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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.

  • You can't drop things from the tower.

  • You'll kill somebody.

  • So, maybe he didn't.

  • He must have surely thought of it.

  • All right.

  • So, the result, had he done it, and had he not had to worry

  • about such spurious effects as air resistance would be that

  • a cannon ball and a feather would fall in exactly the same

  • way, independent of the mass, and the equation would just

  • say, the acceleration would first of all be downward, that's

  • the minus sign, and equal to this constant G.

  • Excuse me.

  • Now, G as a number, it's 10 meters per second per second at

  • the surface of the earth.

  • At the surface of the moon it's something smaller.

  • On the surface of Jupiter it's something larger.

  • So, it does depend 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're dropping it onto

  • but not the mass of the object stopping it.

  • That fact, that gravitational motion, is completely

  • independent of mass is called or it's the simplest version

  • of something that's called the equivalence principle.

  • Why it's called the equivalence principle we'll come to

  • later.

  • What's equivalent to what.

  • At this stage we can just say gravity is equivalent between

  • all different objects independent of their mass.

  • But that is not exactly what is equivalence/inequivalence

  • principle was all about.

  • All right.

  • That has a consequence.

  • An interesting consequence.

  • Supposing I take some object which is made up out of

  • something which is very unrigid.

  • Just a collection of point masses.

  • Maybe let's even say they're not even exerting any forces on

  • each other.

  • It's a cloud, a very diffuse cloud of particles and we watch

  • it fall.

  • Now, let's suppose we start each particle from rest, not all

  • at the same height, and we let them all fall.

  • Some particles are heavy, some particles are light, some of

  • them may be big, some of them may be small.

  • How does the whole thing fall? And the answer is, all of the

  • particles fall at exactly the same rate.

  • The consequence of it is that the shape of this object

  • doesn't deform as it falls.

  • It stays absolutely unchanged.

  • The relationship between the neighboring parts are

  • unchanged.

  • There are no stresses or strains which tend to deform the

  • object.

  • So even if the object were held together but some sort of

  • struts or whatever, there would be no forces on those struts

  • because everything falls together.

  • Okay? The consequence of that is that falling in the

  • gravitational field is undetectable.

  • You can't tell that you're falling in a gravitational field

  • by-- when I say you can't tell, certainly you can tell the

  • difference between free fall and standing on the earth.

  • All right? That's not the point.

  • The point is that you can't tell by looking at your

  • neighbors or anything else that there's a force being

  • exerted on you and that that force that's being exerted on

  • you is pulling downward.

  • You might as well, for all practical purposes, be infinitely

  • far from the earth with no gravity at all and just sitting

  • there because as far as you can tell there's no tendency for

  • the gravitational field to deform this object or anything

  • else.

  • You cannot tell the difference between being in free space

  • infinitely far from anything with no forces and falling

  • freely in a gravitational field.

  • That's another statement of the equivalence principle.

  • >> You say not mechanically detectable? Leonard Susskind:

  • Well, in fact, not detectable, period.

  • But so far not mechanically detectable.

  • >> 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 sort.

  • What's that? Oh, you could certainly tell if you were

  • standing on the floor here you could definitely tell that

  • there was something falling toward you.

  • But the question is, from within 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 whether you're

  • falling and it's, uh-- yeah.

  • If there was something that was not falling it would only be

  • because there was some other force on it like a beam or a

  • tower of some sort holding it up.

  • Why? Because this object, if there are no other forces on

  • it, only the gravitational forces, it will fall at the same

  • rate as this.

  • All right.

  • So, that's another expression of the equivalence principle,

  • that you cannot tell the difference between being in free

  • space far from any gravitating object versus being in a

  • gravitational field.

  • Now, we're gonna modify this.

  • This, of course, is not quite true in a real gravitational

  • field, but in this flat space approximation where everything

  • pulls together, you cannot tell that there's a gravitational

  • field.

  • At least, you cannot tell the difference-- not without

  • seeing the floor in any case.

  • The self-contained object here does not experience anything

  • different than it would experience far from any gravitating

  • object standing still.

  • Or in uniform motion.

  • >> Another question.

  • Leonard Susskind: What's that? Yeah.

  • >> We can tell where we're 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 tell there's no connection between objects? Leonard

  • Susskind: Okay.

  • Here's what you can tell.

  • If you go up to the top of a high building and you close

  • your eyes, and you step off, and go into free fall, you will

  • feel exactly the same-- you will feel weird.

  • I mean, that's not the way you usually feel because your

  • stomach will come up and do some funny things.

  • You know, you might lose it.

  • But the point is, you would feel exactly the same discomfort

  • in outer space far from any gravitating object just standing

  • still.

  • You'll feel exactly the same peculiar feelings.

  • Okay? What are those peculiar feelings due to? They're not

  • due to falling.

  • They're due to not fall-- well. . .

  • [laughing] they're due to the fact that when you stand on

  • the earth here, there are forces on the bottom of your feet

  • which keep you from falling and if the earth suddenly

  • disappeared from under my feet, sure enough, my feet would

  • feel funny because they're used to having that force exerted

  • on their bottoms.

  • You get it.

  • I hope.

  • So, the fact that you feel funny in free fall, uh, is

  • because you're not used to free fall.

  • It doesn't matter whether you're infinitely far from any

  • gravitating objects standing still or freely falling in the

  • presence of a gravitational field.

  • Now, as I said, this will have to be modified in a little

  • bit.

  • There are such things as tidal forces.

  • Those 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 in every point, and that it varies with

  • height.

  • That's due to the finiteness of the earth.

  • But, in the flat space of the-- in the flat earth

  • approximation where the earth is infinitely big pulling

  • uniformly, uh, there is no other effect of gravity that is

  • any different than being in free space.

  • Okay.

  • Again, that's known as the equivalent principle.

  • Now, let's go on beyond the flat space or the flat earth

  • approximation and move on to Newton's theory of gravity.

  • Newton's theory of gravity says 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 toward the other one.

  • Equal and opposite forces and the magnitude of the force--

  • the magnitude of the force of one object on another.

  • Let's characterize them by a mass.

  • Let's call this one 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.

  • All right.

  • Newton's law of force is that the force is proportional to

  • the product of the masses.

  • Making either mass heavier will increase the force.

  • The product of the masses, big M times little m, inversely

  • proportional to the square of the distance between them.

  • Let's call that R squared.

  • Let's call the distance between them R.

  • And there's a numerical constant.

  • This law by itself could not possibly be right.

  • It's not dimensionally consistent.

  • The-- if you work out the dimensions of force, mass, mass

  • and R, it's not dimensionally consistent.

  • There has to be a constant in there.

  • And that numerical constant is called capital G, Newton's

  • constant.

  • And it's very small.

  • It's a very small constant.

  • I'll write down what it is.

  • G is equal to six or 6. 7, roughly, times ten to the minus

  • 11th, which is a small number.

  • So, on the face of it, 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 it's a weak force there's an

  • experiment that you can do.

  • Weak in comparison to other forces.

  • I've done this for classes and you can do it yourself.

  • Just take an object hanging by a string and two experiments.

  • The first experiment, take a little object here and

  • electrically charge it.

  • You electrically charge it by rubbing it on your shirt.

  • That doesn't put much charge on it but it charges it up

  • enough to feel some electrostatic force.

  • Then take another object of exactly the same kind, rub it on

  • your shirt, and put it over here.

  • What happens? They repel.

  • And the fact that they repel means that this will shift.

  • And you'll see it shift.

  • Take another example.

  • Take your little ball there to be iron and put a magnet next

  • to it.

  • Again, you'll see quite an easily detectable deflection of

  • the-- of the string holding it.

  • All right? Next, take a 10,000-pound weight and put it over

  • here.

  • Guess what happens? Undetectable.

  • You cannot see anything happen.

  • The gravitational force is much, much weaker than most other

  • kinds of forces and that's due to-- not due to.

  • Not due to that.

  • The fact that it's so weak is encapsulated in this small

  • number here.

  • Another way to say it is if you take two masses, each of 1

  • kilometer-- not 1 kilometer.

  • 1 kilogram.

  • A kilogram is a good healthy mass, a nice chunk of iron.

  • MM and you separate them from 1 meter, then the force

  • between them is just G and it's 6. 7 times ten to the minus

  • 11th, if you do it with the units being Newton's.

  • Very weak force.

  • But, weak as it is, we feel it rather strenuously.

  • We feel it strongly because the earth is so darn heavy.

  • So, the heaviness of the earth makes up for the smallness of

  • G and so we wake up in the morning feeling like we don't

  • wanna get out of bed because gravity is holding us down.

  • >>

  • [laughing] Leonard Susskind: Yeah? >> Umm, so that force is

  • measuring the force between-- from the large one to the

  • small one or both? Leonard Susskind: Both.

  • Both.

  • They're equal and opposite.

  • Equal and opposite.

  • That's the rule.

  • That's Newton's third law.

  • The forces are equal and opposite.

  • So, the force on the large one due to the small one is the

  • same as the force of the small one on the large one.

  • But it is proportional to the product of the masses.

  • So, the meaning of that is I'm not-- I'm heavier than I'd

  • like to be but I'm not very heavy.

  • I'm certainly not heavy enough to deflect the hanging weight

  • significantly.

  • But I do exert a force on the earth which is exactly equal

  • and opposite to the force that the very heavy earth exerts

  • on me.

  • Okay.

  • Why does the earth accel-- if I drop from a certain height,

  • I accelerate down.

  • The earth hardly accelerates at all, even though the forces

  • are equal.

  • Why is it that the earth-- if the forces are equal, my force

  • on the earth and the earth's force on me are equal, why is

  • it that the earth accelerates so little and I accelerate so

  • much? Yeah.

  • Because the acceleration involves two things.

  • It involves the force and the mass.

  • The bigger the mass, the less the acceleration for the

  • force.

  • So, the earth doesn't accelerate-- yeah, question.

  • >> 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 large-- no, no.

  • It was from Kepler's laws.

  • It was from Kepler's laws.

  • He worked out, roughly speaking-- I don't know what he did.

  • He was secretive and he didn't really tell people what he

  • did.

  • But, umm, the piece of knowledge that he had was Kepler's

  • laws of motion-- planetary motion-- and my guess is that he

  • just wrote down a general force and realized that he would

  • get Kepler's laws of motion for the inverse square law.

  • Umm, I don't believe he had any understand lying theoretical

  • reason to believe in the inverse square law.

  • >> Edmund Halley actually asked him, uh, what kind of force

  • law do you need for conic section orbits and he had almost

  • performed the calculations a year before.

  • Leonard Susskind: Yeah.

  • >> So, yeah.

  • Leonard Susskind: Actually, I don't think-- yeah.

  • >> I think the question-- he asked the question for inverse

  • square laws and I think that Newton already knew the

  • solution was an ellipse.

  • Leonard Susskind: No.

  • It wasn't the ellipse that was there.

  • The orbits might have been circular.

  • It was the fact that the period varies as the three halfs

  • power of the radius.

  • All right? The period of motion is the circular motion has

  • an acceleration toward the center.

  • Any motion of the circle is accelerated toward the center.

  • If you know the period and the radius, then you know the

  • acceleration toward the center.

  • Okay? We could write the-- what's the word? Anybody know

  • what-- if I know the angular frequency of going around in an

  • orbit that's called omega.

  • Anybody know the-- and it's basically just the inverse

  • period.

  • Okay? Omega is roughly the inverse period number of cycles

  • per second.

  • What is the acceleration of a thing moving in a circular

  • orbit.

  • Anybody remember? >> Omega squared R.

  • Leonard Susskind: Omega squared R.

  • That's the acceleration.

  • Now, supposing he sets that equal to some unknown force law

  • F of R and then divides by R.

  • Then he finds 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 it

  • is Kepler's second law? I don't even remember which one it

  • is.

  • It's the law that says that the frequency or the period, the

  • square of the period, is proportional to the cube of the

  • radius.

  • That was the law of Kepler.

  • So, from Kepler's laws he easily could-- or that one law, he

  • 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 you department have

  • a perfectly circular law orbit then the inverse square law

  • was the unique law which would give elliptical orbits.

  • So, it's a two-step thing.

  • >> What happens when the two objects are touching? Do you

  • measure it from the-- Leonard Susskind: Of course, there are

  • other forces on them.

  • If two objects are touching, there are all sorts of forces

  • between them that are not just gravitational.

  • Electrostatic forces, atomic forces, nuclear forces? So,

  • you'll have to modify the whole story.

  • >> As the distance approaches zero-- Leonard Susskind: Yeah.

  • Then it breaks down.

  • Then it breaks down.

  • Yeah.

  • Then it breaks down.

  • When they get so close that other important forces come into

  • play.

  • The other important forces, for example, are the forces that

  • are holding this object and preventing it from falling.

  • These we usually call them contact forces but, in fact, what

  • they really are is various kinds of electrostatic forces

  • between the atoms and molecules on the table and the atoms

  • and molecules in here.

  • So, other kinds of forces.

  • All right.

  • Incidentally, let me just point out if we're talking about

  • other kinds of force laws, for example, electrostatic force

  • laws, then the force-- we still have F equals MA but the

  • force law-- the force law will not be that the force is

  • somehow proportional to the mass times something else but it

  • could be the electric charge.

  • If it's the electric charge, then electrically uncharged

  • objects will have no forces on them and they won't

  • accelerate.

  • Electrically charged objects will accelerate in an electric

  • field.

  • So, electrical forces don't have this universal property

  • that everything falls or everything moves in the same way.

  • Uncharged particles move differently than charged particles

  • with respect to electrostatic forces.

  • They move the same way with respect to gravitational forces.

  • And as repulsion and attraction, whereas gravitational

  • forces are always attractive.

  • Where is my gravitational force? I lost it.

  • Yeah.

  • Here it is.

  • All right.

  • So, that's Newtonian gravity between two objects.

  • For simplicity let's just put one of them, the heavy one, at

  • the origin of coordinates and study the motion of the light

  • one then-- oh, incidentally, one usually puts-- let me

  • refine this a little bit.

  • As I've written it here, I haven't really expressed it as a

  • vector equation.

  • This is the magnitude of the force between two objects.

  • Thought of as a vector equation, we have to provide a

  • direction for the force.

  • Vectors have directions.

  • What direction is the force on this particle? Well, the

  • answer is, it's along the radial direction itself.

  • So, let's call the radial distance R, or the radial vector

  • R, then the force on little m here is along the direction R.

  • But it's also opposite to the direction of R.

  • The radial vector, relative to the origin over here, points

  • this way.

  • On the other hand, the force points in the opposite

  • direction.

  • If we wanna make a real vector equation out of this, we

  • first of all have to put a minus sign.

  • That indicates that the force is opposite to the direction

  • of the radial distance here, but we also have to put

  • something in which tells us what direction the force is in.

  • It's along the radial direction.

  • But wait a minute.

  • If I multiply it by R up here, I had better divide it by

  • another factor of R downstairs to keep the magnitude

  • unchanged.

  • The magnitude of the force is one over R squared.

  • If I were to just randomly come and multiply it by R, that

  • would make the magnitude bigger by a factor of R, so I have

  • to divide it by the magnitude of R.

  • This is Newton's force law expressed in vector form.

  • Now, let's imagine that we have a whole assembly of

  • particles.

  • A whole bunch of them.

  • They're all exerting forces on one another.

  • In pairs, they exert exactly the force that Newton wrote

  • down.

  • But what's the total force on a particle? Let's label these

  • particles the first one, the second one, the third one, the

  • fourth one, dot, dot, dot, dot, dot.

  • This is the Ith one over here.

  • So, I is the running index which labels which particle we're

  • talking about.

  • The force on the Ith particle, let's call it F sub I, and

  • let's remember that it's a vector, it's equal to the sum--

  • now, this is not an obvious fact that when you have two

  • objects exerting a force on the third that the force is

  • necessarily equal to the sum of the two forces, of the two

  • objects.

  • You know what I mean.

  • But it is a fact anyway.

  • Obvious or not obvious it is a fact.

  • Gravity does work that way at least in the Newtonian

  • approximation.

  • With Einstein, it breaks down a little bit.

  • But in Newtonian physics the force is the sum and so it's a

  • sum of all the other particles.

  • Let's write that J not equal to I.

  • That means it's a sum over all not equal to I.

  • So, the force from the first particle.

  • It comes from the second particle, third particle, third

  • particle, and so forth.

  • Each individual force involves M sub I, the force of the Ith

  • particle, times the four, times the mass of the Jth

  • particle.

  • Product of the masses divided by the square of the distance

  • between them, let's call that RIJ squared.

  • The distance between the Ith particle is I and J, the

  • distance between the Ith particle and the Jth particle is

  • RIJ.

  • But then, just as we did before, we have to give it a

  • direction.

  • Put a minus sign here, that indicates that it's attractive,

  • another RIJ upstairs, but that's a vector RIJ, and make this

  • cubed downstairs.

  • All right? So, that says that the force on the Ith particle

  • is the sum of all the forces due to all the other ones of

  • the product of their masses inverse square in the

  • denominator, and the direction of each individual force on

  • this particle is toward the other.

  • All right? This is a vector sum.

  • Yeah? Hmm? The minus indicates that it's attractive.

  • Excellent.

  • >> but you've got the vector going from I to J.

  • Leonard Susskind: Oh.

  • Let's see.

  • That's the vector going from R to I to J.

  • There is a question of the sign of this vector over here.

  • So yeah.

  • You're absolute-- yeah.

  • I actually think it's-- yeah, you're right.

  • You're absolutely right.

  • The way I've written it there should not be a minus sign

  • here.

  • If I put RJI there, then there would be a minus sign.

  • Right? So, you're right.

  • But in any case every one of the forces is attractive and

  • what we have to do is to add them up.

  • We have to add them up as vectors and so there's some

  • resulting vector, some resultant vector, which doesn't point

  • toward any one of them in particular but points in some

  • direction which is determined by the vector sum of all the

  • others.

  • All right in but the interesting fact is, if we combine

  • this, this is the force on the Ith particle.

  • If we combine it with Newton's equations-- let's combine it

  • with Newton's F equals MA equations then this is F.

  • This on the Ith particle, this is equal to the mass of the

  • Ith particle times the acceleration of the Ith particle.

  • Again, vector equations.

  • Now, the sum here is over all the other particles.

  • We're focusing on number I.

  • I, the mass of the Ith particle will cancel out of this

  • equation.

  • I don't wanna throw it away but let's just circle it and now

  • We notice that the acceleration of the Ith particle does not

  • depend on its mass again.

  • Once again, because the mass occurs on both sides of the

  • equation it can be canceled out, and the motion of the Ith

  • particle does not depend on the mass of the Ith particle.

  • It depends on the masses of all the other ones.

  • All the other ones come in, but the mass of the Ith particle

  • cancels the equation.

  • So, what that means is if we had a whole bunch of particles

  • here and we added one more over here, its motion would not

  • depend on the mass of that particle.

  • It depends on the mass of all the other ones but it doesn't

  • depend on the mass of the Ith particle here.

  • Okay? Again, equivalence principle that the motion of a

  • particle doesn't depend on its mass.

  • And again if we had a whole bunch of particles here, if they

  • were close enough together, they would all move in the same

  • way.

  • Before I discuss any more mathematics, let's just discuss

  • tidal forces, what tidal forces are.

  • >> Can I ask one question? Leonard Susskind: Yeah.

  • >> Once you set this whole thing into motion dynamically.

  • Leonard Susskind: Yeah.

  • >> We have all different masses and each particle is gonna

  • be effected by each one? Leonard Susskind: Yes.

  • Yes.

  • >> Every particle in there is going to experience a uniform

  • acceleration? Leonard Susskind: No, no, no.

  • The acceleration is not uniform.

  • The acceleration will get larger when it gets closer to one

  • of the particles.

  • It won't be uniform anymore.

  • It won't be uniform now because the force is not independent

  • of where you are.

  • Now the force depends on where you are relative to the

  • objects that are exerting the force.

  • It was only in the flat earth approximation where the force

  • didn't depend on where you were.

  • Okay? Now, the force varies so it's larger where you're far

  • away-- sorry.

  • It's smaller when you're far away, it's smaller when you're

  • in close.

  • Okay? >> But is it going to be changing in a-- it changes in

  • a vector form with each individual particle.

  • Each one of them is changing position.

  • Leonard Susskind: Yeah.

  • >> So, is the dynamics that every one of them is going

  • towards the center of gravity of the entire-- Leonard

  • Susskind: Not necessarily.

  • I mean, they could be flying apart from each other but they

  • will be accelerating toward each other.

  • Okay? If I throw this eraser in the air with greater than

  • the escape velocity, it's not going to turn around and fall

  • back down.

  • >> Well, the question is, is the acceleration a uniform

  • acceleration or is it changing in dynamics? Leonard

  • Susskind: Changing with what? With respect to what? Time?

  • Oh.

  • It changes with respect to time because the object moves

  • farther and farther away.

  • >> In the two-mass system-- Leonard Susskind: Mm hmm.

  • >> I call that a uniform acceleration.

  • Leonard Susskind: Uniform with respect to what? >> It's not

  • uniform.

  • The radius is changing and it's inversed cubically radiused.

  • Leonard Susskind: Inverse squared.

  • >> Inverse squared.

  • Leonard Susskind: Let's take the earth.

  • Here's the earth, and we drop a small mass from far away.

  • As that mass moves in, its acceleration increases.

  • Why does its acceleration increase in its acceleration

  • increases because the radial distance gets smaller.

  • So, in that sense it's not.

  • All right.

  • Now, once the gravitational force depends on distance then

  • it's not really quite true that you don't feel anything in a

  • gravitational field.

  • You feel something that's to some extent different than you

  • would feel in free space without any gravitational field.

  • The reason is more or less obvious.

  • Here you are-- here's the earth.

  • Now, you, or me, or whoever it is, happens to be extremely

  • tall.

  • Couple of thousand miles tall.

  • Well, this person's feet are being pulled by the

  • gravitational field more than his head.

  • Or another way of saying the same thing is if let's imagine

  • that the person is very loosely held together.

  • >>

  • [laughing] Leonard Susskind: He's more or less a gas of-- we

  • are pretty loosely held together.

  • At least I am.

  • Right.

  • All right.

  • The acceleration on the lower portions of his body are

  • larger than the accelerations on the upper part of his body.

  • So it's quite clear what happens to him.

  • You get stretched.

  • He doesn't get a sense of falling as such.

  • He gets a sense of stretching, being stretched.

  • Feet being pulled away from his head.

  • At the same time, let's-- all right.

  • So let's change his shape a little bit.

  • I just spent a week-- two weeks in Italy and my shape

  • changes whenever I go to Italy.

  • It tends to get more horizontal.

  • My head is here, my feet are here, and now I'm this way.

  • Still loosely put together.

  • All right? Now what? Well, not only does the force depend on

  • the distance but it also depends on the direction.

  • The force on my left end over here is this way.

  • The force on my right end over here is this way.

  • The force on the top of my head is down but it's weaker than

  • the force on my feet.

  • So there are two effects.

  • One effect is to stretch me vertically.

  • It's because my head is not being pulled as hard as my feet.

  • But the other effect is to be squished horizontally by the

  • fact that the forces on the left end of me are pointing

  • slightly to the right and the forces to the right end of me

  • are pointing slightly to the left.

  • So a loosely knit person like this falling in free fall near

  • a real planet or a real gravitational object which has a

  • real Newtonian gravitational field around it will experience

  • a distortion-- will experience a degree of distortion and a

  • feeling of being stretched vertically, being compressed

  • horizontally, but if the object is small enough-- what does

  • small enough mean? Let's suppose the object that's falling

  • is small enough.

  • If it's small enough, then the gradient of the gravitational

  • field across the size of the object will be negligible and

  • so all parts of it will experience the same gravitational

  • acceleration.

  • All right.

  • So tidal forces-- these are tidal forces.

  • These forces which tend to tear things apart vertically and

  • squish them this way.

  • Tidal forces.

  • Tidal forces are forces which are real.

  • You feel them.

  • I mean-- yeah? >> Do you recall if Newton calculated lunar

  • tides? Leonard Susskind: Oh, I think he did.

  • He certainly knew the cause of the tides.

  • Yeah.

  • I don't know to what extent he calculated.

  • What do you mean calculated the-- >> As in this kind of a

  • system with the moon and the sun.

  • Leonard Susskind: Well, I doubt that he was capable-- I'm

  • not sure whether he estimated the height of the deformation

  • of the oceans or not.

  • But I think he did understand this much about tides.

  • Okay.

  • So, that's what's called tidal force.

  • And remember, tidal force has this effect of stretching and

  • in particular if we take the earth -- just to tell you why

  • it's called tidal forces of course is because it has to do

  • with tides.

  • I'm sure you all know the story.

  • But if this is the moon down here, then the moon exerting

  • forces on the earth exerts tidal forces on the earth, which

  • means to some extent it tends to stretch it this way and

  • squash it this way.

  • Well, the earth is pretty rigid so it doesn't form very much

  • due to the moon.

  • But what's not rigid is the layer of water around it and so

  • the layer of water tends to get stretched and squeezed and

  • so it gets deformed into a deformed shell of water with a

  • bump on this side and a bump on that side.

  • All right.

  • I'm not gonna go any more deeply into that than I'm sure

  • you've all seen.

  • Okay.

  • But let's define now what we mean by the gravitational

  • field.

  • The gravitational field is abstracted from this formula.

  • We have a bunch of particles-- >> question.

  • Leonard Susskind: Yeah? >> Don't you have to use some sort

  • of coordinate geometry so that when you have the poor guy in

  • the middle's being pulled by all the other guys on the side.

  • Leonard Susskind: I'm not explaining it right.

  • >> it's always negative, is that what you're saying? Leonard

  • Susskind: No.

  • It's always attractive.

  • All right.

  • So you have-- >> What about the other guys that are pulling

  • upon him from different directions? Leonard Susskind: Let's

  • suppose it's somebody over here and we're talking about the

  • force on this person over here.

  • Obviously there's one force pushing this way and another

  • force pushing that way.

  • Okay? No.

  • They're all opposite to the direction of the object which is

  • pulling on him.

  • All right? That's how this mind of science says.

  • >> well, you kind of retracted the minus sign at the front

  • and reversed the JI.

  • So it's the direction-- Leonard Susskind: We can get rid of

  • the minus sign in the front there by switching this RJ.

  • RIJ and RJI are opposite to each other.

  • One of them is the vector between I and J.

  • I and J.

  • And the other is the vector to J and I, so they're equal and

  • opposite to each other.

  • The minus sign there.

  • Look, as far as the minus sign goes, all that it means is

  • every one of these particles is pulling on this particle

  • toward it as opposed to pushing away from it.

  • It's just a convention which keeps track of attraction

  • instead of repulsion.

  • >> yeah.

  • For the Ith mass-- if that's the right word.

  • Leonard Susskind: Yeah.

  • >> If you look at it as kind of an ensemble wouldn't there

  • be a nonlinear component to it was the I guy, the Ith guy,

  • the Jth guy, then with you compute the Jth guy-- you know

  • what I mean? Leonard Susskind: When you take into account

  • the motion.

  • Now, what this formula is for is supposing you know the

  • positions of all the others.

  • You know that.

  • All right? Then what is the force on one additional one in

  • but you're perfectly right.

  • Once you let the system evolve, then each one will cause a

  • change in motion of the other one and so it because a

  • complicated as you say nonlinear mess.

  • But this formula is a formula for if you knew the position

  • and location of every particle, this would be the force.

  • Okay? Something.

  • You need to solve the equations to know how the particles

  • move.

  • But if you know where they are, then this is the force on

  • the Ith particle.

  • All right.

  • Let's come to the idea of the gravitational field.

  • The gravitational field is in some ways similar to the

  • electric field of an electric charge.

  • It's the combined effect of all the masses everywheres.

  • And the way you define it is as follows: You imagine one

  • more particle, one more particle.

  • You can take it to be a very light particle so it doesn't

  • influence the motion of the others.

  • Add one more particle.

  • In your imagination.

  • You don't really have to add it.

  • In your imagination.

  • And what the force on it is.

  • The force is the sum of the force 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 a thing that you're imagining testing out the

  • gravitational field with.

  • You take a light little particle, and you put it here, and

  • you see how it accelerates.

  • Knowing how it accelerates tells you how much force is on

  • it.

  • In fact, it just tells you how it accelerates.

  • And you can go around and imagine putting it in different

  • places and mapping out the force field that's on that

  • particle.

  • Or the acceleration field since we already know that the

  • force is proportional to the mass.

  • Then we can just concentrate on the acceleration.

  • The acceleration all particles will have the same

  • acceleration independent of the mass.

  • So we don't even have to know what the mass of the particle

  • is.

  • We put something over there, a little bit of dust, and we

  • see how it accelerates.

  • Acceleration is a vector and so we map out in space the

  • acceleration of a particle at every point in space, either

  • imaginary or real particle, and that gives us a vector field

  • at every point in space.

  • Every point in space there is a gravitational field of

  • acceleration.

  • It can be thought of as the acceleration.

  • You don't have to think of it as force.

  • Acceleration.

  • The acceleration of a point mass located at that position.

  • It's a vector that has a direction, it has a magnitude, and

  • it's a function of position.

  • So, we just give it a name.

  • The acceleration due to all the gravitating objects is a

  • vector and it depends on position.

  • Your X means location.

  • It means all of the components of position X, Y, and Z, and

  • it depends on all the other masses in the problem.

  • That is what's called the gravitational field.

  • It's very similar to the electric field except the electric

  • field is force per unit charge.

  • It's the force of an object divided by the charge on the

  • object.

  • The gravitational field is the force on the object divided

  • by the mass on the object.

  • Since the force is proportional to the mass the acceleration

  • field did you want depend on which kind of particle we're

  • talking about.

  • All right.

  • So, that's the idea of a gravitational field.

  • It's a vector field and it 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 work out what it is.

  • We know what the force on the Ith particle is.

  • Right? The force on a particle is the mass times the

  • acceleration.

  • So, if we wanna find the acceleration, let's take the Ith

  • particle to be the test particle.

  • Little i represents the test particle over here.

  • Let's erase the immediate step over here and write that this

  • is MI times AI but let me call it now capital A.

  • The acceleration of a particle at position X is given by the

  • right-hand side.

  • And we can cross out the MI because it cancels from both

  • sides.

  • So, here's a formula for the gravitational field at an

  • arbitrary point due to a whole bunch of massive objects.

  • A whole bunch of massive objects.

  • An arbitrary particle put over here will accelerate in some

  • direction that's determined by all the others and that

  • acceleration is gravitation-- definition.

  • Definition is the gravitational field.

  • Okay.

  • Let's take a little break.

  • We usually take a break at about this time and I recover my

  • breath.

  • To go on, we need a little bit of fancy mathematics.

  • We need a piece of mathematics called Gauss's theorem and

  • Gauss's theorem involves integrals, derivatives,

  • divergences.

  • And we need to spell those things out.

  • They're essential part of the theory of gravity.

  • And much of these things that we've done in the context of

  • the electrical forces, in particular the concept of

  • divergence, divergence of a vector field.

  • So, I'm not going to spend a lot of time on it.

  • If you need to fill in, then I suggest you just find any

  • book on vector calculus and find out what a divergence, and

  • a gradient, and a curl-- we won't do curl today.

  • What those concepts are, and look up Gauss's theorem and

  • they're not terribly hard but we're gonna go through them

  • fairly quickly here since we've done them several times in

  • the past.

  • All right.

  • Imagine that we have a vector field.

  • Let's call that vector field A.

  • It could be the field of acceleration and that's the way I'm

  • gonna use it.

  • But for the moment it's just an arbitrary vector field, A.

  • It dependence on position.

  • When I say it's a field, the implication is that it depends

  • on position.

  • Now I probably made it completely unreadable.

  • A of X varies from point to point.

  • And I want to define a concept called the divergence of a

  • field.

  • Now, it's called a divergence because what it has to do is

  • the way the field is spreading out away from the point.

  • For example, a characteristic situation where we would have

  • a strong divergence for a field is if the field was

  • spreading out from a point, like that.

  • The field is diverging away from the point.

  • Incidentally, after the field is pointing inward, then one

  • might say the field has a convergence but we simply say it

  • has a negative divergence.

  • So, divergence can be positive or negative.

  • And there's a mathematical expression which represents the

  • degree to which the field is spreading out like that.

  • It is called the divergence.

  • I'm gonna write it down and it's a good thing to get

  • familiar with, certainly if you're going to follow this

  • course it's a good thing to get familiar with.

  • But if you're gonna follow any kind of physics course past

  • freshmen physics, the idea of divergence is very point.

  • All right.

  • Supposing the field A has a set of components.

  • The one, two, and three component or we could call them the

  • X, Y, and Z component.

  • Now I'll use X, Y, and Z.

  • X, Y, and Z.

  • Which I previously called X one, X two, and X three.

  • It has components at AX, AY, and AZ.

  • Those are the three components of the vector.

  • Well, the divergence has to do, among other things, with the

  • way the field varies in space.

  • If the field is the same everywheres in space what would

  • that mean? That would men that the field has not only the

  • same magnitude, but the same direction anywheres in space.

  • Then it just points in the same direction everywheres in

  • space with the same magnitude.

  • It certainly has no tendency to spread out.

  • When does a field have a tendency to spread out in when a

  • field varies.

  • For example, it could be small over here, growing bigger,

  • growing bigger, growing bigger.

  • And we might even go in the opposite direction and discover

  • that it's the opposite direction getting bigger in that

  • direction.

  • Now, clearly there's a tendency for the field to spread out

  • from the center here.

  • The same thing could be true if it were varying in the

  • vertical direction or if it was varying 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 just tell you exactly what it is.

  • It is equal to it.

  • The divergence of a field is written this way: Upside down

  • triangle.

  • The meaning of this symbol, the meaning of an upside down

  • triangle is always that it has to do with derivatives, the

  • three derivatives.

  • Derivatives, whether it's the three partial derivatives.

  • Derivative with respect to X, Y, and Z.

  • And this is by definition.

  • The derivative with respect to X of the X component of A

  • plus the derivative with respect to Y of the Y component of

  • A, plus the derivative with respect to Z of the Z component

  • of A.

  • That's definition.

  • What's not a definition is the theorem and it's called

  • Gauss's theorem.

  • >> I'm sorry.

  • Is that a vector or is it-- Leonard Susskind: No.

  • That's a scale of quantity.

  • It's a scale of quantity.

  • Yeah.

  • It's a scale of quantity.

  • So, let me write it.

  • It's the derivative of A sub X with respect to X, that's

  • what this means, plus the derivative of ace of Y with

  • respect of Y, plus the derivative of ace of Z with respect

  • to Z.

  • >> Yeah.

  • So, the arrows you were drawing over there they were just A

  • on the other board.

  • You drew some arrows on the other board that are now hidden.

  • Leonard Susskind: Yeah.

  • >> Those were just A? Leonard Susskind: Yeah.

  • >> Not the divergence.

  • Leonard Susskind: Right.

  • Those were A.

  • And A has a divergence when it's spreading away from a

  • point, but a divergence is itself a scale of quantity.

  • Let me try to give you some idea of what divergence means in

  • a context where you can visualize it.

  • Imagine that we have a flat lake.

  • Just a shallow lake.

  • And the water is coming up from underneath.

  • It's being pumped in from somewheres underneath.

  • What happens if the water's being pumped in.

  • Of course, it tends to spread out.

  • Let's assume that depth can't change.

  • We put a lid over the whole thing so that it can't change

  • its depth.

  • We pump some water in from underneath and it spreads out.

  • Okay? We suck some water out from underneath and it spreads

  • in.

  • It anti-spreads.

  • So, the spreading water has a divergence.

  • Water coming in towards the place where it's being sucked

  • out it has a convergence or a negative divergence.

  • Now, we can be more precise about that.

  • We look down at the lake from above, and we see all the

  • water is moving of course.

  • If it's being pumped in the water is moving.

  • And there is a velocity vector.

  • At every point there is a velocity vector.

  • So, at every point in this lake there's a velocity vector

  • and in particular if there's water being pumped in from the

  • center here, right? Underneath the water of the lake there's

  • some water being pumped in the water's being sucked in from

  • that point.

  • Okay? And there'll be a divergence where the water is being

  • pumped in.

  • Okay if the water is being pumped out then exactly the

  • opposite.

  • The arrows point inward and there's a negative divergence.

  • If there's no divergence, then, for example, a simple

  • situation with no divergence.

  • That doesn't mean the water is not moving.

  • But a simple example of no divergence is the water is all

  • moving together.

  • You know, the river is simultaneous, the lake is moving

  • simultaneously in the same direction with the same velocity.

  • It can do that without any water being pumped in.

  • But if you found that the water was moving from the right on

  • this side and the left on that side, you'd be pretty sure

  • that somewheres in between, water had to be pumped in.

  • Right? If you found the water was spreading out away from a

  • line this way here and this way here, then you'd be pretty

  • sure that some water was being pumped in from underneath

  • along this line here.

  • Well, you would see in another way you would discover that

  • the X component of the velocity has a derivative.

  • It's different over here than it is from over here.

  • The X component of the velocity varies along the X

  • direction.

  • So, the fact that the X component of the velocity is varying

  • along the X direction is an indication that there's some

  • water being pumped in here.

  • Likewise, if you discovered that the water was flowing up

  • over here and down over here, you would expect it in here

  • somewheres some water was being pumped in.

  • So, derivatives of the velocity are often an indication that

  • there's some water being pumped in from underneath.

  • That pumping in of the water is the divergence of the

  • velocity vector.

  • Now, the water, of course, is being pumped in from

  • underneath.

  • So, there's a direction of flow but it's coming from

  • underneath.

  • There's no sense of direction-- well, okay.

  • That's what divergence is.

  • >> I have a question.

  • The diagrams you already have on the other board behind you?

  • Leonard Susskind: Yeah.

  • >> With the arrows? Leonard Susskind: Yeah.

  • Leonard Susskind: If you take, say, the right-most arrow and

  • you draw a circle between the head and tail in between, then

  • you can see the in and the out.

  • Leonard Susskind: Mm hmm.

  • >> The in arrow and the out arrow of a certain right in

  • between those two.

  • And let's say that the bigger arrow's created by a steeper

  • slope of the streak.

  • Leonard Susskind: No, this is faster.

  • >> It's going faster.

  • >> Okay.

  • And because of that, there's a divergence there that's

  • basically it's sort of the difference between the in and the

  • out.

  • Leonard Susskind: That's right.

  • That's right.

  • If we draw a circle around here or we would see that-- the

  • water is moving faster over here than it is over here, more

  • water is flowing out over here than is coming in over here.

  • Where's it coming from? It must be coming in.

  • The fact that there's more water flowing out on one side

  • than is coming in from the other side must indicate that

  • there's a net inflow from somewheres else and the somewheres

  • else would be from the pumping water from underneath.

  • So, that's the idea of divergence.

  • >> Could it also be because it's thinning out? Could that be

  • a crazy example? Like, the lake got shallower? Leonard

  • Susskind: Yeah.

  • Well, okay.

  • I took-- so, let's be very specific now.

  • I kept the lake absolutely uniform height and let's also

  • suppose that the density of water-- water is an

  • incompressible fluid.

  • It can't be squeezed.

  • It can't be stretched.

  • Then the velocity vector would be the right think to think

  • about there.

  • Yeah.

  • You could have-- no, you're right.

  • You could have a velocity vector having a divergence because

  • the water is-- not because water is flowing in but because

  • it's thinning out.

  • Yeah, that's possible.

  • But let's keep it simple.

  • And you can have-- the idea of a divergence makes sense in

  • three dimensions just as much as two dimensions.

  • You just have to imagine that all of space is filled with

  • water and there are some hidden pipes coming in, depositing

  • water in different places so that it's spreading out away

  • from points in three dimensional space.

  • Three dimensional space, this is the definition for the

  • divergence.

  • If this were the velocity vector at every point you would

  • calculate this quantity and that would tell you how much new

  • water is coming in at each new point in space.

  • So, that's the divergence.

  • Now, there's a theorem which the hint of the theorem was

  • just given by Michael there.

  • It's called Gauss's theorem and it says something very

  • intuitively obvious.

  • You take a surface, any surface.

  • Take any surface or any curve in two dimensions and now

  • suppose there's a vector field-- vector field point.

  • Think of it as the flow of water.

  • And now let's take the total amount of water that's flowing

  • out of the surface.

  • Obviously there's some water flowing out over here and of

  • course we wanna subtract the water that's flowing in.

  • Let's calculate the total amount of water that's flowing out

  • of the surface.

  • That's an integral out of the surface.

  • Why is it an integral in because we have to add up the flows

  • of water outward when the water is coming inward that's just

  • negative flow, negative outward flow.

  • We add up the total outward flow by breaking up the surface

  • into little pieces and asking how much flow is coming out

  • from each little piece here? How much water is passing out

  • through the surface? If the water is incompressible,

  • incompressible means its density is fixed and furthermore,

  • the depth of the water is being kept fixed.

  • There's only one way that water can come out of the surface

  • and that's if it's being pumped in, if there's a divergence.

  • The divergence could be over here, could be over here, could

  • be over here, could be over here.

  • In fact, anywheres there is a divergence will cause an

  • effect in which water will flow out of this region here.

  • So, there's a connection.

  • There's a connection between what's going on on the boundary

  • of this region, how much water is flowing throughout

  • boundary on one hand, and what the divergence is on the

  • interior.

  • There's a connection between the two and that connection is

  • called Gauss's theorem.

  • What it says is that the integral of the divergence in the

  • interior, that's the total amount of flow coming in from

  • outside, from underneath the bottom of the lake, the total

  • integrated-- now, by integrated, I mean in the sense of an

  • integral.

  • The integrated amount of flow in, that's the integral of the

  • divergence.

  • The integral over the interior in the three dimensional case

  • it would be 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-- is equal to the total

  • amount of flow that's going out through the boundary.

  • Now, how do we write that? The total amount of flow that's

  • flowing outward through the boundary we break up-- let's

  • take the three dimensional case.

  • We break up the boundary into little cells.

  • Each little cell is a little area.

  • Let's call each one of those little areas D sigma.

  • D sigma, sigma stands for surface area.

  • Sigma is the Greek letter.

  • Sigma stands for surface area.

  • This three dimensional integral over the interior here is

  • equal to a two dimensional integral, the sigma over the

  • surface and it is just the component of A perpendicular to

  • the surface.

  • It's called A perpendicular to the surface D sigma.

  • A perpendicular to the surface is the amount of flow that's

  • coming out of each one of these little boxes.

  • Notice, incidentally, if there's a flow along the surface it

  • doesn't give rise to any fluid coming out.

  • It's only the flow perpendicular to the surface, the

  • component of the flow perpendicular to the surface which

  • carries 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 coming out per unit

  • time, for example and that has to be equal to the amount of

  • fluid that's being generated in the interior by the

  • divergence.

  • This is Gauss's theorem.

  • The relationship between the integral of the divergence and

  • the interior of some region and integral over the boundary

  • where it's measuring the flux-- the amount of stuff that's

  • coming out from the boundary.

  • Fundamental theorem.

  • And let's see what it says now.

  • Any questions about Gauss's theorem here? You'll see how it

  • works.

  • I'll show you how it works.

  • >> Now, you mentioned that the water is compressible.

  • Is that different from what we were given with the

  • compressible fluid.

  • Leonard Susskind: Yeah.

  • You could-- if you had a compressible fluid you would

  • discover that the fluid out in the boundary here is all

  • moving inwards in every direction without any new fluid

  • being formed.

  • In fact, what's happening is the fluid's getting squeezed.

  • But if the fluid can't squeeze, if you can not compress it,

  • then the only way that fluid could be flowing in is if it's

  • being removed somehow from the center.

  • If it's being removed by invisible pipes that are carrying

  • it off.

  • >> So that means the divergence in the case of water would

  • be zero would be integrated over a volume? Leonard Susskind:

  • If there was no water coming in it would be zero.

  • If there was a source of the water-- divergence is the same

  • as its source.

  • Source of water is-- source of new water coming in from

  • elsewhere is. . .

  • Right.

  • So, with the example of the two dimensional lake, the source

  • is water flowing in from underneath, the sink, which is the

  • negative of a source, is the water flowing and in the two

  • dimensional example this wouldn't be a two dimensional

  • surface integral.

  • It would be the integral in here equal to a one dimensional

  • surface equal to the surface coming out.

  • Okay.

  • All right.

  • Let me show you how you use this.

  • Let me show you how you use this and what it has the do with

  • what we've said up 'til now about gravity.

  • I think-- I hope we'll have time.

  • Let's imagine that we have a source it could be water but

  • let's take three dimensional case, there's a divergence of a

  • vector field, let's say A.

  • There's a divergence of a vector field, dell dot A, and it's

  • concentrated in some region of space.

  • It's a little sphere in some region of space that has

  • spherical symmetry.

  • In other words, it doesn't mean that the divergence is

  • uniform over here but it means that it has the symmetry of a

  • sphere.

  • Everything is symmetrical with respect to rotations.

  • Let's suppose that there's a divergence of the fluid.

  • Okay? Now, let's take-- and it's restricted completely to be

  • within 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 or a certain

  • total divergence, an integrated divergence is occurring with

  • nice spherical shape.

  • Okay.

  • Let's see if we can use that to figure out what the A field

  • is.

  • That's dell dot A in here and now let's see can we figure

  • out what the field is elsewhere outside of here? So, what we

  • do is we draw a surface around there.

  • We draw a surface around there and now we're going to use

  • Gauss's theorem.

  • First of all, let's look at the left side.

  • The left side has the integral of the divergence of the

  • vector field.

  • All right.

  • The vector field or the divergence is completely restricted

  • to some finite sphere in here.

  • What is-- incidentally, for the flow case, for the fluid

  • flow case, what would be the integral of the divergence?

  • Does anybody know? It really was the flow of a fluid.

  • It'll be the total amount of fluid that was flowing in per

  • unit time.

  • It would be the flow per unit time that's coming 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

  • bigger than this region.

  • Why? Because the integral over the divergence of A is

  • entirely concentrated in this region here and there's zero

  • divergence on the outside.

  • So, first of all, the left-hand side is independent on the

  • radius of this outer sphere, as long as the radius of the

  • outer sphere is bigger than this concentration of divergence

  • here.

  • So, it's a number.

  • Although it's a number.

  • Let's call that number M.

  • No, no.

  • Not M.

  • Q.

  • That's the left-hand side.

  • And it doesn't depend on the radius.

  • On the other hand, what is the right side? Well, there's a

  • flow going out and if everything is nice and spherically

  • symmetric then the flow is going to go radially outward.

  • It's going to be a pure, radially outward directed flow if

  • the flow is spherically symmetric.

  • Radially outward directed flow means that the flow is

  • perpendicular to the surface of the sphere.

  • So, the perpendicular component of A is just the magnitude

  • of A.

  • That's it.

  • It's just the magnitude of A and it's the same everywheres

  • on the sphere.

  • Why is it the same? Because everything has spherical

  • symmetry.

  • Now, in spherical symmetry, the A that appears here is

  • constant over this whole sphere.

  • So, this integral is nothing but the magnitude of A times

  • the area of the total sphere.

  • All right? If I take an integral over a surface, a spherical

  • surface like this, on something that doesn't depend on where

  • I am in the sphere, then it's just you can take this on the

  • outside, the magnitude of the field and the integral D sigma

  • is just the total surface area of the sphere.

  • What's the total surface area of the sphere? >> Four thirds

  • pi R.

  • Leonard Susskind: No third.

  • Just four pi.

  • Four pi R squared.

  • Oh, yeah.

  • Four pi R squared times the magnitude of the field is equal

  • to 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 that look familiar? It's a vector field.

  • It's pointed radially outward.

  • Well, it's pointed radially outward if the divergence is

  • positive.

  • If the divergence is positive, it's pointed radially outward

  • and its magnitude is one of R squared.

  • It's exactly the gravitational field after a point particle

  • of the center.

  • >> It's the magnitude of A.

  • Leonard Susskind: Yeah.

  • That's why we have to put a direction in here.

  • You know what this R is? It's a unit vector pointing in the

  • radial direction.

  • It's a vector of unit length pointed in the radial

  • direction.

  • Right? So, it's quite clear from the picture that the A

  • field is pointing radially outward.

  • That's what this says here.

  • In any case, the magnitude of the field that points radially

  • outward, it has magnitude Q, and it falls off like 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-- a

  • concentrated divergence of the gravitational field.

  • Concentrated in some very little small volume.

  • Think of it, if you like, you can think of it as the

  • gravitational field, the flow field, the velocity field of a

  • fluid that's spreading out.

  • Oh, incidentally, of course, I've got the sign wrong here.

  • The real gravitational acceleration points inward which is

  • an indication that this divergence is negative.

  • The divergence is more like a convergence sucking fluid in.

  • So the Newtonian gravitational field is isomorphic, is

  • mathematically equivalent, or mathematically similar, to a

  • flow field to a flow of water or whatever other fluid where

  • it's all being sucked out 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 go,

  • like, one over R squared.

  • That's a useful analogy.

  • That is not the say that space is a flow or anything.

  • It's a mathematical analogy that's useful to understand the

  • one over R squared force law that it is mathematically

  • similar to a field of velocity flow from the flow that's

  • being generated right at the center of a point.

  • Okay.

  • That's a useful observation.

  • But notice something else.

  • Supposing now, instead of having the flow concentrated at

  • the center here, supposing the flow is concentrated over a

  • sphere that is bigger but the same total amount of flow.

  • It would not change the answer.

  • As long as the total amount of flow is fixed, the way that

  • it flows out through here is also 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 whether it's a spread out mass, or whether it's a

  • spread out mass this big, as long as you're outside the

  • object and as long as the object is spherically symmetric,

  • in other words, as long as the object is shaped like a spear

  • and you're outside of it, outside of it, outside of where

  • the mass distribution is, then the gravitational field of it

  • doesn't depend on whether it's a point, it's a spread out

  • object, whether it's denser at the center and less dense on

  • the outside, less dense at the center and more dense on the

  • outside.

  • All it depends on is the total amount of mass.

  • The total amount of mass is like the total am of flow coming

  • into the-- that theorem is very fundamental and important to

  • thinking about gravity.

  • For example, supposing we are interested in the motion of an

  • object near the surface of the earth but not so near 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 by this point, it's attracted

  • by this point, it's attracted by that point.

  • It's close to this point, it's far from this point.

  • It sounds like a hellish problem to figure out what the

  • gravitational effect on this point is.

  • But no.

  • This tells you the gravitational field is exactly the same

  • as if the same total mass was concentrated right at the

  • center.

  • Okay? That's Newton's theorem.

  • It's a marvelous theorem.

  • It's a great piece of luck for him because without it he

  • couldn't have solved 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 one over R squared force law and

  • only the one over R squared force law wouldn't have been

  • true if it'd be R cubed, R to the fourth, over R to the

  • seventh.

  • With the one over R squared force law, a spherical

  • distribution of mass behaves exactly as if all the mass was

  • concentrated right at the center as long as you're 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's spherical in shape,

  • behaves if it were a point mass.

  • >> So, if you're down in a mine shaft that doesn't hold?

  • Leonard Susskind: That's right.

  • If you're down in a mine shaft it doesn't hold.

  • But, that doesn't mean that you can't figure out what's

  • going on.

  • You can figure out what's going on.

  • I don't think we'll do it tonight.

  • It's a little too late.

  • But yes, we can work out what would happen in a mine shaft.

  • But that's right.

  • It doesn't hold in a mine shaft.

  • For example, supposing you dig a mine shaft right down

  • through 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 pulling toward the

  • center? Not much.

  • Certainly much less than if all the mass were concentrated

  • right at the same theory.

  • You've got the-- it's not even obvious which way the force

  • is but it's toward the center.

  • But it's very small.

  • You displace away from the earth a little bit.

  • There's a tiny, tiny force.

  • Much, much less than as if all the mass were squashed

  • towards the center.

  • So, right.

  • It doesn't work for that case.

  • Another interesting case is supposing you have a shell of

  • material.

  • To have a shell of material, think about a shell of source,

  • fluid flowing in.

  • Fluid is flowing in from the outside onto this blackboard

  • and all the little pipes are arranged on a circle like this.

  • What does the fluid flow look like in different places?

  • Well, the answer is, on the outside it looks exactly the

  • same as if everything were concentrated on a point.

  • But what about in the interior? What would you guess?

  • Nothing.

  • Nothing.

  • Everything is just flowing out away from here and there's no

  • flow in here at all.

  • How can there be? Which direction would it be? And so there'

  • s no flow in here.

  • >> Wouldn't you have the distance argument? Like, if you're

  • closer to the surface of the inner shell-- Leonard Susskind:

  • Yeah.

  • >> Wouldn't that be more force towards that? No.

  • See, 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 divergency.

  • But there is no divergency here.

  • So, the net integrated field coming out is zero.

  • No field on the interior of the shell.

  • Field on the exterior of the shell.

  • So, the consequence is that if you made a spherical shell of

  • material like that, the interior would be absolutely

  • identical like it would be if there was no gravitational

  • material there at all.

  • On the other hand, on the outside, you would have a field

  • which would be absolutely identical to what happens at the

  • center.

  • Now, there is an analogue of this in the general theory of

  • relativity.

  • We'll get to it.

  • Basically what it says is the field of anything, as long as

  • it's spherically symmetric on the outside, looks like

  • identical to the field of a black hole.

  • I think we're finished for tonight.

  • Go over divergence and Gauss's theorem.

  • Gauss's theorem is central.

  • There would be no gravity without Gauss's theorem.

  • >>

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