字幕表 動画を再生する 英語字幕をプリント [music playing] This program is brought to you by Stanford University. Please visit us at Stanford.edu. Leonard Susskind: Gravity. Gravity is a rather special force. It's unusual. It has difference in electrical forces, magnetic forces, and it's connected in some way with geometric properties of space, space and time. But-- 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, uh, begin with Newton's equations. The first equation, of course, is F equals MA. Force is equal to mass times acceleration. Let's assume that we have a reference, a frame of reference that means a set of coordinates and that was a set 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. Uh, 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 a vector acceleration and so forth 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 cannot change. Now, of course, the mass of this cup of coffee here can change. It's lighter now but it only changes because mass 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 dimensions on the blackboard 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 in. X three 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 X one, X two, and X three exactly. X one, and X two, and X three 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 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 just call it A. The acceleration is just equal-- the components of it are equal to the second derivatives of the coordinates with respect to time. That's what acceleration is. The first derivative of position is called velocity. Okay. We can take this thing component by component. X one, X two, and X three. 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 summery 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 how Galileo understood it. Actually, I'm not sure how much of these mathematics Galileo did or didn't understand. Uh, 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 everywheres. This is not true, of course, if the earth is curved then gravity will point toward the center. But the flat space approximation, gravity points down. Down everywheres always in the same direction. And second of all, perhaps a little less obvious but nevertheless true, 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