sees the atom slowing down, she sees the emission of photons getting slower and slower,  the photons getting longer and longer wavelength, smaller and smaller frequency,  and eventually his photons become so monumentally long wavelength that they can't be seen at all.  The energy of them gets diminished to negligible. Yeah.  How would she measure the temperature of the surface right above the horizon?  Well, to do that, she has to send down a thermometer and make sure the thermometer doesn't fall into the black hole.  So to keep it from falling through the black hole, she'd send down a thermometer, measure the temperature,  pull it back up and see what it recorded. But that's not what she's doing with Bob.  And the same incidentally, same thing here.  If Alice, so here would be the experiment. Alice has a very long rope, a long cable.  On the end of the cable is a thermometer. She throws it out hard enough that it's beyond the escape velocity from a local cluster of galaxies  and waits a long time for it to migrate out toward the horizon, her horizon.  But she doesn't let it go through the horizon. She then pulls it back in.  And after she's pulled it back in, she sees what the temperature that it recorded was.  She will also see a high temperature, but we'll come to that.  Horizons are always hot in a sense, but we'll come to that.  This is so far the classical description of the sitter space.  Full of horizons. There's horizons everywhere. No matter where Alice is, there's somebody, she's passing through somebody's horizon.  So wherever you are, you're passing through somebody's horizon. Say goodbye because you won't see them again.  I'm trying to understand what it looks like in the untransformed, in the little thing.  They're at the same distance.  I'm going to leave that for you to do an exercise. It's an exercise to see what it looks like.  No, it's a good exercise. Just undo what I did.  I'm going to just say that my assumption is they're not moving apart. The x is not expanding there.  No, no, no, no, no. The whole metric is blowing up, including the x parts of it.  So the distance between points, if you keep the points at fixed coordinate separation, here Bob and Alice are at fixed x's.  The distance between them is getting bigger and bigger.  It goes to e to the infinity.  That's right. The distance between them blows up when they get up to, right, that's another fact of the sitter space.  Now, something you can work out is you can compare the distances between these two points,  or the distance between these two points, between the distance comparing.  How far apart are these two? How far apart are these two?  What do you think the answer is?  Are they growing? Are they shrinking?  No, it's staying the same. Staying the same.  If you draw a horizontal line, work it out. Here's the metric.  Draw a horizontal line here and ask for the distance, let's say, from the center to the horizon over here.  And compare it with the distance from the center to the horizon over here.  Compare it with the distance from the center to the horizon over here. You'll find they're all the same.  The coordinate distance is shrinking, but the metric is increasing, and the distance to the horizon is always the same.  That is connected with the fact that the Hubble constant is constant,  and so the distance to where the speed of light is, where the things are moving away with the speed of light is constant.  So Alice doesn't see things changing with time. Everything stays the same.  She just sees a horizon around her, and that horizon stays at a fixed distance from her.  So would she be seeing the projection of all things in the past be projected onto that horizon space?  Because they'll never go away, right?  They'll never go away, but she sees it harder and harder to see them because they get redshifted.  Now, that doesn't take into account the Hawking radiation coming from the horizon, but what you say is right.  Okay, so assuming that we don't pass through each other's horizons, we'll meet here next week.
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