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.
