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  • 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

  • on an object is independent of where you put it.