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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.

And 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 X two 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 everywheres, in other

words, mass times G does vary from place to place.

That's this fact that gravity doesn't depend on where you

are in 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 D 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 the

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 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. "

>>

[laughing] Leonard Susskind: Now, I don't know.

Maybe the doge or whoever they called the guy at the time

said, no, no Galileo.