Einstein's General Theory of Relativity | Lecture 1

This program is presented by Stanford University. Visit us at Stanford.edu. Leonard Susskind: Gravity. Gravity is a pretty special force. Is unusual. It has difference in electrical forces, magnetic forces and is somehow connected with geometric properties of space, space and time. But... and that connection is, of course, the

general

theory

of

relativity

. Before we begin, we will not discuss the

general

theory

of

relativity

tonight. We will deal with gravity in its oldest and simplest mathematical form. Well, maybe not the oldest and simplest gravity, but Newtonian. And going a little further than Newton, certainly nothing that Newton wouldn't have recognized or couldn't have grasped (Newton could grasp anything), except some ways of thinking about it that wouldn't be found in Newton's actual work.

But it's still Newtonian gravity. Newtonian gravity is set up in a way that is useful for moving on to the general theory. Well. Let's start with Newton's equations. The first equation, of course, is F equals MA. Force is equal to mass times acceleration. Suppose we have a reference, a reference frame that means a set of coordinates and that was a set of clocks, and that reference frame is what is called an inertial reference frame. An inertial reference frame simply means one that, if there are no objects around that exert forces on a particular... let's call it a test object.

A test object is simply some object, a small particle or whatever, that we use to test the various fields, force fields, that might be acting on it. An inertial system is one in which, when there are no objects around it exerting forces, that object will move with uniform motion without acceleration. That's the idea of ​​an inertial reference frame. So if you're in an inertial frame of reference and you have a pen and you let it go, it stays there. It does not move. If you give it a push, it will move away with uniform speed. That is the idea of ​​an inertial reference frame and in an inertial reference frame the basic Newtonian equation number one.

I always forget which law is which. There is Newton's first law, second law, and third law. I can never remember which is which. But they all practically boil down to F equals mass times acceleration. This is a vector equation. I hope people know what a vector is. Uh, a three vector equation. We will come later to the four vectors where space and time come together in space-time. But for now, space is space and time is time. And vector means something that is a pointer in a direction of space, has a magnitude, and has components. So, component by component, the X component of force is equal to the mass of the object multiplied by the To indicate a vector acceleration and so on, I'll try to remember to put an arrow over the vectors.

Mass is not a vector. Mass is simply a number. Every particle has a mass, every object has a mass. And in Newtonian physics mass is conserved. Can not change. Now, of course, the mass of this cup of coffee here can change. It is now lighter but it only changes because the mass is transported from one place to another. So, you can change the mass of an object by hitting a piece of it, but if you don't change the number of particles, you change the number of molecules, etc., then mass is a conserved and invariant quantity. So, that's the first equation.

Now let me write that another way. The other way we imagine that we have a coordinate system, an X, a Y and a Z. I don't have enough dimensions on the board to draw Z. It doesn't matter. X, Y and Z. Sometimes we just call them X one, X two and X three. I guess I could draw it. X three is around here somewhere. X, Y and Z. And a particle has a position, which means it has a set of three coordinates. Sometimes we will exactly summarize the collection of the three coordinates X one, X two and X three.

X one, X two and X three are components of a vector. They are components of the particle's position vector. The position vector of the particle I will often call small r or large R depending on the particular context. R means radius, but radius simply means the distance between the point and the origin, for example. We're really talking now about something with three components, X, Y and Z, and it's the radial vector, the radial vector. This is the same as the components of the vector R. Okay. Acceleration is a vector that is made up of the time derivatives of X, Y, and X, or X one, X two, and X three.

So for each component, for each component, one, two or three, the acceleration, which let me indicate, let's call it A. The acceleration is simply equal, its components are equal to the second derivatives. of the coordinates with respect to time. That's acceleration. The first derivative of position is called velocity. Well. We can take this component by component. X one, X two and X three. The first derivative is speed. 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 speed.

And so Newton's equations are summarized (not summarized but rewritten) as the force on an object, whatever it is, component by component, is equal to the mass multiplied by the second derivative of the position component. So, that's the summary of... I think it's Newton's first and second laws. I can never remember what they are. Newton's first law, of course, is simply the statement that if there are no forces, then there is no acceleration. That is Newton's first law. Equal and opposite. Good. And this sums up both the first and second laws. I never understood why there was a first and a second law.

It seemed to me that it was a law, F equals MA. Alright. Now, let's start even before Newton with Galilean gravity. Gravity as Galileo understood it. Actually, I'm not sure how much of this mathematics Galileo understood or didn't understand. He certainly knew what acceleration was. He measured it. I don't know if he had... he certainly didn't have calculus, but he knew what acceleration was. So what Galileo studied was the motion of objects in the Earth's gravitational field in the approximation that the Earth is flat. Now, Galileo knew that the Earth was not flat but he studied gravity in an approach where you never get very far from the surface of the Earth.

And if you don't go too far from the surface of the earth, you could also consider that the surface of the earth is flat and the meaning of that is twofold. First of all, the direction of gravitational forces is the same everywhere. Of course, this is not true; If the Earth is curved, then gravity will point towards the center. But in the flat space approximation, gravity points downward. Everywhere always in the same direction. And secondly, maybe a little less obvious but true, the approximation where the Earth is infinite and flat, goes on forever, infinite and flat, the gravitational force does not depend on how high up you are.

The same gravitational force here as here. The implication of this is that the acceleration of gravity, the force apart from the mass of an object, the acceleration on an object is independent of where it is placed. And then Galileo realized it or not. Again, I don't know exactly what Galileo knew or didn't know. But what he said was the equivalent of saying that the force of an object in approximating flat space is very simple. First of all, it has a single component that points downwards. If we take the upward direction of things as positive, then we would say that the force is... let's just say that the component of the force in the X two direction, the vertical direction, is equal to minus... the minus simply means that The force is downward and is proportional to the mass of the object multiplied by a constant called gravitational acceleration.

Now the fact that it is constant everywhere, in other words, the mass multiplied by G varies from place to place. That is the fact that gravity does not depend on where you are in the approximation of flat space. But the fact that force is proportional to the mass of an object is not obvious. In fact, for most forces, this is not true. For electric forces, force is proportional to electric charge, not mass. And so gravitational forces have a special character: the strength of the gravitational force of an object is proportional to its mass. This almost completely characterizes gravity.

That's what's special about gravity. Force is proportional to mass. Well, if we combine F equals MA with the force law (this is the force law), then what we find is that mass times acceleration D b X, now this is the vertical component, times DT squared equals minus- - that's the negative point - MG point. That's all. Now, the interesting thing that happens in gravity is that mass cancels out on both sides. That's what's so special about gravity. The mass cancels on both sides. And the consequence of that is that the motion of the object, its acceleration, does not depend on the mass... it does not depend on anything related to the particle.

The particle, object... I will use the word particle. I don't necessarily mean a point particle, a baseball is a particle, an eraser is a particle, a piece of chalk is a particle. That the motion of an object does not depend on the mass of the object or anything else. The result of this is that if you take two objects of very different mass and let them fall, they fall in exactly the same way. Galileo did that experiment. I don't know if he actually threw something from the Leaning Tower of Pisa or not. It's not important. He rolled down an inclined plane.

I don't know if he really did it or not. I know the myth is that he didn't do it. I find it very difficult to believe that he didn't do it. I have been to Pisa. Last week I was in Pisa and took a look at the Leaning Tower of Pisa. Galileo was born and lived in Pisa. He was interested in gravity. How he couldn't think of leaving something from the Leaning Tower is beyond me. You look at that tower and say: "That tower is used for one thing: leaving things." Leonard Susskind: Now, I don't know.

Maybe the Doge or whatever they called the guy at the time said, no, not Galileo. You can't drop things from the tower. You will kill someone. So maybe he didn't do it. He surely he must have thought of that. Alright. So if you had done so and not had to worry about such spurious effects as air resistance, the result would be that a cannonball and a feather would fall in exactly the same way, regardless of mass, and the equation would be Just Let's say, the acceleration would be first down, that's the minus sign, and equal to this constant G.

Excuse me. Now, G as a number, is 10 meters per second per second on the surface of the earth. On the surface of the moon it is somewhat smaller. On the surface of Jupiter it is somewhat larger. So it depends 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 drop it on, but not the mass of the object stopping it. That fact, that gravitational motion is completely independent of mass, is called or is the simplest version of something called the equivalence principle.

We will see why it is called the equivalence principle later. What is equivalent to what. At this stage we can simply say that gravity is equivalent between all different objects regardless of their mass. But that's not exactly what the principle of equivalence/inequivalence is about. Alright. That has a consequence. An interesting consequence. Suppose I take an object that is made of something that is not rigid. Just a collection of point masses. Maybe we'll even say that they're not even exerting any force on each other. It is a cloud, a very diffuse cloud of particles and we see it fall.

Now, suppose we start each particle from rest, not all at the same height, and let them all fall. Some particles are heavy, some are light, some may be large, and some may be small. How does everything fall? And the answer is that all particles fall at exactly the same rate. The consequence of this is that the shape of this object does not deform when falling. It remains absolutely unchanged. The relations between neighboring parties do not change. There are no stresses or deformations that tend to deform the object. So even if the object was held together by some kind of struts or whatever, there would be no forces on those struts because everything falls together.

Well? The consequence of this is that falling into the gravitational field is undetectable. You can't tell you're falling into a gravitational field... when I say you can't tell, you can certainly tell the difference between free fall and staying on Earth. Alright? That's not the point. The point is that you can't tell, by looking at your neighbors or anything else, that a force is being exerted on you and that force being exerted on you is pulling you down. You could also, for all practical purposes, be infinitely far from Earth without any gravity and just sitting there because, as far as you can see, the gravitational field has no tendency to deform this object or anything else.

You cannot distinguish between being infree space infinitely far from anything without forces and freely falling in a gravitational field. That is another statement of the equivalence principle. You say it's not mechanically detectable? Leonard Susskind: Well, actually, undetectable, period. But so far it cannot be detected mechanically. 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 kind. What's that? Oh, you could certainly tell that if you were standing on the ground here, you could definitely tell that something was falling towards you. But the question is, from inside this object by itself, without looking at the floor, without knowing that the floor was-- Something that wasn't moving.

Leonard Susskind: Well, you can't tell if you're falling and it's... yeah. If there was something that didn't fall, it would only be because there was some other force on it, like a beam or tower of some kind, holding it up. Because? Because this object, if there are no other forces on it, only gravitational forces, will fall at the same rate as this one. Alright. So that's another expression of the equivalence principle, that you can't distinguish between being in free space, far from any gravitational object, and being in a gravitational field. Now let's modify this. This, of course, is not entirely true in a real gravitational field, but in this approximation of flat space where everything comes together, you cannot say that there is a gravitational field.

At least, you can't tell the difference, at least not without seeing the ground. The autonomous object here experiences nothing different than it would experience away from any still gravitating object. Or in uniform motion. Another question. Leonard Susskind: What is that? Yes. We can know where we are accelerating. Leonard Susskind: No, you can't tell when you're accelerating. Well, you can... you can't feel... isn't that because you can say that there is no connection between the objects? Leonard Susskind: Okay. This is what you can say. If you climb to the top of a tall building and close your eyes, get off and free fall, you will feel exactly the same: you will feel weird.

I mean, that's not how you normally feel because your stomach bloats up and does some weird things. You know, you could lose it. But the thing is, one would feel exactly the same discomfort in outer space, far from any gravitating object that was simply standing still. You will feel exactly the same peculiar feelings. Well? What are these peculiar feelings due to? They are not due to a fall. They shouldn't fall... well. . . They are due to the fact that when you are here on earth, there are forces on the bottom of your feet that prevent you from falling and if the earth suddenly disappeared under my feet, sure enough, my feet would feel funny because they are used to that force. is exerted on their butts.

You get it. Wait. So the fact that you feel strange in free fall is because you are not used to free fall. It doesn't matter whether you are infinitely far from gravitational objects that are stationary or freely falling in the presence of a gravitational field. Now, as I said, this will have to be modified a little. There are things like tidal forces. These 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 at each point and that it varies with height. This is due to the finitude of the earth.

But, in the flat space of... in the flat Earth approximation, where the Earth is infinitely large and pulls uniformly, there is no other effect of gravity that is different from being in free space. Well. Again, this is known as the equivalent principle. Now, let's move beyond the flat space or flat Earth approximation and move on to Newton's theory of gravity. Newton's theory of gravity says that 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 towards the other.

Equal and Opposite Forces and Force Magnitude: The magnitude of the force of one object on another. Let's characterize them by a mass. Let's call this 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. Okay. Newton's force law is that force is proportional to the product of masses. Making either mass heavier will increase strength. The product of the masses, large M times small m, inversely proportional to the square of the distance between them. Let's call that R squared. Let's call R the distance between them.

And there is a numerical constant. This law alone could not be correct. It is not dimensionally consistent. The... if you calculate the dimensions of force, mass, mass and R, it's not dimensionally consistent. There has to be a constant there. And that numerical constant is called capital G, Newton's constant. And it is very small. It is a very small constant. I'll note what it is about. G is equal to six or 6. 7, approximately, multiplied by ten to the power of negative 11, which is a small number. So, at first glance, it seems that gravity is a very weak force.

Umm, you might not think that gravity is such a weak force, but to convince yourself that it is a weak force, there is an experiment you can do. Weak compared to other forces. I have made this for classes and you can do it yourself. It is enough to take an object hanging from a rope and two experiments. The first experiment, take a small object here and charge it electrically. You charge it electrically by rubbing it on your shirt. That doesn't put much charge on it, but it charges it enough to feel some electrostatic force. Then take another object of exactly the same type, rub it on your shirt and put it here.

What happens? They repel each other. And the fact that they repel each other means that this will change. And you will see how it changes. Let's take another example. Take your little ball there to make it iron and put a magnet next to it. Again, you will see a fairly detectable deflection of the rope holding it. Alright? Next, take a 10,000 pound weight and place it here. Guess what happens? Undetectable. You can't see anything happening. The gravitational force is much, much weaker than most other types of forces and that is because... not because. Not for that. The fact that it is so weak is summed up here in this small number.

Another way to say it is if you take two masses, each 1 kilometer, not 1 kilometer. 1 kilogram. A kilogram is a good healthy mass, a good lump of iron. MM and you separate them 1 meter, then the force between them is just G and it is 6.7 times ten to the power of negative 11, if you do it with the units in Newton. Very weak strength. But, weak as it is, we feel it quite intensely. We feel it strongly because the earth is very heavy. So the weight of the Earth compensates for the smallness of G and that's why we wake up in the morning feeling like we don't want to get out of bed because gravity is holding us down.

Leonard Susskind: Yes? Umm, so that force measures the force between... from the big one to the small one or both? Leonard Susskind: Both. Both. They are equal and opposite. Equal and opposite. That's the rule. That is Newton's third law. The forces are equal and opposite. So the force on the big one due to the small one is the same as the force of the small one on the big one. But it is proportional to the product of the masses. So the meaning of that is that I'm not heavier than I'd like, but I'm not very heavy.

I'm certainly not heavy enough to significantly deflect the hanging weight. But I exert a force on the Earth that is exactly equal and opposite to the force that the very heavy Earth exerts on me. Well. Why is the Earth accelerating? If I fall from a certain height, I accelerate downwards. The Earth barely accelerates, even though the forces are equal. Why does the Earth... if the forces are equal, my force on the Earth and the Earth's force on me are equal, why does the Earth accelerate so little and I accelerate so much? Yes. Because acceleration implies two things.

It involves force and mass. The greater the mass, the smaller the acceleration of the force. So the earth doesn't accelerate... yes, ask. 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, umm, what was the key... it was big... no, no. It was from Kepler's laws. It was from Kepler's laws. He worked out, generally speaking... I don't know what he did. He was secretive and didn't really tell people what he did. But, umm, the knowledge that he had was Kepler's laws of motion, planetary motion, and I guess he just wrote down a general force and figured out that he would get Kepler's laws of motion for the inverse square law. .

Umm, I don't think he had any lying theoretical reason to believe in the inverse square law. Edmund Halley actually asked him what kind of force law is needed for conic section orbits and he had almost done the calculations a year earlier. Leonard Susskind: Yes. So, yes. Leonard Susskind: Actually, I don't think... yeah. I think the question... he asked the question about inverse square laws and I think Newton already knew that the solution was an ellipse. Leonard Susskind: No. It wasn't the ellipse that was there. The orbits could have been circular. It was the fact that the period varies like the three halves of the radius.

Alright? The period of motion is circular motion that has an acceleration toward the center. Any movement of the circle is accelerated towards the center. If you know the period and the radius, then you know the acceleration toward the center. Well? We could write... what's the word? Does anyone know what? If I know the angular frequency of rotating in an orbit called omega. Does anyone know... and it's basically the reverse period. Well? Omega is approximately the number of cycles per second of the inverse period. What is the acceleration of an object moving in a circular orbit? Does anyone remember?

Omega squared R. Leonard Susskind: Omega squared R. That's the acceleration. Now, suppose you equate that to some unknown force law F of R and then divide it by R. Then you find 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, is it Kepler's second law? I don't even remember which one it is. It is the law that says that the frequency or period, the square of the period, is proportional to the cube of the radius.

That was Kepler's law. So, from Kepler's laws you could easily... or from that law, you could easily reduce 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 that if one had a perfectly circular law orbit, then the inverse square law was the only law that would give elliptical orbits. So, it's a matter of two steps. What happens when the two objects touch each other? Is it measured from the... Leonard Susskind: Of course, there are other forces on them. If two objects touch each other, there are all kinds of forces between them that are not just gravitational.

Electrostatic forces, atomic forces, nuclear forces? Then you will have to modify the entire story. As the distance approaches zero... Leonard Susskind: Yeah. Then it breaks down. Then it breaks down. Yes. Then it breaks. When they get so close that other important forces come into play. The other important forces, for example, are the forces that hold this object and prevent it from falling. We generally call them contact forces, but in fact what they really are are various types of electrostatic forces between the atoms and molecules on the table and the atoms and molecules here. So, other types of forces.

Alright. By the way, let me point out that if we are talking about other types of force laws, for example, electrostatic force laws, then the force... we still have F equals MA but the force law... the force law It won't be so the force is somehow proportional to mass multiplied by something else, but it could be electric charge. If it is electric charge, then objects without electric charge will have no forces on them and will not accelerate. Electrically charged objects will accelerate in an electric field. So electric forces don't have this universal property that everything falls or everything moves in the same way.

Discharged particles move differently than charged particles with respect to electrostatic forces. They move in the same way with respect to gravitational forces. And like repulsion and attraction, while gravitational forces are always attractive. Where is my gravitational force? I lost it. Yes. Here you have it. Alright. So, that's Newtonian gravity between two objects. For simplicity, let's just place one of them, the heavy one, at the origin of the coordinates and study the motion of the light one and then... oh, by the way, as they say... let me refine this a little. As I wrote it here, I haven't really expressed it as a vector equation.

This is the magnitude of the force between two objects. Thinking of it as a vector equation, we have to provide a direction for the force. Vectors have directions. In what direction is the force on thismotion. Now, what this formula is for is assuming that you know the positions of all the others. You know it. Alright? So what is the force on an additional one? But you're absolutely right. Once you let the system evolve, each will cause a change in the motion of the other and will therefore become a complicated non-linear mess, as you say. But this formula is a formula so that if you knew the position and location of each particle, this would be the force.

Well? Something. You need to solve the equations to find out how the particles move. But if you know where they are, then this is the force on the I-th particle. Alright. Let's get to the idea of ​​the gravitational field. The gravitational field is in some ways similar to the electric field of an electric charge. It is the combined effect of all masses everywhere. And the way you define it is this: you imagine one more particle, one more particle. You can consider it a very light particle so that it does not influence the movement of the others.

Add one more particle. In your imagination. You don't really need to add it. In your imagination. And what is the force on him. Force is the sum of the forces 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 something you imagine testing the gravitational field with. You take a small light particle, put it here and watch it accelerate. Knowing how it accelerates tells you how much force is exerted on it. In fact, it only tells you how it accelerates.

And you can imagine putting it in different places and mapping the force field that's on that particle. Or the acceleration field since we know that force is proportional to mass. Then we can focus on acceleration. Acceleration All particles will have the same acceleration regardless of mass. So we don't even need to know what the mass of the particle is. We put something there, a little bit of dust, and we see how it accelerates. Acceleration is a vector, so we plot in space the acceleration of a particle at each point in space, whether an imaginary or real particle, and that gives us a vector field at each point in space.

At every point in space there is a gravitational acceleration field. It can be considered acceleration. You don't have to think of it as strength. Acceleration. The acceleration of a point mass located in that position. It is a vector that has direction, magnitude and is a function of position. So, we just give it a name. The acceleration due to all gravitating objects is a vector and depends on position. Your X means location. It means all the components of positions X, Y and Z, and depends on all the other masses in the problem. That is what is called gravitational field.

It is very similar to the electric field except that the electric field is force per unit charge. It is the force of an object divided by the charge of the object. The gravitational field is the force on the object divided by the mass on the object. Since force is proportional to mass, the desired acceleration field depends on the type of particle we are talking about. Alright. That's the idea of ​​a gravitational field. It is a vector field and 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 find out what it is. We know what the force is on particle I. Okay? The force on a particle is the mass multiplied by the acceleration. So if we want to find the acceleration, let's take the Ith particle as the test particle. The little i represents the test particle here. Let's delete the immediate step here and write that this is MI multiplied by AI, but let me call it capital A now. The acceleration of a particle in position X is given by the right side. And we can cross out the MI because it cancels out on both sides.

So here's a formula for the gravitational field at an arbitrary point due to a bunch of massive objects. Lots of massive objects. An arbitrary particle placed here will accelerate in some direction determined by all the others and that acceleration is gravitation, definition. The definition is the gravitational field. Well. Let's take a little break. We usually take a break around this time and I catch my breath. To continue, we need some fancy math. We need a piece of mathematics called Gauss's theorem and Gauss's theorem involves integrals, derivatives and divergences. And we need to explain those things. They are an essential part of the theory of gravity.

And a lot of these things we've done in the context of electrical forces, particularly the concept of divergence, divergence of a vector field. So I'm not going to spend much time on it. If you need completion, I suggest you look up any book on vector calculus and find out what a divergence, a gradient, and a curvature are; today we will not do curvatures. What those concepts are, look up Gauss's theorem and they're not very difficult, but we'll go over them pretty quickly here since we've done them a few times in the past. Alright. Let's imagine that we have a vector field.

Let's call that vector field A. It could be the acceleration field and that's how I'll use it. But at the moment it is just an arbitrary vector field, A. It depends on the position. When I say it is a field, what I imply is that it depends on the position. Now I probably made it completely unreadable. A of X varies from one point to another. And I want to define a concept called divergence of a field. Now, it's called divergence because what it has to do with is the way the field extends away from the point.

For example, a typical situation where we would have strong divergence for a field is if the field extended from a point like that. The field moves away from the point. By the way, after the field points inward, you could say that the field has a convergence, but we simply say that it has a negative divergence. So the divergence can be positive or negative. And there is a mathematical expression that represents the degree to which the field is extending like this. It's called divergence. I'm going to write it down and it's good to get familiar with it.

Certainly, if you are going to follow this course, it is good to familiarize yourself with it. But if you're going to take any kind of physics course after freshman physics, the idea of ​​divergence is very important. Alright. Suppose field A has a set of components. Components one, two and three or we could call them components X, Y and Z. Now I will use X, Y and Z. X, Y and Z. Which I previously called X one, X two and X three. It has components in AX, AY and AZ. Those are the three components of the vector.

Well, 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 that the field not only has the same magnitude, but also the same direction anywhere in space. So it just points in the same direction everywhere in space with the same magnitude. It certainly has no tendency to spread. When does a field tend to extend and when does it vary? For example, it could be small here, growing bigger, bigger, bigger. And we might even go in the opposite direction and discover that it is the opposite direction, getting bigger in that direction.

Now, there is clearly a tendency here for the field to spread out from the center. The same could be true if it varied in the vertical direction or if it varied 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 tell you exactly what it is. It's equal to that. The divergence of a field is written this way: Inverted triangle. The meaning of this symbol, the meaning of an inverted triangle is always that it has to do with the derivatives, the three derivatives.

Derivatives, whether the three partial derivatives. Derived with respect to X, Y and Z. And this is by definition. The derivative with respect to What is not a definition is the theorem and is called Gauss's theorem. I'm sorry. Is it a vector or is it... Leonard Susskind: No. That's a scale of quantities. It is a scale of quantities. Yes. It is a scale of quantities. So let me write it down. It's the derivative of A sub that you were drawing there were just A's on the other board. You drew some arrows on the other board that are now hidden.

Leonard Susskind: Yes. Were those just A's? Leonard Susskind: Yes. Not the divergence. Leonard Susskind: Right. Those were A. And A has a divergence when it moves away from a point, but a divergence is itself a scale of quantity. Let me try to give you an idea of ​​what divergence means in a context where you can visualize it. Let's imagine we have a flat lake. Just a shallow lake. And the water rises from below. It's being pumped from somewhere below. What happens if water is pumped? Of course, it tends to spread. Suppose the depth cannot change. We put a lid on everything so it can't change its depth.

We pump some water from below and it spreads. Well? We suck some water from below and it spreads. It is anti-spread. So the water that spreads has a divergence. The water that enters the place where it is being sucked has a negative convergence or divergence. Now we can be more precise about it. We look at the lake from above and see that, of course, all the water is moving. If it is being pumped the water is moving. And there is a velocity vector. At each point there is a velocity vector. So at every point in this lake there is a velocity vector and in particular if you pump water from the center here, right?

Some water is pumped under the lake water and sucked from that point. Well? And there will be a divergence in where the water is pumped. Well, if the water is pumped, then exactly the opposite. The arrows point inwards and there is a negative divergence. If there is no divergence, then, for example, a simple situation without divergence. That doesn't mean the water isn't moving. But a simple example of non-divergence is that water moves all together. You know, the river is simultaneous, the lake moves simultaneously in the same direction with the same speed. You can do it without pumping water.

But if you find that the water is moving to the right on this side and to the left on that side, you'll be pretty sure that somewhere in between, water had to be pumped out. . Good? If you found that the water was spreading away from a line from here to here and from here to here, then you would be pretty sure that some water was being pumped from below along this line here. Well, you would see another way you would discover that the X component of velocity has a derivative. It's different here than from here. The X component of the velocity varies along the X direction.

So the fact that the X component of the velocity varies along the Likewise, if you found that water was flowing up here and down here, you would expect that somewhere some water was being pumped out. So the derivatives of the velocity are often an indication that some water is being pumped from below. This pumping of water is the divergence of the velocity vector. Now the water, of course, is pumped from below. So, there is a direction of flow but it comes from below. There's no sense of direction... well, that's fine. That is the divergence. I have a question.

Do you already have the diagrams on the other board behind you? Leonard Susskind: Yes. With the arrows? Leonard Susskind: Yes. Leonard Susskind: If you take, say, the rightmost arrow and draw a circle between the head and the tail, then you can see the entrance and the exit. Leonard Susskind: Hmmm. The entry arrow and the exit arrow of a certain right between those two. And let's say the larger arrow is created by a steeper slope of the streak. Leonard Susskind: No, this is faster. It goes faster. Okay. And because of that, there is a divergence that is basically a kind of difference between the inside and the outside.

Leonard Susskind: That's right. That's how it is. If we draw a circle around here or we would see that... the water moves faster here than here, more water leaves here than enters here. Where does it come from? It must be coming in. The fact that there is more water flowing on one side than entering on the other side must indicate that there is a net inflow from somewhere else and the other place would be water pumped from below. So, that's the idea of ​​divergence. Could it also be because he is losing weight? Could that be a crazy example?

Did the lake become shallower? Leonard Susskind: Yes. Well, okay. I took... so let's get very specific now. I kept the lake at an absolutely uniform height and let's also assume that the density of the water...water is an incompressible fluid. It cannot be squeezed. It can't be stretched. So the velocity vector would be the right thing to think aboutover there. Yes. You could have... no, you're right. You could have a velocity vector that has a divergence because the water is not flowing in but because it is thinning. If that's possible. But let's keep it simple. And you can have... the idea of ​​a divergence makes as much sense in three dimensions as it does in two dimensions.

You just have to imagine that the entire space is full of water and that there are some hidden pipes that come in and deposit water in different places so that it spreads away from points in three-dimensional space. three-dimensionalspace, this is the definition of divergence. If this were the velocity vector at each point, you would calculate this quantity and that would tell you how much new water enters each new point in space. So that's the divergence. Now, there's a theorem that Michael just hinted at there. It's called Gauss's theorem and it says something very obvious and intuitive.

You take a surface, any surface. Take any surface or curve in two dimensions and now suppose that there is a vector field: a vector field point. Think of it like the flow of water. And now let's take the total amount of water flowing out of the surface. Obviously some water flows out, and of course we want to subtract the water coming in. Let's calculate the total amount of water flowing from the surface. That's an integral off the surface. Why is it an integral? Because we have to add the water flows out when the water comes in, that is simply a negative flow, a negative flow out.

We add up the total outflow by dividing the surface into small pieces and asking how much flux comes out of each small piece here. How much water comes out of the surface? If water is incompressible, incompressible means that its density is fixed and, furthermore, the depth of the water remains fixed. There is only one way that water can leave the surface and that is if it is pumped in, if there is a divergence. The divergence could be over here, it could be over here, it could be over here, it could be over here. In fact, anywhere there is a divergence it will cause an effect where water will flow out of this region.

So, there is a connection. There is a connection between what happens at the boundary of this region, how much water flows along the boundary on one side, and what the divergence is inside. There is a connection between the two and that connection is called Gauss's theorem. What it says is that the integral of the divergence inside, that's the total amount of flow coming in from outside, from below the bottom of the lake, the total integrated... now, by integrated, I mean in the sense of an integral one. The integrated amount of inflow, that is the integral of the divergence.

The integral over the interior in the three-dimensional case would be the 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) equals the total quantity of flow leaving across the boundary. Now how do we write that? The total amount of flux that flows outward through the boundary we break...let's take the three-dimensional case. We divide the boundary into small cells. Each little cell is a small area. Let's call each of those small areas D sigma. D sigma, sigma means surface area.

Sigma is the Greek letter. Sigma means surface area. This three-dimensional integral over the interior here is equal to a two-dimensional integral, the sigma over the surface, and it's just the component of A perpendicular to the surface. It is called A perpendicular to the surface D sigma. One perpendicular to the surface is the amount of flow coming out of each of these little boxes. Note, by the way, that if there is a flow along the surface it does not result in any fluid coming out. It is just the flow perpendicular to the surface, the component of the flow perpendicular to the surface that transports the fluid from the inside to the outside.

So we integrate the perpendicular component of the employee over the surface, that's still the sigma here. That gives us the total amount of fluid that comes out per unit of time, for example, and it has to be equal to the amount of fluid that is generated inside due to the divergence. This is Gauss's theorem. The relationship between the integral of the divergence and the interior of some region and the integral over the boundary where it measures the flow: the amount of matter leaving the boundary. Fundamental theorem. And let's see what he says now. Any questions about Gauss's theorem here?

You will see how it works. I'll show you how it works. Now, you mentioned that water is compressible. Is that different from what they gave us with the compressible fluid? Leonard Susskind: Yes. You could... if you had a compressible fluid, you would find that the fluid at the boundary here moves inward in all directions without any new fluid forming. In fact, what happens is that the liquid is squeezed out. But if the fluid cannot be compressed, if it cannot be compressed, then the only way that fluid could flow in is if it is somehow extracted from the center.

If it is being removed by invisible pipes that take it away. Does that mean that the divergence in the case of water would be zero and would be integrated into a volume? Leonard Susskind: If no water came in, it would be zero. If there was a source of water, the divergence is the same as its source. The source of water is... the source of new water that comes from other places is. . . Good. So in the two dimensional lake example the fountain is the water flowing from below, the sink which is the negative of a fountain is the flowing water and in the two dimensional example this would not be a two dimensional surface. comprehensive.

The integral here would be equal to a one-dimensional surface equal to the surface that comes out. Well. Alright. Let me show you how to use this. Let me show you how you use this and what it has to do with what we've said so far about gravity. I think... I hope we have time. Let's imagine we have a source, it could be water, but let's take a three-dimensional case, there is a divergence of a vector field, say A. There is a divergence of a vector field, point A, and it is concentrated in some region of space. .

It is a small sphere in some region of space that has spherical symmetry. In other words, it doesn't mean that the divergence is uniform here, but rather that it has the symmetry of a sphere. Everything is symmetrical with respect to rotations. Suppose there is a divergence of the fluid. Well? Now, let's take... and it's completely restricted to being in 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 is produced or a certain total divergence, a divergence integrated with a nice spherical shape.

Well. Let's see if we can use that to find out what field A is. That's Dell point A here and now let's see, can we find out what the field is somewhere else outside of here? So what we do is we draw a surface around there. We draw a surface there and now we are going to use Gauss's theorem. First, let's look at the left side. The left side has the integral of the divergence of the vector field. Alright. The vector field or divergence is completely restricted here to a finite sphere. What is... by the way, for the case of flow, for the case of fluid flow, what would be the integral of the divergence?

Someone knows? It really was the flow of a fluid. It will be the total amount of fluid that flowed per unit of time. It would be the flow per unit of time that passes through the system. But whatever it is, it doesn't depend on the radius of the sphere as long as the sphere, this outer sphere here, is larger than this region. Because? Because the integral over the divergence of A is completely concentrated in this region here and there is zero divergence outside. So first of all, the left side is independent of the radius of this outer sphere, as long as the radius of the outer sphere is greater than this divergence concentration here.

So, it's a number. Although it is a number. Let's call that number M. No, no. No M.Q. That's the left side. And it doesn't depend on the radius. On the other hand, which is the right side? Well, there is a flow coming out and if everything is nice and spherically symmetrical then the flow will go radially outward. It will be a pure flow, directed radially outward if the flow is spherically symmetric. Flow directed radially outward means that the flow is perpendicular to the surface of the sphere. So the perpendicular component of A is simply 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 here is constant throughout this sphere. So this integral is nothing more than the magnitude of A multiplied by the area of ​​the total sphere. Alright? If I take an integral over a surface, a spherical surface like this, into something that doesn't depend on where I am on the sphere, then you can take this on the outside, the magnitude of the field and the integral D. sigma is just the surface area total of the sphere.

What is the total surface area of ​​the sphere? Four thirds pi R. Leonard Susskind: No third. Only four pi. Four pi R squared. Oh yeah. Four pi R squared times the magnitude of the field equals 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 it look familiar to you? It is a vector field. It is pointing radially outward. Well, it points radially outward if the divergence is positive. If the divergence is positive, it points radially outward and its magnitude is R squared.

It is exactly the gravitational field after a point particle from the center. It's the magnitude of A. Leonard Susskind: Yes. That's why we have to put an address here. Do you know what this R is? It is a unit vector that points in the radial direction. It is a vector of unit length pointed in the radial direction. Good? So it's pretty clear from the image that field A points radially outward. That's what it says here. In either case, the magnitude of the field pointing radially outward has magnitude Q and falls as 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 divergence of the gravitational field. Concentrated in a very small volume. Think about it, if you want, you can think of it as the gravitational field, the flow field, the velocity field of an expanding fluid. Oh, by the way, of course, I was wrong about the sign here. The actual gravitational acceleration points inward, which is an indication that this divergence is negative. The divergence is more like a fluid-sucking convergence.

So the Newtonian gravitational field is isomorphic, it is mathematically equivalent, or mathematically similar, to a flow field, to a flow of water or any other fluid where everything is being sucked in 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 be, like, one over R squared. That's a useful analogy. That's not to say that space is a flow or anything like that. It is a mathematical analogy that is useful to understand the force law on R square which is mathematically similar to a flow velocity flow field that is generated right at the center of a point.

Well. That's a useful observation. But notice something else. Now suppose that, instead of having the flow concentrated here in the center, suppose that the flow is concentrated over a sphere that is larger but with the same total amount of flow. I wouldn't change the answer. As long as the total amount of flow is fixed, the way it flows through here will also be 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 if it is an extended mass, or if it is an extended mass of this size , as long as you are outside the object and as long as the object is spherically symmetrical, in other words, as long as the object is spear-shaped and you are outside of it, outside of it, outside of where the mass distribution is. is, then its gravitational field does not depend on whether it is a point, it is a dispersed object, whether it is more dense in the center and less dense on the outside, less dense in the center and more dense on the outside.

All it depends on is the total amount of dough. The total amount of mass is like the total amount of flow coming in... that theorem is very fundamental and important for thinking about gravity. For example, suppose we are interested in the motion of an object near the Earth's surface, but not so close 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 to this point, it is attracted to this point, it is attracted to that point.

It's close to this point, it's far from this point. It seems like a hell of a problem to figure out what the gravitational effect is at this point. But not. This tells you that the gravitational field is exactly the same as if the same total mass were concentrated right in the center.Well? That's Newton's theorem. It is a wonderful theorem. It is a great luck for him because without it he would not have been able to solve his equations. He knew. He had an argument, it may have been essentially this argument. I'm not sure what argument he made.

But he knew that with the law of force greater than R squared and only the law of force greater than R squared it would not have been true if it were R cubed, R to the fourth, over R to the seventh. With the force law greater than R squared, a spherical distribution of mass behaves exactly as if all the mass were concentrated right in the center, as long as you are 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 has a spherical shape, behaves as if it were a point mass.

So, if you are in a mine shaft that can't hold up? Leonard Susskind: That's right. If you're in a mine shaft, it can't hold up. But that doesn't mean you can't figure out what's going on. You can find out what's going on. I don't think we'll do it tonight. Its a bit late. But yes, we can find out what would happen in a mine shaft. But that's correct. It can't hold up in a mine shaft. For example, let's say you dig a mine shaft right in 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 exerted toward the center? Bit. Certainly much less than if the entire mass were concentrated on the same theory. You have... it's not even obvious which direction the force is in, but it's towards the center. But it is very small. You move a little away from the land. There is a very, very small force. Much, much less than if the entire dough was squished toward the center. So, correct. It doesn't work for that case. Another interesting case is to suppose that you have a shell of material. To have a layer of material, think of a source layer, a fluid that flows inward.

The fluid flows from the outside into this slate and all the little tubes are arranged in a circle like this. What does fluid flow look like in different places? Well, the answer is that on the outside it looks exactly the same, as if everything were concentrated in one point. But what happens inside? What would you guess? Nothing. Nothing. Everything just flows out of here and there is no flow here at all. How can there be? What direction would it be? And so there's no flow here. Wouldn't you have the distance argument? For example, if you are closer to the surface of the inner layer...

Leonard Susskind: Yes. Wouldn't that be more force towards that? No. Look, 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. But there is no divergence here. So the resulting net integrated field is zero. No field inside the shell. Field on the outside 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 like if there wasn't any gravitational material there.

On the other hand, outside we would have a field absolutely identical to what occurs in the center. Now, there is an analogy to this in the general theory of relativity. We'll get to it. Basically what it says is that the field of anything, as long as it is spherically symmetrical on the outside, appears identical to the field of a black hole. I think we're done for tonight. Review divergence and Gauss's theorem. Gauss's theorem is central. There would be no gravity without Gauss's theorem. the above program is sponsored by Stanford University. Visit us at Stanford. education.