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 sounds 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 the
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 oh point three 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 oh point three 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.
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 fre- 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 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 s- 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 -- 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, and 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 ten billion 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 Humason 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 t- 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 w- 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 T H 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.
At MIT we have three world experts, Professors Alan Guth,
w- made major contributions to this concept,
cosmology, we have Ed Bertschinger and we have Scott
Burles. If we take Hubble's law at face
value, I can calculate how far the edge of our visible universe
is. Which is the horizon.
We call that the horizon. I can calculate what the
maximum distance is that we can look.
D maximum can be found by making the velocity C.
So that the galaxies are moving -- we are moving away from the
galaxies, the galaxies are moving away from us -- with the
speed of light. And so you would find then that
D max is C divided by H.
That is a distance. And you will find then,
no surprise, if you use the modern number,
that that distance is fourteen billion light-years.
We can never see beyond that. Because if V equals C then beta
becomes one and if beta becomes one lambda prime becomes
infinitely large, you have an infinite amount
of red shift, and F prime becomes zero.
So the electromagnetic radiation has no frequency
anymore and so there's no energy anymore in the photons.
So that is then the edge of our universe, of our visible
universe. You can never see beyond that.
So now comes a reasonable question.
How far have we been able to see into the universe?
And to my knowledge 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 -- the distance.
But there is no question that the -- that object is probably
at a distance of something like thirteen billion light-years.
Very very far away from us, near the edge of our universe.
I will show you an object that is also believed to be near the
edge of the universe. It comes up in the next slide.
The distance is roughly twelve billion light-years.
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 that is twelve billion light-years away from us?
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,
our eyes I think can only see up to sixty-five hundred.
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, and there is little doubt that this object is also
near the edge of our visible universe.
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 its 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 galaxy.
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 both Allen Guth and Ed
Bertschinger and also Scott Burles are the experts in this
area and they happen to be one of our best
teachers. So you can't lose there.
Now comes a key question and that is will our universe expand
forever? If the universe expands
forever, we call that an open universe, that's just a name.
It's also possible that our universe will come 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. And the answer to whether or
not our universe is open or closed would then depend on 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
megaparsecs. 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 H zero 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, the universe would be open.
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 tomatoes or
potatoes. It could be electromagnetic
radiation. Because according to Einstein E
equals MC squared. So any form of energy
represents mass. 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,
whether it's open or whether it is closed, is something that's
emotionally an important issue for us.
If the universe is open and it will expand forever then stars
will all burn out and the universe will
become a cold, dead and boring place.
If however the universe is closed, the expansion will come
to a halt, it 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 be closed, 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 it's open 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 eight oh two reunion.
Same place same time same people, perhaps see you then.