Full Text of Einstein’s General Theory of Relativity Lecture 1 by Leonard Susskind at Stanford University. Recorded September 22, 2008.
Listen to the MP3 Audio here: MP3 – Einstein’s General Theory of Relativity – Lecture 1
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.
Leonard Susskind: What’s that? Yeah.
Question: We can tell where we’re accelerating.
Leonard Susskind: No, you can’t tell when you’re accelerating.
Question: Well, you can — you can’t feel — isn’t that because there’s no connection between objects?
Leonard Susskind: Okay. Here’s what you can tell. If you go up to the top of a high building and you close your eyes, and you step off, and go into free fall, you will feel exactly the same — you will feel weird. I mean, that’s not the way you usually feel because your stomach will come up and do some funny things. You know, you might lose it. But the point is, you would feel exactly the same discomfort in outer space far from any gravitating object just standing still. You’ll feel exactly the same peculiar feelings.
Okay? What are those peculiar feelings due to? They’re not due to falling. They’re due to not fall – well they’re due to the fact that when you stand on the earth here, there are forces on the bottom of your feet which keep you from falling and if the earth suddenly disappeared from under my feet, sure enough, my feet would feel funny because they’re used to having those forces exerted on their bottoms. You get it. I hope. So, the fact that you feel funny in free fall is because you’re not used to free fall. It doesn’t matter whether you’re infinitely far from any gravitating objects standing still or freely falling in the presence of a gravitational field.
Now, as I said, this will have to be modified in a little bit. There are such things as tidal forces. Those tidal forces are due to the fact that the earth is curved and that the gravitational field is not the same in — the same direction in every point, and that it varies with height. That’s due to the finiteness of the earth. But, in the flat space of the — in the flat earth approximation where the earth is infinitely big pulling uniformly, there is no other effect of gravity that is any different than being in free space. Okay. Again, that’s known as the equivalence principle.
Now, let’s go on beyond the flat space or the flat earth approximation and move on to Newton’s theory of gravity. Newton’s theory of gravity says every object in the universe exerts a gravitational force on every other object in the universe. Let’s start with just two of them. Equal and opposite. Attractive. Attractive means that the direction of the force on one object is toward the other one. Equal and opposite forces and the magnitude of the force — the magnitude of the force of one object on another. Let’s characterize them by a mass. Let’s call this one little m. Think of it as a lighter mass and this one, which we can imagine as a heavier object, we’ll call it big M.
All right. Newton’s law of force is that the force is proportional to the product of the masses. Making either mass heavier will increase the force. The product of the masses, big M times little m, inversely proportional to the square of the distance between them. Let’s call that R squared. Let’s call the distance between them R. And there’s a numerical constant. This law by itself could not possibly be right. It’s not dimensionally consistent. If you work out the dimensions of force, mass, mass and R, it’s not dimensionally consistent. There has to be a numerical constant in there. And that numerical constant is called G, Newton’s constant. And it’s very small. It’s a very small constant. I’ll write down what it is. G is equal to six or 6.7, roughly, times ten to the minus 11th, which is a small number. So, on the face of it, it seems that gravity is a very weak force. You might not think that gravity is such a weak force, but to convince yourself it’s a weak force there’s an experiment that you can do. Weak by comparison with other forces. I’ve done this for classes and you can do it yourself. Just take an object hanging by a string and two experiments. The first experiment, take a little object here and electrically charge it. You electrically charge it by rubbing it on your sweater. That doesn’t put very much electric charge on it but it charges it up enough to feel some electrostatic force. And the take another object of exactly the same kind, rub it on your shirt, and put it over here. What happens? They repel. And the fact that they repel means that this string will shift. And you’ll see it shift.
Take another example. Take your little ball there to be iron and put a magnet next to it. Again, you’ll see quite an easily detectable deflection of the string holding it.
All right? Next, take a 10,000-pound weight and put it over here. Guess what happens? Undetectable. You cannot see anything happen. The gravitational force is much, much weaker than most other kinds of forces and that’s due to — not due to that. The fact that it’s so weak is encapsulated in this small number here.
Another way to say it is if you take two masses, each of 1 kilometer — not 1 kilometer. 1 kilogram. A kilogram is a good healthy mass, a nice chunk of iron. MM and you separate them from 1 meter, then the force between them is just G and it’s 6.7 times ten to the minus 11th, the units being Newton’s. Very weak force. But, weak as it is, we feel it rather strenuously. We feel it strongly because the earth is so darn heavy. So, the heaviness of the earth makes up for the smallness of G and so we wake up in the morning feeling like we don’t want to get out of bed because gravity is holding us down.
Question: So that force is measuring the force between — from the large one to the small one or both?
Leonard Susskind: Both. They’re equal and opposite. Equal and opposite. That’s the rule. That’s Newton’s third law. The forces are equal and opposite. So, the force on the large one due to the small one is the same as the force of the small one on the large one. But it is proportional to the product of the masses. So, the meaning of that is I’m not — I’m heavier than I’d like to be but I’m not very heavy. I’m certainly not heavy enough to deflect the hanging weight significantly. But I do exert a force on the earth which is exactly equal and opposite to the force that the very heavy earth exerts on me.
Okay. Why does the earth — if I drop from a certain height, I accelerate down. The earth hardly accelerates at all, even though the forces are equal. Why is it that the earth — if the forces are equal, my force on the earth and the earth’s force on me are equal, why is it that the earth accelerates so little and I accelerate so much? Yeah. Because the acceleration involves two things. It involves the force and the mass. The bigger the mass, the less the acceleration for a given force. So, the earth doesn’t accelerate — yeah, question.
Question: How did Newton arrive at that equation for the gravitational force?
Leonard Susskind: I think it was largely a guess. But it was an educated guess. And what was the key — it was large — no, no. It was from Kepler’s laws. It was from Kepler’s laws. He worked out, roughly speaking — I don’t know what he did. He was rather secretive and he didn’t really tell people what he did. But the piece of knowledge that he had was Kepler’s laws of motion — planetary motion — and my guess is that he just wrote down a general force and realized that he would get Kepler’s laws of motion for the inverse square law. I don’t believe he had any underlying theoretical reason to believe in the inverse square law.
Question: Edmund Halley actually asked him, what kind of force law do you need for conic section orbits and he had almost performed the calculations a year before.
Leonard Susskind: Yeah. Actually, I don’t think — yeah.
Question: I think the question — he asked the question for inverse square laws and I think that Newton already knew the solution was an ellipse.
Leonard Susskind: No. It wasn’t the ellipse that was there. The orbits might have been circular. It was the fact that the period varies as the three halfs power of the radius. All right? The period of motion is the circular motion has an acceleration toward the center. Any motion in the circle is accelerated toward the center. If you know the period and the radius, then you know the acceleration toward the center. Okay? We could write the — what’s the word? Anybody know what — if I know the angular frequency of going around in an orbit that’s called omega. And it’s basically just the inverse period. Okay? Omega is roughly the inverse period number of cycles per second. What is the acceleration of a thing moving in a circular orbit. Anybody remember? Omega squared R.
Omega squared R. That’s the acceleration. Now, supposing he sets that equal to some unknown force law F of R and then divides by R. Then he finds omega as a function of the radius of the orbit. Well, let’s do it for the real case. For the real case, inverse square law, F of R is one of R squared so this would be one of R cubed and in that form it is Kepler’s second law? I don’t even remember which one it is. It’s the law that says that the frequency or the period, the square of the period, is proportional to the cube of the radius. That was the law of Kepler. So, from Kepler’s laws he easily could — or that one law, he could easily deduce that the force was proportional to one of R squared. I think that’s probably historically what he did.
Then, on top of that he realized if you didn’t have a perfectly circular law orbit then the inverse square law was the unique law which would give elliptical orbits. So, it’s a two-step thing.
Question: What happens when the two objects are touching? Do you measure it from the –
Leonard Susskind: Of course, there are other forces on them. If two objects are actually touching each other, there are all sorts of forces between them that are not just gravitational. Electrostatic forces, atomic forces, nuclear forces? So, you’ll have to modify the whole story.
Question: As the distance approaches zero –
Leonard Susskind: Yeah. Then it breaks down. Then it breaks down.
Then it breaks down. When they get so close that other important forces come into play. The other important forces, for example, are the forces that are holding this object and preventing it from falling. These we usually call them contact forces but, in fact, what they really are is various kinds of electrostatic forces between the atoms and molecules on the table and the atoms and molecules in here. So, other kinds of forces.
All right. Incidentally, let me just point out if we’re talking about other kinds of force laws, for example, electrostatic force laws, then the force — we still have F equals MA but the force law — the force law will not be that the force is somehow proportional to the mass times something else but it could be the electric charge. If it’s the electric charge, then electrically uncharged objects will have no forces on them and they won’t accelerate. Electrically charged objects will accelerate in an electric field. So, electrical forces don’t have this universal property that everything falls or everything moves in the same way. Uncharged particles move differently than charged particles with respect to electrostatic forces. They move the same way with respect to gravitational forces. And as repulsion and attraction, whereas gravitational forces are always attractive. Where is my gravitational force? I lost it.
Here it is. All right. So, that’s Newtonian gravity between two objects. For simplicity let’s just put one of them, the heavy one, at the origin of coordinates and study the motion of the light one then — incidentally, one usually puts — let me refine this a little bit. As I’ve written it here, I haven’t really expressed it as a vector equation. This is the magnitude of the force between two objects. Thought of as a vector equation, we have to provide a direction for the force. Vectors have directions. What direction is the force on this particle? Well, the answer is, it’s along the radial direction itself. So, let’s call the radial distance R, or the radial vector R, then the force on little m here is along the direction R. But it’s also opposite to the direction of R. The radial vector, relative to the origin over here, points this way. On the other hand, the force points in the opposite direction. If we want to make a real vector equation out of this, we first of all have to put a minus sign. That indicates that the force is opposite to the direction of the radial distance here, but we have to also put something in which tells us what direction the force is in. It’s along the radial direction. But wait a minute. If I multiply it by R up here, I had better divide it by another factor of R downstairs to keep the magnitude unchanged. The magnitude of the force is one over R squared. If I were to just randomly come and multiply it by R, that would make the magnitude bigger by a factor of R, so I have to divide it by the magnitude of R. This is Newton’s force law expressed in vector form.
Now, let’s imagine that we have a whole assembly of particles. A whole bunch of them. They’re all exerting forces on one another. In pairs, they exert exactly the force that Newton wrote down. But what’s the total force on a particle? Let’s label these particles the first one, the second one, the third one, the fourth one, dot, dot, dot, dot, dot. This is the Ith one over here. So, I is the running index which labels which particle we’re talking about. The force on the Ith particle, let’s call it F sub I, and let’s remember that it’s a vector, it’s equal to the sum — now, this is not an obvious fact that when you have two objects exerting a force on the third that the force is necessarily equal to the sum of the two forces, of the two objects. You know what I mean. But it is a fact anyway. Obvious or not obvious it is a fact. Gravity does work that way at least in the Newtonian approximation. With Einstein, it breaks down a little bit. But in Newtonian physics the force is the sum and so it’s a sum of all the other particles. Let’s write that J not equal to I. That means it’s a sum of all not equal to I. So, the force on the first particle doesn’t come from the first particle, it comes from the second particle, third particle, fourth particle, and so forth. Each individual force involves M sub I, the force of the Ith particle, times the four, times the mass of the Jth particle. Product of the masses divided by the square of the distance between them, let’s call that RIJ squared. The distance between the Ith particle is I and J, the distance between the Ith particle and the Jth particle is RIJ. But then, just as we did before, we have to give it a direction. Put a minus sign here, that indicates that it’s attractive, another RIJ upstairs, but that’s a vector RIJ, and make this cubed downstairs. All right? So, that says that the force on the Ith particle is the sum of all the forces due to all the other ones of the product of their masses inverse square in the denominator, and the direction of each individual force on this particle is toward the other.
All right? This is a vector sum. Yeah? Hmm? The minus indicates that it’s attractive.
Question: But you’ve got the vector going from I to J.
Leonard Susskind: Oh. Let’s see. That’s the vector going from R to I to J. There is a question of the sign of this vector over here. So yeah. You’re absolute — yeah. You’re right. You’re absolutely right. The way I’ve written it there should not be a minus sign here. If I put RJI there, then there would be a minus sign. Right? So, you’re right. But in any case every one of the forces is attractive and what we have to do is to add them up. We have to add them up as vectors and so there’s some resulting vector, some resultant vector, which doesn’t point toward any one of them in particular but points in some direction which is determined by the vector sum of all the others. All right but the interesting fact is, if we combine this, this is the force on the Ith particle. If we combine it with Newton’s equations — let’s combine it with Newton’s F equals MA equations then this is F. This on the Ith particle, this is equal to the mass of the Ith particle times the acceleration of the Ith particle. Again, vector equations. Now, the sum here is over all the other particles. We’re focusing on number I. I, the mass of the Ith particle will cancel out of this equation. I don’t want to throw it away but let’s just circle it and now put it over on the side. We notice that the acceleration of the Ith particle does not depend on its mass again. Once again, because the mass occurs on both sides of the equation it can be canceled out, and the motion of the Ith particle does not depend on the mass of the Ith particle. It depends on the masses of all the other ones. All the other ones come in, but the mass of the Ith particle cancels out of the equation. So, what that means is if we had a whole bunch of particles here and we added one more over here, its motion would not depend on the mass of that particle. It depends on the mass of all the other ones but it doesn’t depend on the mass of the Ith particle here.
Okay? Again, equivalence principle that the motion of a particle doesn’t depend on its mass. And again if we had a whole bunch of particles here, if they were close enough together, they would all move in the same way. Before I discuss any more mathematics, let’s just discuss tidal forces, what tidal forces are.
Question: Once you set this whole thing into motion dynamically. We have all different masses and each particle is going to be affected by each one?
Leonard Susskind: Yes.
Question: Every particle in there is going to experience a uniform acceleration?
Leonard Susskind: No, no, no. The acceleration is not uniform. The acceleration will get larger when it gets closer to one of the particles. It won’t be uniform anymore. It won’t be uniform now because the force is not independent of where you are. Now the force depends on where you are relative to the objects that are exerting the force. It was only in the flat earth approximation where the force didn’t depend on where you were. Okay? Now, the force varies so it’s larger when you’re far away — sorry. It’s smaller when you’re far away, it’s larger when you’re in close.
Question: But is it going to be changing in a — it changes in a vector form with each individual particle. Each one of them is changing position.
Leonard Susskind: Yeah.
Question: So, is the dynamics that every one of them is going towards the center of gravity of the entire –
Leonard Susskind: Not necessarily. I mean, they could be flying apart from each other but they will be accelerating toward each other. Okay? If I throw this eraser into the air with greater than the escape velocity, it’s not going to turn around and fall back down.
Question: Well, the question is, is the acceleration a uniform acceleration or is it changing in dynamics?
Leonard Susskind: Changing with what? With respect to what? Time? It changes with respect to time because the object moves farther and farther away.
Question: In the two-mass system — I call that a uniform acceleration.
Leonard Susskind: Uniform with respect to what?
Question: It’s not uniform. The radius is changing and it’s inversed cubically radiused.
Leonard Susskind: Inverse squared.
Let’s take the earth. Here’s the earth, and we drop a small mass from far away. As that mass moves in, its acceleration increases. Why does its acceleration increase and its acceleration increases because the radial distance gets smaller. So, in that sense it’s not.
All right. Now, once the gravitational force depends on distance then it’s not really quite true that you don’t feel anything in a gravitational field. You feel something which is to some extent different than you would feel in free space without any gravitational field. The reason is more or less obvious. Here you are — here’s the earth. Now, you, or me, or whoever it is, happens to be extremely tall. Couple of thousand miles tall. Well, this person’s feet are being pulled by the gravitational field more than his head. Or another way of saying the same thing is if let’s imagine that the person is very loosely held together. He’s more or less a gas of – we are pretty loosely held together. At least I am.
All right. The acceleration on the lower portions of his body are larger than the accelerations on the upper part of his body. So it’s quite clear what happens to him. You get stretched. He doesn’t get a sense of falling as such. He gets a sense of stretching, being stretched. Feet being pulled away from his head. At the same time, let’s — all right. So let’s change his shape a little bit. I just spent a week — two weeks in Italy and my shape changes whenever I go to Italy. It tends to get more horizontal. My head is here, my feet are here, and now I’m this way. Still loosely put together.
All right? Now what? Well, not only does the force depend on the distance but it also depends on the direction. The force on my left end over here is this way. The force on my right end over here is this way. The force on the top of my head is down but it’s weaker than the force on my feet. So there are two effects. One effect is to stretch me vertically. It’s because my head is not being pulled as hard as my feet. But the other effect is to be squished horizontally by the fact that the forces on the left end of me are pointing slightly to the right and the forces to the right end of me are pointing slightly to the left. So a loosely knit person like this falling in free fall near a real planet or a real gravitational object which has a real Newtonian gravitational field around it will experience a distortion — will experience a degree of distortion and a degree of being stretched vertically, being compressed horizontally, but if the object is small enough — what does small enough mean? Let’s suppose the object that’s falling is small enough. If it’s small enough, then the gradient of the gravitational field across the size of the object will be negligible and so all parts of it will experience the same gravitational acceleration. So tidal forces — these are tidal forces. These forces which tend to tear things apart vertically and squish them this way. Tidal forces. Tidal forces are forces which are real. You feel them.
Question: Do you recall if Newton calculated lunar tides?
Leonard Susskind: Oh, I think he did. He certainly knew the cause of the tides. I don’t know to what extent he calculated. What do you mean calculated the —
Question: As in this kind of a system with the moon and the sun.
Leonard Susskind: Well, I doubt that he was — I’m not sure whether he estimated the height of the deformation of the oceans or not. But I think he did understand this much about tides. So, that’s what’s called tidal force. And remember, tidal force has this effect of stretching and in particular if we take the earth — just to tell you why it’s called tidal forces of course is because it has to do with tides. I’m sure you all know the story. But if this is the moon down here, then the moon exerting forces on the earth exerts tidal forces on the earth, which means to some extent it tends to stretch it this way and squash it this way. Well, the earth is pretty rigid so it doesn’t deform very much due to the moon. But what’s not rigid is the layer of water around it and so the layer of water tends to get stretched and squeezed and so it gets deformed into a deformed shell of water with a bump on this side and a bump on that side. All right. I’m not going to go any more deeply into that than I’m sure you’ve all seen.
But let’s define now what we mean by the gravitational field. The gravitational field is abstracted from this formula. We have a bunch of particles –
Question: Don’t you have to use some sort of coordinate geometry so that when you have the poor guy in the middle’s being pulled by all the other guys on the side. I’m not explaining it right. But it’s always negative, is that what you’re saying?
Leonard Susskind: No. It’s always attractive.
Question: What about the other guys that are pulling upon him from different directions?
Leonard Susskind: Let’s suppose it’s somebody over here and we’re talking about the force on this person over here. Obviously there’s one force pushing this way and another force pushing that way. They’re all opposite to the direction of the object which is pulling on him.
Question: Well, you kind of retracted the minus sign at the front and reversed the JI. So it’s the direction –
Leonard Susskind: We can get rid of the minus sign in the front there by switching this RJ. RIJ and RJI are opposite to each other. One of them is the vector between I and J. I and J. And the other is the vector from J and I, so they’re equal and opposite to each other. The minus sign there. Look, as far as the minus sign goes, all it means is every one of these particles is pulling on this particle toward it as opposed to pushing away from it. It’s just a convention which keeps track of attraction instead of repulsion.
Question: For the Ith mass — if that’s the right word. If you look at it as kind of an ensemble wouldn’t there be a nonlinear component to it was the I guy, the Ith guy, the Jth guy, then you compute the Jth guy — you know what I mean?
Leonard Susskind: When you take into account the motion. Now, what this formula is for is supposing you know the positions of all the others. You know that. All right? Then what is the force on one additional one but you’re perfectly right. Once you let the system evolve, then each one will cause a change in motion of the other one and so it becomes complicated as you say nonlinear mess. But this formula is a formula for if you knew the position and location of every particle, this would be the force.
Okay? Something. You need to solve some equations to know how the particles move. But if you know where they are, then this is the force on the Ith particle.
All right. Let’s come to the idea of the gravitational field. The gravitational field is in some ways similar to the electric field of an electric charge. It’s the combined effect of all the masses everywhere. And the way you define it is as follows: You imagine one more particle, one more particle. You can take it to be a very light particle so it doesn’t influence the motion of the others. Add one more particle. In your imagination. You don’t really have to add it. In your imagination. And what the force on it is. The force is the sum of the force due to all the others. It is proportional. Each term is proportional to the mass of this extra particle. This extra particle which may be imaginary is called a test particle. It’s a thing that you’re imagining testing out the gravitational field with. You take a light little particle, and you put it here, and you see how it accelerates. Knowing how it accelerates tells you how much force is on it. In fact, it just tells you how it accelerates. And you can go around and imagine putting it in different places and mapping out the force field that’s on that particle. Or the acceleration field since we already know that the force is proportional to the mass. Then we can just concentrate on the acceleration. The acceleration all particles will have the same acceleration independent of the mass. So we don’t even have to know what the mass of the particle is. We put something over there, a little bit of dust, and we see how it accelerates. Acceleration is a vector and so we map out in space the acceleration of a particle at every point in space, either imaginary or real particle, and that gives us a vector field at every point in space. Every point in space there is a gravitational field of acceleration. It can be thought of as the acceleration. You don’t have to think of it as force. Acceleration. The acceleration of a point mass located at that position. It’s a vector that has a direction, it has a magnitude, and it’s a function of position. So, we just give it a name. The acceleration due to all the gravitating objects is a vector and it depends on position. Your X means location. It means all of the components of position X, Y, and Z, and it depends on all the other masses in the problem. That is what’s called the gravitational field. It’s very similar to the electric field except the electric field is force per unit charge. It’s the force of an object divided by the charge on the object. The gravitational field is the force on the object divided by the mass on the object. Since the force is proportional to the mass the acceleration field doesn’t depend on which kind of particle we’re talking about.
All right. So, that’s the idea of a gravitational field. It’s a vector field and it varies from place to place. And, of course, if the particles are moving, it also varies in time. If everything is in motion, the gravitational field will also depend on time. We can even work out what it is. We know what the force on the Ith particle is. Right? The force on a particle is the mass times the acceleration. So, if we want to find the acceleration, let’s take the Ith particle to be the test particle. Little i represents the test particle over here. Let’s erase the intermediate step over here and write that this is MI times AI but let me call it now capital A. The acceleration of a particle at position X is given by the right-hand side. And we can cross out the MI because it cancels from both sides. So, here’s a formula for the gravitational field at an arbitrary point due to a whole bunch of massive objects. A whole bunch of massive objects. An arbitrary particle put over here will accelerate in some direction that’s determined by all the others and that acceleration is gravitation — definition. Definition of the gravitational field.
Okay. Let’s take a little break. We usually take a break at about this time and I recover my breath.
To go on, we need a little bit of fancy mathematics. We need a piece of mathematics called Gauss’s theorem and Gauss’s theorem involves integrals, derivatives, divergences. And we need to spell those things out. They’re essential part of the theory of gravity. And much of these things that we’ve done in the context of the electrical forces, in particular the concept of divergence, divergence of a vector field. So, I’m not going to spend a lot of time on it. If you need to fill in, then I suggest you just find any book on vector calculus and find out what a divergence, and a gradient, and a curl — we won’t do curl today. What those concepts are, and look up Gauss’s theorem and they’re not terribly hard but we’re going to go through them fairly quickly here since we’ve done them several times in the past.
All right. Imagine that we have a vector field. Let’s call that vector field A. It could be the field of acceleration and that’s the way I’m going to use it. But for the moment it’s just an arbitrary vector field, A. It depends on position. When I say it’s a field, the implication is that it depends on position. Now I probably made it completely unreadable. A of X varies from point to point. And I want to define a concept called the divergence of a field. Now, it’s called a divergence because what it has to do is the way the field is spreading out away from a point. For example, a characteristic situation where we would have a strong divergence for a field is if the field was spreading out from a point, like that. The field is diverging away from the point. Incidentally, after the field is pointing inward, then one might say the field has a convergence but we simply say it has a negative divergence. So, divergence can be positive or negative. And there’s a mathematical expression which represents the degree to which the field is spreading out like that. It is called the divergence. I’m going to write it down and it’s a good thing to get familiar with, certainly if you’re going to follow this course it’s a good thing to get familiar with. But if you’re going to follow any kind of physics course past freshmen physics, the idea of divergence is very important.
All right. Supposing the field A has a set of components. The one, two, and three component or we could call them the X, Y, and Z component. Now I’ll use X, Y, and Z. X, Y, and Z. Which I previously called X1, X2, and X3. It has components at AX, AY, and AZ. Those are the three components of the field. Well, the divergence has to do, among other things, with the way the field varies in space. If the field is the same everywhere in space what would that mean? That would mean the field has both not only the same magnitude, but the same direction everywhere in space. Then it just points in the same direction everywhere with the same magnitude. It certainly has no tendency to spread out. When does a field have a tendency to spread out, when the field varies. For example, it could be small over here, growing bigger, growing bigger, growing bigger. And we might even go in the opposite direction and discover that it’s in the opposite direction getting bigger in that direction. Then clearly there’s a tendency for the field to spread out away from the center here. The same thing could be true if it were varying in the vertical direction or if it was varying in the other horizontal direction. And so the divergence, whatever it is, has to do with derivatives of the components of the field. I’ll just tell you exactly what it is. It is equal to it. The divergence of a field is written this way: Upside down triangle. The meaning of this symbol, the meaning of an upside down triangle is always that it has to do with derivatives, the three derivatives. Derivatives, whether it’s the three partial derivatives. Derivative with respect to X, Y, and Z. And this is by definition. The derivative with respect to X of the X component of A plus the derivative with respect to Y of the Y component of A, plus the derivative with respect to Z of the Z component of A. That’s definition.
What’s not a definition is the theorem and it’s called Gauss’s theorem.
Question: Is that a vector or is it –
Leonard Susskind: No. That’s a scale of quantity. It’s a scale of quantity. It’s a scale of quantity. So, let me write it. It’s the derivative of A sub X with respect to X, that’s what this means, plus the derivative of ace of Y with respect of Y, plus the derivative of ace of Z with respect to Z.
Question: Yeah. So, the arrows you were drawing over there they were just A on the other board. You drew some arrows on the other board that are now hidden. Those were just A? Not the divergence.
Leonard Susskind: Right. Those were A. And A has a divergence when it’s spreading out away from a point, but the divergence is itself a scale of quantity. Let me try to give you some idea of what divergence means in a context where you can visualize it. Imagine that we have a flat lake. Just a shallow lake. And the water is coming up from underneath. It’s being pumped in from somewhere underneath. What happens if the water’s being pumped in. Of course, it tends to spread out. Let’s assume that depth can’t change. We put a lid over the whole thing so that it can’t change its depth. We pump some water in from underneath and it spreads out. Okay? We suck some water out from underneath and it spreads in. It anti-spreads. So, the spreading water has a divergence. Water coming in towards the place where it’s being sucked out it has a convergence or a negative divergence. Now, we can be more precise about that. We look down at the lake from above, and we see all the water is moving of course. If it’s being pumped in, the water is moving. And there is a velocity vector. At every point there is a velocity vector. So, at every point in this lake there’s a velocity vector and in particular if there’s water being pumped in from the center here, right? Underneath the bottom of the lake there’s some water being pumped in, the water will spread out away from that point. Okay? And there’ll be a divergence where the water is being pumped in. Okay if the water is being pumped out, then exactly the opposite. The arrows point inward and there’s a negative divergence. If there’s no divergence, then, for example, a simple situation with no divergence. That doesn’t mean the water is not moving. But a simple example with no divergence is the water is all moving together. You know, the river is simultaneous, the lake is moving simultaneously in the same direction with the same velocity. It can do that without any water being pumped in. But if you found that the water was moving from the right on this side and the left on that side, you’d be pretty sure that somewhere in between, water had to be pumped in. Right? If you found the water was spreading out away from a line this way here and this way here, then you’d be pretty sure that some water was being pumped in from underneath along this line here.
Well, you would see it in another way you would discover that the X component of the velocity has a derivative. It’s different over here than it is from over here. The X component of the velocity varies along the X direction. So, the fact that the X component of the velocity is varying along the X direction is an indication that there’s some water being pumped in here. Likewise, if you discovered that the water was flowing up over here and down over here, you would expect it in here somewheres some water was being pumped in. So, derivatives of the velocity are often an indication that there’s some water being pumped in from underneath. That pumping in of the water is the divergence of the velocity vector. Now, the water, of course, is being pumped in from underneath. So, there’s a direction of flow but it’s coming from underneath. There’s no sense of direction — well, okay. That’s what divergence is.
Question: I have a question. The diagrams you already have on the other board behind there with the arrows — If you take, say, the right-most arrow and you draw a circle between the head and tail in between, then you can see the in and the out. The in arrow and the out arrow of the circle right in between those two. And let’s say that the bigger arrow’s created by a steeper slope of the stream.
Leonard Susskind: No, this is faster.
Question: It’s going faster. Okay. And because of that, there’s a divergence there that’s basically it’s sort of the difference between the in and the out.
Leonard Susskind: That’s right. If we draw a circle around here or we would see that – the water is moving faster over here than it is over here, more water is flowing out over here than is coming in over here. Where’s it coming from? It must be pumped in. The fact that there’s more water flowing out on one side than is coming in from the other side must indicate that there’s a net inflow from somewhere else and the somewhere else would be from the pumping water from underneath. So, that’s the idea of divergence.
Question: Could it also be because it’s thinning out? Could that be a crazy example? Like, the lake got shallower?
Leonard Susskind: Yeah. Well, okay. I took — so, let’s be very specific now. I kept the lake having an absolutely uniform height and let’s also suppose that the density of water — water is an incompressible fluid. It can’t be squeezed. It can’t be stretched. Then the velocity vector would be the right thing to think about there. You could have — no, you’re right. You could have a velocity vector having a divergence because the water is — not because water is flowing in but because it’s thinning out. Yeah, that’s also possible. But let’s keep it simple. And you can have — the idea of a divergence makes sense in three dimensions just as well as two dimensions. You simply have to imagine that all of space is filled with water and there are some hidden pipes coming in, depositing water in different places so that it’s spreading out away from points in three dimensional space. In three dimensional space, this is the expression for the divergence. If this were the velocity vector at every point you would calculate this quantity and that would tell you how much new water is coming in at each new point in space. So, that’s the divergence.
Now, there’s a theorem which — the hint of the theorem was just given by Michael there. It’s called Gauss’s theorem and it says something very intuitively obvious. You take a surface, any surface. Take any surface or any curve in two dimensions and now suppose there’s a vector field — vector field point. Think of it as the flow of water. And now let’s take the total amount of water that’s flowing out of the surface. Obviously there’s some water flowing out over here and of course we want to subtract the water that’s flowing in. Let’s calculate the total amount of water that’s flowing out of the surface. That’s an integral over the surface. Why is it an integral because we have to add up the flows of water outward when the water is coming inward that’s just negative flow, negative outward flow. We add up the total outward flow by breaking up the surface into little pieces and asking how much flow is coming out from each little piece here? How much water is passing out through the surface? If the water is incompressible, incompressible means its density is fixed and furthermore, the depth of the water is being kept fixed. There’s only one way that water can come out of the surface and that’s if it’s being pumped in, if there’s a divergence. The divergence could be over here, could be over here, could be over here, could be over here. In fact, anywhere where there is a divergence will cause an effect in which water will flow out of this region here. So, there’s a connection. There’s a connection between what’s going on on the boundary of this region, how much water is flowing through the boundary on one hand, and what the divergence is on the interior. There’s a connection between the two and that connection is called Gauss’s theorem. What it says is that the integral of the divergence in the interior, that’s the total amount of flow coming in from outside, from underneath the bottom of the lake, the total integrated — now, by integrated, I mean in the sense of an integral — the integrated amount of flow in, that’s the integral of the divergence. The integral over the interior in the three dimensional case it would be integral DX, DY, DZ over the interior of this region of the divergence of A — if you like to think of A as the velocity field that’s fine — is equal to the total amount of flow that’s going out through the boundary.
Now, how do we write that? The total amount of flow that’s flowing outward through the boundary we break up — let’s take the three dimensional case. We break up the boundary into little cells. Each little cell is a little area. Let’s call each one of those little areas D sigma. D sigma, sigma stands for surface area. Sigma is the Greek letter. Sigma stands for surface area. This three dimensional integral over the interior here is equal to a two dimensional integral, the sigma over the surface and it is just the component of A perpendicular to the surface. It’s called A perpendicular to the surface D sigma. A perpendicular to the surface is the amount of flow that’s coming out of each one of these little boxes. Notice, incidentally, if there’s a flow along the surface it doesn’t give rise to any fluid coming out. It’s only the flow perpendicular to the surface, the component of the flow perpendicular to the surface which carries fluid from the inside to the outside. So, we integrate the perpendicular component of the flow over the surface, that’s still the sigma here. That gives us the total amount of fluid coming out per unit time, for example and that has to be equal to the amount of fluid that’s being generated in the interior by the divergence. This is Gauss’s theorem. The relationship between the integral of the divergence and the interior of some region and integral over the boundary where it’s measuring the flux — the amount of stuff that’s coming out through the boundary. Fundamental theorem.
And let’s see what it says now. Any questions about Gauss’s theorem here? You’ll see how it works. I’ll show you how it works.
Question: Now, you mentioned that the water is incompressible. Is that different from what we were given with the compressible fluid.
Leonard Susskind: Yeah. You could — if you had a compressible fluid you could discover that the fluid out in the boundary here is all moving inwards in every direction without any new fluid being formed. In fact, what’s happening is the fluid’s getting squeezed. But if the fluid can’t squeeze, if you cannot compress it, then the only way that fluid could be flowing in is if it’s being removed somehow from the center. If it’s being removed by invisible pipes that are carrying it off.
Question: So that means the divergence in the case of water would be zero would be integrated over a volume?
Leonard Susskind: If there was no water coming in it would be zero. If there was a source of the water — divergence is the same as its source. Source of water is — source of new water coming in from elsewhere is — Right. So, with the example of the two dimensional lake, the source is water flowing in from underneath, the sink, which is the negative of a source, is the water flowing out and in the two dimensional example this wouldn’t be a two dimensional surface integral. It would be the integral in here equal to a one dimensional surface integral coming out.
All right. Let me show you how you use this. Let me show you how you use this and what it has the do with what we’ve said up ’til now about gravity. I think — I hope we’ll have time. Let’s imagine that we have a source, it could be water but let’s take three dimensional case, there’s a divergence of a vector field, let’s say A. There’s a divergence of a vector field, dell dot A, and it’s concentrated in some region of space. It’s a little sphere in some region of space that has spherical symmetry. In other words, it doesn’t mean that the divergence is uniform over here but it means that it has the symmetry of a sphere. Everything is symmetrical with respect to rotations. Let’s suppose that there’s a divergence of the fluid. Okay? Now, let’s take — and it’s restricted completely to be within here. It could be strong near the center and weak near the outside or it could be weak near the center and strong near the outside but a certain total amount of fluid or a certain total divergence, an integrated divergence is occurring with nice spherical shape.
Let’s see if we can use that to figure out what the A field is. That’s dell dot A in here and now let’s see can we figure out what the field is elsewhere outside of here? So, what we do is we draw a surface around there. We draw a surface around there and now we’re going to use Gauss’s theorem. First of all, let’s look at the left side. The left side has the integral of the divergence of the vector field. All right. The vector field or the divergence is completely restricted to some finite sphere in here. What is — incidentally, for the flow case, for the fluid flow case, what would be the integral of the divergence? Does anybody know? if it really was the flow of a fluid. It’ll be the total amount of fluid that was flowing in per unit time. It would be the flow per unit time that’s coming through the system. But whatever it is, this integral doesn’t depend on the radius of the sphere as long as the sphere, this outer sphere here, is bigger than this region. Why? Because the integral over the divergence of A is entirely concentrated in this region here and there’s zero divergence on the outside. So, first of all, the left-hand side is independent of the radius of this outer sphere, as long as the radius of the outer sphere is bigger than this concentration of divergence here. So, it’s a number. Although it’s a number. Let’s call that number M. No, no. Not M. Q. That’s the left-hand side. And it doesn’t depend on the radius.
On the other hand, what is the right side? Well, there’s a flow going out and if everything is nice and spherically symmetric then the flow is going to go radially outward. It’s going to be a pure, radially outward directed flow if the flow is spherically symmetric. Radially outward directed flow means that the flow is perpendicular to the surface of the sphere. So, the perpendicular component of A is just the magnitude of A. That’s it. It’s just the magnitude of A and it’s the same everywhere on the sphere. Why is it the same? Because everything has spherical symmetry. Now, in spherical symmetry, the A that appears here is constant over this whole sphere. So, this integral is nothing but the magnitude of A times the area of the total sphere. All right? If I take an integral over a surface, a spherical surface like this, of something which doesn’t depend on where I am in the sphere, then just you can take this on the outside, the magnitude of the field and the integral D sigma is just the total surface area of the sphere. What’s the total surface area of the sphere? Four thirds pi R.
Leonard Susskind: No third. Four pi R squared. 0Oh, yeah. Four pi R squared times the magnitude of the field is equal to Q. So, look what we have. We have that the magnitude of the field is equal to the total integrated divergence divided by four pi. Four pi is just a number, times R squared. Does that look familiar? It’s a vector field. It’s pointed radially outward. Well, it’s pointed radially outward if the divergence is positive. If the divergence is positive, it’s pointed radially outward and its magnitude is one of R squared. It’s exactly the gravitational field of a point particle at the center here.
Question: It’s the magnitude of A.
Leonard Susskind: Yeah. That’s why we have to put a direction in here. You know what this R is? It’s a unit vector pointing in the radial direction. It’s a vector of unit length pointing in the radial direction. Right? So, it’s quite clear from the picture that the A field is pointing radially outward. That’s what this says here. In any case, the magnitude of the field that points radially outward, it has magnitude Q, and it falls off like one over R squared. Exactly like the Newtonian field of a point mass. So, a point mass can be thought of as a concentrated divergence of the gravitational field right at the center. A point mass. A literal point mass can be thought of as a concentrated – a concentrated divergence of the gravitational field. Concentrated in some very little small volume. Think of it, if you like, you can think of the gravitational field as the flow field, the velocity field of a fluid that’s spreading out. Oh, incidentally, of course, I’ve got the sign wrong here. The real gravitational acceleration points inward which is an indication that this divergence is negative. The divergence is more like a convergence sucking fluid in. So the Newtonian gravitational field is isomorphic, is mathematically equivalent, or mathematically similar, to a flow field to a flow of water or whatever other fluid where it’s all being sucked out from a single point and, as you can see, the velocity field itself or in this case the field, the gravitational field, the velocity field would go, like, one over R squared. That’s a useful analogy. That is not to say that space is a flow or anything. It’s a mathematical analogy that’s useful to understand the one over R squared force law that it is mathematically similar to a field of velocity flow from a flow that’s being generated right at the center of a point. Okay. That’s a useful observation.
But notice something else. Supposing now, instead of having the flow concentrated at the center here, supposing the flow is concentrated over a sphere which was bigger but the same total amount of flow. It would not change the answer. As long as the total amount of flow is fixed, the way that it flows out through here is also fixed. This is Newton’s theorem. Newton’s theorem in the gravitational context says that the gravitational field of an object, outside the object is independent of whether the object is a point mass at the center or whether it’s a spread out mass, or whether it’s a spread out mass this big, as long as you’re outside the object and as long as the object is spherically symmetric, in other words, as long as the object is shaped like a spear and you’re outside of it, outside of it, outside of where the mass distribution is, then the gravitational field of it doesn’t depend on whether it’s a point, it’s a spread out object, whether it’s denser at the center and less dense on the outside, less dense in the inside, more dense on the outside. All it depends on is the total amount of mass. The total amount of mass is like the total amount of flow coming into the — that theorem is very fundamental and important to thinking about gravity. For example, supposing we are interested in the motion of an object near the surface of the earth but not so near that we can make the flat space approximation. Let’s say at a distance two, three, or one and a half times the radius of the earth. Well, that object is attracted by this point, it’s attracted by this point, it’s attracted by that point. It’s close to this point, it’s far from this point. It sounds like a hellish problem to figure out what the gravitational effect on this point is. But no. This tells you the gravitational field is exactly the same as if the same total mass was concentrated right at the center. Okay? That’s Newton’s theorem. It’s a marvelous theorem. It’s a great piece of luck for him because without it he couldn’t have solved his equations. He knew. He had an argument, it may have been essentially this argument. I’m not sure exactly what argument he made. But he knew that with the one over R squared force law and only the one over R squared force law wouldn’t have been true if it was one over R cubed, one over R to the fourth, one over R to the seventh. With the one over R squared force law, a spherical distribution of mass behaves exactly as if all the mass was concentrated right at the center as long as you’re outside the mass. So, that’s what made it possible for Newton to easily solve his own equations. That every object, as long as it’s spherical in shape, behaves if it were a point mass.
Question: So, if you’re down in a mine shaft that doesn’t hold?
Leonard Susskind: That’s right. If you’re down in a mine shaft it doesn’t hold. But, that doesn’t mean that you can’t figure out what’s going on. You can figure out what’s going on. I don’t think we’ll do it tonight. It’s a little too late. But yes, we can work out what would happen in a mine shaft. But that’s right. It doesn’t hold in a mine shaft.
For example, supposing you dig a mine shaft right down through the center of the earth and now you get very close to the center of the earth. How much force do you expect to be pulling toward the center? Not much. Certainly much less than if all the mass were concentrated right at the center. You’ve got the — it’s not even obvious which way the force is but it’s toward the center. But it’s very small. You displace away from the center of the earth a little bit. There’s a tiny, tiny force. Much, much less than as if all the mass was squashed towards the center. So, right. It doesn’t work for that case.
Another interesting case is supposing you have a shell of material. To have a shell of material, think about a shell of source, fluid flowing in. Fluid is flowing in from the outside onto this blackboard and all the little pipes are arranged on a circle like this. What does the fluid flow look like in different places? Well, the answer is, on the outside it looks exactly the same as if everything were concentrated on a point. But what about in the interior? What would you guess? Nothing. Everything is just flowing out away from here and there’s no flow in here at all. How can there be? Which direction would it be? And so there’s no flow in here.
Question: Wouldn’t you have the distance argument? Like, if you’re closer to the surface of the inner shell — wouldn’t that be more force towards that?
Leonard Susskind: No. See, you use Gauss’s theorem. Let’s use Gauss’s theorem. Gauss’s theorem says okay, let’s take a shell. The integrated field coming out of that shell is equal to the integrated divergence in here. But there is no divergence in here. So, the net integrated field coming out is zero. No field on the interior of the shell. Field on the exterior of the shell. So, the consequence is that if you made a spherical shell of material like that, the interior would be absolutely identical to what it would be if there was no gravitational material there at all.
On the other hand, on the outside, you would have a field which would be absolutely identical to what happens at the center. Now, there is an analogue of this in the general theory of relativity. We’ll get to it. Basically what it says is the field of anything, as long as it’s spherically symmetric on the outside, looks like identical to the field of a black hole. I think we’re finished for tonight. Go over divergence and Gauss’s theorem. Gauss’s theorem is central. There would be no gravity without Gauss’s theorem.