Stanford University. OK. For some reason, I have a set of notes here from last week. Did you ever get notes from last week? Here they are, if you want them.

And then I made a mistake. In this week's notes, I wrote them on pads, which are probably too long to scan in one piece. Is that OK? Just to figure it out, OK. I inherited a whole bunch of yellow pads

from the mathematics department. It seems mathematicians don't like writing on long pads, so I got them, and about 25 of them from the mathematics department. Yeah, right. That's hard to scan, isn't it? Push your paper.

OK. We've got a couple of minutes. Let's begin with some questions just for the next couple of minutes.

I don't think so. I mean, one idea was that it was smaller and that it was a kind of torus that was just

periodically repeated itself, so that the large part of it that we were looking at was just repetitions of the same thing, or another way of saying that we were looking at ourselves through the back door. That has consequences. That has observable consequences.

You might think it'd be real easy. You just look out and you see, no, it doesn't work that way. It's much harder than that. And people have looked for periodicity in astronomical and cosmological observations,

and there is no evidence and some counter evidence that it's not periodic like that. And if it's not, I don't see how it could be smaller than the observed part. So it's an interesting question, but I don't think so. OK. Yeah.

So far I haven't, I'm not sure that I've mentioned dark energy, but so far we have not talked about dark energy. We will. But the universe is expanding. The universe is definitely expanding, but the expansion do, the consequence of dark energy

is not that the universe is expanding. It is, but I mean, it's not the, universe was expanding before anybody knew about dark energy. Dark energy just makes it expand faster.

Yeah. Well, no, yeah, no. If you were holding your friend's hand out in space,

your distance between you and your friend would not expand, would be a general expansion. The general expansion produces a kind of very, very mild repulsive force between everything and everything else.

You can think of it that way. It's a way to think about it, that everything has a little bit of repulsion relative to everything else, basically proportional to the Hubble constant. But that, with you holding your friend's hand,

that very, very tiny repulsion between the two of you is more than made up for, vastly more than made up for, by just the attractive force of you holding onto your friend's hand. So as long as you didn't let go, now I'll tell you a little more in a minute.

As long as you didn't let go, you would not participate in the general expansion, you as a couple, so to speak. In fact, there's probably enough of other kinds of forces between you, besides holding your hand, that they would overwhelm this tiny, tiny tendency to separate.

That's because you're closer together than the average distance between your atoms, your atoms and your friend's atoms are closer together. They have more attraction. In particular, holding your hands will keep you from separating.

And so you're not part of this general expansion. The same thing is true of the solar system. The solar system is largely held together. Well, first let's talk about an atom. What about an atom? Do atoms, why did you come to the solar system? Why not an atom? An atom also is embedded in space. If that space is expanding, why isn't the atom also expanding?

And the answer is the electrostatic forces, more than overwhelm, by enormous amount, more than overwhelm the general tendency to expand. Now, what you can say is that in an expanding universe,

let's even take the case of a accelerated expansion, that this tiny little bit of outward force will tend to modify the atom a little bit. It will tend to make the atom just a tiny, tiny bit bigger. The atom will have to be slightly expanded.

But other than that, and that's a tiny, tiny effect, but it will not cause the atom to fly apart. It's not strong enough to cause the atom to fly apart. Same is true of the solar system. Solar system is held together by gravity. The gravitational pull of the sun

is simply much, much larger than this tendency to expand. So for that reason, and yeah. The tendency to expand.

In order to do what? So you need to do. It turns ever so slightly, but in the expansion of space. Yep.

Energy conservation in an expanding universe is different than energy conservation in the static universe. Energy conservation, there's several ways to think about it,

but let me give you the simplest way to think about it. Energy conservation is a consequence of time translation invariance. In other words, if everything is time independent, meaning to say space, and then the time,

and all experiments would reproduce exactly the same effect if they were later or earlier, the consequence of that is energy conservation. On the other hand, if the basic setup called the background, space-time itself, if it's not static,

if it's expanding, or if it's changing with time, then energy conservation doesn't apply. There is no energy conservation in a world where the parameters of the world are time dependent. In this case, the radius of the universe is time dependent.

Energy conservation is not quite what you thought it is. And we're going to come to it. We're going to use energy conservation. We're going to use it in a certain form, but it doesn't say that the net amount of energy in the universe is fixed.

That is not what it says. It basically says that changes of energy in the universe translate into kinetic energy of expansion. So there's a back and forth between changes of energy and expansion. And you don't have to ask where the energy came from.

It came from the expansion. On small scales, it falls apart.

On large scales, as far as we can tell, universe is homogeneous and isotropic. The air in this room is largely homogeneous and isotropic. Every time somebody opens a door, of course, a draft blows in. But if we kept the room closed, and you kept your mouth shut,

and the air in the room would become highly homogeneous, but not on every scale. On tiny, tiny scales, what you see is atoms, molecules, and so forth, one over here, one over here. It's not homogeneous. In fact, on even bigger scales, there are fluctuations that take place.

Density fluctuates. Sometimes it's a little more over here, a little less over here. So whether something is homogeneous or not is a scale-dependent question. The universe is not homogeneous on small scales.

And small scales means hundreds of millions of light years. On scales bigger than hundreds of millions of light years, everything seems to be distributed uniformly. I'm not sure I'm answering your question, but I'm answering somebody's question. If the universe is expanding, how come I can't find a parking spot on campus?

Oh, yes. That, yeah, yeah, no, that's one we're working on. I'm following three types of closed-flat, multiple solutions.

What does that mean mathematically in terms of the quantities in, say, Einstein's field? OK, it's a statement about geometry. Einstein's equations are the dynamics of how things change with time and so forth. This is a pure statement of geometry. If you classify the geometries, which

are in some sense the same everywhere, that every point is the same as every other point. And what that means, I'll tell you what it means exactly. Given a space, how do you describe a space? You describe a space by a metric.

You write a metric for the space, some g mu nu of x. Or g, it's only space. It's not space and time. So let's call it g m n of x. And x now stands for all of the coordinates. That represents some geometry. Now, you can make a coordinate transformation.

Imagine you make a coordinate transformation. The point x equals 0, which was the origin of the coordinates, this was x equals 0, that now has a new value. The new coordinates could be called y. What was originally x equals 0 is no longer the origin.

y equals 0 might be over here. And so this corresponds to some kind of transformation which replaces the origin with some new origin. Now, when you do that, the metric, when you do a coordinate

transformation, the metric transforms. It transforms into a new metric. We can call it g prime m n of y. And this is the metric in the y coordinates. Typically, the metric in the y coordinates will have a different form than the metric in the x coordinates.

That could be for two reasons. Two reasons. First of all, it could be because the space itself is different at this point than this point. And so if we transform our coordinates to this point, we

might discover that the curvature is larger over here or something different over here, in which case the metric will have a different look to it when you express it in terms of y than it did in terms of x. Or it might be not because the points are different, but simply because you use screwy coordinates over here and some other screwy coordinates over here.

A homogeneous space means, one, that you can find a coordinate transformation which will replace any given point as origin, let's say, any given point by any other point in such a way that the form of the metric

is identical both before and after the transformation. For example, for example, let's just take flat space. Flat space has a metric which is the s squared equals the x squared, let's just take two dimensions, the x

squared, the x1 squared plus the x2 squared. x1 squared plus the x2 squared, where x1 and x2 are the coordinates in the blackboard. Now supposing I make a translation of coordinates, that means that y, y doesn't stand for x and y, it stands for

a new set of coordinates, which is just equal to x at y1 is x1 plus a shift. Let's call it A1, A does not stand for any other A that

we've used so far in this course. And y2 is equal to x2 plus A2. This is a shift. This is a shift which, if this is the x-coordinates, then the y-coordinates are shifted by a vector A.

What is the metric in terms of y-coordinates? Well, we can do it by simply re-expressing dx1 in terms of dy1, but dy1 is the same as dx1. If I differentiate, if I make a little change in y, it will

be equal to the little change in x, but since this is a constant, it doesn't contribute anything. Likewise, for dy2. So, for this kind of transformation, ds2 is equal also to dy12 plus dy22, has exactly the same form as

the metric had in terms of x's. The implication of that is that the neighborhood of the origin, x equals 0, has exactly the same properties as the neighborhood of the origin of y equals 0, the

y-coordinates. In other words, these two points have exactly the same property. Now, since A could be anything, there's a coordinate transformation which takes the origin over here to any other point, to any other point, whatever,

that preserves the form of the metric, and that's what's called a homogeneous space, whose properties are everywhere the same. So, if there exists a coordinate transformation between x and y that takes the origin to any other point, such that the form of the metric in terms of y

is the same as in x, then the space is called homogeneous, and it means it's everywhere the same. A sphere, take a sphere. We could put the origin, the south pole.

The south pole could be the origin. You know what the form of the metric is. We wrote it down last time. I'll write it down again for you. The r squared plus sine squared r times, let's say, d phi squared, or d theta squared. I've forgotten what I called it. D theta squared, that's the metric of the sphere.

Now let's make a coordinate transformation by rotating the sphere. Rotating the sphere means choosing some other point, the point over here, for example. Instead of measuring r from the south pole, we measure r from the east pole.

Here's the east pole over here. We measure r from that point, and instead of measuring angles about the south pole, we measure angles about the east pole. This is a coordinate transformation. It's a little nasty and complicated to write down. How does the distance from this point depend on the

coordinates as measured about here? A little bit complicated, but when you figure it out, you'll find that the metric, when written in terms of the coordinates relative to this point, has exactly the same form as relative to this point.

Another, I won't bother writing it down, it'll be exactly the same form, except that the meaning of r and theta will be distances measured from here instead of here, and angles measured about that point instead of about that point. Form of the metric is the same. And since you can choose your coordinate transformation to

take this point to any point, it tells you that every point on the sphere is basically the same as every other point on the sphere. One test which works very well in two dimensions for a

uniform geometry is that the curvature should be the same everywheres. That's not good enough in other dimensions, but the basic idea that the geometry is the same everywheres is called a homogeneous geometry. This is also true, it's true of the flat plane, it's

true of the sphere that it's homogeneous, and it is also true of the hyperbolic plane, of the negative curved sort of analog of the sphere where you replace signs by hyperbolic signs. There are no others.

Yeah, yeah, yeah, good, right, right. There, no, okay, yes, it is true that the torus is translation invariant is the right word.

Translation invariant means you can transform any point to any other. But, as a matter of fact, it happens not to be isotropic. Why is it not isotropic? It's not isotropic because there are definite, remember I told you what a torus is, it's a rectangle with periodic boundary conditions, and the axes are preferred.

If you go out here, you come out here, so this axis is preferred, that axis is preferred, and an axis like that is basically different. Observations would be different along 45 degrees than

along, so, yes, it's true, it's a homogeneous space, but it's not an isotropic space, good. So, good.

Now, of course, the idea that space is homogeneous and isotropic kind of has a status somewheres between being a postulate, and therefore highly questionable on the one hand, and somewheres in between that and an

observational fact. So, on certain scales, it really does look homogeneous, but on scales so big that we can't see them, we don't know. Okay, one other remark before we go on, since this comes up over and over and over again, I just want to, and

you'll be tired of hearing this, but there's often an enormous amount of confusion when you say such things as, for example, space has a spherical shape, right? People get into their heads, the idea that it really is

like a balloon, and namely that it has an inside and an outside, and they ask questions, what happens if you move away from the balloon, or into the balloon? No, you'll have to learn to think of geometries as

having their own intrinsic shape that has to do with what happens if you move around in the geometry, and not having to do with some imaginary, possibly real, or imaginary, additional directions that you can move away from the space.

For example, one of the most confusing kinds of spaces is a one-dimensional space, a one-dimensional space, let's say a closed one-dimensional space, a closed and finite one-dimensional space.

What is a closed, here's a closed and finite one-dimensional space. Here's another one, here's another one, happens to be a square, OK? From the point of view of the intrinsic geometry, all

that counts is measuring distance along the geometry. No concept, as far as the intrinsic geometry goes, of moving away from the surface. So in fact, how many different kinds of one-dimensional spaces are there from the intrinsic point of view?

The answer is one. All of them are identical, well, not quite, they're not identical to each other. There's a one-parameter family of them, and the parameter is how far around you have to go to come back to the same place. So if this circle here happens to have the same

circumference as this peanut shape over here, then intrinsically they are identical. They're identical, and in fact, are they curved? They are not curved, and to see that they're not

curved, think of them as strings, not super strings, not string theory strings, just as strings. Without stretching the strings, you can always deform them to make any piece of them straight. Just pull and straighten it out.

One intrinsic fact about them is that they are closed, they come back to themselves. But since any piece of it could be straightened out without stretching, without deforming the metric, the dimensional spaces are all flat.

They have no curvature. The fact that they're drawn curved, that has to do with the way you drew them in two dimensions. So you have to learn to think about the intrinsic geometry. It's hard, and there's a reason that it's hard.

A line segment is also a one-dimensional space. It has a different property than the closed one-dimensional space. It's topologically different, it has end points, but the only thing that characterizes it is the distance between

the end points. This line segment is the same as that line segment. That's right, only if it's embedded.

Now, I emphasize this over and over again when explaining some of these things to a general audience. I think I probably said it to you before, but what is it, after all, that is special about three dimensions? So I typically ask the question, do you think that

you can visualize a five-dimensional space? Visualize. Visualize means close your eyes and see it. And everybody says, no, I can't do that. Well, I can't do it either. Four? Not really. No, I can use a trick to help me visualize it, but direct visualization I cannot.

Now, I say close your eyes and see if you can see a sphere. Let's not take a sphere, let's take a cube. Can you visualize a cube? Yeah, I can visualize the cube. I can see its cubical nature, I can see its three dimensionality, and then we go down with dimension. Let's go down to two dimensions. What I want you to do is to visualize an abstract

two-dimensional space. Can you do that? And everybody says, sure. And I say, what do you see? I say, oh, I see a curved surface. And my response to that is, yes, you see a curved surface, but the only way you can visualize it is by visualizing it as embedded in three dimensions, unless you

have some brain different than mine. Can you even visualize a one-dimensional space? Sure, I can see a line. No, what you see is that line possibly embedded on a piece of paper, on two dimensions, or possibly embedded in three dimensions.

And even a point, an abstract point you cannot visualize without seeing it suspended in three dimensions. What is it that's special about three dimensions? Is there something really mathematically special? No. It's your architecture. Your brain architecture evolved for the purpose of navigating around in three dimensions.

And so it's not surprising that your ability at visualizing is hung up at three dimensions. That doesn't mean that three dimensions are in any way special. Of course, every dimensionality has its own special features, but three dimensions is not special.

And as mathematically sophisticated people, we just say, if you want to discuss two dimensions, you make an X and a Y. If you want to discuss three dimensions, you make an X and a Y and a Z. If you want to discuss four dimensions, you make an X, a Y, a Z, and a W. And I like to joke that

you can get all the way to 26 dimensions and so forth. Okay, so I just want to remind you over and over again that when you, I won't do it again.

I will not do it again, but when we talk about the geometry of a space, we're talking about the intrinsic geometry and not the way it's embedded for purposes of visualization in some higher dimensional geometry. There is, yeah, right, except unless there are more.

The more would be very small, too hard to see. Yeah. If I drew this, if I, all right, let's, that's right.

That's right. But assume that light rays only propagate along the surface. If it's a one dimensional world that we're living in, which we're not, but if it's a one dimensional world, doesn't matter how we draw it, as long as it's one dimensional.

But assume that it has the properties sort of as an optical fiber, that light only propagates internally along it, and no such thing as messages are getting off the axis and back onto it, that everything that takes place, takes place on the line, or on the line

segment. With that stipulation, there would be no difference between this and the straight line segment. There would be identical, yeah, yeah.

No, two dimensional surfaces cannot be, in general, cannot be deformed without stretching them to make them flat. It cannot, okay?

The curvy line segment here, the right term is that the intrinsic curvature is zero, the extrinsic curvature is not zero. Extrinsic curvature means what you naively think about when you say this is a curved line.

And it has to do with the way the line is placed into higher dimensions. But, as you said, from the point of view of little creatures that live on this line and can't see anything off the line, they have no experience off the line, no experience of anything, and so forth. As far as they're concerned, there's only the line.

So that's the way you should think about it. And not ask whether the line is embedded in the... Now, you know, if these creatures could be fooled, they might literally live on a optical fiber and think everything is moving along that optical fiber.

But no doubt there would be some things that they could do. For example, they could invent a laser within their own fiber which emitted high enough frequencies, gamma rays, and I assure you that gamma rays do not stay

in an optical fiber. They will jump across. So it's conceivable that these creatures could discover that they really are living in a bigger space with more dimensions. I don't think that's going to happen with us. Nothing like that.

But the experimental facts at the moment tell us that we're living in a three-dimensional world, and intrinsic geometry, the intrinsic geometry being one

of three kinds. Right. Right. Right. That is a good...

That's right. That is a good... Right. That's exactly right. And it's a useful example. A piece of paper that you take and curve it like this is no less flat from the intrinsic point of view than

the flat piece of paper that you started with. All the relationships within the surface are the same as they were before you bent it like this. So that's as flat as... But this is not true of a sphere or a hemisphere.

I'll take a hemisphere. You'll notice that this can easily be flattened without stretching it. This cannot be flattened without stretching it. So two-dimensional surfaces can be curved. One-dimensional surfaces or one-dimensional lines, curves, cannot be curved.

I always have difficulty with converses. Let's see. A gravitational field implies curvature.

Yes. Yes. Yes. Yes. As I mentioned to you before, curvature is a measure of tidal forces. And tidal forces have to be due to something.

In general relativity, they have to be due to masses, and therefore they are the tidal gravitational fields due to some masses. Yes. OK. That was a long but very, very, I thought, hopefully

helpful to you. Longer question period. OK, let's move on now. Let's accept that space is isotropic and homogeneous, and therefore is one of the three candidate type

spaces. Give them names, very colorful names. The colorful names are K equals one, K equals zero, and K equals minus one. K stands for curvature.

Curvature equals zero, that's flat space. K equals one, that's positively curved space, and it's the analog of a sphere. But of course we are not talking about a two-dimensional sphere. Space is not a two-dimensional sphere.

It could be a three-dimensional sphere. Now again, keep in mind, when I say that space is either flat or a sphere or a hyperbolic plane and it's completely homogeneous, that of course is not completely true. It's definitely not completely true. It could be true for the average properties of the space

over distance scales of big enough to average things out. It's a statement comparable to saying the Earth is a sphere. Or the surface of the Earth is a sphere. But of course the surface of the Earth has bumps on it, it has mountains, it has valleys, it has this and that.

So it's certainly not a perfect sphere. And however, how big a distance do you have to think about before mountains, valleys, all that sort of stuff average out? 50 miles?

Let's see, Mount Everest is what, seven miles high? OK, so on a scale of 100 miles by 100 miles, the Earth looks pretty flat. On a scale of 1,000 miles by 1,000 miles, it looks very flat. So this idea of whether something is flat or not is a scale-dependent idea.

Now of course, the Earth did not have to be flat even on scales of 1,000 miles. It could have been shaped like a cigar. Then on no scale would it have been thought to be flat. So there's some content in saying that the Earth on big enough scales is homogeneous and isotropic, the surface of

it, and that it looks spherical. Same is true of cosmology. OK, so now let's, I think we discussed last time a little bit, I think we started to discuss space-time geometry. We're now moving from Newton, once we start to talk about

curved geometry, of course we've moved away from Newton, and we're doing relativity, general relativity. General relativity starts with a metric. It starts with a metric, and we are going to make an assumption about the space-time metric.

We're going to make the assumption that space and time aren't mixed with each other. Aren't mixed with each other in the metric. In other words, the metric has a form that looks like this, minus dt squared.

Remember, in relativity, the time component of the metric always comes in with a minus sign, and the space component of the metric comes in with a plus sign. Some scale factor, which depends on time in general, has

to do with the expansion of the universe and so forth, times a metric of one of three types. One of three types, k equals, I erased k equals one, k equals zero, and k equals minus one, but the three metrics which can be here are, first of all, k equals

zero. That's flat space. Just plain old dx squared plus dy squared plus dz squared. That's k equals zero.

And as far as we're concerned now, that's just its name, k equals zero. There is k equals plus one. That means the positively curved space. And we can either write that it is d omega three squared, or we can write it out in detail. We can write it out that it's equal to dr squared plus sine

of r squared d theta squared. Oh, sorry, wait a minute. No, no, no, I made a mistake, didn't I? d omega two squared.

So that's k equals one. We could have also written this one, the flat space, in a similar form, still doing k equals zero, we could have

written it as dr squared plus r squared d omega two squared. This is three-dimensional polar coordinates. Same space, but in three-dimensional polar coordinates, where r equals zero is your location, and omega two is just the angular world around you.

This is also just k equals zero. Same thing, no change. And finally, k equals minus one, which is the hyperbolic Escher angels and devils world, except in three

dimensions, and that's dr squared plus hyperbolic sine squared of r times the same d omega two squared. And they all look similar, but qualitatively they are, well, especially quantitatively, but qualitatively they're fairly different.

This a of t squared is called the scale factor. The distance between fixed points, by fixed points I mean points with fixed coordinates in space, the

distance between two points will in general be proportional to a times some characteristic difference. For example, on the three-dimensional sphere, there might be some angular distance here, some, I don't

know, let's just call it delta, delta of some angle. But this delta of angle is just some fixed value, which is the distance on the unit sphere. Likewise, the velocity between those same two points is the same thing, except time differentiated.

The ratio of the velocity to the distance, Hubble's law, v equals a dot over a times the distance, just dividing these two. And the thing about a dot over a, that's not a

constant in general, it could be a constant, by constant I mean independent of time, could be a constant, but in general it's not a constant, but it doesn't depend on position. That's the sense in which the Hubble constant, a dot over a, is a constant. It doesn't depend on where you are, although it may

depend on when you are. OK, so there we are with the basic geometry, the basic setup. Now what we want to do with it, now we're all set up. Last quarter we learned about Einstein's field equations.

We learned that they're complicated as hell, that actually writing them down in detail is a real pain in the butt. And we're not going to write them down, but write down the general form of them.

But how do you do it? You write down the metric, in this case the space-time metric, full space-time metric, incidentally in case you were wondering, this multiplies this, was that clear?

Yeah, OK. Good. You write down the space-time metric, you calculate the Einstein tensor, you set it equal to whatever it's supposed to be set equal to, and out of that operation comes equations for how A varies with time. That's the goal.

Now, how do you calculate the Einstein field tensor on the left-hand side, the energy momentum tensor on the right-hand side, Einstein's field equations, you have to calculate some Christoffel symbols, that's a nuisance, they have derivatives of all kinds and so forth, and

there's a lot of Christoffel symbols, in this case there aren't too many actually, but it's a nuisance, and I'm not going to do it on the blackboard. We're just going to outline what the basic idea is, and then I'll mentally plug in the Einstein field equations

and spell out the answer. Not the answer for what the geometry of what A is, but what equations A satisfies. Okay, so let's just remind ourselves, the Einstein field equations have a left-hand side and a right-hand side,

like any equation. The left-hand side has to do with geometry. If you remember, and if you don't remember, it's not going to be terribly important, but if you remember, the left-hand side is called the Einstein tensor, it's built out of the curvature tensor, I'm not going to go into detail

about what the curvature tensor is, it's got the Ricci curvature, it's got two indices, it's a tensor, and I chose the indices to be upstairs indices, that's some curvature tensor, and minus one-half the metric g mu

nu times the scalar curvature R, and as I said, it will not be terribly important what the details of this is, just notice the left-hand side has the curvature tensor, and the right-hand side has, anybody remember what's on the

right-hand side? Anybody remember? The energy momentum tensor, but it also has eight pi g

divided by seven, right? No, you know it's not seven, it's three. I'll just remind you, this was the same factor that appeared in Newton's equations, and the energy momentum tensor T mu nu.

Both sides are a tensor, therefore if it's true in any frame, it's true in every frame, and this is a good tensor law of physics. First question, let's start with the right-hand side. T mu nu contains a complex of things, which include the

density of energy, the flow of energy, the flux of energy, the density of momentum, and the flux of momentum, different components of it. In particular, the time-time component is the one that

we're going to fix on. The time-time component, eight pi g over three, times T time-time, T naught naught. The time-time component is the energy density. So let's call the energy density rho, we've been

calling it rho previously, let's just call it rho, it stands for the energy, the ordinary energy in matter, whatever kind of material this energy momentum is describing. Incidentally, the right-hand side is completely sensitive to the kind of material that's in the universe.

Is it particles? Is it electromagnetic radiation? Is it something else? It knows about the material nature of the ingredients that are making up the universe. The left-hand side has nothing to do with it, the left-hand side is geometry.

So on the right-hand side of the time-time equation is the energy density, and on the left side is something that involves curvature. Now if you go back to the definition of the curvature, work it out, you'll discover that there are two

contributions to it, one of which has second derivatives with respect to the coordinates, and the other has first derivatives squared, or quadratic things in the first derivatives. Fortunately for us, this particular combination, when

you take the time-time components, only has one of the two. The things with second derivatives, that you have to differentiate the metric twice, these things are made up out of derivatives of the metric, the things which involve differentiating the metric twice actually cancel

between these two for the time-time component, not for the space components, but for the time-time component. So on the right-hand side here, the left-hand side, excuse me, things are strictly proportional to squares of first derivatives, quadratic things of first

derivatives. That's one thing. The second fact is that the energy momentum, that the Einstein tensor here has two contributions. One comes from derivatives with respect to space.

The other comes from derivatives with respect to time. The things which have to do with only derivatives with respect to space couldn't care less that there's time in the problem. What do they have to do with it? They have to do with the curvature of space, just the curvature of space itself.

Here it is. This has positive curvature, this has zero curvature, and this has negative curvature. So the curvature of space is one contribution in here. The other has to do with the way space is changing with

time, but the only way in which space changes with time is through the factor of A of T here. So there's going to be one factor which will have to do with time derivatives of A squared. One term in this equation here will have to do, will

have a factor of A dot, some kind of factor with A dot squared in it. And if you work it out, guess what you find? It's A dot squared over A squared. Bit of a nuisance, but you can calculate it. A dot squared over A squared.

The other term which has to do with the curvature of space itself comes in with a plus sign. And think about for a moment how curved a space is as a

function of its radius. If the Earth were 1,000 times bigger than it is, I think we would all agree that it would be less curved, at least locally, that it would look flatter to us. If the Earth was a marble, it would be small, but its curvature would be large.

The curvature of a surface scales in a certain way as its radius, and in fact it's one over the radius squared. The curvature of a sphere, for example, is one over the radius squared, and it either comes in, well, it's

proportional to one over A squared, and it either is positive if space is a sphere, zero if space is flat, minus one if space is a hyperboloid.

So it means that there's a K here. And K is plus, minus one, or zero. So it's just a placeholder here that tells you which of the three kinds of spaces you're talking about. This is what Einstein's equations boil down to.

In fact, let's just switch this to the other side. Let's just switch it to the other side so it becomes minus K over A squared. But this equation is absolutely identical to the

Newtonian version if we think of rho as the mass density for a, it's the equations that we've already explored. The only thing new is that we now have an interpretation of this term. Do you remember what this term stood for in the Newtonian example?

It stood for the energy, whether the energy was positive, negative, or zero. It had to do with whether you were above or below the escape velocity. That's the same exact term here, but it has to do with

something entirely different or something on the face of it very, very different, namely the curvature of space. Same equation, somewhat different physical interpretation. You might ask, why does this general theory of relativity, which contains, among other things, the

ingredients of special relativity and so forth, why is it the same as Newton's equations? And the reason is basically that if you look, let's suppose the universe is curved, but we just look on a small little piece of it. A small little piece of it, we can't tell that it's

curved, but look at all the galaxies in there. The way these galaxies move for a small enough piece here must be the curvature can't be important for a small enough piece. Once we realize that the curvature can't be small and

that the equations are exactly the same for a small piece or a big piece, we realize that we must somehow reproduce Newton's equations, but in any case we do. The Newtonian version is correct, but we do have to

keep and remember that in Newton's physics this stood for the mass density. I'm setting the speed of light equal to one, so mass E equals MC squared is just E equals M. Energy and

momentum, same units. This was the density of ordinary mass, ordinary rest mass. It was assumed in Newton's equations that everything is moving much slower than the speed of light, and if I have a collection of particles all moving much

slower than the speed of light, the energy density is just the density of mass. On the other hand, these equations are more general. They follow from Einstein's equations, and for example they do apply to situations where particles may be moving fast relative to each other. In fact, they even apply if the energy density here is

due to photons, massless radiation moving with the speed of light. So the Newtonian equations wouldn't know what to do with photons. The Einstein equations know what to do with photons or radiation, but otherwise the equations look very similar.

OK. Yes. Two questions. In that metric where we assume that space and time are separated, does that correspond to some physical

restriction, or is that just the way things are? Yeah, it's a consequence of isotropy and homogeneity. If it's isotropic, and it stays isotropic, and

homogeneous, and it stays homogeneous, there really is no alternative. We could prove that, but I won't prove it tonight, but it is a consequence of that. You could actually choose any of them. For example, you could choose the space-space component.

What you would get would again be like Newton, except instead of being, this is the energy equation from Newton, is the F equals MA equation. The F equals MA equation does have second time derivatives. It has A double dot.

There are a linear combination of the space-space and the space-time and the time-time equations, which instead of looking like the energy equation, really do look like the Newton F equals MA equations. But they're all equivalent. The point is they're all equivalent.

They better be equivalent. Why is it that you don't need more than just one of them? Well, for this simple example, there's only one function to calculate. It's A of T. More general context where geometry may be wavy and fluctuate and do other things,

you may need all of the equations because there are just a lot of functions to compute. Here, for this case, the only unknown is A of T. It's only one function, and so really you only need one equation. You could have picked any one of them. Good.

You could not have picked the mixed space-time equations. Then you would have just got zero equals zero. But if you take the space-space components, you'll get the same equations back. Okay.

Let me just remind you one more set of facts, and then I want to discuss, so we'll take a little break, and we'll discuss the equation of state and how it determines information about this rho.

What do I mean by information about rho? I mean how rho itself varies as a function of the scale factor. There's not much you can do with this equation unless you know something more about rho. Well, we do know more things about rho.

For example, if rho is just made of ordinary particles just sitting there, and the universe expands, it's quite clear that the density of energy decreases, and we even know how. I'll write it down in a minute. If it's radiation, it decreases in another way. So how rho depends on A depends on the nature of the material that's making up the energy.

But it's clear that in order to solve this equation, to even think about it as having any content, we have to know something about how rho depends on A. If we know how rho depends on A, then this just becomes an equation,

an ordinary differential equation for A as a function of time. So I'm just going to write down now. We're going to take a little break. But before, I want to write down two examples. The two examples are the ones I just mentioned. Here they are.

They have names. One of them is called matter dominated. And it simply corresponds to ordinary particles moving slowly relative to the mesh. Particles which are not moving so fast that we have to worry either about relativity or even kinetic energy very much.

Their energy, if you're standing next to one of these particles, its energy relative to you is simply its mass. It's called matter dominated. And it's the case where the energy density rho is equal to some constant rho naught, let's call it, divided by A cubed.

Incidentally, what is A? What is the meaning of A? Let's take the case, let's take the spherical case.

In the spherical case, the meaning of A is extremely clear. It's the radius of the universe at any given instant, the radius of the sphere. Let's take that case. A has a definite meaning. Of course, we have to provide some units. We could measure meters, let's say in meters.

Then in meters, A is simply the radius of the universe. And what is rho naught? Rho naught is just the density at the time when A was equal to one. At the time when the universe had a radius of one meter, now, we don't want to go that far, really.

Maybe a mega parsec would be a better idea, but just conceptually, conceptually, the meaning of rho, the meaning of rho naught is that it is the density, whatever that density was, when the universe was one meter large. That's a tunable thing.

You can change it. It's sort of initial conditions. So, rho is a constant, but every time you double the radius of the universe, or change the scale of the universe by rescaling A, the density changes by the cube power of A.

So that's straightforward. Question? Yeah.

Three times as large as the initial one. Okay, so that's... In the flat case, which is infinite, A itself has no invariant meaning. Nothing to compare it with. In the round case, you can compare it with the radius of the whole universe.

Also, in the negatively curved space, there's a natural definition of the radius of a hyperbolic geometry. Okay. The other case that we talked about was radiation dominated.

Rho is rho naught divided by A fourth. And we talked at length about the difference between these two.

The fact that if you had a bit of both, that very early on, this would be dominant, because one over A to the fourth is likely to be bigger than one over A cubed. Late times, this is dominant. We talked about that. No change. Everything is exactly the same as before.

But one new piece of information. If you remember, the sign of K here determined whether the universe is going to continue to expand linearly, or whether it's going to re-collapse.

So what we find now is a correlation. Incidentally, this will change a little bit when we come to talk about dark energy. But up till now, with these forms of energy, if K is positive, that's the sphere case, that corresponds to the situation where the universe re-collapses.

If it's flat, then it's as if every galaxy was exactly at the escape velocity, so it continues to expand, ever slowing down and slowing down, and asymptotically coming to rest, but it doesn't re-collapse. But it's sort of a knife edge.

And the last case is K negative, in which case that corresponds to being above the escape velocity. And in that case, at late times, A just continues to increase linearly as a function of time. Alright, so as I said, we didn't waste our time by doing the Newtonian case.

The Newtonian case and the Einstein case are very, very correlated. Let's take a break for a few minutes. Alright, we want to understand these kind of equations better.

If the only possibilities were matter-dominated and radiation-dominated, we might not care very much to give a more general understanding of these equations.

What's the point of a general understanding when there's only two cases? But of course, there are many cases. There are many things in between matter-dominated and radiation-dominated, and not only in between, but more extreme. There's a whole range of possible behaviors like this, and they are important.

They're important to understand. And so we want a deeper understanding of the connection for different kinds of material. Let's call them material, for different kinds of material,

of how the energy density would change as a function of scale factor. What is it that you need to know? Now we're going to go through this in two steps. We're going to go through it in two steps. I don't know that we'll finish tonight, but the important ingredient,

the important ingredient in determining how rho depends as a function of A is the equation of state, is what is called the equation of state.

We'll have two steps. The first will be to assume an equation of state, and I will tell you exactly what I mean by an equation of state. To assume an equation of state, and from that derive equations like that. The second, so the first step will be to assume an equation of state

and derive the appropriate formula of this type. The second part will be to derive for different kinds of materials the equation of state. I don't think we'll get to both of them tonight.

Tonight I'm going to assume the equation of state for what? For different, we're just going to write down equations of state, and I'll tell you what they correspond to, but the next time we will derive the relationship between the equation of state and the kind of material we're talking about. All right, what do I mean by an equation of state?

Well, an equation of state is basically a thermodynamic idea, and it's the relationship between thermodynamic variables describing a system. For our purposes, temperature will not play any big role.

Temperature is not going to be the important thing. The important thing is going to be pressure and energy density, and in particular, the equation of state is a relationship between energy density and pressure. Now, even for ordinary, even for an ordinary gas,

it is quite clear that there's a connection, ordinary gas now, of moving particles. It's quite clear that there's a connection in this room, for example, between the energy density and the pressure.

The higher the energy density, and what energy density are we talking about? Basically the kinetic energy of molecules. There's the MC squared, which is dominant by far, much bigger than anything else, but let's forget the MC squared for a moment just to get the idea. There's the kinetic energy of motion of the molecules,

and the kinetic energy is proportional to the square of the velocity and so forth. Those same molecules bounce off the walls of the room and exert pressure. It's pretty clear that the faster the molecules move, the bigger the pressure on the wall is going to be, so it's clear there's a connection between energy density and pressure.

And usually the way it goes, although there are some exceptions to it, is the higher the energy density, the higher the pressure. The examples that are usually studied most frequently,

and in fact these cover pretty much the ground of interest to cosmologists, can be described by very simple equations of state. The equation of state is the equation which tells you what the pressure is,

P, as a function of the energy density. And the usual equation of state that cosmologists study over and over is the equation of state that says that the pressure is a constant called W,

W is a constant, times the energy density rho. Now, in many cases, of course, it's not really true that the pressure is a strictly linear function of the energy density, but as I said, just more or less by accident, the interesting cases have this form.

Pressure is some number times rho. I'll tell you right now what W is for the two cases of interest here. Later we will come back and derive it. For the matter dominated case, the matter dominated case is basically the case where the molecules in the room,

if we wanted to think about the room, are at rest. They're moving very slowly compared to anything else, or they're moving very slowly compared to the speed of light. The energy is mostly just the MC squared energy.

So in first approximation, you can just say the molecules are at rest in the room, and when the molecules are at rest in the room, the pressure on the walls is zero. The only thing that creates pressure is collisions with the wall, and so for the matter dominated world, the pressure is equal to zero,

or very, very, very small compared with the energy density. The big, huge E equals MC squared energy density and essentially a negligible pressure because things are moving slowly. All right, so W is equal to zero is matter dominated.

I don't need to derive that. That was obvious, that W equals zero is matter dominated. The harder case, which, as I said, I don't think we'll get to tonight, but next time, and it's an important thing to understand, is radiation.

And that case is W equals one third. Where did the third come from? Three dimensions.

Why does the number of dimensions come in? Well, okay, we'll work that out. Yes, it is three dimensions. Pressure equals one third rho for radiation, or W equals one third, and W equals zero for matter dominated.

Now, what does the equation of state have to do with anything? What I'm going to do is use the equation of state to derive how the energy density changes as a function of scale factor. And what do we do? We use the simplest kind of thermodynamic identities.

Supposing we have a box of gas, a box of material of some sort. It could be gas, it could be liquid, it could be whatever it happens to be. It could be molecules, it could be radiation, it doesn't matter. And it has some energy in it. The energy, of course, the total energy is equal to the energy density times the volume of the box.

So the box has volume V. The energy density is rho. And now let's imagine changing the volume of the box a little bit. It doesn't really matter whether you change the volume of the box

sort of isotropically in all directions or any other direction. But let's imagine changing it equally in all directions. Then how much does the energy change? What's the change in energy if you change the volume by amount dV?

Anybody know the answer? Not from this equation. From the work done by the pressure on the walls of the box.

PdV. Pressure dV. But is it positive or negative? If the volume expands, what happens to the energy in the box? It goes down. You do work, it does work on the box. And therefore the work on the box means a diminution of the energy in the box.

So it's minus PdV. That's all we need to use. That's enough, that's the basic identity. There's another term. Anybody know what the other term is?

For thermodynamics? TdS. Temperature times the change in entropy. When variations are slow and the universe is expanding slowly by comparison with any other kind of time scale in the gas,

when changes take place slowly, entropy doesn't change. That's a rule called adiabatic change. For our purposes, the change in entropy is zero, and this is the formula. dE equals minus PdV. Okay, now the energy is rho times V.

Let's calculate dE. We just take the differential of energy, and that's equal to rho times the change in volume plus the volume times the change in rho.

Both are changing. We change the volume. Changing the volume will change the energy density in some way, in some as yet unspecified way. Changing the volume will change rho, but we don't know how yet. However, whatever the rule is,

dE equals rho dV plus Vd rho. That's the left-hand side, dE. On the right-hand side, we have minus PdV. Well, we have two terms here with dV. Let's group them together.

Let's put them over on the left-hand side and keep them together because they both have the same differential dV. On the left-hand side, we have Vd rho, and on the right-hand side, we have two terms. First of all, a minus sign. Let's put a parenthesis around it and a dV.

What goes inside the parenthesis? P and rho. P plus rho because this has to go over to the other side. P plus rho. But now, if we didn't know any connection between P and rho, we'd be kind of stuck.

We wouldn't know what to do about it with this. But let's assume that P is known in terms of rho. And in particular, let's take the very simple case where P is W times rho. Now we can write this as minus 1 plus W rho dV.

1 is just the rho dV. The W is the PdV. So now we have something we can work with. We have an equation involving two variables.

It's going to be a differential equation, volume, and energy density. Let's rewrite it. Let me just rewrite it on this board for a minute. Vd rho is equal to minus 1 plus W.

Now that's just a number. 1 plus W is just a number. Rho dV. And now let's regroup the equation so that on one side I have everything involving rho, and on the other side I have everything involving volume.

To do that, I just divide by rho to get all the rho dependence on the left. Let's do it. D rho over rho. And divide by volume, we've got rid of rho on the right. Now I divide by the volume,

and here's our equation. D rho over rho. What is that? Indeed, it's the differential of the logarithm of rho. D rho over rho is the differential of log rho. What about dV over V?

Differential of log V. So, we can integrate this equation, and it just says that rho is proportional, sorry, it says that logarithm of rho is equal to 1 minus 1 plus W

logarithm of V. You can add a constant, a numerical constant to it. Let's just put the constant over here.

Now, how do you solve this for rho in terms of V? You just exponentiate it. What does it say? It says that rho is equal, the minus sign here gives you a 1 over volume to the power 1 plus W.

With again, a constant up here. Different constant. Just take the logarithm of both sides, logarithm of rho on the left, and on the right, minus 1 plus W times log of V.

But volume, the volume of this box, if we think of it as a box which is expanding with the general expansion of the universe, the volume of the box is just basically A cubed.

It's proportional to A cubed. Apart from a constant, it's proportional to A cubed. So, what this is telling us now is how rho varies with the scale factor. Let's write down the equation. It says that rho is equal to some constant, which we'll worry about another day, divided by A, the scale factor,

to the 3 times 1 plus W. That's what thermodynamics, or that's what the thermodynamics of a nice homogeneous material would tell us. That's how rho varies with scale factor.

Constant being some constant. And again, you can say that the constant is just the value of the energy density when A is equal to 1. Okay, let's see what it says. Supposing W is equal to 0. That's the case of matter dominance. If W is equal to 0, that's the top case up there.

This just says rho is equal to a constant over A cubed. That's this. What if W is equal to one-third? 1 plus one-third is four-thirds.

Four-thirds times 3 is just 4. So, that's this equation over here. So, you see, there is a general framework in which these things emerge, and really what you need to know is W. To describe a cosmology based on some sort of energy,

what you need to know is the equation of state, and very little more, there are very little other than that, the equation of state in the form pressure is W times energy density.

Any questions? I think this is a natural place to stop because I think if we go past this, I'll overload you. Next, yeah. Go ahead. In the radiation case, what is the radiation mostly? Is it the microwave background?

It's mostly microwave background, yes. Almost all of it. By an overwhelming factor, it's the microwave background. Other photons, even sunlight and starlight, and only a tiny, tiny fraction of it. Yeah, right. So, it's almost all the CMB, the cosmic microwave background,

and today, at the present time, it's a very, very small fraction of the energy in ordinary atoms, which in turn is a somewhat small fraction

of the dark matter. So, most of it is dark matter. Some fraction of that, I don't remember exactly something, 10, 20% is protons, neutrons, atoms, and so forth, and then a very, very tiny fraction of that, roughly 1,000 cubed, 1,000 cubed, a billionth of it,

1,000 times 1,000 times 1,000, yeah, a billionth of it is radiation today. But if you run it backward into past times, at some point, this becomes bigger than this,

and it becomes radiation dominated. So, the early universe was radiation dominated. Late universe today, if it were not for dark energy, if it were not for cosmological constant, it would be matter dominated. Right. And again, if we didn't worry about the cosmological constant,

we would say that this parameter k is correlated with the future history of the universe, whether it collapses or continues to expand. Now, I emphasize that that is wrong because of one other ingredient, and that's dark energy.

Well, I can tell you, all right, we've gone far enough that I can tell you a little about dark energy. If we don't care where the equations came from, then cosmological constant, cc, or dark energy, or vacuum energy, all the same thing,

cosmological constant, dark energy, vacuum energy, all the same thing, is w. Let's see what we get. W equals minus one. W equals minus one. A little bit odd, isn't it? That the pressure and the energy density

have the opposite sign. How odd is that, incidentally? Can you think of any situation in which pressure might have the opposite sign of energy density? Yeah, I can give you one. Just where the box, let's replace the box

just by a line interval now in one dimension, where the box, there's a box, it's just a line interval, well, there's a line interval, and the physics is that the two ends of the line interval are held together by a spring.

Crazy, that's not a reasonable description of energy density, but it does have the property that the pressure is negative. Why is the pressure negative? Because it's pulling together. It's not pushing apart. So the pressure is negative. We call it tension. When pressure is negative, it's called tension. That's a tendency to pull together.

But the bigger you stretch it, the more the potential energy. So it has the property that increasing the potential energy makes the pressure more and more negative.

No, usually pressure, it can, it can, it can. Yeah, it can under certain circumstances. It can go in that direction.

As things get closer and closer to absolute zero, you might think, well, it just corresponds to springs. Yeah, but there's also some quantum uncertainty energy, so it can go either way. But yes, it is possible for pressure to be negative, and it is possible for pressure to increase in the negative sense

as the energy density goes up. That's what W equals minus one means. Why such a tension should exist, that's another question, but let's just examine its consequences. If W is equal to minus one, let's see, here we are over here.

W equals minus one, what do we get? Rho is constant. Rho is constant. That's the nature of dark energy. It does not change when you expand the size of the box. That's the character of it.

The energy density in a box doesn't depend on how big the box is. Just changing the size of the box doesn't do anything to the energy density. It doesn't change the energy in the box, but it doesn't change the density. Why? Because that energy density is a property of empty space,

and empty space doesn't dilute when you stretch it. Okay, but let's just, well, I'll tell you what. I think I don't want to go through this tonight. Next time, we will examine more critically the reason why W's have these various values

and then study the behavior of the universe under the various kinds of conditions, in particular, study what happens if we have dark energy.

That raises new things that we haven't seen before. I think for tonight, I'll take on a couple of questions, but then you're getting tired. Dark matter as well as regular matter. Yes.

And black holes and everything else. Yep. Anything which can be regarded as non-relativistic particles, but a particle can be a black hole, it can be a planet, it can be a star, it can be anything that's just a chunk of material that is moving slowly relative to the ambient background

and whose energy is mostly in the form of its mass.

Absolutely. Same equation. Yeah, exactly the same equation. The only thing was that the interpretation of this had to do with being above or below the escape velocity. Other than that, the equation is exactly the same. I'll remind you, for the case where K equals zero, I remember the answer very good.

Very well, excuse me. In that case, for the matter-dominated world, A went as T to the two-thirds. Is that right? I think T to the two-thirds. For the radiation-dominated world, it went to square root of T. That's still true here, if this is zero.

That's the flat-space case. If it's not flat, then you have to correct it. But go back to the previous lectures, and these were the equations we studied. In order to have both a repulsive force between all objects and an attractive force like gravity,

I assume that the law regarding the distance between them has to be different. The law is different than what? I understand that gravity is an inverse square law. So what is the nature of the repulsive force that dark energy produces?

You can think of it just as an expansion of the universe, which we will work out. But it is, at least at the Newtonian level, you can mock it up by just saying that the gravitational force between every pair of particles has an additional term in it, which is linear with distance.

A force proportional to distance, a very small coefficient, and we'll go through that. But you can mock it up, you can mock up the effect of the vacuum energy of the cosmological constant by saying that every particle has a force with every other particle proportional to the distance.

If the cosmological constant is positive, then it's repulsive. And if the cosmological constant is negative, then it's attractive. Needless to say, if it's attractive, then it will cause the universe to collapse even faster

than it would have without the cosmological constant. If it's repulsive, it gives a chance for continued expansion even though the universe might be closed. So we'll work that out. Good. Okay. If there are no more questions, go home.

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