Full Text of Einstein’s General Theory of Relativity Lecture 1 by Leonard Susskind at Stanford University. Recorded September 22, 2008.
Leonard Susskind – Felix Bloch Professor of Theoretical Physics, Stanford University
Gravity. Gravity is a rather special force. It’s unusual. It has different thin electrical forces, magnetic forces, and it’s connected in some way with geometric properties of space — space and time. And that connection is, of course, the general theory of relativity.
Before we start, tonight for the most part we will not be dealing with the general theory of relativity. We will be dealing with gravity in its oldest and simplest mathematical form. Well, perhaps not the oldest and simplest but Newtonian gravity. And going a little beyond what Newton, certainly nothing that Newton would not have recognized or couldn’t have grasped — Newton could grasp anything– but some ways of thinking about it which would not be found in Newton’s actual work. But still Newtonian gravity. Newtonian gravity is set up in a way that is useful for going on to the general theory.
Okay. Let’s begin with Newton’s equations. The first equation, of course, is F equals MA.
F = ma
Force is equal to mass times acceleration. Let’s assume that we have a frame of reference, that means a set of coordinates and a collection of clocks, and that frame of reference is what is called an inertial frame of reference. An inertial frame of reference simply means one which if there are no objects around to exert forces on a particular – let’s call it a test object. A test object is just some object, a small particle or anything else, that we use to test out the various fields — force fields, that might be acting on it. An inertial frame is one which, when there are no objects around to exert forces, that object will move with uniform motion with no acceleration. That’s the idea of an inertial frame of reference.
And so if you’re in an inertial frame of reference and you have a pen and you just let it go, it stays there. It doesn’t move. If you give it a push, it will move off with uniform velocity. That’s the idea of an inertial frame of reference and in an inertial frame of reference the basic Newtonian equation number one — I always forget which law is which. There’s Newton’s first law, second law, and third law. I never can remember which is which. But they’re all pretty much summarized by F equals mass times acceleration. This is a vector equation. I expect people to know what a vector is. A three-vector equation. We’ll come later to four-vectors where when space and time are united into space-time. But for the moment, space is space, and time is time. And vector means a thing which is a pointer in a direction of space, it has a magnitude, and it has components.
So, component by component, the X component of the force is equal to the mass of the object times the X component of acceleration, Y component Z component and so forth. In order to indicate that something is a vector acceleration I’ll try to remember to put an arrow over vectors. The mass is not a vector. The mass is simply a number. Every particle has a mass, every object has a mass. And in Newtonian physics the mass is conserved. It does not change. Now, of course, the mass of this cup of coffee here can change. It’s lighter now but it only changes because mass has been transported from one place to another. So, you can change the mass of an object by whacking off a piece of it but if you don’t change the number of particles, change the number of molecules and so forth, then the mass is a conserved, unchanging quantity. So, that’s first equation.
Now, let me write that in another form. The other form we imagine we have a coordinate system, an X, a Y, and a Z. I don’t have enough directions on the blackboard to draw Z. I won’t bother. There’s X, Y, and Z. Sometimes we just call them X1, X2, and X3. I guess I could draw it in. X3 is over here someplace. X, Y, and Z. And a particle has a position which means it has a set of three coordinates. Sometimes we will summarize the collection of the three coordinates X1, X2, and X3 incidentally. X1, and X2, and X3 are components of a vector. They are components of the position vector of the particle. The position vector of the particle I will often call either small r or large R depending on the particular context. R stands for radius but the radius simply means the distance between the point and the origin for example. We’re really talking now about a thing with three components, X, Y, and Z, and it’s the radial vector, the radial vector. This is the same thing as the components of the vector R.
All right. The acceleration is a vector that’s made up out of a time derivatives of X, Y, and Z, or X1, X2, and X3. So, for each component — for each component, one, two, or three, the acceleration — which let me indicate, let’s just call it A. The acceleration is just equal — the components of it are equaled to the second derivatives of the coordinates with respect to time. That’s what acceleration is. The first derivative of position is called velocity. We can take this to be component by component X1, X2, and X3. The first derivative is velocity. The second derivative is acceleration. We can write this in vector notation. I won’t bother but we all know what we mean. I hope we all know what we mean by acceleration and velocity. And so, Newton’s equations are then summarized – not summarized but rewritten — as the force on an object, whatever it is, component by component, is equal to the mass times the second derivative of the component of position. So, that’s the summary of — I think it’s Newton’s first and second law. I can never remember which they are.
Newton’s first law, of course, is simply the statement that if there are no forces then there’s no acceleration. That’s Newton’s first law. Equal and opposite. Right. And so this summarizes both the first and second law. I never understood why there was a first and second law. It seemed to me that it was one law, F equals MA.
All right. Now, let’s begin even previous to Newton with Galilean gravity. Gravity as how Galileo understood it. Actually, I’m not sure how much of these mathematics Galileo did or didn’t understand. He certainly knew what acceleration was. He measured it. I don’t know that he had the — he certainly didn’t have calculus but he knew what acceleration was. So, what Galileo studied was the motion of objects in the gravitational field of the earth in the approximation that the earth is flat. Now, Galileo knew that the earth wasn’t flat but he studied gravity in the approximation where you never moved very far from the surface of the earth. And if you don’t move very far from the surface of the earth, you might as well take the surface of the earth to be flat and the significance of that is two-fold. First of all, the direction of gravitational forces is the same everywhere. This is not true, of course, if the earth is curved then gravity will point toward the center. But in the flat space approximation, gravity points down. Down everywhere is always in the same direction. And second of all, perhaps a little less obvious but nevertheless true, in the approximation where the earth is infinite and flat, goes on and on forever, infinite and flat, the gravitational force doesn’t depend on how high you are. Same gravitational force here as here. The implication of that is that the acceleration of gravity, the force apart from the mass of an object, the acceleration on an object is independent of where you put it. 2And so Galileo either did or didn’t realize — again, I don’t know exactly what Galileo did or didn’t know. But what he said was the equivalent of saying that the force of an object in the flat space approximation is very simple. It, first of all, has only one component, pointing downward. If we take the upward sense of things to be positive, then we would say that the force is — let’s just say that the component of the force in the X2 direction, the vertical direction, is equal to minus — the minus simply means that the force is downward — and it’s proportional to the mass of the object times a constant called the gravitational acceleration.
Now, the fact that it’s constant everywhere, in other words, mass times G doesn’t vary from place to place. That’s this fact that gravity doesn’t depend on where you are in the flat space approximation. But the fact that the force is proportional to the mass of an object, that is not obvious. In fact, for most forces, it is not true. For electric forces, the force is proportional to the electric charge, not to the mass. And so gravitational forces are at a special the strength of the gravitational force of an object is proportional to its mass. That characterizes gravity almost completely. That’s the special thing about gravity. The force is proportional itself to the mass.
Well, if we combine F equals MA with the force law — this is the law of force — then what we find is that mass times acceleration the second X, now this is the vertical component, by DT squared is equal to minus — that is the minus – MG period. That’s it. Now, the interesting thing that happens in gravity is that the mass cancels out from both sides. That is what’s special about gravity. The mass cancels out from both sides. And the consequence of that is that the motion of an object, its acceleration, doesn’t depend on the mass — doesn’t depend on anything about the particle. The particle, object– I’ll use the word particle. I don’t necessarily mean a point small particle, a baseball is a particle, an eraser is a particle, a piece of chalk is a particle. That the motion of an object doesn’t depend on the mass of the object or anything else. The result of that is that if you take two objects of quite different mass and you drop them, they fall exactly the same way. Galileo did that experiment. I don’t know whether he really threw something off the Leaning Tower of Pisa or not. It’s not important. He did balls down an inclined plane. I don’t know whether he actually did or didn’t. I know the myth is that he didn’t. I find it very difficult to believe that he didn’t. I’ve been in Pisa. Last week I was in Pisa and I took a look at the Leaning Tower of Pisa. Galileo was born and lived in Pisa. He was interested in gravity. How it would be possible that he wouldn’t think of dropping something off the Leaning Tower is beyond my comprehension. You look at that tower and say, “That tower is good for one thing: Dropping things off. ”
Now, I don’t know. Maybe the doge or whoever they called the guy at the time said, no, no Galileo. You can’t drop things from the tower. You’ll kill somebody. So, maybe he didn’t. He must have surely thought of it.
All right. So, the result, had he done it, and had he not had to worry about such spurious effects as air resistance would be that a cannon ball and a feather would fall in exactly the same way, independent of the mass, and the equation would just say, the acceleration would first of all be downward, that’s the minus sign, and equal to this constant G. Excuse me. Now, G is a number, it’s 10 meters per second per second at the surface of the earth. At the surface of the moon it’s something smaller. On the surface of Jupiter it’s something larger. So, it does depend on the mass of the planet but the acceleration doesn’t depend on the mass of the object you’re dropping. It depends on the mass of the object you’re dropping it onto but not the mass of the object stopping it. That fact, that gravitational motion, is completely independent of mass is called or it’s the simplest version of something that’s called the equivalence principle. Why it’s called the equivalence principle we’ll come to later. What’s equivalent to what. At this stage we can just say gravity is equivalent between all different objects independent of their mass. But that is not exactly what the equivalence – an equivalence principle was about. That has a consequence. An interesting consequence.
Supposing I take some object which is made up out of something which is very unrigid. Just a collection of point masses. Maybe let’s even say they’re not even exerting any forces on each other. It’s a cloud, a very diffuse cloud of particles and we watch it fall. Now, let’s suppose we start each particle from rest, not all at the same height, and we let them all fall. Some particles are heavy, some particles are light, some of them may be big, some of them may be small. How does the whole thing fall? And the answer is, all of the particles fall at exactly the same rate. The consequence of it is that the shape of this object doesn’t deform as it falls. It stays absolutely unchanged. The relationship between the neighboring parts are unchanged. There are no stresses or strains which tend to deform the object. So even if the object were held together by some sort of struts or whatever, there would be no forces on those struts because everything falls together.
The consequence of that is that falling in the gravitational field is undetectable. You can’t tell that you’re falling in a gravitational field by — when I say you can’t tell, certainly you can tell the difference between free fall and standing on the earth. That’s not the point. The point is that you can’t tell by looking at your neighbors or anything else that there’s a force being exerted on you and that that force that’s being exerted on you is pulling downward. You might as well, for all practical purposes, be infinitely far from the earth with no gravity at all and just sitting there because as far as you can tell there’s no tendency for the gravitational field to deform this object or anything else. You cannot tell the difference between being in free space infinitely far from anything with no forces and falling freely in a gravitational field. That’s another statement of the equivalence principle.
Question: You say not mechanically detectable?
Leonard Susskind: Well, in fact, not detectable, period. But so far not mechanically detectable.
Question: Well, would it be optically detectable?
Leonard Susskind: No. No. For example, these particles could be equipped with lasers. Lasers and optical detectors of some sort. What’s that? Oh, you could certainly tell if you were standing on the floor here, you could tell that there was something falling toward you. But the question is, from within this object by itself, without looking at the floor, without knowing that the floor was—
Question: Something that wasn’t moving.
Leonard Susskind: Well, you can’t tell whether you’re falling and it’s, uh — yeah. If there was something that was not falling it would only be because there was some other force on it like a beam or a tower of some sort holding it up. Why? Because this object, if there are no other forces on it, only the gravitational forces, it will fall at the same rate as this.
All right. So, that’s another expression of the equivalence principle, that you cannot tell the difference between being in free space far from any gravitating object versus being in a gravitational field. Now, we’re going to modify this. This, of course, is not quite true in a real gravitational field, but in this flat space approximation where everything moves together, you cannot tell that there’s a gravitational field. At least, you cannot tell the difference — not without seeing the floor in any case. The self-contained object here does not experience anything different than it would experience far from any gravitating object standing still or in uniform motion.