Today I want to talk with you about Doppler
effect, and I will start with the Doppler effect of sound
which many of you perhaps
remember from your high school physics.
If a source of sound moves towards you or
if you move towards a source of sound,
you hear an increase in the pitch.
And if you move away from each other you hear
a decrease of a pitch.
Let this be the transmitter of sound,
and this is the receiver of sound,
it could be you, your ears
And suppose this is the velocity of the transmitter
and this is the velocity of the receiver.
And v should be larger than zero if the velocity
is in this direction.
And in the equations what follow, smaller
than zero it is in this direction.
The frequency that the receiver will experience,
will hear if you like that word,
that frequency I call f prime.
And f is the frequency as it is transmitted
by the transmitter.
And that f prime is f times the speed of sound
minus v receiver
divided by the speed of sound
minus v of the transmitter
So this is known as the Doppler shift equation.
If you have Volume one of Giancoli you can
look it up there as well.
Suppose you are not moving at all.
You are sitting still.
So v receiver is zero.
But I move towards you
with one meter per second.
If I move towards you then f prime will be
larger than f.
If I move away from you with one meter per
second then f prime will be smaller than f.
The speed of sound is three hundred forty
meters per second.
So if f which is the frequency that I will
produce is four thousand hertz,
then if I move to you with one meter per second,
which I'm going to try to do,
then the frequency that you will experience
is about four thousand twelve hertz.
It's up by 0.3 percent.
Which is that ratio one divided by three forty.
And if I move away from you with
one meter per second,
then the frequency that you will hear
is about twelve hertz lower.
So you hear a lower pitch.
About 0.3 percent lower.
I have here a tuning fork.
Tuning fork is four thousand hertz.
I will bang it and I will try to move my hand
towards you one meter per second roughly.
That's what I calculated it roughly is.
Move it away from you, towards you, away from
you, as long as the sound lasts.
You will hear the pitch change from four thousand
twelve to three thousand nine hundred eight-eight.
Very noticeable.
(sound from the tuning fork)
Have you heard it? Who has heard clearly the
Doppler shift, raise your hands, please? OK.
Chee chee chee chee it's very clear.
Increased frequency and then when I move
my hands away a lower pitch.
Now you may think that it makes no difference
whether I move towards you,
or whether you move towards me.
And that is indeed true if the speeds are
very small compared to the speed of sound.
But it is not true anymore when we approach
the speed of sound.
As an example, if you move away from me with
the speed of sound, you will never hear me.
Because the sound will never catch up with
you, and so f prime is zero.
And you can indeed confirm that
with this equation.
But if I moved away from you
with the speed of sound,
for sure the sound will reach with you.
And the frequency that you will hear is only
half of the one that I produce.
So there's a huge asymmetry.
Big difference whether I move
or whether you move.
So I now want to turn towards
electromagnetic radiation.
There is also a Doppler shift
in electromagnetic radiation.
If you see a traffic light red and you approach
it with high enough speed,
you will experience a higher frequency
and then you will see the
wavelengths shorter than red and
and you may even think it's green.
You may even go through that traffic light.
To calculate the proper relation between f
prime and f requires special relativity.
And so I will give you the final result.
f prime is the one that you receive.
f is the one that is emitted by the transmitter.
And we get here then one minus beta divided
by one plus beta to the power one-half.
And beta is v over c,
c being the speed of
light, and v being the speed,
the relative speed between
the transmitter and you.
If beta is larger than zero,
you are receding
from each other in this equation.
If beta is smaller than zero,
you are approaching each other.
You may wonder
why we don't make a distinction now between the transmitter on the one hand, the velocity,
and the receiver
on the other hand.
There's only one beta.
Well, that is typical for special relativity.
What counts is only relative motion.
There is no such thing as absolute motion.
The question are you moving relative to me
or I relative to you is an illegal question
in special relativity.
What counts is only relative motion.
If we are in vacuum then
lambda equals c divided by f,
and so lambda prime
equals c divided by f prime.
Lambda prime is now the wavelength
that you receive
and lambda is the wavelength
that was emitted by the source.
So I can substitute in here, in this f,
c divided by lambda
which is more commonly done.
So this Doppler shift equation for electromagnetic
radiation is more common given in terms of lambda.
But of course the two are identical.
And then you get now one plus beta upstairs
divided by one minus beta to the power one-half.
The velocity there if I'm completely honest
with you is the radial velocity.
If you are here
and here is the source of emission
and if the relative velocity
between the two of you were this,
then it is this component,
this angle is theta,
this component which is v cosine theta,
which we call the radial velocity,
that is really the velocity
which is in that equation.
Police cars measure your speed with radar.
They reflect the radar off your car
and they measure the change in frequency
as the radar is reflected
That gives a Doppler shift
because of your speed
and that's the way they determine
the speed of your car,
to a very high degree of
accuracy.
You can imagine that in astronomy Doppler
shift plays a key role.
Because we can measure the radial velocities
of stars relative to us.
Most stellar spectra show discrete frequencies,
discrete wavelength,
which result from atoms and molecules
in the atmosphere of the stars.
Last lecture I showed you
with your own gradings
a neon light source
and I convinced you that there were
discrete frequencies
and discrete wavelengths
emitted by the neon.
If a particular discrete wavelength,
for instance
in our own laboratory,
would be five thousand Angstrom,
I look at the star,
and I see that 
that wavelength is longer,
lambda prime is larger than lambda,
then I conclude --
lambda prime is larger than lambda,
that means the wavelength the way I observe it
is shifted
towards longer wavelength,
is shifted in the direction of the red,
and we call that red shift.
It means that we are
receding from each other.
If however I measure lambda prime
to be smaller than lambda,
so lambda prime smaller than lambda,
we call that blue shift
in astronomy,
and it means that we are
approaching each other
And so we make reference to the direction
in the spectrum where the lines are moving.
I can give you a simple example.
I looked up for the star Delta Leporis
what the red shift is.
There is a line that most stars
show in their spectrum
which is due to calcium,
it even has a particular name,
I think it's called the calcium K line,
but that's not so important, the name.
In our own laboratory lambda is known to a
high degree of accuracy,
is three nine three three
point six six four Angstroms.
We look at the star and
we recognize without a doubt
that that's due to calcium in the
atmosphere of the star
and we find that lambda
prime is one point two nine eight Angstroms
higher than lambda.
So lambda prime is larger than lambda.
So there is red shift and so
we are receding from each other.
I go to that equation.
I substitute lambda prime
and lambda in there,
and I find that beta equals
plus three point three times ten to the minus four.
The plus for beta indeed confirms
that we are receding,
that our relative velocity is away from each other,
and I find therefore
that the radial velocity
I stress it is the radial component
of our velocity
is then beta times c and that turns out
to be approximately
ninety-nine kilometers per second.
So I have measured now the relative velocity,
radial velocity, between the star and me,
and the question whether the star is moving
away from me or I move away from the star
is an irrelevant question, this is always
a relative velocity that matters.
How can I measure the wavelength shifts so
accurately that we can see the difference
of one point three Angstroms
out of four thousand?
The way that it's done is that you observe
the starlight and you make a spectrum and
at the same time
you make a spectrum of light sources in the laboratory
with well-known
and well-calibrated wavelength.
Suppose there were some neon in the atmosphere
of a star.
Then you could compare the neon light the
way we looked at it last lecture.
You could compare it with the wavelength that
you see from the star
and you can see very very small shifts.
You make a relative measurement.
So you need spectrometers
with very high spectral resolution.
So there was a big industry
in the early twentieth century
to measure these relative velocities of stars.
And their speeds were typically a hundred,
two hundred kilometers per second.
Not unlike the star
that I just calculated for you.
Some of those stars relative to us are approaching.
Other stars are receding in our galaxy.
But it was Slipher in the nineteen twenties
who observed the red shift of some nebulae
which were believed at the time to be in our
own galaxy and he found that they were
had a very high velocity of up to fifteen hundred
kilometers per second,
and they were always moving away from us.
And it was found shortly after that that these
nebulae were not in our own galaxy
but that they were galaxies in their own right.
So they were collections of about a hundred billions
stars just like our own galaxy.
And so when you take a spectrum of those galaxies,
then of course you get the average of millions
and millions of stars, but that still would
allow you then to calculate the red shift,
the average red shift, of the galaxy, and
therefore uh its velocity.
And Hubble, the famous astronomer after which
the Hubble space telescope is named,
and Humanson made a very courageous attempt to measure also the distance to these galaxies.
They knew the velocities.
That was easy because they knew the red shifts.
The distance determinations in astronomy is
a can of worms.
And I will spare you the details about the
distance determinations.
But Hubble made a spectacular discovery.
He found a linear relation between the velocity
and the distances.
And we know this as Hubble's law.
And Hubble's law is that the velocity is a
constant which is now named after Hubble,
capital H, times d.
And the modern value for H,
the modern value for H is seventy-two
kilometers per second per megaparsec.
What is a megaparsec?
A megaparsec is a distance.
In astronomy we don't deal with inches,
we don't deal with kilometers,
that is just not big enough,
we deal with parsecs and megaparsecs.
And one megaparsec is three point two six
times ten to the six light-years.
And if you want that in kilometers, it's not
unreasonable question,
it's about three point one times
ten to the nineteen kilometers.
So I could calculate for a specific galaxy
that I have in mind,
I can calculate the distance
if I know the red shift.
I have a particular galaxy in mind
for which lambda prime
for which lambda prime is
one point zero zero three three times lambda.
So notice again that the wavelength that I
receive is indeed longer than lambda,
so there is a red shift.
I go to my Doppler shift equation
which is this one.
I calculate beta.
One equation with one unknown,
can solve for beta.
And I find now that v is
five thousand kilometers per second.
Very straightforward, nothing special,
very easy calculation.
But now with Hubble's law
I can calculate what d is.
Because d now is the velocity which is five
thousand kilometers per second
divided by that seventy-two and that then is approximately sixty-nine megaparsec.
Again we have the distance if we do it in
these units in megaparsecs.
That's about two hundred and twenty-five
million light-years.
And so the object is about two hundred and
twenty-five million light-years away from us.
So it took the light two hundred and twenty-five
million years to reach us.
So when you see light from this object you're
looking back in time.
And if you have a galaxy which is twice as
far away as this one,
then the velocity would be twice as high.
And they're always receding relative to us.
I'd like to show you now some
spectra of three galaxies.
Can I have the first slide, John?
All right, you see here a galaxy
and here you see the spectrum of that galaxy.
That may not be very impressive to you.
The lines that are being recognized
to be due to calcium K
and calcium eight are these two dark lines.
Some of you may not even be able to see them.
And this is the comparison spectra taken in
the laboratory.
These lines are seen as dark lines,
not as bright lines.
We call them absorption lines.
They are formed in the atmosphere of the star.
Why they show up as dark lines and not as
bright lines is not important now.
I don't want to go into that.
That's too much astronomy.
But they are lines and that's what counts.
And these lines are shifted towards the red
part of the spectrum by a teeny weeny little bit.
You see here this little arrow.
And the conclusion then is that in this case
the velocity of that galaxy is
seven hundred twenty miles per second which translates into eleven hundred fifty kilometers per second,
and so that brings this object if you believe
the modern value for Hubble constant
at about sixteen megaparsec.
This galaxy is substantially farther away.
No surprise that it therefore also looks smaller
in size, and notice that here the lines have shifted.
These lines have shifted substantially further.
And if I did my homework
using the velocity
that they claim,
which they can do with
high degree of accuracy
because you can calculate
lambda prime divided by lambda,
those measurements can be
made with enorm accuracy,
I find that this object is about three hundred five megaparsecs away from us,
so that's about twenty times
further away than this object.
So the speed is also about twenty times higher
of course because there's a linear relationship.
And if you look at this one which is even
further away, then notice that these lines
have shifted even more.
The next slide shows you what I would call
Hubble diagram.
It was kindly sent to me by Wendy Freedman
and her coworkers.
Wendy is the leader of a
large team of scientists
who are making observations with
the Hubble space telescope.
You see here distance
and you see here velocity
in the units that we used in class,
kilometers per second.
Forget this part.
That's not so important.
But you see the incredible
linear relationship.
And Wendy concluded that Hubble's constant
is around seventy-two.
It could be a little lower,
it could be a little higher.
She goes out all the way to
four hundred megaparsecs
with associated velocities of about
twenty-six thousand kilometers per second.
That's about nine percent
of the speed of light.
So beta is about one-tenth.
So for this object lambda prime divided by
lambda would be about one point one.
With a ten percent shift in the wavelength.
Hubble, who published his data in the twenties,
his whole data set when he concluded
that there was a linear relation
had only objects with velocities less than
eleven hundred kilometers per second.
And eleven hundred kilometers per second
is this point here.
So Hubble had only points -- there are not
even any in Wendy's diagram, which are here.
And he concluded courageously that there was
this linear relationship.
And you see it has stood the acid test.
We still believe it is linear.
The only difference was that Hubble's distances
were very different from what we believe today.
They were about seven times smaller.
So Hubble constant was different for him but
the linear relationship was there.
OK, that's enough for this slide.
So now comes a sixty-four dollar question,
why do all galaxies which are far away,
why do they move away from us?
Well, I can uh suggest
a very simple picture to you.
We are at the center of the universe and there
was a huge explosion a long time ago.
We refer to that explosion as the Big Bang.
And since we are at the center
where the explosion occurred,
the galaxies which obtained the largest speed
in the explosion are now the
farthest away from us.
Now assume that this explosion
is the correct idea.
Assume that there was a Big Bang.
Then I can ask the question now
when did it occur?
I can now turn the clock back
and I can do the following.
I can take two objects which are a distance
d apart today
but they were together when
the universe was born at the Big Bang.
And let's assume that they have been going
away from each other
always with the same velocity.
Let's assume that now for simplicity.
So if they always went away with the same
velocity from each other
then the distance that they are now today
is their velocity times the time t
which is then the age of the universe.
But we also know with Hubble's law that the
velocity v is H times d.
And we assume that these velocities are the
same now for simplicity.
You multiply these two equations with each
other and you find immediately that the age
of the universe is one over H.
And that indeed has the unit of time.
If you take H, the one that we believe in
nowadays, and you calculate one over H,
and you work in M K S units, you'll find that
Tage is about fourteen billion years,
I'll first give it to you in seconds,
it's about four point three times
ten to the seventeen seconds
And that is about fourteen billion years.
So with this picture in mind, the universe
would be about fourteen billion years old,
but because of the gravitational attraction
of these galaxies, they attract each other,
you may expect that the speed of the galaxies
was larger in the past,
and therefore the speed that we have
we assume that the speed
doesn't change is not quite accurate,
and so maybe the universe is a little younger,
maybe twelve billion years or so.
We know from theoretical calculations that
the oldest stars in our own galaxy
are about ten billion years old.
Therefore the universe cannot be younger than
ten billion years.
And there is general consensus
in the community
that our universe is probably
twelve to fourteen billion years old.
Now the whole issue of this deceleration that
I mentioned
as the galaxies moved away from each other
is at the heart of research in
cosmology.
And in fact it is now believed that very early
on in the universe there was first acceleration
followed by deceleration,
and maybe again acceleration.
That is quite mysterious.
Frontier research is going on in this area.
And at MIT we have three world experts,
Professors Alan Guth, who made major
contributions to this concept cosmology,
we have Max Tegmark
and we have Scott Hughes.
So now comes a reasonable question.
How far have we been able
to see into the universe?
And to my knowledge, this year,
which means in 2002,
the record holder
is a galaxy for which lambda prime over lambda
is seven point five six.
Was published only two months ago.
Seven point five six.
Now at such very large
values of red shift,
general relativity becomes very important.
And the equation that we derived here was
derived for special relativity.
And so with very high
values of red shift
like lambda prime over lambda
seven point five six,
you cannot reliably calculate the
velocities using that equation.
And so you cannot use that velocity
then and shove it into
Hubble's law and find the distance.
But there is no question that that
object emitted the light that we see now
about thirteen billion years ago.
It's very very far away from us.
I will show you an object,
It comes up in the next slide.
The light that we see now was emitted
about twelve billion years ago.
So for one,
when you look at that object,
there it is,
it doesn't look very impressive
but what do you expect
from an object so far away?
It's a quasar,
which is a very peculiar galaxy.
It uh emits emission lines,
the spectra do
not show these dark lines
that I showed you earlier,
but they actually have emission lines,
and the light that you see here
was emitted some twelve billion years ago.
And now comes the spectrum from this object
in the next slide.
This was published last year
by Scott Anderson
and his coworkers,
University of Washington in Seattle.
I have collaborated with Scott
on many projects.
So here you see the spectrum of that quasar
that you just saw.
And here you see a line,
an emission line,
at roughly seven --
seventy-eight hundred Angstroms.
And there are all reasons
to believe that this
in the frame of reference of that quasar
was the Lyman alpha line which
is emitted by hydrogen,
which is twelve hundred
and sixteen Angstroms.
Now we have here five thousand, four thousand,
three thousand, two thousand, one thousand,
so here is roughly where
the wavelength lambda is,
and here is lambda prime.
Lambda prime is six point four one
times larger than lambda.
He mentions five point four one,
but Z,
is what astronomers
in general quote
is lambda prime divided by
lambda minus one,
so the ratio lambda prime over lambda
is six point four one.
Absolutely amazing that you can make such
accurate measurements,
such incredible beautiful data,
and this line is all the way
in the infrared,
you cannot see this
with your naked eye anymore,
you can see up to about 7000 Angstrom.
So the twelve sixteen line was in the UV,
shifts all the way into the infrared,
and this allows astronomers then to measure the
value lambda prime over lambda,
galaxies that we see now
those that emitted their light about
twelve to thirtheen billion years ago,
are now at a distance from us
which is about twice as far away
as twelve to thirteen billion light years,
thus they are at about
twenty five billion light years away
that's the result of
the expansion of the universe
yet the age of our universe
is very close to one
divided by the Hubble constant,
which is about fourteen bilion years.
I calculated that by assuming for simplicity
that the expansion of the universe was constant in time
the reason why the fourteen billion years
is still quit accurate
is an accident.
It turns out that the acceleration
and the decelleration of space
approximately cancelled each other out.
The horizon of our universe,
is at an even larger distance
than twenty five billion light years.
It's about forty five billion light years
beyond that distance radiation
hasn't had time to reach us.
That's enough, John, thank you.
I'd like to return to the Big Bang,
to the explosion some
twelve or fifteen billion years ago.
And I'd like to raise the question, are we
at the center of that explosion?
Are we really at the center of our universe?
That cannot
be of course.
It's an incredible arrogance.
It would be too egocentric.
I know that we all think very highly of ourselves,
but this cannot be.
We are nothing in the framework
of the total universe.
We cannot possibly be at the center.
So how do we reconcile this now
with what we observe?
Imagine that you were a raisin in a raisin bread.
Quite a promotion, from a human being to a
raisin in a raisin bread.
And I put you in an oven.
And the raisin bread,
the dough is going to expand.
All raisins will see other raisins
move away from each other,
and the larger the distance to your raisins
the larger the speed will be.
And each raisin will think that
they are very special.
Suppose here this is you, one raisin,
and here's another raisin,
and here's another raisin.
After a certain amount of time all distances
have doubled.
So this one is here.
And this one is here.
So you can immediately see
that when you look at this one,
that it's velocity is substantially
lower than that one.
This is twice as far away,
you will see twice as high a speed.
But this raisin will look at this one.
And it will also conclude that this raisin
relative to this one has a higher velocity
than this raisin has relative to this one.
So all of them will think that they are special
and you as a raisin would
come up with Hubble's law.
You would conclude that the velocity
of your other raisins
are linearly proportional to the distance.
There is an analogy which is
even nicer than raisin bread,
and that analogy is
with Flatlanders.
A Flatlander is someone who lives
on a two-dimensional world.
He happens to live on the
surface of a balloon.
And light travels only along
the surface of the balloon.
So the two-dimensional world is curved in
the third dimension,
but the Flatlanders cannot
see in the third dimension.
They can only see
the second dimension.
So here you have such a world.
So here are the galaxies.
Flat world.
And the universe is curved in the third dimension
which these Flatlanders cannot see.
And when you blow this balloon up,
the galaxies
move away from each other,
and the farther the galaxies
are away from each other,
the higher the velocity.
This model works actually quite well
and I want to pursue that
in my next calculations.
Let me first try to bring this
universe to a halt.
Because I don't want the universe to collapse
again.
Ooh.
OK.
I succeeded.
So you can pursue this idea very nicely
and you can see that
the Flatlanders would draw
quite amazing conclusions.
Here is that balloon.
The balloon has a radius R.
Here is one galaxy.
And here is another galaxy.
And they are a distance S apart.
I will call that later d.
But now I want to call it S.
You will see why.
A little later in time,
the universe has expanded,
this galaxy is here
and this galaxy is here.
And this distance now is R plus dR
and so this distance now between
the two galaxies is S plus dS.
And it follows immediately
from the geometry
that S plus dS divided by S
is R plus dR divided by R.
Simple high school geometry.
I can work this out.
I get S R plus R dS
is SR plus S dR.
I lose this SR.
I divide by dT.
dS dT is the velocity with which these two
galaxies move away from each other.
That's they what they would
measure in their universe.
So there is a v here.
It's clear that S is the
distance between them.
I will call that d again now.
So that is d.
And then I have one over R times dR dT.
One over R
I will write this a little higher.
dR over R.
No no no we had dR dT.
So now I have one over R dR dT.
And look at this.
I have v equals d times something.
And that something at a given moment
in time has a unique value.
R of the balloon has a unique value.
And dR dT which is the expansion velocity
also has a unique value.
And so it's immediately obvious that in this
universe this is Hubble's constant.
And this Hubble's constant is 
a function of time.
It is changing with time.
And it's obvious that it should
change in time.
No reason why it shouldn't do the same in
our own universe.
Because R in the past
was much smaller.
So even if you take an expansion velocity
which is constant,
if R is smaller in the past,
then H was larger in the past.
And that is the reason why if you
ever see a quote of H
to be seventy-two kilometers
per second per megaparsec,
there's always
a little zero here.
And the zero means now.
The zero means not a billion years from now
and not a billion years ago.
We really don't know what it was
a billion years ago.
Now don't get --
don't carry this analogy
between the two-D balloon and the --
our own universe too far.
But it gives some interesting insights.
It is suggestive of the idea that our own
three-dimensional space
may be curved in the fourth
dimension that we cannot see.
This is very fascinating
and I would advise
you if you are interested in this area
that you take a course in cosmology.
You should also take one in general relativity.
It will open a whole new world for you.
And Allen Guth, Scott Hughes and
Max Tegmark are the experts in this area
and they also happen to be
very good teachers.
So you can't lose there.
Now comes a key question,
and that is will
our universe expand forever?
It's also possible that our universe
will comes to a halt.
That means that H, Hubble's constant, will
become zero, that everything will stand still,
no relative motion anymore, which then will
be followed by collapse.
And so all the red shifts will then come to
zero and will turn to blue shifts.
It's the same idea, the same question, when
you throw up an apple,
will the apple come back
or will the apple not come back.
It depends on the speed of the apple and on
the gravitational field of the earth,
and we all know that if
you throw it fast enough,
about eleven kilometers per second
in the absence of atmosphere,
the apple would never
come back.
Now if only gravity played
the key role in our universe,
then we can do
a very simple calculation.
The answer to the question
whether our universe will expand for ever
or whether it will collapse
would then only depend on the
the average density of the universe.
And when I say average density then you have
to think in terms of a big scale.
You don't think in terms of Cambridge.
That's not representative for the average
density of the universe.
Nor is our solar system.
Nor is our galaxy.
But you have to think probably on the scale
of a few hundred million parsecs.
Maybe five hundred megaparsecs.
And so I bring you
out now into the universe.
Here is the universe.
And these are galaxies.
And here is a sphere
which has a radius R
and that's on a scale of
about five hundred megaparsesc.
So rho, the average density,
is representative for the universe.
And here let's suppose you were here,
or I can take any part in the universe,
there's nothing special about it,
and you see here a galaxy and that galaxy moves
away from you with a velocity v.
That galaxy has a mass little m.
The mass inside here, capital M
inside this sphere,
is four-thirds pi R cubed times rho.
It's the average density, right?
Now we know from Newton that the
force that this galaxy will experience
is only determined by the
mass inside this sphere
and not by the mass
outside the sphere.
And so if I want to calculate whether these
two objects will forever move away from each other
or whether they will fall back to each
other then all I have to make sure that I
make the total energy zero,
the sum of the kinetic energy
and the potential energy must be zero.
So one-half mv squared of this object,
it must be m M G divided by R.
That is when the total energy is zero.
We will expand forever and ever and ever
and it will never come back.
Little m cancels out.
Capital M I can write
four-thirds pi R cubed rho.
Here comes my G and here comes R.
Notice that the R cubed upstairs
becomes R squared.
And so if I have an R squared here
and I have a v squared here,
remember that v divided by R,
that is Hubble's constant.
Because R is D, it's the distance between
us and the galaxy.
And so v squared divided by R squared is the
Hubble constant as we measure it today squared.
And so you'll find then from this simple result
that rho as it should be today,
that's why I put a little zero there,
is three divided by eight pi --
I get a G there, and I get Hzero squared.
And so this tells me that if the density,
the average density of our universe,
is larger than this value, then our universe
will come to a halt and will collapse.
And we can calculate that value.
Because we know H zero, we think we know,
we know G, and so you will find then
I'll write it down here,
that rho zero is about
ten to the minus twenty-six kilograms per cubic meter.
And so if rho is smaller than this amount
then we will continue to expand forever.
If the mean density right now is larger than
that amount, then we will
the expansion will come to a halt,
red shift will become
blue shifts, and we will collapse again.
The matter here, this matter density,
doesn't have to be galaxies or gass
or tomatoes or potatoes.
It could be dark matter.
So don't think of it necessarily as this being
the stars and galaxies and tomatoes.
It is generally believed today that the expansion
of our universe will not come to a halt and collapse.
But our views could change.
Enormous development has been going on in
the last ten years
and you can read about that
in the New York Times.
Almost every month
you will read something
about the enormous progress
that's being made in cosmology.
And of course the idea of whether or not the
universe will expand forever,
is something that's emotionally
an important issue for us.
If the universe will expand
forever then stars will all burn out and the
universe will become a cold,
dead and boring place.
On the other hand, if the expansion
will come to a halt,
the universe will collapse.
And it will end up with what we call the Big Crunch
as opposed to the Big Bang.
And it will be hot, there will be fireworks,
it will be like the early days of the Big Bang.
Temperatures of billions of degrees.
I'd like to read a poem from Robert Frost
which he wrote in nineteen twenty.
It's called Fire and Ice.
Some say the world will end in fire,
some say in ice.
From what I've tasted of desire,
I hold with those who favor fire.
But if it had to perish twice,
I think I know
enough of hate
to know that for destruction ice is also great.
And would suffice.
There are many people who want our universe
to come to a halt and collapse.
Probably for emotional reasons,
maybe for religious reasons,
maybe it's more esthetic,
maybe it's more reassuring,
maybe it's more romantic.
I don't know.
But if that is not the case than
the end is not very spectacular.
T. S. Eliot wrote, "This is the way the world ends
not with a bang but a whimper."
Now it is conceivable that the expansion
of the universe will come to a halt
and that the universe
will ultimately collapse.
We will have a big crunch.
And it is even conceivable that a new universe
will then be born afterwards.
That there will be a new Big Bang.
And if the evolution of that universe were
a carbon copy,
exact carbon copy of the present universe,
a few thousand billion years from
now we may have a great 8.02 reunion.
Same place same time same people,
perhaps see you then.
