sábado, 26 de novembro de 2011

3. Space, Time, and Gravity - Sean Carroll - Dark Matter, Dark Energy: The Dark Side of the Universe

Sean says at the beginning that this is actually his favorite lecture. Now I know exactly what he means. I've never had the Einstein/Newton comparison made quite so clearly to me. The topic I always got tripped up on, is now made into something understandable. Relativity is now also my favorite subject, instead of the usual stumbling block. Sean is as much a physicist as cosmologist. If he can describe relativity so well, I can't wait for the actual astronomy aspects.

OK, enough of the hype. The lecture does proceed at a fast pace, not that he talks fast, but he goes through the concepts pretty quickly. I often wonder if any given lecturer could grasp all the information presented by someone else in a different field. Unfortunately the course guide does a rather poor job of conveying this lecture, so a second viewing wouldn't hurt, or reading this review might also work!

The concepts of Newton's "absolute" changing to "relative" in special relativity, and Newton's "fixed" changing to "dynamic" in general relativity is hammered home quite effectively. Special relativity is further clarified by contrasting Newton's concept of time being distinct from space, into Einstein's time being like space. Also Newton's limit of simply moving forward in time changes into Einstein's speed limit of light. General Relativity is clarified from Newton's concept of gravity as a force into Einstein's concept of space-time. The traditional representation of curved space-time as a marble on a sheet is used, but I've heard others critique this as flawed in the same way as Sean dissed the balloon and raisin bread analogies!

In any case, until now I have never seen a lecture actually spell out Einstein's field equation,

Rμν - ½Rgμν = (8πG)Tμν

The curved space-time equals energy and momentum, implying not only Newton's mass creates gravity but everything does. This infers we can use gravity to detect mass and energy, or the pull of dark matter and the acceleration of dark energy. This dynamic space-time allows one to rephrase the distance velocity relation as more space coming into existence between galaxies. New questions of why, how and where could be asked of space-time, although still not answered.

Sean Carroll conveys difficult scientific concepts at an appropriate pace. Compared to the recent lectures of Steven Pollock on Classical Physics and even Alex Filippenko's massive Intro to Astronomy, this course really sets a new level of standard in the TTC library of courses.

This is perhaps Sean's favorite lecture. We shouldn't hold him to this, since he may say that other lectures in the future may be his favorite! This is the lecture in which we take the expanding universe that we talked about last time, and we ask what it really means. We won't answer that question until the next lecture, but this question is going to take us into trying to understand how space and time work.

This is the real thing that made Einstein quite a famous international celebrity. He is the person who figured out how space and time work. He figured out how they're related to gravity. This is something in which Einstein did better than Isaac Newton, which is a hard thing to do!

So we'll go from what Newton did, up through Einstein, explaining along the way how we get there. It's really is a set of deep ideas about the structure of space and time that are counter-intuitive. So we'll be thinking pretty hard about why things are like that, and why they're one way than another.

So lets start just by thinking about what the ideas of space and time are supposed to be. Space and time are what help you locate things. Space tells you where things are, and time tells you when things are. So lets think about space to begin. Scientists like to say that space is three-dimensional. What does that mean? It means if you want to locate the position of an object, in principle you need to give three numbers. The way we can do that can be different, dependent on the circumstances. So for example, the three numbers might be the height of something, the length of something, and the breadth of something.

Once you get those three numbers, you've specified its dimensions entirely. If you only get two numbers, it wouldn't be enough. So we say that space is three-dimensional. Saying the same thing from a different way, if you want to meet someone for coffee, you need three numbers to say where to meet them. One way of thinking about that, would be latitude, longitude, and height above ground. Ordinarily we're lucky enough to know the height above ground for the people we're going to meet, yet you might tell them which of the two cross-streets you'll meet at!

So it takes three pieces of information to locate something in the universe, and that's what it means to say the universe is three-dimensional. Space is the arena in which things play out. Space is where you locate things, how two things come together, and how things are distributed all throughout the universe.

Time, on the other hand, tells us when things happen. So say we'll be meeting that friend of ours for coffee and give them the three numbers that say where we're going to meet them. That is not enough. We'll not successfully meet them if we give them only these three numbers, since we need a fourth number, which is when we're going to meet them.

So it takes four numbers to specify a unique point in the universe, the three numbers to tell us where we are in space, and the one to tell us what time it is. One could think about all these four numbers as being one, four-dimensional thing, called spacetime. For along time, people didn't think that way, because there was no point! Space is something very clearly different than time. Part of the revolution of relativity as Einstein understood the universe, is to realize that space and time are in fact two different aspects of the same thing, a single four-dimensional thing that we call spacetime.

So what we'll do in this lecture is start with Isaac Newton, who was probably without a doubt the greatest physicist who ever lived. One of the things he did was to put together a very sensible picture of space and time. In Newton's picture of space and time, they are both absolute, fixed structures of the universe. They are the stage in which the drama of physics plays itself out.

Now Einstein comes along with his collaborators also, since he wasn't just alone. By 1905 he put the finishing touches on something called Special Relativity. This was a replacement for Newton's notions of space and time. Special Relativity first said that space and time are different aspects of the same four-dimensional thing called spacetime. So in Special Relativity there was still a fixed structure, yet the thing that was fixed was one, four-dimensional spacetime, not separately the three dimensions of space, and the one dimension of time.

Then Einstein still thinks he's not done yet, and tries to incorporate gravity into his theory of Special Relativity, and realizes they are incompatible. He eventually throws away Special Relativity and replaces it with something better, which we call General Relativity. It's just as good as before, but now it also includes gravity.

Yet the most profound difference is that now, nothing is fixed. Nothing is given to you ahead of time in General Relativity. The very structure of space and time themselves are dynamical, so they can change. Spacetime has a geometry and a curvature which we actually interpret as gravity.

That's the short version of the story, so lets go slowly through the longer version. Here we see a picture of Sir Isaac Newton, who as we said, is arguably the greatest physicist and/or mathematician of all time. He invented calculus among other things, and among the pieces of physics he invented was the fact that when you pass white light through a prism, it separates into colors. He invented the law of gravity that we know and love, the inverse square law. The gravitational force between two objects goes down as they get further away. If their separation is doubled, the force of gravity decreases by a factor of four, which is the square of the factor of two for which we doubled their distance.

So this inverse square law of gravity will play a very crucial role in what is to come, but it's not what we'll be talking about in this lecture right now. What we care about today, right now, is what Newton said about space and time. So we think that space and time are so immediately obvious, that there's nothing more to say. That's where things happen, since space tells us how to locate things and time tells us when they happen. What more could one need to know?

One of the great features of the Newtonian revolution in physics, was being quantitative, attaching equations to things. It wasn't good enough to just say that space is where things happen. Now you needed to have some equations and mathematical structure. So Newton did that by telling us what it meant to say that space was an absolute, three-dimensional set of points, and that time is a different absolute one-dimensional set of points.

The basic picture of the Newtonian universe is shown here in a diagram. It's taking the four dimensions we need to slice up the universe, and slicing them up int moments of constant time. Sadly on a two-dimensional picture we can't draw all four dimensions of spacetime, so the two-dimensional planes that we've compressed diagonally in the picture, are supposed to represent all three dimensions of space.

So this picture should not be intimidating. It's a fairly straightforward representation of the universe, the universe according to Isaac Newton. We know better now, and we'll get to just why that is, but this is how spacetime worked according to Newton. There was space and there was time. So what we see here are little clocks, where each clock tells a different time located in a different moment of space.

So the Newtonian universe is one where space happens over and over again, at different moments in time. At every moment objects are going to be located in slightly different places. So we call that motion through space. Things change as a function of time. So that's how Newton's universe worked.

So the point we're emphasizing about Newtonian spacetime is that it is fixed and absolute. So what does that mean? Those are two different words. Fixed means it never changes. There is this structure that space and time are staying in, and don't change into something else. The rate at which time flows does not depend on where you are. There's one fixed thing called the flow of time, and it's the same all throughout the universe. That's what fixed means.

Absolute means it's the same for everybody. The rate at which time flows, or the distance between two objects, doesn't depend on who you are, where you are, or how you're moving through space. It's just a notion that for example, it took three seconds between this event and that event. Everyone agrees on the fact that it takes three seconds between these events, no matter where they are, or how they're looking at it, as long as they take accurate observations.

These notions will change, once we get to relativity. Absolute in the Newtonian sense, as opposed to relative. So the absolute fact that a certain object has a certain length, is going to turn into a relative fact in relativity. That means that the length of an object will depend on who is measuring it, possibly depends on where they are, and certainly depends on how fast they're moving. "Fixed" is what we'll get rid of when we go from Special Relativity to General Relativity, where spacetime is dynamical and can change, taking on different forms, no longer being absolutely fixed.

So to understand the change from Isaac Newton's absolute space and time, to Albert Einstein's relative notion of space and time, we have to understand how we'd go about operationally making sense of Newtonian spacetime? In other words it's one thing for Sean to say that there are moments of time that are the whole three-dimensional universe of space, and those three-dimensional pieces of space repeat themselves over and over again. Yet what does that mean and how does one make sense of it?

The way you do that is in something you see in spy movies all the time. You synchronize your watches. So when we say that there's a moment of time that extends throughout the universe, what that means is that you have a clock right here that says a certain time, and we can set up clocks all throughout space that agree with this clock. So when this clock says that it's 3:00, then it's 3:00 here and everywhere else! These are just a set of words. How do we make them real?

What we do is, first to set clocks throughout all the universe. That the easy part! We're doing thought experiments here, and these are not real experiments. We have a clock sitting at every point in space. Yet we have to synchronize our clocks, so we take the clock that we have here. We take for granted that we have good clocks, so these are clocks that run at the same rate. They are reliable clocks.

So we take the clock we have here and leave it. Yet then we take another clock and set it to say the same thing as our first clock, and then this second clock that agrees with the first one, we move throughout the universe. We move this clock as shown in the diagram from before, and move it to all the other clocks. When we get to another clock, we align them so they are reading the same time. By that procedure, we synchronize our clocks all throughout the universe. It doesn't matter what the clocks are doing, how they're moving, or where they are. When we do this, we take our one clock and make sure that every other clock, all throughout the universe, agrees with it, and then we are done. We've synchronized our watches as it were, all throughout the universe.

That's what Isaac Newton says happens. There's only one rule in Newton's universe, which says you have to move forward in time. In other words, if it's 3:00 here, then when Sean's watch goes one hour to the future, he will be at 4:00 no matter where he is in the universe, no matter what he's done in the meantime.

Then Einstein and his friends came along in the early 20th century and said, "Actually it's not like that!" We need to replace the absolute notions of space and time that Newton had, with relative notions of space and time. In particular the fundamental insight of Einstein can be summed up in the phrase, "Time is like space."

So what is that supposed to mean? Suppose you measured the distance between two points. You have a path that goes from one place to another. Well you have to actually go down that path, and measure the distance along it. That distance could be different, depending upon the path you take.

Einstein says that time is like that. The rate at which time flows, depends on what you do. This is really hard to get through your brain, because it's very non-Newtonian. In some sense, even though Newtonian mechanics is a great triumph of the human imagination, it's still quite intuitive and makes sense to us, once we understand it.

Yet Relativity is very counter-intuitive. Einstein says that the amount of time you feel elapsing is personal, so it's not a fact of the universe, it's a fact of what you do. It can be different for two different observers, even if they start and end at the same point.

So lets see how that works. Imagine that you do in Einstein's universe, that is to say the universe in which we actually live, what you tried to do in Newton's universe. You try to synchronize your clocks all throughout space. So you take the clock that you leave here, that's reading a certain time, you take your movable clock and make them agree, then send out your movable clock to synchronize all the other clocks all throughout space.

You think you've done it, but then you bring your clock back to where it started, and you notice that the two clocks don't say the same time anymore. It's not true that this clock, even though it's a perfectly good clock, clicked at the same rate as it traveled throughout space, synchronizing all the other ones. In fact, it turns out that in Special Relativity, there is no way to make that happen. You didn't make a mistake, since it's just a feature of space and time, that when a clock moves around, it measures a different amount of time than a clock that stays still.

This seems weird to us, because to make the effect obvious, you need to move close to the speed of light, which is the magical velocity in the theory of relativity, at which all these weird things become obvious. Since we're always in our every day lives moving much slower than the speed of light, we don't notice them. We think its very strange for someone to say that the amount of time that clicks off on our wristwatch, depends on how we walk, depends on the journey we take through space and time.

Yet once Einstein tells us that time is kind of like space, there is a way of thinking about it that makes perfect sense. So in a new diagram, we draw an analogy between distances and times, because Einstein says the two things work the same way. You draw two different curves that connect two points in space. One curve is a straight line and the other is a sort of curvy thing that goes back and forth.

Nobody in the world is surprised with the fact that the length along those two curves is different, even though they begin and end at the same point. Why should they be the same? They move in different ways! So Einstein says that time works in the same way.

Take two points in spacetime. Take two events, two points where you're both located in space and time. Two people begin at the same point, and they end at the same point, but they take different journeys. One just sits there not moving, and one zips around. In space, it was the case that the shortest distance between two points was a straight line. Any curvy path you went along, gave you a longer distance.

Einstein says that a very similar thing happens, with one interesting twist. The longest time that elapses between two moments is if you just sat there and waited for time to go by you. Yet if you zip off and come back, you will have experienced less time than your friend who stayed behind. The shorter time is the path through spacetime that zips around!

There had to be some difference, because we know time is not exactly like space, but Einstein is telling us that deep down, space and time work in similar ways. Just like the the distance around a path depends on how you move, the time that elapses on your wristwatch, depends on how you move through spacetime. So that's the fundamental difference between spacetime in Einstein's universe and spacetime in Newton's universe.

Yet remember there was a rule in Newton's universe, where you had to move forward in time. That rule gets replaced in Special Relativity, and in fact is more stringent than the one in Newton's universe. Newton's universe allowed you to move as fast as you wanted, as long as you kept moving forward in time. Yet in Einstein's universe, the speed of light is special. So if there's an event in spacetime, and you want to get somewhere else in spacetime, you can only plausibly get there if you move slower than the speed of light.

So if you imagine drawing a picture of spacetime, and you have an event where you imagine all the light rays that are leaving that event, those rays represent the ultimate speed limit. They represent the velocity faster than which you cannot travel. So there is no allowed trajectory through space and time, that moves faster than the speed of light. Everything you're allowed to do, moves up in that diagram, and eats up more time than space, if you like.

So that is the fundamental rule in Special Relativity. Newton's rule is one has to move forward in time, Einstein's rule is one has to move slower than the speed of light. That set of things, faster than which you cannot go, is called a light cone. The set of all points connected to your original event, by things moving at the speed of light. So the only thing you need to remember is that whatever the speed of light is, where you are in the universe, you need to move more slowly than that. That is the fundamental rule.

So this was a replacement as we mentioned, for Newton's concept of space and time. Newton was the greatest physicist who had ever lived. He invented not only this notion of space and time, but also the theory of gravity and many other things. So Einstein knew that if he's going to replace Newtonian space and time, he would have to reproduce the other successes that Newton had.

You had a very good theory of how gravity worked, which was able to predict how planets moved around the sun, or the moon around the earth. This theory of Newton's was verified to high accuracy over and over again by experiments. Yet here comes Einstein saying that it was not right. So clearly you have to reconcile the new theory of spacetime, Special Relativity, with what Newton thought gravity was. Gravity was the inverse square law, where the force was proportional to the mass of the thing that is pulling you, and inversely proportional to the mass between the two objects.

So after inventing Special Relativity in 1905, Einstein devoted himself to this task of trying to figure out how to reconcile gravity with his new notion of spacetime. It took him ten years to do it, and the way he ultimately did it was he realized you needed to throw away what Newton had said about gravity. The way that gravity was going to work in relativity, is just very different than how it works in Newtonian mechanics.

So here we see a picture of Einstein, which we've all seen before. Yet usually these are when he was in his older ages, and when he had let himself go a little bit! He's got the hair going up and he's wearing a sweatshirt for days and days! This is a picture from 1912 when Einstein was young, a sharp dressed man, and somebody was combing his hair. He was thinking about the nature of space and time and how to make it compatible with gravity. Ultimately he hit on a thought experiment that provided the key to understanding how gravity would work in the context of relativity.

The thought experiment actually traces itself back to Galileo, the famous Italian physicist who lived just before Newton, who really invented a lot of the framework that Newton made sharp and quantitative and attached equations to. Galileo was one of the best physicists of all time. One of the many thing he did, was to understand that gravity was universal. The famous experiment that he may or may not have done, he probably didn't but serves as a nice visual image, was to drop objects off of the leaning tower of Pisa, and notice that no matter what the objects are made of, no matter what the mass is, they always fall at the same rate. In other words, gravity as a force has a very strange characteristic where everything responds to it in the same way.

So that's different from, say, the force of electricity where the electromagnetic force has things called positive charges and negative charges. They move in different ways and in opposite directions in an electric field. Yet gravity has everything moving the same way. It has this feature of universality where everything responds to it identically.

So Einstein, because he's smarter than us, realized that this was the key to understanding how gravity works, the fact that it works the same on everything. What that meant was that gravity was undetectable! That's a bit of a surprise to say this, since it seems pretty obvious that if we let something fall, we see the force of gravity at work.

Yet what Einstein says was that imagine you're in a box (or elevator) so that you can't see outside. Someone asks you the question if there was gravity in that box, or in that room? Well you can drop things and see them falling, so you say yes there is gravity.

Yet Einstein says that you might be very far away from any gravitational field. It might be that you're in a rocket, being accelerated at a tremendous rate through interstellar space where there's no gravity around. In that rocket, if you drop things, they will fall in exactly the same way that they would fall if there were gravity! This is not true for electricity, since we can detect an electric field by dropping a positive particle and a negative particle, then seeing them move in different directions. Yet since everything moves the same way under the influence of gravity, you can never be sure in a small region of spacetime inside a tiny little box, if there is actually any gravity at all!

So again, to us we might just say, "Well that's interesting." Yet to Einstein, he says, "That's the key. If everything responds to gravity in the same way, then gravity is not a force at all, but a feature of spacetime itself." In fact, that very feature of gravity, is the curvature of spacetime itself. Einstein's most brilliant insight was to say that spacetime has a geometry.

By 1915 he came up with General Relativity which described how the geometry of spacetime, how the curvature of space and time itself, manifests itself in what we observe as gravity. So according to Einstein we have a picture of something like the following, although it's a somewhat colorful conception of what is going on.

We have the sun that is warping the space around it. The usual analogy used here, which is pretty good, is that you put a bowling ball in a rubber sheet, that distorts the shape of the rubber sheet around it. A marble that you rolled by the bowling ball would then be deflected by the curvature of the rubber sheet.

Einstein is saying is that this is what's happening in spacetime. That is what gravity really is. The gravitational force that we attach to the sun, is really the warping of the geometry of space near the sun. Earth is just doing its best to move in a straight line. Yet there are no straight lines because the geometry itself is curved. So what we see the earth do, is to orbit the sun. This orbiting of the earth around the sun, is just the earth's response to the geometry of space and time. This was Einstein's brilliant idea.

Now again it's not enough to just say words like that, since you need to make it quantitative. So Einstein has an equation known as Einstein's equation. This is not E=mc², although it's a very good and famous equation that is Einstein's, which we'll talk more about later, but what is known to physicists as Einstein's equation is the equation for gravity. We see it here, again not because we're going to understand it in any great detail, but it's nice to see just what it looks like, to get an appreciation for the art and poetry that are these physicist's equations.

Rμν - ½Rgμν = (8πG)Tμν

We see a left-hand side and a right hand side, so that part is easy. The left-hand side explains the way in which spacetime is curved. It's a set of numbers, in fact it's a 4x4 matrix, a little array of 16 numbers whose values are telling us by how much spacetime is curved. If spacetime is flat, just like a tabletop with no geometry, then all those numbers will be zero.

On the right-hand side, we have stuff. We have (8πG) which are just constants of nature, just fixed numbers. Then we have Tμν which is the energy and momentum of the stuff in the universe. Anything that is in the universe, comes along with energy, some substance, some heat, some pressure and things like that. Einstein is telling us in very specific equations that all of those things contribute to the curvature of spacetime.

Newton was telling us was that what makes gravity is mass, yet Einstein is telling us that what makes gravity is everything, every form of energy which includes mass but also includes heat, pressure, temperature, and stress. All of these contribute to gravity in some way.

So this equation, Einstein's equation, is the one that should be famous. It's a little bit more intimidating than E=mc², a little bit less immediately relevant to our lives, but it's a tremendous accomplishment to conceptualize gravity as a feature of spacetime itself, as a feature of the curvature of space and time.

So we're all allowed to ask, "Who cares?" Why should if matter that we use the words "gravity is the curvature of space and time" rather than "gravity is a force stretching through space and time?" Is there any difference between these two sets of ideas? Well there are differences, and they will make a big difference to us in our quest to understand cosmology, dark matter, and dark energy.

The first implication of the claim that gravity is the curvature of spacetime, is that universality, the fact that everything responds to gravity in the same way, implies as its converse that everything creates gravity in the same way. There's a universal coupling between the curvature of space and time, and the stuff in the universe. In other words, you can't hide from gravity. If there is stuff in the universe, if it has mass and energy, if it exists in any substantial or physical way, it will give rise to a gravitational field.

So Einstein is giving us a surefire technique for detecting absolutely everything in the universe! There is nothing that can hide from us, as long as we can detect gravitational fields throughout the cosmos. You can't hide from gravity, so you can go the other way by detecting gravity, and can infer that there must be stuff. If you see a gravitational field pointing in some direction, there must be some stuff there, causing that gravitational field, even if you don't see the stuff.

This will be the technique that we use to infer the existence of dark matter and dark energy. Dark matter we will say, must be there because we will see stuff being pulled in some direction where there's not enough ordinary stuff to explain it. Dark energy has a more subtle effect, which is that it makes the universe accelerate. We'll talk about how this acceleration of the universe is a manifestation of the curvature of spacetime in the way that would be caused by dark energy.

The second implication of the motto that "gravity is the curvature of spacetime," is that spacetime is dynamical. Spacetime can change as a function of time. It need not be the same in the past as in the future. So it gives us a changed viewpoint on how we would think about the fact that galaxies, for example, are moving away from us.

Hubble said that we have a velocity for galaxies that is larger if they are further away. The same phenomenon makes sense according to Einstein, yet we attach slightly different words to it. This is why, when we're being careful, we refer to the apparent velocity of distant galaxies, not the actual velocity. In the real, careful way of understanding things, a la Einstein, the galaxies aren't moving. They are sitting there, located in space. What's happening is that space is expanding. Spacetime itself is dynamical and can change, so what's happening is not that the galaxies have a velocity through space, but that the space in between galaxies is growing, stretching, getting bigger. More and more space is coming into existence.

This is a way of thinking about things that Einstein gives us, that allows us to ask questions which just wouldn't have made sense to Isaac Newton. We can ask for example, where did space and time come from? What happened at the beginning? Did space and time get created, or is there something before what we know think of as the Big Bang?

To Newton, with an absolute, fixed, spacetime, these kinds of questions really don't make sense. You don't even ask where spacetime came from, because it was always there! General Relativity is saying that space and time are dynamical and can change. You're at least allowed to ask the question of why they're here, and why did they come into existence? We won't be surprised to hear that we don't yet know the answers to these questions, yet we're hopeful that our understanding of dark matter and dark energy would be part of the clues that help us answer them eventually.

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