ASTR 160: Frontiers and Controversies in Astrophysics

Lecture 11

 - Special and General Relativity (cont.)


The lecture begins with a comprehensive overview of the historical conditions under which Einstein developed his theories. Of particular impact were the urgent need at the turn of the 19th century to synchronize clocks around the world; Einstein’s position at a patent office; and a series of experiments that he himself carried out. In 1905 Einstein published three papers that are still considered the greatest papers in the field of physics. The lecture then moves to General Relativity and how it encompasses Newton’s laws of gravity. A visual demonstration shows how space-time undergoes curvature when mass is introduced. Class ends with a question-and-answer period on a variety of topics in Special Relativity.

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Frontiers and Controversies in Astrophysics

ASTR 160 - Lecture 11 - Special and General Relativity (cont.)

Chapter 1. Lessons from Einstein’s Discovery of Relativity [00:00:00]

Professor Charles Bailyn: We were talking about relativity. And where we ended up last time–we were talking about special relativity and all the weird things that happen to space and time. And the question arose, you know, what prompted Einstein to think these thoughts? And we talked a little bit about the fact that the speed of light is always the same no matter what the velocity of the person observing it and how that–then, if you take that literally, it tells you that space and time are weird, and are not constant and absolute the way you might think they are in a Newtonian sense, but vary according to the velocity of the observer and in these various ways having to do with that parameter gamma.

And so, Einstein’s great stroke of genius was not to try and fix up these little problems with the experiment in some, sort of, patchwork kind of way, or to try and invoke a minor change in the laws of nature, but to recognize that this was a very big deal, and that would involve changing fundamentally how we think space and time work.

So, this is a very famous fable of science, you know. This is the thing where Einstein is a young man, and he’s sort of a rebel. And he refuses to take his exam and he pisses off his professors. And the consequence is he doesn’t get a good job. And he gets this job at the patent office right, in Bern, Switzerland, where he labors away in obscurity. And then, suddenly, in a blaze of glory in 1905, he publishes three papers, one of which is about relativity, one of which is the start of quantum mechanics, and the other of which proves that atoms exist. And so, these are three of the greatest papers of all time, published in a single year, by some clown who’s a clerk in a patent office. All right. Very big deal. This happened in 1905, two years ago it was the hundredth anniversary of this. This was declared Einstein Year, and all sorts of fuss was made about it.

So, there are various morals that can be drawn from this. And I’m going to draw several of them in what you might call increasing order of sophistication. So, the low sophistication moral, moral number one, here, is, you know, about the genius in obscurity. A genius in obscurity can revolutionize science.

This is a very dangerous way to approach the Einstein story. Because it–you know, it’s really easy to convince yourself that you are a genius in obscurity. Many people have convinced themselves that they are geniuses in obscurity. I know about this, because they send me email. And they explain to me in their email that they send me–and this happens to, you know, any scientist, particularly someone working in astrophysics or fundamental physics of some kind–they explain to me why Einstein was wrong. And indeed, why all science known to date is wrong, and they are right. And then, they say, you know, nobody believed Einstein either–as if that was somehow meant to make you believe them. They tend to write these things all in capital letters. There’s a kind of pathology. When they used to write–when they used to do it on pencil and paper, they would write in such a way that you–they covered every inch of the paper. And, you know, they’d write something, then they’d write around the margins, back to the top. There’s some psychological thing–I don’t know what it is. But the legend of Einstein prompts a whole bunch of people to think that they too might be patent clerks. Yes?

Student: How do people find you?

Professor Charles Bailyn: Oh, well, look. You can look up the astronomy department website of Harvard, Yale and Princeton, and you’ll find a whole bunch of people, I suppose. I also have published papers on black holes, and things that made their way into the media. I’ve actually had a fall-off on this, because I haven’t been, you know, quoted opining about black holes in The New York Times recently. But ten years ago that–well I’ll explain that in a couple weeks. There was an interesting thing that happened. In any case, you know, if you want to publicize your theory, what you do is you find a bunch of fairly prominent people in the field and you send them a big screen saying, you know, you will regret it if you do not appreciate my genius.

Student: Do you ever read it? Do you read them?

Professor Charles Bailyn: Sometimes I do. We have a little file in the library called the Crank File, in the Astronomy Department library. And from time to time, you know, we assign graduate students to figure out why they’re wrong. We haven’t done that lately.

Sometimes people amuse themselves by answering them. This can get you into various kinds of trouble. There was the gentleman with the theory of the Universe, which he illustrated in colored pencils, while he was serving his time at a medium security institution in Washington State. And we thought, you know, what a great way to–you know, much better way to be spending your time in prison than some other things you could think of. So, we sent him a bunch of textbooks and stuff. Then we started getting letters back, saying, after I finish my sentence, I’m coming to study with you, and explaining why he was innocent of the crime of which he had been committed. And so, at a certain point we started returning things, you know. No, terribly sorry, this individual has left Yale. We don’t know where he is, whoever it was. This is Jerry Orosz who carried on this conversation for a while. And so, you have to be a little bit careful.

And then–oh there was the guy–This was amazing. There was a guy who showed up at the start of the semester. He’s dressed in a suit and tie, carrying a briefcase, walks into the department office and says, “Where’s my classroom? I’m supposed to teach cosmology.” And he looked very respectable. The secretary thought, not with us, you’re not, so sent him over to the Physics Department. He asked the same question of the Physics Department. He had just walked in off the street with a briefcase full of his textbook. And he was very annoyed that we hadn’t assigned him a classroom and assigned students to come and learn, you know, his particular theory of cosmology. And in the end, he had to be escorted off the premises, and asked never to return.

So yeah, all kinds of patent clerks in the world, but most of them aren’t Einstein. That’s kind of the moral. And the reason you can tell–you asked whether I read these things. It’s very easy to tell whether there’s anything that you might be useful to read. Because what these people tend to say when they are not worth reading is, they tend to say that all previous science is wrong. I am revolutionizing things. Einstein is wrong, they say.

But Einstein didn’t say that Newton was wrong. That wasn’t what happened. That wasn’t how a revolution works. You revolutionize science only if the new theory encompasses what was previously known–the previous theories. And this is this business of the parameter, gamma, where under appropriate circumstances, gamma is one. And under those circumstances, you reproduce the theory that was already in existence. And so, you know, there’s been hundreds and hundreds of years of Newtonian physics, all of which is basically right, one way or another. So whatever your new idea is has to encompass all the things that were right about the old idea. And so, you don’t overturn Newton. What you do is, you demonstrate that Newton’s theory is only a part of a bigger theory. And then, the bigger theory is the new theory and it encompasses new kinds of data and explains new kinds of things, while simultaneously not overturning all the things that you know are right about the old theory.

And this is the thing that the people who wish they were Einstein, but aren’t, don’t understand. And so, anytime you see something where you’re–where they’re going to revolutionize all of science, it’s certainly wrong. Because there’s a large fraction of science in current existence that is correct. And you have to understand that. And you have to encompass it and go beyond that–but it’s not going to get overturned. The planets aren’t going to stop going in Keplerian orbits because somebody has a new idea. And so, it has to encompass the old idea.

Recently, in conjunction, in part with Einstein Year, two years ago, the historians of science have been rethinking what’s going on with Einstein, and what happened in that patent office. And there’s a new kind of idea that actually, it was–it wasn’t that he was working in obscurity and some–and, you know, flipping burgers or something–that his work in the patent office was very important to the way he developed his theory. And that one of the reasons Einstein did what he did was he was working in a patent office and not in a university. Because, what was he doing in the patent office?

Well, it turns out that in the early part of the twentieth century, one of the big set of inventions that was coming through were ways of synchronizing clocks. It turn–because railroads had just been invented. And until then, you had had a situation where every city had its own time. Every city had a little observatory. They would figure out when the Sun was directly overhead. They would proclaim that to be noon. They’d fire off a whistle. Everyone would set all their clocks. But that meant–but they didn’t have time zones. And that meant that every little town had a different time system. There was–Boston time was different from Hartford time, was different from New York time by 22.5 minutes, or whatever the appropriate amount would be. And that’s all fine, if you move between these cities slowly. If you start running trains, you’ve got to have–the train system has to have a unified system of time, otherwise they run into each other. And there were crashes and terrible things.

And so, then, you have to have–okay. Then you have to have Boston and New York having the same time. So, how do you synchronize those clocks? Well, you’ve just invented the telegraph, so you send an–you send an electromagnetic signal at the speed of light from Boston to New York, and then back again, in order to synchronize your clocks. And there were various inventions that Einstein was reviewing in the patent office to accomplish this. And so, in his job, in his day job, he spent all his time thinking about clocks and time, and signals moving at the speed of light. So, perhaps it isn’t surprising that he had deep insights into these kinds of things.

So, the current feeling among, at least, some historians of science, is that the patent office was, in fact, key to Einstein’s thinking. So, he didn’t come out of obscurity at all. He came out of the only place where the surroundings were such that you would have thought these thoughts. This is associated with a historian of science named Peter Galison, among others, who wrote a book called Einstein’s Clocks and Poincare’s Maps to try and demonstrate how the practical work these people were doing influenced their theoretical work. Yes?

Student: Was it triggered by one event?

Professor Charles Bailyn: Well no, he–was he triggered by one particular event? What prompted him to think about this was a whole series of experiments that had been carried out–all of which, as I mentioned last time, one way or another, demonstrated that this business that the speed of light was constant in all reference frames. And that there was no ether through which was the medium that was imagined to exist through which electromagnetic waves were supposed to pass. And there had been a series of experiments to try and detect slight changes in the speed of light when you put it on a rail–when you put it on a train, right? It’s all about trains and elevators, which had just been invented. And–but it wasn’t one particular triggering event. There was this whole range of things.

And Einstein, it should be said, was, by far, not the only person thinking about this. We call the transformations Lorentz transformations, because there was a guy named Lorentz who thought them up. Mathematicians like Poincare, physicists like Minkowski–there were a whole bunch of people thinking about this, and it was an urgent problem in physics. And so that had to be dealt with. And if Einstein hadn’t existed, it might have taken a little longer, but we would have gotten special relativity in the end.

That’s not necessarily true of general relativity, as I’ll explain in a second. General relativity–if Einstein hadn’t existed, general relativity might not have happened until the 1960s, because it was not prompted by a range of experimental results. The other bunch of experimental results that were prominent at the time were the atomic physics experiments that lead to quantum mechanics, which Einstein had a hand in, as well.

Chapter 2. General Relativity as a Theory on Gravity [00:13:27]

Okay, so, now let me move on to general relativity. That’s a good moment to make this transition. So, general relativity, known as GR, for short, is now a theory of gravity. So, the part of Newtonian physics that is overthrown, or encompassed, as I was saying, by special relativity, are the Laws of Motion. Now, general relativity has the law–those same laws of motion, but also has embedded in it a theory of gravity, which is fundamentally different from the way Newton thought about gravity. So, Newtonian gravity is thought of as a force. It’s a particular force. It’s the force of gravity. There’s a little equation, the force of gravity is equal to the gravitational constant, G, times the mass of one object, times the mass of the other object, divided by the distance between them squared.

And the odd thing–and this was sort of the starting point of Einstein’s thought. The odd thing is these masses. And because–remember, somebody asked me last time, what is mass? And I gave you a whole thing about the inertial mass. How much it resists being pushed. And that comes about from F = ma. And so, the kind of mass that I defined last time is the inertial mass–how much it resists being pushed.

And so, conceptually–the mass that’s in Newton’s equations of gravity–conceptually, that’s a different kind of mass. It doesn’t have to be the same number. This is gravitational mass, because it’s defined differently. It’s defined not by how much an object resists being pushed by a force–any force, but by how much the object creates a particular kind of force. So, in this, it’s how much it resists forces. In this, it’s how much the force is created. And these are conceptually different things.

It turns out that to the best experimental evidence that you can muster, which by now is many, many decimal places, these two kinds of mass are exactly the same. This has an experimental consequence that was known already to Galileo in the seventeenth century. Look what happened. Supposing M1 is the Earth, and M2 is some object that you’re dropping from a high tower. So, the story is, Galileo goes to the top of the Tower of Pisa. He drops a cannonball, or something, and watches it fall.

Then what happens? You substitute Fgravity into the equation of motion. And what do you get? So, you’re using a particular kind of force. So, you get GM of the object, M of the Earth, divided by distance squared, where the distance is the distance between the centers of the objects–so, this is basically the distance to the center of the Earth–is equal to the mass of the object, times its acceleration.

And, look. If this mass is the same as that mass, they cancel. What does that mean? So, you can do this. It means the motion, the acceleration of the object, doesn’t depend on its mass. The acceleration of the object will be the same for any object because, if it has greater inertia, it also generates, by exactly a proportional amount, a greater force. And so, everything falls from the tower at the same rate. And that’s what Galileo did. He brought up a cannonball, and a wooden ball, and a whole bunch of other objects, dropped them, and discovered that the Greeks had been wrong. The Greeks imagined the heavier the thing is, the fast it falls. But the only reason you think that is because of dropping things like feathers, which do fall kind of slowly and erratically. That has nothing to do with gravity. That’s air pressure, right? And so, if you drop things that are heavy enough so that air pressure isn’t important, they fall at the same rate. So, that is the experiment that proves that gravitational mass is the same as inertial mass.

All other forces don’t have this consequence. There is, for example–well, let me not draw on the corner here. I’ll be just like the cranks. You know, I’ll start drawing all the way around the edge of the thing, here. So, let us go on. Let’s see. There is, for example, electrical force, which has a very similar kind of equation. That’s a constant, times the charge of one object, times the charge of another object, divided by the distance between them squared. That looks almost identical to what the gravitational force looks like.

But, because the charge of an object isn’t equal to its mass, different objects respond differently to–when put in–when an electric force is exerted on them. If you compare a proton, which is a mass of subatomic particle of charge one, with a positron, which is a low-mass particle of charge one. And you put the same amount of force on them, the positron will move much more, because it has so much less mass. And so, different objects respond in different ways to all forces except gravity.

So, the unusual thing about gravity–so, gravity is weird, because the gravitational mass turns out to be exactly equal, at least, to the best possible measurements, to the inertial mass. And there’s no excuse within Newtonian theory to explain why that might be, and why all objects move–therefore, move the same when put in a gravitational field.

Okay, so, this bugged Einstein. This is the kind of thing that bothered him. And so, to explain this and other related, sort of conceptual issues, he promulgated a completely different theory–namely–so. Einstein says–okay.

So, gravity, Einstein says, is not a force. That’s not the right way to think about it. And so, what–how does this work, then? All right, so point one. Objects move in a straight line, when they are not in a straight line, in the absence of a force. This is Newton’s first law, or something. If you’re moving at a constant velocity, you will continue moving in the same direction, at the same speed, unless something happens; like, a force is applied to you. If gravity isn’t a force, that should still be true. Objects are moving in straight lines all the time, whether there’s a gravita–whether there’s gravity around or not.

That seems very odd, because we know perfectly well that things–that when you have a gravitational force, and you’re in an orbit, you go around in a circle, not in a straight line at all. So, how does Einstein reconcile this with reality? And that’s part two of the theory.

Chapter 3. Space-Time Curvature [00:21:30]

Part two of the theory is that the presence of mass–inertial mass, now–creates curvature in space-time, space and time. Now, how does that help? That helps because you have to then go back and think: what do you mean by going in a straight line if you’re in a curved space? I’ll give you some examples of this that you can see in a minute. What you mean by a straight line is the shortest distance between two points.

Now, we know from experience–or many of us know from experience–that on a curved spaced the shortest difference between two points is not exactly what you would expect it to be. Has anyone flown on an airplane from here to Asia, for example? Fly from New York to Beijing. So, let’s make a map, a flat map. So, here’s a flat space–north, east, west, south. Here’s New York. Here’s Beijing. And you might imagine, if things were flat, that the way you would get–and the quickest way that the airlines would save money, you know, because they don’t–is that they would fly directly east. Because it’s more or less at the same latitude, New York and Beijing.

But that isn’t what happens. That isn’t what happens at all. What you do, if you fly that route, or if you go the other way to Europe, wherever it is that you’re going is, you fly this way. You go way up north. You look down halfway through, there’s the Artic. Why do you do that? It’s because the Earth isn’t flat. The Earth is a sphere. And so, what it really looks like is this. Here’s the Earth. Here’s New York. Here’s Beijing, and you fly this way, because that’s the shortest distance across a–the surface of the sphere, between two points. I have a sphere somewhere.

Let’s see, here’s a sphere [holds up a basketball]. And this line here–think of this as the equator. And if you go from here to here, it would be the shortest distance between the two points–would be to go right across the equator. But we’re not going on the equator. We’re in the northern hemisphere. So now, I’m going to tip this upward so that these two things are both in the northern hemisphere. And now this line goes up and across. And if I tip it all the way up, so that you would be dealing with two cities on the opposite side of the world, as New York and Beijing, close enough, it goes–the closest line between them doesn’t go around this way, it goes straight up over the top. So, if you project that onto a flat map, it looks like this. But in fact, on a sphere, on the surface of a sphere, that is the shortest distance between those two points. It’s the exact same distance as it would be in the equator. You just tip the thing up, and it’s clear that this is the long way around.

So, if you have a curved space and you’re trying to get, in the shortest way, from one point to another, and you don’t know you’re on a curved space, you interpret it as a flat space, then you will interpret the path that you take as curved. And so, Einstein’s statement is that the presence of mass doesn’t exert a gravitational force. That’s the wrong way to think about it. What it does is, it creates a curvature in the space-time. And so, then, objects traveling from one place to another move in straight lines, defined as the shortest distance between those two points. But if we’re not taking the curvature into account, it looks like it’s going around in circles. All right. Tough concept.

Chapter 4. Q&A on Special Relativity [00:25:45]

Professor Charles Bailyn: The sheet is supposed to represent space-time. Space-time is–and now, I’ve got to explain that, right? So, what is space time? Because it’s space-time that’s doing the curving. It’s–you can think of it as a four-dimensional coordinate system, where the four dimensions are the three familiar spatial dimensions, X, Y, Z, and T for time. And you can understand where any given event, where an event is something that happens, can be identified with four numbers. It’s spatial position–that’s three numbers, and the time at which it occurs.

Now, in order to get these things into the same units, you have to actually represent this as the speed of light times time, because that–multiplying by the speed of light turns time units into space units, because this is miles per hour, or space per time. So, one year of time is the equivalent of one light-year of distance. That’s how it converts. And so, one interesting thing–and, of course, the sheet, there, was a two-dimensional space, and only had X and Y that could identify any point on the sheet. So, the metaphor is to use two spatial dimensions to represent what is actually a four–conceptually, a four-dimensional space. That’s why it’s a metaphor and not something real. Yes?

Student: [Inaudible]

Professor Charles Bailyn: Yes, yes. Absolutely correct. Does gravity bend time, too? It has an effect on time similar to time dilation when you’re going really fast. If you get really close to a black hole, your clocks will slow down. Yeah, yes?

Student: Why is that?

Professor Charles Bailyn: Why is that? Let me–next time, I’ll write down the Schwarzschild metric, and I’ll try and answer that question. Let me go on, here, for a minute, and point out an important thing about this agreement, here, which is–that means the Earth, in its motion, for example, in one year–the time coordinate moves one light-year, which is 1016 meters. The space coordinate moves 2π times the semi-major axis, which is 1 Astronomical Unit. That’s 2π times 1.5 times 1011 meters. That’s something like 1012 meters.

So, the Earth moves 10,000 times further in time than it moves in space in a year. And that means that the Earth’s motion in space-time is almost precisely a straight line, going straight forward in time. So, here’s time. Here’s, I don’t know, the X-axis. And you could imagine that as time goes on, the Earth, you know, moves around the Sun. So its X-axis –so its X-coordinate goes back and back–back and forth. Here’s the Sun, which, you know, wiggles a little bit, much less than the Earth, as we discussed.

But this picture, here, is inaccurate, because in fact this distance here, which is half a year, is 10,000 times longer than this distance here, which is half an orbit. And so–and so, in fact, the Earth’s motion is represented by something that’s almost exactly a straight line–so much so that I couldn’t possibly draw the wiggle in a way that you could see it. So, here’s the Earth going forward in time. And the wiggle is really small. That’s not surprising, because the gravitational force of the Sun is small enough so that the Earth’s orbit is correctly described by Newtonian physics.

And so, this is one of those situations where you have a Newtonian physics in relatively low gravitational fields. And as the gravitational fields get stronger, weird relativistic effects kick in. But that doesn’t happen for the Earth. So there’s–the number is this. Schwarzschild radius divided by distance, is the measure of how relativistic an orbit is. And if this number is small, then you get correct answers from the Newtonian theory. And we can work it out. The Schwarzschild radius of the Sun, 3 x 103 meters. The distance from the Earth to the Sun, 1.5 x 1011 meters. That’s 2 x 10-8 and that’s a small number. So, Earth works fine in a Newtonian sense. Yes?

Student: Is that R the radius of the orbit or the radius of the [inaudible]

Professor Charles Bailyn: That is the distance of the moving object from the center of the gravitating thing that’s making the curvature happen. Yeah. So, yes is the answer. It’s the radius of the orbit. Yes?

Student: Was the theory of general relativity a theory that was just promulgated, or does it actually have equations associated with it?

Professor Charles Bailyn: Oh yes. Does it have equations associated with it? You bet. You don’t want to see them. There’s a branch of mathematics called differential geometry, because–what happens is, this curvature stuff is all geometry. And the problem is that the further away you get from an object, the less the space is curved. So, the amount on the sphere, on the surface of the sphere–it’s curved the same way everywhere. So, the equations that describe the curvature are the same at any point on a sphere. But you can imagine something that starts out with strong curvature and tails off. And then the amount of curvature at any given point is different from any other point. So, you have to describe it with a differential amount of curvature, and then integrate over the path that things take. This gets heinous, and we’re not going to go there. But I will show you, next time, some specific examples of some of these equations, because it’ll help explain things like why space and time are reversed inside an event horizon and stuff.

But, you bet, there’s a whole mathematical theory. In fact, Einstein was not the world’s greatest mathematician and it took him until 1917 to get all this worked out. He had to go and get himself some math tutoring at the University of Göttingen and–so that he could understand the geometric properties. So, yeah, there’s a big mathematical theory behind it. Yes?

Student: So, if gravity is not a force, how can you use F = ma?

Professor Charles Bailyn: F = ma is a general equation of motion. It doesn’t have to be the force of gravity. There are many other forces in the world.

Student: But when you’re setting F = ma to GMM or [inaudible]

Professor Charles Bailyn: Right, so, that’s the Newtonian idea, that you set F = ma equal to GMM over D2. And the thing that prompted Einstein’s thought was the fact that the two different kinds of masses on each side of the equation turned out to be equal. And you can see why this explains that. Because, if what’s happening is that you curve space, and then things follow a trajectory dictated by the curvature of that space, then it makes perfect sense that all objects follow the same trajectory, regardless of their mass. Because they’re not actually being forced to move anywhere. They’re just going in straight lines, and straight lines are the same no matter what your mass is. And so, that’s why this kind of curvature provides an explanation for what, in the Newtonian approximation, is the equality of the inertial and the gravitational mass. How we doing? Oh, we’ve got some more time, good. Go ahead.

Student: Would a clock really slow down as it moved towards the [inaudible]

Professor Charles Bailyn: Yeah.

Student: [inaudible] would you have a perception of [inaudible]

Professor Charles Bailyn: Well, of course, you are a clock, right? You’re–no, you wouldn’t notice. But what would happen is this. Supposing you’re falling into a black hole, or something. You look at your clock. You feel your pulse, you know. Your pulse measured by your clock has not changed. But you look back out into the world that is distant from the black hole, and things are moving really fast out there. And so, if you had two synchronized, identical clocks at the start of your voyage, and you keep an eye on what’s happening out here, you will discover that, all of a sudden, it looks like your clock is moving slow. And, all of a sudden, you know, people are aging and dying back on Earth, while, you know, it’s only been a week in your perception.

And so, turn it around. You’re sitting on Earth watching your friend fall through–fall into a black hole. And it looks like he’s slowing down, and slowing down, and slowing down and maybe–and never really quite gets to the event horizon, even in an infinite amount of time. But the perception of the person falling through is that they just fall through. It’s just, the whole Universe is evolved in the time that they’ve done it.

So, it’s one of these weird things where time turns out–how fast time passes turns out to be a property of who’s doing the measuring. Very strange concept, and very counterintuitive to the way we live our life. Because we all think–because we live in low gravitational fields at low speeds, we think time is an absolute thing that ticks onward for everybody in the same way, but it turns out it isn’t true.

Yes? And again, I’ll show you an equation that–actually, this is much easier to deal with in equations than it is conceptually, and I’ll show you an equation next time which may help.

Student: Can you explain about the names behind the two theories of relativity? I mean, it doesn’t seem like the general theory is a generalization of the special theory.

Professor Charles Bailyn: It actually is, because it incorporates aspects of the special theory. And it’s a generalization in terms of what’s called the metric. The metric is the mathematical formula that describes the curvature. And the special theory of relativity is the special case where the metric is flat. And then, the reason general relativity is a generalization is because it allows other–many other kinds of metrics. Or, at least, that’s one way of thinking about it. Yes sir.

Student: [Inaudible]

Professor Charles Bailyn: Yeah well I–too small for me to draw.

Student: [Inaudible]

Professor Charles Bailyn: The wiggle is from the motion of the Earth around the Sun. And it’s caused by–the mass of the Sun warps the space–

Student: [Inaudible]

Professor Charles Bailyn: Yeah, yeah, yeah. It’s just–all I meant in the second one–sorry, I didn’t explain this clearly. All I meant in the second one was that this one is inaccurate in terms of scale.

Student: Okay, oh I see.

Professor Charles Bailyn: And that the scale, the true scale, is such that this length ought to be 10,000 times longer than that. And under those circumstances you’d never be able to see it. That’s all I meant by that. Yeah?

Student: What happens to time when you’re caught traveling–you mentioned this, but what happens to time when you’re traveling close to the speed of light?

Professor Charles Bailyn: Slows down yeah. Yes?

Student: [Inaudible]

Professor Charles Bailyn: So, how does, how does this work? Right. We are, you know–you can explain the motion of the Earth in terms of these so-called space-time diagrams, where one–you take the four-dimensional thing. You project it onto two dimensions. You can write a space-time diagram for yourself, and you can explain all of the motion and all of the things that happen by such a curved thing. So, what happens if you jump, for example? In the Newtonian theory, you jump, you ought to keep going straight up, except that there is a force, which pulls you back down. So, that we understand.

What happens–how is that explained in general relativity? You jump and you go in a straight line, but the straight line is through a curved space–curved because of the presence of the Earth. And so, it turns out that the straight line looks like this. It goes up and comes back down again, in just the same way that the straight line between New York and Beijing goes over the North Pole. It goes up and comes back down again, because space-time is curved due to the gravity of the Earth.

So, something that we perceive to be a curved trajectory–jumping up, falling back down–if you project it into space-time, is actually the closest distance between two events. So, there’s two alternative explanations of these things.

In the situation where you’re nowhere near a black hole, it turns out that, mathematically, they’re exactly equivalent. If they’re exactly equivalent, it’s much easier to calculate the Newtonian way, and much easier to think about it. So, until you get into a situation where the Schwarzschild radius is comparable to the distances involved, you’re way better off thinking Newtonian, because we’re just better at it. Yes?

Student: I understand that in order to get units that you can line up in any way, uniquely, you need to multiply time by a velocity. But I’m a little confused about–I mean I know that c is this really important number–

Professor Charles Bailyn: Yeah.

Student: - but if the only reason why the wiggles in the graph of motion are really tiny is because c is so enormous. I mean -

Professor Charles Bailyn: Couldn’t you pick another velocity?

Student: Right.

Professor Charles Bailyn: Yeah.

Student: What makes that an accurate scaling?

Professor Charles Bailyn: Okay, so, the reason that’s an accurate scaling is, one of the things that comes out of the theory of relativity is that there is a natural unit. There’s a natural set of units. And the natural set of units are the units in which these crucial constants, of which c is one and G is another, and H–the Planck constant, which governs quantum mechanics, are all equal to one. And that turns–those are called Planck units, and they have–and the conversion between different kinds of energy, mass to energy, things like that, all work out as equalities in the particular set of units, where all those constants come out to be one.

So, there is a reason why that particular velocity is chosen, and it’s not surprising, if you had to pick a velocity, that you would end up with the speed of light, which has this sort of profound importance. Okay, we’ll continue next time.

[end of transcript]

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