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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

• 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