Newtonian Cosmology and the Expanding Universe Explained

[Music] Stanford University okay let's uh let's

start this uh this quarter's subject is cosmology cosmology is of course a very old subject uh it

uh goes back thousands of years but I'm not going to tell you about thousands of

years of cosmology but I say thousands of years I'm talking about the Greeks of course uh but we're not going to go here

back thousands of years we're going to go back at most to some time in

the second quarter second quarter of the uh 20th century when Hubble discovered that the

universe is expanding but uh let's just say a few words about the science of cosmology the

science of cos mology is new it or at least what we know about I a minute ago I said it was very old yes in a sense

but the modern subject of cosmology is very new it really dates to well after Hubble it dates

um to the discovery of the Big Bang the 3 degree microwave radiation that was discovered as uh as the remnant of the

big bang and that happened you know sometime in the 60s so I was I was a student I was a young student and

before that um cosmology was in a certain sense less like physics and more like uh

being um Natural Science like what a naturalist does studies this kind of thing studies that kind of thing you

find a funny Star over there you find a Galaxy over there that looks a little weird you classify

you name things you measure things to be sure but the accuracy with which things were known was so poor that it was

extremely difficult to be precise about it and it's only fairly recently that physicists physicists were always

involved but they were involved because many of the things that you see many of these strange creatures funny Stars uh

galaxies and so forth of course are physical systems and to describe them properly they have angular momentum they

have all the things that physical systems have uh there's chemicals out there and so uh physical chemists were

involved but thinking of the universe as a physical system as a system to study mathematically and um with a set of uh

physical principles and a set of equations of course there were always sets of equations way back but wrong

equations right equations and accurate equations things

which agreed with observation that's relatively new um more or less more or less over the

history of my career in physics which is 50 years something like that and that's what we're going to

study we're going to study the universe as a system in other words a universe as a

system that we can study with equations so if you don't like equations you're in the wrong

place all right so where do you start you start with some observations now the first observation

which may not really turn out to be absolutely true for reasons uh that uh it's not

absolutely true but it looks like it's approximately true is that the universe is what is called

isotropic isotropic means that when you look in that direction or that direction or that direction or that direction now

of course if you look right at a star it looks a little different than if you look at the away from the Star but on

the whole averaging over patches in the sky and looking out far enough so that you get away from the immediate

foreground of our own Galaxy the universe looks pretty much the same in every direction

Direction okay that's called isotropic same in every

direction right now if the universe is isotropic with one exception that I'll describe in a moment if it's isotropic

around us then you can bet with a high degree of confidence that it's also pretty close to being

homogeneous homogeneous doesn't mean it's the same in every direction it means it's the same in every place if

you went out over there and you looked around from uh from uh uh 16 galaxies over and you looked around what you

would see you would see about the same thing you saw here so first of all what's the argument for that why does

being isotropic which means the same in every direction tell you anything about why it would be the same if you moved

away to a very distant place and the argument is very simple um imagine that there's some

distribution of galaxies you know incidentally at least in the first part

of uh of this study here it's not going to matter very much whether what we're talking about are whe whether we call

them galaxies or whether we just call them particles they're just effectively Mass

points distributed throughout uh space for the moment I might even lapse into calling them particles from time to the

time now you must mean when I say particles I mean literally galaxies but uh but unless I otherwise specify okay

so the universe has a lot of them anybody know how many galaxies are within the visible

uh about 100 billion 10 to the 11th just there some nice numbers to keep track of incidentally it's a good idea to keep

track of a few numbers within what we can see within what we can see with telescopes uh out to as far

as astronomy takes us about 10 to the 11th galaxies each Galaxy about 10 to the 11 Stars altogether 10 to the 22

Stars if each star has roughly 10 planets that's 10 to the 23 avagadro's number of uh planets out there

a mole right planetary planetary mole right all right

now imagine that when we're over here and every direction that you look in it looks pretty much the same well then I

maintain that not only must it be the same in every direction but it must be the same from place to place what would

it mean for it not to be the same from place to place well if it's isotropic the the only way it could not be

homogeneous is if it uh if it somehow formed rings of some sort it's got to be such that would that it looks the same

in every direction but it's not the yeah shells I think somebody said shells it have the geometry of some sort of

shell-like structure why it doesn't literally mean shells it just

means yeah so if that was the case if that were the

case and you went someplace else and you looked around clearly it wouldn't look isotropic anymore so for it to look

isotropic unless by accident we just happen to be at the center of the universe if we happen to be at the very

center where everything just accidentally or not accidentally Maybe by Design happens to be nice and

rotationally symmetric about us if we don't want to believe that then we have to believe it's pretty much the same

everywhere and that it's homogeneous so homogeneous means that as far as we can

see space is uniformly filled on the average with particles uniformly filled

okay that's called the cosmological principle now you can't why is it true well how could it not be true it's the

cosmological principle right and uh sometimes people argue like that it's true because it's

been observed to be true to some to some degree of approximation now as was mentioned in some media that I don't

know how to evaluate some astronomers apparently claim to see structures out there which are so big if the Blackboard

here was the whole visible Universe they would stretch across great big patches of it and that seems to be a little bit

counter to this idea of uh complete uniformity and of course certainly the the idea of complete uniformity is not

exact just the fact that there are galaxies means the not the same over here and over here and in fact there are

clusters of galaxies and superclusters of galaxies so it it appears it's not really homogeneous but it tends to come

in of clusters which on some big enough scale like a billion light years roughly maybe a little bit less if you average

over that much it looks homogeneous okay so that's the basic fact that we're going to begin

with now what's the first step in formulating a physics problem know your variables that's a good yep

know your variables usually the answer is sharpen your

pencil okay after you sharpen your pencil you the next is know your variables but um a good step I'm not

sure whether it comes after that or before that is Professor could the um uniformity of

the cosic micro be an argument oh you bet you bet you bet but we're going back I'm

purposefully going back a few decades uh to sometime around uh the 60s or something like that 50s

60s 40s uh uh the the idea of the cosmological principle was put forward before people had any real right to uh

to put it Forward they just said oh well let's just say it's homogeneous we'll call it the cosmological principle and

if people ask us why it's true it's because it's a principle all right but then you know with more and more

astronomical investigation and then finally the cosmic microwave background really nailed it and in some sense uh

the primordial distribution of matter was extremely smooth extremely smooth but we'll get to that all right so here

we have a uniform gas if you like it's a uniform gas and that gas is

interacting it's a gas of particles it's interacting each particle is interacting with the other particles now galaxies on

the whole are not electrically charged they're electrically neutral but they're not gravitationally

neutral they interact through Newtonian gravity and that's the only important force on big enough scales on big enough

scales where matter tends to be electrically neutral uh the only really important

force is gravity right so gravity is p either pulling all this stuff together or it's

doing something to it but it's a little bit confusing what uh what

happens to this point over here does it accelerate toward the center because at the center there's a whole bunch of

matter there or does it accelerate out to here because after all there's as much matter out there as there is on

this side in fact it sort of looks like it oughtn't to move anywhere it ought to just stay there

because there's as much on one side is on the other side right so it'll just stay there well what about this one over

here same thing because every place is the same as every other place so the natural thing to guess is that the

Universe must be just static it must just sit there because nothing has any net force on it and so

there's nothing pulling it one way or another that's wrong we're going to work out tonight the actual Newtonian

equations of uh of cosmology but you may have heard that the expanding Universe

somehow fit together especially well and wasn't really understood until general relativity until Einstein that is simply

false uh it may be so historically I mean in terms of years yes it is true that the expanding

Universe was not understood until after Einstein had created the general theory of relativity that is a fact about dates

it's not a fact at all about logic Newton could have done the expanding Universe since Newton didn't

do it we are going to do it here the way Newton should have done it if only Newton was a little bit

smarter okay all right so the first thing know your know your variables for sure but the first step is

usually to introduce a set of coordinates introduce a set of coordinates into a

problem and that means exactly what it always means take space and Rule it into

coordinates three dimensions for sure but I'm only going to draw two in other words introduce fictitious

a fictitious grid of coordinates now what shall we take for the distance between

neighboring lattice points on this grid we could take it to be 1 meter we could take it to be 10 meters we could take it

to be a million meters we could take it whatever we like but there's a smarter thing to do than to just fix the

distance between the points the smarter thing to do is to imagine these points of been

chosen so that the grid points always pass through the same galaxies in other words

that the galaxies here provide a grid provide a

grid in such a way that no matter what happens since the galaxies are nice and uniform no matter what happens this

galaxy over here here will always be at that point on the grid that Galaxy over here will always be at that point on the

grid and that means that if the universe indeed either expands or contracts the grid has to expand or or let me say it

differently if the galaxies are moving relative to each other perhaps away from each other or closer to each other then

the grid moves with them let's choose coordinates so that the galaxies are sort of Frozen in the

grid now it's not obvious you can do that it's not obvious you can do that if the galaxies were such that some were

moving this way over here some were moving that way over here some were moving that way over here a sort of

random kind of motion then there would be no way to fix the coordinates by attaching them to the

uh to the galaxies because even at a point different ones would be moving in different ways but that's not what you

see when you look out at the at the heavens what you see is that they're moving very

coherently exactly as if they were embedded in a grid with the grid perhaps expanding

perhaps Contracting we'll come to that but the whole grid being sort of Frozen any motion that takes place is

because the grid is either expanding in size or Contracting in size that's an observation about the

relative motion of nearby galaxies galaxies over here and over here which are relatively nearby are not moving

with tremendous velocity relative to each other they're moving in a nice coherent way as I said so that uh so we

can choose coordinates we'll call them X Y and Z standard names for coordinates x y X Y and

Z but x y and z are not measured in length they're not measured in length because the length of a grid cell may

change with time okay so we've labeled the galaxies by where they are in a grid and now we

can ask the question Let's uh let's say the distance let's take two point let's start with two

points separated by an X distance over here let's call that X distance Delta X how far apart are they well I don't

know how far apart they are yet but I'm now going to postulate that the distance between them the actual distance in

meters or in some physical unit that you measure with a ruler could be a light year on a side it could be a million

light years on a side but a ruler that the actual distance is proportional to Delta X

the distance between these two people over here is half the distance between these two is a third the distance

between these two so it's proportional to Delta x

times a parameter that's called the scale parameter the scale parameter may or may

not be just a constant it may just be a constant if it were just constant then the distance

between galaxies fixed in the grid would stay constant with time but it may also be

time dependent so let's allow that that would say the distance between two galaxies let's say this is Galaxy a

this is Galaxy B the distance from A to B is a of T * Delta xab where Delta X is the X distance is the coordinate

distance between them let me write it more generally if we have two galaxies or arbitrary

positions on the grid then the distance between them

daab is equal to a of T the same a of T and then the square root of Pythagoras's Theorem Delta x^2 plus Delta y^ 2 plus

Delta z^2 in other words you measure the distance along the Grid in grid units

and then multiply it by AF of T to find the actual physical distance between the two

points as I said AF may or may not be constant in time well of course it's not if it was constant in time that would

mean literally the galaxies were just Frozen in space and they didn't move and that's not what we see we see them

moving apart from each other okay so let's calculate now the velocity

between Galaxy a and Galaxy B here's the distance between Galaxy a and this of course should be Delta AB the distance

the coordinate grid distance let's just use the simpler formula up here let's forget Pythagoras and just take them to

be along the x- axis it doesn't really matter okay here's

daab what's the velocity between what's the relative velocity of the ab galaxies it's just the time derivative of this

right just the time derivative of the distance is the velocity so the velocity between A and B is just equal to the

time derivative and the only thing that's changing for A and B A and B are fixed in the grid so Delta

X is not changing that's fixed the only thing that's changing perhaps is a so the velocity is just the time derivative

of a a DOT means the time derivative of a a DOT time Delta

X all I've done is differentiate this formula with respect to time now I can write that the ratio of the Velocity to

the distance I'll leave out the a well let's let's put them in a the ratio of the Velocity to the distance is just the

ratio of a DOT to a notice that Delta X canel out well

that's interesting it means that the ratio of the Velocity to the distance doesn't depend on which pair of galaxies

we're talking about every pair of galaxies no matter how far apart no matter how close no matter what angle

they're oriented in the relative velocity between the two of them relative either separation or the

opposite of Separation the ratio of the Velocity to the distance is a DOT over a what the what what's the name for this

thing anybody know the Hubble constant it's called the Hubble

constant let's call it h now is there any reason why it should be a constant what do we mean when we

say it's a constant there's no reason for it to be independent of the time and in fact it's

not what we find here is that it's independent of X it doesn't matter where you are it doesn't matter which two

galaxies you're talking about the same Hubble constant at a given time so the Hubble the Hubble constant is a kind of

misnomer the Hubble Hubble the Hubble parameter the Hubble function the Hubble function is

independent the position but depends on time and now I just write this in the standard form that the velocity between

any two galaxies in the universe is equal to the same Hubble

parameter times the distance between them and that's the derivation of the Hubble law excuse me question um don't

you sort of start out almost assuming what you're trying to show by saying the D equals one function AF that's

independent yes yes yes indeed absolutely yeah I mean you would never would have written

this if Hubble hadn't discovered that the that the that the Hubble law was right but the other The Hub law is in

some sense not all that surprising all it's you know some witty person said you shouldn't be surprised that the fastest

horse goes the furthest okay right uh the faster you move the far far farther you go so

that's all this thing says however it's

interesting the connection between this formula and the Hubble formula is as you point out a close one but as what it

says is everything is moving on a grid and it's the grid itself whose size scale may may or may not be changing

with time but of course it is changing with time and the Hubble constant is just the

ratio of the time derivative of a to a itself okay that's that's the facts those are the that that's the facts

those are the facts as Hubble discovered them and as theoretical cosmologists then had something to work with

all right let's uh let's say a few more things about this what about the mass within a region

let's take a region of size Delta X Delta y Delta Z and now I mean a region which is big

enough so that uh I don't know what happened to my universe I had my universe here but

uh big enough so that we can average over the small scale structure how much mass

is in there okay well that the amount of mass that's in there is going to be proportional to

Delta X Delta y Delta Z the bigger the region that you take the more mass and so let's just say the amount of mass or

call it new new is nothing but the amount of mass per unit volume of the grid but volume not being measured in

meters but being measured in X all right so let's say that's the mass in a given region of coordinate

volume Delta X Delta y Delta Z on the other hand what's the actual volume of that region let's take this volume of

the same region the volume of the same region is not Delta X Delta y Delta Z why because the distance along the x-

axis and the Y AIS and the z- axis is not Delta X it's a * Delta X so that means the volume of that same

cell that same cell is a cubed time Delta X Delta y Delta Z right that's because the length along

the XA is a * Delta x a * Delta y a * Delta Z and so now let's write a formula for the density of mass the density I

mean the physical density of mass now how much mass is there per cubic kilometer or cubic Lightyear or whatever

units we haven't specified units yet later on we'll specify units uh meters are fine meters seconds and kilograms

are fine mass measured in kilograms volume me measured in cubic met what's the density so let's write let's call

the density a standard terminology for density is row I don't know where it came from row stands for density let's

write it over here density and density means the number of kilogram per cubic meter if you like

it's the ratio of the mass to the volume it's the ratio of the mass to the volume and it's just as new here divided

by a cu M that's a that's a formula that we have nude ided a cubed now the amount of

mass in each cell in here stays fixed why does it stay fixed because the galaxies move with the grid so the

amount of mass in a given region in the grid stays the same that's just something I've called

new the Greek letter new and I divided it by the volume to get the density and of course if a changes

with time the density changes with time that's obvious if the universe grows the density decreases if the universe

collapses the density increases and so this is a Formula that we will use from time to

time all right so far we've done nothing that uid himself couldn't have done all right we didn't

even need Newton yet now enters Newton and Newton says look let's not play games let's forget all this all

right we'll take into account that the universe is homogeneous and all that stuff but Newton was a very very

self-centered person he always believed that he was at the center of the universe and so it was very natural for

him to take the perspective that I Sir Isaac Newton am at the

origin now of course we know and Newton also would have known that if he's clever he'll get the same equations no

matter where he places himself but there's nothing wrong with choosing the grid such that Newton and we are at the

center of the grid right then surrounding Newton and

moreover Newton will also say I'm not moving not moving why well I'm standing still so Newton is at rest at the center

of the universe as far as uh for for mathematical purposes and now he wants to ex and and of course we're talking

about on a scale so that everything is nice and uniformly distributed now he looks out to a

distant Galaxy he looks out the Galaxy over

here and he wants to know how that Galaxy moves well that Galaxy moves under the

assumption of Newton's equations Newton's equations say that everything gravitates with everything

else right but there's something special about Newton's there's a theorem Newton knew

this theorem in fact it's called Newton's theorem what Newton's theorem

says is that if you want to know what the gravitational force on a system is given that everything is isotropic

doesn't even have to be homogeneous for this given that everything is isotropic you want to know the gravitational force

in a frame of reference like I've drawn here you want to know the gravitational force on that particle then draw a

sphere with that particle on the sphere centered at the origin and then take all of the mass

within that sphere and pretend that it's just sitting at the

origin Just Pretend We're not we're not literally moving it just pretend that the only only mass in the universe

within this sphere is at the origin and what about the outside the mass is on the

outside ignore it Newton's theorem says that the force on a particle in a isotropic world like this all comes

from the sphere inside the uh the radius of the particle and nothing from the outside

I think we may have proved that in previous classes in classical mechanics I don't remember but it's a true it's a

true theorem it's a true theorem and it's the reason that we here uh in evaluating the gravitational field on

this pen here why we can pretend that all of the mass of the earth is at the center of the

earth when I evaluate the gravitational field here uh keeping in mind that the Earth

is a sphere keeping in mind in mind that it's pretty uniform and so forth I can just pretend that all of the mass was at

the center of the earth until of course the pen hits the floor then they'll say no the mass wasn't the but until it hits

the floor pretend that all the mass was concentrated at the center and furthermore the masses

outside Beyond this even though there's a lot more out there incidentally there's a

lot of mass out there I'm not talking about the ceiling of the the building I'm talking about all the galaxies out

there there's a lot more but the pen doesn't feel them only feels the things on the inside of the sphere so Newton

says what I'm going to do is I'm going to take this galaxy which

is at a certain distance away what's its distance here its distance is

D its distance is the square root of x² + y^2 + z^

2 X2 and y^2 and z^2 that's the coordinates of this point over here time a the distance from the center can you

read the is it easy to read the red I don't know why I started the red I just uh lapsed is the red

readable okay all right squ X2 + y s + e s that's Pythagoras and you multiply by a to find the actual distance I can call

that let's call that D equals a of T and let's just call this thing here r capital r r is not measured in meters

it's just square root of x s + y sare + z^ S that's the distance from the center to the Galaxy in question now Newton's

equations are about forces and accelerations so the first thing is let's calculate the acceleration of X of

the point x of the Galaxy at the point x relative to the origin well first the velocity the

velocity is V is equal to a DOT of T * R and what about the acceleration the

acceleration is just differentiate again acceleration is equal to a double dot of T time R do we have to worry

about whether R is changing with time no because the Galaxy is at a fixed point in this expanding lattice R

was fixed for that Galaxy and so this is the acceleration we could multiply it by the

mass of the Galaxy if we wanted to but I don't need to it's just the acceleration and what are we going to set that equal

to we're going to s that set that equal to the acceleration that we would get from all the gravitating material inside

here okay so let's see what how much first question is how much mass is in there well let's just call it the

mass the mass that's this is the mass that's inside this sphere the formula that we're going to

compare this with is Newton's gravitational formula force is equal to mass time Mass which mass is this little

one here G that's the Galaxy the mass of the big one which one is

that that's all a mass on the inside and the distance

between well the distance squared and I'm missing couple of things two things I'm

missing Newton's gravitational constant 6.7 * 10us 11 in some units I'm missing one more thing anybody know what it is

minus sign the minus sign indicates that the force is attractive pulling in all right that's the convention Force

pulling in is is counted as negative Force pushing out is counted as positive all right this is the force of gravity

on a particle of mass m what is the acceler ation of gravity the acceleration of gravity is just drop the

mass just drop this out to get the mass here the the acceleration is the force

per unit mass and right that this is the acceleration M minus mg / c^2 that's the acceleration of that uh that g what's

that don't you want to use the small no okay no no no I divided out the small

M okay good so that's the acceleration due to the presence of all this mass in the interior here and that had better be

equal to Able dot of T * R God knows where this is going but we're just following our

nose writing equations and you know that's always the way do it you you start out with some

physical principles you write down the equations and then you blindly follow them for a

way until uh until you need to think again so we we're on autopilot now we're just doing equations let's let me

rewrite it down here a do R is equal to minus m g over D2 okay let's uh let's plug in this guy over here the distance

is a * R so maybe we can maybe maybe who knows at some point we might have actually discovered something that looks

interesting at the moment it's just a blind uh a of t^2 or just a s let's just call

it a s a 2 time d^2 no a 2 time R 2 right okay now excuse me but I'm just going to

I'm going to divide by R here see I I secretly know where I'm going right maybe you do too but that's

all right a r Cub I divide it and I'm going to divide by another

a makes this a cubed okay now

this is good this this will do but next question what's the volume of the sphere let's write the volume of the sphere

this is this is Newton's equations Now volume of the sphere what's the

volume 4/3 4/3 pi now is it R cubed no D cubed

which means a cubed * R cubed right because distance is really a * R

that's the actual physical volume when I say the volume I mean the volume is measured in some standard unit like

meters that's a volume now look here we have a cub * R cubed here let me write that as volume 3 over

4 Pi volume is equal to a cub R cub oh maybe I'm being dumbb maybe I

shouldn't yeah that's that's dumb let's not do that let's just look at this formula

here notice that we have a cub R cubed downstairs let's multiply by four over 3 Pi or divide by 4 over 3 pi and multiply

by 4 over 3 Pi 4/3 pi what I did here I undid

here but now I have M over the volume what's M over the volume the density o something something nice may be

happening a do over a is equal to minus 4/3 piun Newton's constant times the ratio

of the mass in that sphere to the volume volum in that sphere and that is the density

now that's a nice equation notice that it really doesn't depend on R anymore if we know what the

density of the universe is and the density of the universe does not depend on where you

are the density of the universe does not depend on R the left hand side R has dropped out

the right hand side no memory of R it means this equation is true for every Galaxy no matter how far

away same equation had we done a different galaxy we would have gotten the same equation the only way that this

equation had any memory of which Galaxy we were talking about was because of R but R dropped of the equation that's of

course a good thing because if we want to think of a as something which doesn't depend on where you

are then it had better be that uh that it drops out so Newton uh confirms what he might have expected that the equation

for a is a universal equation for all galaxies yeah seems like something seems a little off because um

we picked the origin it was arbitrary supposedly it was that again it we would have gotten exactly the same thing no

matter what origin we picked but I mean it says the whichever origin we pick pick then we get the force going toward

that origin right that's right and so something doesn't add up there well it has added up but the answer doesn't

depend on which uh no you have to no no no the point is you have to do the the transformation carefully you have to do

the transformation carefully you go to another origin and in your frame Newton could have said let me work this out

from my frame of reference which I will put myself at the origin but let me study now the relative motion of some

galaxies relative to some moving some Galaxy which is moving he would have found exactly the

same equations but he would have had to do the transformation carefully so we finessed that and got

away from it by just putting ourselves at the center but as you can see the final formula doesn't care where you are

it confirms the fact that nothing really depended on which Galaxy we thought of as our

home but the direction of the force would if if we really look at the direction of the gravitational force

that's always toward the origin right there's a relative Force you the right way to think about it is really a

relative Force no yeah yeah yeah yeah yeah yeah yeah yeah yeah yeah yeah yeah yeah in in this way of thinking about it

the force is always toward the origin right but had we stationed ourselves on some other Galaxy that was moving and

did all the Transformations remember when you go to a a a moving frame there are fake forces

inertial forces fake forces that you have to add in so from the point of view of this guy over

here this galaxy over here has a Force which could be thought of as being Tor here plus a fake Force the fake Force

being an inertial force due to his acceleration but we got around that by just saying let's position ourselves at

the center no acceleration no velocity we just sit at the center so the only test question the only

question is do we get an answer which doesn't depend on who we are and which Galaxy we're on right okay so that

that's that's part of main message of this the answer doesn't depend on which Galaxy you're on and so it really didn't

depend on Newton's assumption that he was at the center can we go back to the equation for Mass would anything have

changed if we had assumed that new was not a constant oh yes yes yes new things would have

changed if new was not a constant could we say we've kind of made it happen by making an

assumption made what happen um the end result me a con in

space yes to say that it's a constant in space is the principle that the universe is homogeneous hinges on absolutely

everything hinges on the homogeneity of the universe right and that the right the N right that the number of the mass

per unit volume is the same everywhere in space okay yes everything hinged on that and uh

okay so here's one equation it's a central fundamental equation of cosmology and it's a differential

equation it's an equation for how a changes with time now there's a number of things to look at but the first

interesting thing to look at is it's impossible to have a universe which is static static means that a doesn't

change with time unless it's empty empty means row equals zero only if it is empty so that this

side is zero can the time derivative of a or the second time derivative in this case be

zero so we derive the fact that the universe is not

static all right one more thing we could do to make this uh to make this an equation that we could

solve is to replace row by the constant new / a cub now new is literally a constant it's the number

of galaxies times the mass of a galaxy uh in a unit coordinate volume it doesn't change with time because the

galaxies are frozen in the grid and so we could write this equation one more

step able dot not surprising that there's an able dot why is there an able dot because

because Newton's equations are about acceleration and uh not surprising that there's an able dot equals -4

over3 < * gtimes the density but the point is now the density is not a constant new is a

constant but new over a cub is not because a is changing with time and so we better put that in here new / a cubed

okay so there's a lot of constants here minus the minus sign is a constant 4 pi over 3 G Newton is a constant and we can

pick new also to be a constant so everything out here is constant a is not constant but we have a kind of

differential equation it is a differential equation a kind of equation of motion in terms of one constant 4 Pi

G new over 3 we have an equation of notion for the scale factor for the scale factor a is a function of

time uh who was the first to discover this equation it was actually discovered in

the context of the general theory of relativity it was discovered um I think fredman Pro Alexander fredman in

uh um before he got himself killed in World War I I think was it using the the general theory of relativity was

consistent with what Einstein should have done but it's perfectly possible there nothing in it that wasn't just

good old Newtonian mechanics okay yeah do you need to keep Thea and the denominator on the left

hand side well I can multiply it if I like a con a s sure you can do that it's

just it's traditional to write it this way it's just a tradition yeah so does that negative sign not tell us anything

about whether we're expanding or Contracting it doesn't tell us whether it's expanding or Contracting okay so

let me explain why let me write forget that now now we just have the

Earth let's compare this with something else we have the Earth and we have a particle over here let's put it on the

xaxis on the x-axis all right there's an equation for that particle it's the same equation let's

call it now um let's call it X but X doesn't stand for these coordinates now it just stands for the standard position

coordinate the height above the Earth that satisfies some equation x dot is equal to the gravitational force

whatever the gravitational force is uh mg um over x^2 minus uh that's it something like that okay

does this equation this what this equation tells us is that the particle is accelerating toward the Earth the

minus sign tells us that the acceleration is toward the Earth but whether it whether it's moving

away from the earth or toward the Earth is a question of velocity not acceleration is the velocity that way or

is it that way well you can imagine somebody over here taking this particle and ejecting it that way it will have a

positive velocity it will be moving away from the earth you could also Imagine the same person pushing it that way

it'll be moving toward the Earth X is decreasing but the acceleration will be the same in either case the velocity

will have a negative acceleration which means if it's going this way it'll turn around or may turn around if it's going

this way it'll increase the in velocity um whether it turns around or not depends on what initial conditions

and the initial conditions or whether it's above or below the escape Velocity but in either case the acceleration is

toward the Earth so knowing that the acceleration is toward the Earth as it is for this pen does not tell me whether

it's moving up or moving down it can move up and then move down and uh and you get the point okay so no

this equation doesn't tell us whether the universe is expanding or Contracting but it tells us that the second

derivative is negative so it means that even if it's expanding it's tending to slow down if it's expanding it's tending

to slow down if it's Contracting it's tending to speed up its contraction there is an analog here of

whether you are above or below the escape Velocity and we'll come to it all right so uh I was asked a question which

uh I will just point out um all sorry art but I'm going to I'm going to use your name art asked me well he looked at

this and said this is negative and he looked at what is this he looked at this and said this is

positive if the universe is expanding H is positive how come this one's negative but that's because he didn't read

carefully there's two Dots here and only one dot here this is velocity this is acceleration not hard

for acceleration to be negative you know you're you're in your uh in your Ferrari and you're going down

the uh down the autoban or whatever and uh you uh press down on the brake your acceleration is negative but

your velocity is positive right okay you're slowing down but you're still going

ahead now in fact the universe is not slowing down this we're make we're we're really doing what Newton would have done

and what all cosmologists thought the right thing to do was until about 15 years

ago okay so 15 this

is Newton's model of the universe and it is the model that would have been called the standard model or

close to it the standard model of the universe until until the accelerated Universe was

discovered this is the decelerated universe you see but the universe accelerates so

there's got to be something else in this equation well there is there are several other things in that equation but we'll

come to them uh some parts don't do have to do with

Einstein okay let's talk about

um not cosmology but just particles rocks stones

thrown upward from the surface of the Earth equations are very similar but let's just examine them for

a minute and uh take home a couple of lessons about it here's the Earth and we might as well

think of it as a point because Newton uh Newton proved the theorem that said we can think of it as a point we're outside

we're we're above the surface of the Earth so and here's right here's the Earth here's our particle over here oh

no where should I put it put it over here no put it over here x-axis put it on the

x-axis and what are its equations the equations of Newton's equations but there's

actually a more useful version of Newton's equations which is just energy conservation just energy conservation

let's write down the energy of this particle over here and write

down that it's conserved in fact it's a uh it's a more useful equation than this one over here energy the energy equ or

more useful than f equals ma what is the energy of this particle here it's moving outward it has some velocity the

velocity could be negative it could be moving inward but what is the total energy of this particle the total energy

of it is its kinetic energy plus its potential energy kinetic energy one half the mass of the particle not the mass of

the Earth the mass of the particle times its velocity squared which we could call X do squ if

we wanted well let's just leave it as velocity squared for moment but what about the potential energy remember the

potential energy potential energy is minus Little M Big M Newton's constant

divided by what R not R squ just R say it again x x x yeah X

C now this can be positive or negative believe it or not the energy does not have to

be positive for example supposing this particle over here is at rest I don't know how it got there it

got there it's an initial condition it got here at at some time T it's at rest but at a positive value of

x x is really always positive it really stands stands for the distance from this from the earth not from the uh not the x

coordinate so X is always positive this is always negative this can be zero if the uh if

the particle is at rest and so the energy is negative in that case the energy can also be

positive supposing we now take the same particle at the same position but give it a velocity if the velocity is big

enough then this can outweigh that this can outweigh that simply when

if I write the equation down I'll write it down in a minute when this is bigger than this when the kinetic energy is

bigger than the potential energy and then the total energy is positive now if the

total energy is positive this thing cannot turn around it cannot you might say well let's see

this particle could go out and turn around why can't it turn around if the

total energy is positive incidentally energy of course is conserved so whatever the energy is at one instant

it's the energy at every instant energy is conserved let's suppose it turned around

at that point what would its velocity be at that point zero so what would its energy

be negative right so therefore if it turns around it's negative the energy is negative if it

doesn't turn around the energy is positive energy equals zero is a sort of edge of a parameter

space if the energy is positive the particle just keeps going and going and going it

escapes if the energy is zero that's exactly the escape Velocity we'll ask later whether it escapes or not if it's

exactly at zero what is the escape Velocity the escape Velocity is the solution of the equation

that this is equal to zero so let's write it out 12 v^2 I'm dropping the M because it cancels from both

sides to Little M 12 V ^2 is equal to Big M Big G divided by X and now we can just

multiply by two and that gives us a formula for the escape Velocity that's the formula for

the escape Velocity when the energy is exactly equal to zero in exactly the same

manner the universe can be above the escape Velocity below the escape Velocity or at

the escape Velocity we we're going to work that out in a

minute but all it means is if it's above the escape Velocity it means that initially at some point the outward

expansion was large enough that it doesn't turn around if it's below the escape Velocity then the universe turns

around and Rec contracts okay so that's the main reason for showing you this and the escape

Velocity is kind of the edge the escape Velocity is also the velocity at which the energy is equal to zero keep that in

mind escape velocity same thing as energy equal to zero all right now let's

concentrate on this particle over here now for all practical purposes this particle over

here all it knows is that it's moving in the gravitational field of a point mass at the center where the point mass is

capital M so for all practical purposes we can replace this problem over

here by this one over here in fact it's exactly the same

problem so let's work out the energetics the kinetic the potential energy of this

particle and keep in mind that it's constant it's constant because for all practical purposes this particle is

moving a exactly as it would be if all there was was a mass at the center and in that case energy would be constant so

we can just lift the things that I wrote before but let's uh let's work them out can I a Qui question yeah isn't a

capital m in this case changing As you move out further and further outside this continuous density no no uh no the

whole the whole thing is growing but the but remember the grid everything moves with the grid everything moves to the

grid the only thing that's changing is a the amount of mass in this sphere stays fixed in other words the number of

galaxies that this fellow over here sees within this sphere stays fixed right good okay so no so we don't have to

worry about the mass changing all right let's work out now the energy the kinetic well the kinetic plus potential

energy in the in Newton's frame in Newton's frame we'll work out the K all right so what is it 1 12 mv^2 again 12

the mass of this galaxy times the velocity squared but that's a do squar R 2 right same R where is

it same R uh same yeah same r d is equal to a * R distance is a * R velocity is a do * are this is 12 mv^

2 and then minus Little M Big M G divided

by distance right just divided by distance that's the potential energy m m mg and what is the distance the distance

is a Time R right let's do the and that's equal the

energy of this uh this guy here that's its energy now for Simplicity and because

it's uh because of Simplicity and also because I'm getting a little tired I think I will just do tonight the case

where the energy is exactly equal to zero what does that correspond to exactly the critical escape velocity

that case the other case is just as easy but let's um let's do that case all right

so that's the case where the universe is sort of just on the edge not clear whether it's going to

turn around and fall back or whether it's going to keep going uh and it's on the edge of the the

cusp of doing one or the other all right so we're going to set this equal to zero let's work out that equation let's

work out that equation using the various things we know okay first thing to do

is to get rid of the Little M here why should we get rid of the Little M because it appears in both terms here

and the whole thing's equal to zero so I divide it out I'll also multiply by

two next I'm going to divide by R 2 you know why why am I dividing by r s I want to get R cubed down here because I know

that R cubed is has to do with a volume and the volume I'm going to get a density I'm I'm trying to get this thing

in terms of density all right so I divide by R SAR and and that gives me an R Cub

downstairs that's nice because there's a mass here and an R cubed downstairs it looks like I'm getting the density but

not quite because the volume of the sphere is a cub * R cub not a * R cubed so what do I do I just divide the

equation by another a s a cubed okay

that's good ah a cub * R

Cub a cub * R Cub what do I do next well if I'm smart I will multiply this by 4 over 3 * pi

that'll literally make this a volume but I'm doing something illegal unless I multiply it here also 4 over 3

* pi equals zero right equals zero we're almost there

let me rewrite it a DOT over a squared remember what a do over a is

it's the Hubble constant so this is the square of the I take it back it's not it's not a constant the Hubble

thingy a DOT over a squ that's the Hubble thingy squared and that's equal to I'm just

transposing this to the right hand side there's an 8 pi 3 famous 8 2 * 4 is 8 8 piun over

3 there's a g and now there's an M divided by the volume of the

sphere that's why that's why I went to this effort here to put another couple of factors of A and R downstairs so that

I would get a divided by the volume of the sphere and that's row that's the mass density Row the actual mass

density a do over a^ 2 = 8 piun / 3 G * row that is the Freedman equation that's the freedoman

equation the way that it's usually written it's equivalent to this equation this one over here is energy

conservation also set the energy equal to zero this one over here is Newton's equations but the same physics the same

physics the Newton version of it the conservation of energy version of it this one is more

useful as I said it's called the Freedman equation it's not completely General

because we did set the energy to zero we did set it to just exactly the critical escape

velocity so this universe is not going to recollapse but it's

gonna what what does happen if you shoot something out at exactly the escape Velocity what happens to its motion as

time goes on doesn't it slow to zero at Infinity yeah it just asymptotically gets slower and slower and slower but it

never turns around this universe will ASM totically get get slower and slower and slower in its expansion but never

turn around for the same reasons okay so that's our that's our Freedman

equation I'd like to solve it but I don't know enough yet the reason I don't know enough is because there's row over

here and I don't know what to do with row except we do know what to do with row remember the equation that

row is equal to the constant new incidentally the constant new can be said to be anything you want it does it

doesn't it's not it's yeah okay it's the mass per unit coordinate volume by changing your coordinates you can change

the amount of mass that's in your in a coordinate volume so actually new never really comes into anything important but

row is equal to new / a cubed remember that okay so we can now write an even more useful version of this a do a

2ar is equal to 8 piun over 3 G new and new is a constant new does not

change with time divided by a cubed I think we have it right all of this junk here is just a

constant 8 Pi new over 3 * G is just a constant in fact I could if I like have chosen new so that 8 Pi go new over 3 is

just the number one there's not nothing interesting in it the the basic equation the basic equation the basic

form of the equation is just that a DOT over a squared is equal to some constant but

let's just choose that constant to be one just for Simplicity is 1/ a cub if we can solve this equation we can if we

can solve this equation we can solve this one it's uh not hard to go from one to the other

so we'd like to see how to solve this equation now notice first of all that the right hand side is always

positive in fact it never quite goes to zero no matter how big a gets it's always

positive as a gets really really big it gets smaller and smaller so that tells

us that a do over a never becomes equal to zero a do equals a do equals z would be the universe turning around that

would be the place where the universe turned around when the when the expansion rate inre went to zero so this

tells us the expansion rate never goes to zero Hubble constant never changes sign or at least the square of the

Hubble constant never goes to zero so if it doesn't go to zero it can't change sign and but it does slow down the

Hubble constant gets smaller and smaller and smaller with time so it's as if the universe just gets tired of

expanding but it never gets tired enough to stop okay we can try to solve this

um I think I will just take the it's getting late and I get tired about this time so I will take the easy way of

solving it but we will come back to these kind of equations we'll come back to these kind and this this type of

equation this is absolutely when I say not this type of equation this type of equation is

absolutely Central to all of cosmology and we can solve them we can solve them quite easily let's just look for a

solution of a particular type okay or look for a solution rather than to solve the equations let's look see if we can

find a solution where a is some constant times time to some power we don't know that that's a that

that that we can solve it this way but we can try we can take a trial solution a equals a proportional to T would just

what would a proportional to T mean that would just mean a grows in proportion to time in a very simple way we don't

expect that to be right and it's not because the thing slows down but we can look for a solution of this type so

let's try it out let's see if we can if we can use the equation to see what whether we can solve for C and

P okay so what's a DOT a DOT is c p t to the P minus1 right that's just

differentiation now a do over a that's easy we just have to divide by a so we have to divide this by c t to the

P C's cancel neat the constant here cancels and what's t to the p minus1 over t to the p

p over T right that's the left hand side p over T oh sorry we have to square

it this p^2 over t^2 that's the left side

p^2 sorry p^2 over t^2 now what about 1/ a cub let's see what that

is 1 over a cubed that's 1 / C cubed t to the 3

p do I have that right I do all right now we can read off how to match the two sides let's get rid

of this over here and let's match the two sides in the denominator we have a power

we also have a power over here this is 1 over T ^2 this is 1 over T to 3 p but I haven't told you what p is yet so if we

want to match I've just said let the let's look for a solution of the form CT to the

p and see if we can figure out what C and P have to be well the first thing we learned

is that 3 p had better equal two otherwise these things can't match there's no way that t to 4th here can

match t^2 here so the first thing we learn is that 3 p has to equal 2 we'll come back to it in a

minute all right that will guarantee that the t^2 and the t^2 agree on this side on the other hand we also have to

match the constant and the constant tells us that

p^2 = 1 / C Cub so that tells us there was really only one constant that we had to worry

about either P or C once we know p and we do know P we know P from here and therefore we know