Professor Charles
Bailyn: Okay,
welcome back for more
cosmology.
What I want to do today is
quickly review what we were
doing about magnitudes and make
a comment or two about the
problem set,
and then, go back and talk
about the implications of the
Hubble Law and the Hubble
Diagram,
which are formidable,
to put it mildly.
Okay, magnitudes.
There's a couple of these
magnitude equations.
I'm just going to write them
down.
The first of them looks like
this.
And this equation is used--okay.
So, this equation is used to
relate magnitudes of two
different objects to each other.
So, we've got two different
objects.
And it can be used for either
kind of magnitude--either
absolute or apparent magnitude,
just so long as you don't mix
them.
So, it's two different objects,
but only one of the magnitudes.
One kind of magnitude.
And depending on which kind of
magnitude you use,
this brightness ratio--it's
either the ratio of how bright
it looks or the ratio of how
bright it is--whatever's
appropriate.
Now, on the help sheet on the
web, I have this equation in a
somewhat different form,
and it's important to realize
that it's the exact same
equation.
Watch this.
Let's see.
Let's multiply both halves by -
2/5 which is - 0.4.
So, this is -0.4
(M_1 –
M_2) = log
(b_1 /
b_2).
And then, let's take 10 to the
power of that.
That gets rid of the log.
And this is now the form that
it is on the help sheet on the
web.
So, it's exactly the same
equation, just expressed
differently.
And you can use either form,
whichever is more convenient.
Okay.
The other equation looks like
this.
5 log (D/10 parsecs).
And this relates one object,
but it relates both kinds of
magnitude to each other.
So, the first one is two
different objects,
but only one of the magnitudes.
The other is one object and it
relates the two different kinds
of magnitudes to each other,
and to the distance to the
object.
And as you can see,
this--both of these equations,
actually, have three unknowns.
One, two, three.
That means you've got to know
two things in order to find out
the third.
And that brings me to the
comment I want to make about
problem 2a on the current
problem set.
You are asked in this problem
to determine the difference
between the absolute magnitude
of one kind of star,
called Type 1 Cepheid,
so I label it C1.
And another kind of star,
Type 2 Cepheids,
which I label C2.
And if you're asked to--and
this difference is called,
I don't know,
delta M_C or
something like that.
And having been asked to do
this, the logical thing that you
might try to do is say,
all right, I'm going to use one
or the other of these
equations--I'm not sure,
in advance, which,
to compute this one.
Then I'm going to compute this
one.
And I'm going to subtract the
two, and that's going to give me
the answer.
That approach will fail.
Okay?
That doesn't work in this
particular case,
because you don't actually have
enough information to compute
either one of these things.
You do have enough information
to compute the difference.
And let me just give you a very
brief hint on how you might go
about doing that.
Let's see.
Let me take a new piece of
paper here.
Write down
m_C1 -
M_C1 = 5 log
(D_C1 /
10parsecs).
And now write the exact same
equation down for C2,
where the two different
distances are the distances you
get by assuming one or the other
kinds of these magnitudes.
This is equation one.
This is equation two.
Now, let's subtract.
1 – 2.
So, then, you get
m_C1 -
m_C2 -
[M_C1 -
M_C2]
= 5 (log D_C1
- log D_C2).
Okay, now.
Here's the trick.
Turns out for reasons that you
had better tell me--and TFs
[teaching fellows],
take note that we really want
them to say why this is true,
now that I've told them it is.
This is zero.
The two apparent magnitudes are
the same.
And so, that means that this
side of the equation is what you
want.
It's the difference between
these two magnitudes.
And then, over here,
you have to use one of these
log rules: log (x) - log
(y),
if you remember back to
eleventh grade,
is log (x / y).
And if you use that,
it turns out that you have
information elsewhere in the
problem, which will tell you
what you need to know about the
distances.
And so, in this way,
you can solve for the
difference without actually
being able to determine either
one of these two things.
So, we'll just leave it at that
for the moment.
If you have problems there's
the usual forum,
there's the usual office hours,
but ponder this.
That's basically how the
problem has to go.
Okay?
Problems with magnitudes?
Okay.
If you do have some,
let us know,
because this is going to be
critical for solving,
basically, problems for the
whole rest of the class.
All right.
Yes go ahead.
Student: What is the
location--the last thing in
green-- Professor Charles
Bailyn: This?
Student: --5 log--is it
over log?
Professor Charles
Bailyn: This is log
(D_C1 / 10
parsecs) - log
(D_C2 / 10
parsecs).
That's just subtracting the two
right-hand sides of this
equation.
But then, you get to do the log
thing and divide them instead,
which is this to this on this
log subtraction.
Okay.
Let me remind you why we're
putting ourselves through this
pain.
Okay?
Recall why we started doing
this in the first place.
The goal was to figure out how
to measure the Hubble Diagram.
The Hubble Diagram is this
diagram of velocity,
which you can measure by
redshift versus distance.
And the whole reason we
embarked on this adventure in
magnitudes was because that's a
critical component in how you
determine the distance.
But if you've got a bunch of
galaxies and you measure these
two quantities for each one of
them,
what you discover--what Hubble
discovered--what Edwin Hubble
discovered many years ago,
is that they line up.
You get this perfect;
well, not quite perfect,
but close to it--this beautiful
straight line,
if you measure a bunch of these
things.
And the way you represent that
straight line is with this
equation, where H is
Hubble's Constant.
And so, that's the purpose of
all this magnitude stuff,
is to be able to determine the
y-axis of this plot.
What I now want to do is talk
about the implications of this
observational fact that galaxies
line up on this line.
It turns out,
this is one of the most
profound plots in all of
astrophysics,
and possibly all of science.
Because what this implies is,
first of all,
that the Universe is expanding,
and hence, it's the basis for
the whole Big Bang Theory of
cosmology.
And by performing relatively
simple calculations using this
quantity H,
you can determine the age of
the Universe,
and the ultimate fate of the
Universe.
Not bad for a relatively simple
algebraic equation.
Yes?
Student: [Inaudible]
Professor Charles
Bailyn: Huh?
Student: [Inaudible]
Professor Charles
Bailyn: Oh,
fate.
I'll get back to that.
The big question in cosmology
is, you know--the Universe is
now expanding.
The question is,
will it continue to expand,
in which case,
the Universe just gets sparse
and cold and boring,
and expands forever until
there's, you know,
one pathetic hydrogen atom
every cubic megaparsec of space.
Or, alternatively,
it could stop,
slow down, and recollapse into
something called a Big Crunch,
which is sort of the Big Bang
run backwards,
and basically the whole thing
turns into a massive black hole.
These are--you know,
this is ending in fire or in
ice, I guess.
And it can be computed in ways
we'll describe later.
Okay, so, here's what I want to
do.
I want to start understanding
how this plot and this little
equation gives you all these
wonderful things.
I'm going to go on for a little
while, then we'll pause,
and we'll do one of these Q
& A sessions,
because this is sort of the
heart of the Big Bang Theory.
And so, we'll do one of these
things that we did when we were
talking about relativity,
where you talk to each other
and come up with questions.
So, if you've got questions
along the way,
by all means,
ask them, but we will have a
specific moment a little ways
down the line where we actually
pause and do this on purpose.
So, everybody,
keep thinking as we go along,
what are your questions?
What don't you understand or
what questions could you ask to
understand more than what I've
just told you?
Okay.
Here we go.
Imagine a one-dimensional
Universe, just because it's easy
for me to write down.
And here's our one-dimensional
Universe.
It's all strung out on a line.
Here's the line.
And it's got a bunch of
galaxies on it.
Let's label these galaxies,
A, B,
C, D,
E, and F.
And these galaxies are spaced
evenly, let us imagine.
And we'll give them coordinates.
So this is at 0,1,
2, and so forth.
Okay?
Now, next thing we're going to
do: the Universe is going to
double in size.
So, we're just going to stretch
the thing.
The whole thing is going to get
stretched.
So, here's our Universe.
And now A,
B, C,
D, E and F
are further apart by a factor of
two.
A, B,
C, D, E,
F. And if A,
we imagine stays at the same
coordinate--if our coordinate
system starts with A,
this means B is now at 2.
C is 4--6,8 and 10.
Okay.
And let us imagine that it
takes one time unit--one year or
something like that,
for this doubling to take
place.
Now, we're going to ask,
if you sit--if you live in
galaxy A,
if you live on planet A,
and you observe the distance
and velocity of all these other
galaxies, what's it going to
look like?
So, observer on A.
And so, we're going to observe
a particular galaxy--one of
these other galaxies.
We're going to write down the
distance.
We'll choose the distance at
the start, because it's going to
change.
Then we're going to evaluate
how that galaxy has moved,
and then the distance changed.
And then, over here,
we're going to get the
velocity.
The velocity is going to be the
change in distance divided by
the change in time,
which we've defined to be one
time unit.
Okay?
So, galaxy B.
Galaxy B starts at a
distance of 1 away from us,
because it starts 1.
A starts at 0.
It moved from 1 to 2,
and that gives it a change in
distance of 1 and therefore its
velocity over this time,
it's changed in distance by 1,
it's taken 1 time unit.
The velocity is 1 divided by 1,
equals 1.
Okay?
The algebra is easy.
Okay, so, how about C?
C starts 2 away.
Its motion--it goes from 2 away
to 4 away, and so,
the change in its distance is
2.
And since it takes 1 time unit
to do, its velocity is 2/1,
which is equal to 2.
See, the algebra is simple,
but I screw it up.
Bethany is laughing at me and
well she might,
but I caught myself.
All right.
And so on down the line.
I could repeat this simple
exercise.
D, E,
F start at 3,4,
and 5.
Goes from 3 to 6,
from 4 to 8,
from 5 to 10,
and their velocities are 3,4,
and 5, respectively.
And so, if I plot distance
versus velocity,
I'll get points lined up just
like this.
And so, basically,
what happens is this.
If you take a set of points on
a coordinate system and you
simply stretch the coordinate
system,
what happens is that the
further away you start,
the greater the stretch is.
And so, there is a correlation
between how far away you start
and how fast the thing recedes
from you.
Now, this is true regardless of
which point you sit on.
Let's imagine that we sit on
point--that the observer is on
point E.
So, on point E, we're
now going to--which galaxy,
initial distance,
motion, velocity.
So, this is the same
plot--chart as before.
E--let's look at
F, starts at a distance
of 1 because it's 1 unit away.
The motion--okay,
at the start it's--let's take a
quick look at how this is set up
here.
Yes, E is at 4.
So, at the start,
the distance is between 4 and
5, which is 1.
And it goes to a distance of 8,
which is where E ends
up, and 10, which is where
F ends up.
So that is a change of 2.
No a change of 1, right?
Because it's gone from 1 away
to 2 away, and the velocity is
1.
If we look at D,
it starts at a distance of 1,
goes--starts at 4 to 3,
that's one separation.
Goes to 8 to 6,
that's a separation of 2.
Delta D is again equal
to 1.
Velocity is equal to 1.
Let's look, for example,
at B.
That starts 3 away.
And it starts from 4--B,
C, yeah--from 4 to 1.
And it goes from 8 to 2.
So, that's a difference of 6
here, a difference 3 there.
And so delta D is equal
to 3.
And so, the velocity is once
again 3.
And so, you get the exact same
plot with different galaxies,
because you're sitting in a
different place.
So, it doesn't matter which
galaxy you're sitting on.
You see the exact same ratio of
distance to velocity and you
create the exact same Hubble
flow no matter which galaxy you
sit on in this little toy
Universe.
So, that's the key point--that
if you take a coordinate system
and you expand it,
you naturally get this
relationship between distance
and velocity.
Or to turn it around,
if you observe this
relationship between distance
and velocity,
then what you're looking at is
a system in which all the
coordinates are--in which you've
simply stretched the coordinate
system.
Okay.
Now, this gives rise to--this
analogy, with these stretching
one-dimensional lines gives rise
to two questions people have,
which I like to call
un-questions,
because they're actually
questions that arise because of
the analogy,
not because of the way the
Universe works.
One question is,
Q1: "Where is the center?"
You know, here's your line.
It's expanding.
But somewhere in the middle
here, around C or
D in our thing,
is the center away from which
everything is expanding.
So that's one question.
And the second question is,
"What is it expanding into?"
You know, here you have a
little Universe and it's moving
outwards.
And so, what's going on over
here?
What was there before the
Universe moved into it?
These kinds of questions.
And those kinds of questions
come about because this is
actually a bad analogy--this
straight line Universe.
So, let me give you a slightly
better one.
We'll stick with the
one-dimensional Universe,
but now we'll do it this way.
Here's a one-dimensional
Universe.
You have to stay on the line.
So here's A,
B, C, D, E, F,
whatever.
And it's going to expand.
And it's going to expand into
something that looks like this:
A, B, C, D,
E, F. And all of what we
just did about the velocity,
and so forth,
remains the same.
But, notice that this system is
unbounded.
There's no edge.
There's no edge.
There's no place where you can
say, this is the end of the
Universe, because if you
traveled around it you'd just
come back to where you were,
and therefore,
there's also no center.
And where does it expand into?
It expands into a dimension
that, if you're a
one-dimensional creature,
you can't experience because
the whole thing is being pushed
out.
But if you're forced to live on
this circle you can't even--you
have no comprehension of what it
expands into.
It expands into a higher
dimension.
But all of this stuff about,
you know, velocity and
distance, remains basically the
same.
Here's a two-dimensional
analogy.
Let's see, this is--so,
I made a little diagram of the
Old Campus [an area of the Yale
campus].
Here's Linsley–Chittenden [a
classroom building]
where we're sitting right now,
yes?
Here's the statue of Abraham
Pierson.
This is the gate between Durfee
and Wright [two undergraduate
dorm buildings].
Here's Phelps Gate [a classroom
building and the entrance to Old
Campus].
Here's Vanderbilt [a dorm
building].
This is Harkness Tower and
here's Starbucks.
Okay, that's all that's
important, right?
So, you following me with that?
And what I did was,
I took this little picture and
I took it to the Xerox machine
and I blew it up by 20%.
So, here's the exact same
diagram blown up by 20%.
So, now, supposing we're
sitting in Linsley-Chittenden,
which we happen to be doing,
and the Universe expanded by
20%--or, our little corner of
the Universe expanded by 20%,
here's what would happen.
Now, notice what's happening.
Every object in the Universe is
moving away from us.
See?
Here's where Harkness was,
and now it's moved a little
further, in a straight line away
from us.
Here is Pierson,
and he's moved a little
further, straight away from us.
And here's Phelps,
and it's moved a little
further, straight away from us.
And let me erase those lines,
because what I want to
demonstrate is that if you're
anywhere else in this Universe,
the exact same thing happens.
Here we're now sitting on that
statue, and Linsley-Chittenden
is moving away from us.
Harkness is moving away from us.
Phelps is moving away from.
Starbucks is moving away from
us, and so forth.
Similarly, if you're sitting in
Starbucks, waiting for students
to come by or something,
the exact same thing happens.
And now, because the distances
are greater, you can see the
effect that the velocity is
greater at greater distances.
If I'm looking down at
Vanderbilt, it moves straight
away from me,
but only a little bit.
If I'm looking all the way
across Old Campus,
this gate moves a lot away from
me.
And so, once again,
you have a situation in which
the further away
some--everything is moving
straight away from you,
but the further away it is,
the faster it's moving away,
right?
And that's just a consequence
of the fact that you have taken
this geometry and expanded it.
And so, wherever you sit in an
expanding geometry,
every object you see will be
moving directly away from you.
And the further away it is,
the faster it will be moving,
which is Hubble's law.
Oh, and one other thing about
this nice analogy,
here.
Let us imagine for a second
that this tiny piece of a tiny
Universe is actually not a flat
plane,
but is sitting on a curved
surface, which is curved all the
way round into a big ball.
That's actually not so hard to
imagine because it's true.
This sits on the surface of the
Earth.
And so, what is happening when
this thing blows up by 20% is,
basically, somebody has taken a
valve to the Earth and has blown
the Earth up by a factor of 20%.
And that would have this effect.
And it would have the exact
same effect everywhere else on
the surface of the Earth.
And the Earth,
you know--where is the center
of the surface of the Earth?
You can answer the question:
"Where is the center of the
Earth?"
But you can't answer the
question of where is the center
of the surface of the Earth.
Because wherever you sit,
whether you're sitting at
Starbucks or in Phelps Gate or,
you know, in Los Angeles
somewhere or wherever,
if they blow the Earth up by
20% you're going to see this
exact same effect.
Everything will be moving away
from you.
The further away something is
the faster it will be moving.
So that's the one--yes,
go ahead.
Student: Someone in the
back apparently [inaudible]
Professor Charles
Bailyn: Yeah,
talk.
Student: If something's
far enough away from you would
it appear to be moving at the
speed of light?
Professor Charles
Bailyn: Yes,
yes, good question.
If something's far away from
you, will it appear to be moving
at the speed of light?
Yes, it will,
and that's one of the
fundamental differences between
the motion of an object due to
what's called the Hubble Flow,
due to the expansion,
and ordinary motion of objects.
Now, if something's moving
faster than the speed of light,
of course, you can't see it,
because the light from that is
redshifted down to greater than
infinite wavelengths.
So, the photons don't have any
energy left.
But, let me come back to the
question after I do one more
thing.
That's a good question.
Here's the thing I want to do.
So, we've had the
one-dimensional case,
the circle.
We've had the two-dimensional
case, the expanding sphere.
Of course, what we want is the
three-dimensional space.
Okay, here we are in three
dimensions.
Someone is expanding the
Universe, so everywhere we look,
everything is going away from
us, and the further away it is,
the faster it's going.
What's it expanding into?
Well, that, we have a little
more trouble visualizing,
right?
Because in one dimension,
you can visualize this circle
expanding onto the plane.
In two dimensions,
you can imagine this spherical
surface expanding.
In three dimensions,
we can't imagine what it's
expanding into.
That's beyond us.
And so, having had this failure
of the imagination,
what do you do?
You resort to mathematics.
That's what we always do.
And so, imagine that every
object has a position,
which is denoted by three
coordinates,
three spatial coordinates
x, y,
and z. But now,
let's imagine that every
object's position has this
coordinate system times a scale
factor,
which is a function of T.
So, it's a scale factor times a
coordinate position.
And there are two ways that
things can change their
position.
One is, they can move;
they can change their x,
y, z position.
This is the equivalent of
somebody walking across the Old
Campus.
You walk from Starbucks to
Phelps Gate, or something like
that, and you change your x,
y, z coordinate position by
moving through space.
So, changes in position,
which is to say,
velocity, can be accomplished
in two ways.
One is motion through the
coordinate system--that is to
say, changing your x,
y, and z.
This is called peculiar motion.
That's a jargon.
And it's called peculiar
because one object can have a
different peculiar motion from
every other.
It's peculiar in the
old-fashioned sense meaning
specific to one object.
So, you have a peculiar motion
that's all your own.
I have one that's all mine as
we move through x,
y, and z.
But the other is just the
effect of the change.
And in particular,
in the case of the current
Universe, the increase in the
scale factor.
And these two kinds of velocity
are conceptually different from
each other.
Because you don't have to do
anything to change your position
in this way.
You just sit there.
You don't expend any energy.
You don't have any requirement
to expend energy or to exert a
force or to do any of these
things that we ordinarily do to
change our position.
You just sit there and the
Universe expands you,
or expands your position.
And that's why,
going back to your question,
that's why it's possible for
this kind of velocity to turn
out to be greater than the speed
of light,
if it's far enough away,
whereas, it's not possible
here.
What the effect of having this
kind of velocity be faster than
the speed of light does,
is it makes the object
impossible to see,
because photons coming off them
would be redshifted into
oblivion.
And so, you can't actually see
them.
And this imposes a kind of
cosmic event horizon,
similar in kind to the event
horizons around black holes.
You can't see events on the
other side of these horizons.
All right, let me go backwards
in time.
Back in time to when this scale
factor A of T is
equal to 0.
So, if the scale factor is
expanding, you know,
the Old Campus,
whatever, is expanding,
and you reverse time and you
think about what happened long
ago, it must have been smaller.
So this A factor must
have been smaller.
And if you go back a sufficient
amount in time,
you go back to the point where
A is equal to 0.
So then, what does the Old
Campus look like?
Looks like this, right?
It's all been--imagine I take a
Xerox machine and I de-magnify
the thing down to 0% of its
original size,
or 0 plus epsilon,
perhaps.
What does that little diagram
look like?
It looks like a tiny little dot.
A single point,
except it isn't a single point.
What it is, is it's many points
superposed on each other.
And not only is the Old Campus
in there--imagine you've taken
this sphere that is the Earth
and collapsed its radius down to
0.
Not only the Old Campus is
there, but so is the rest of New
Haven, so is Connecticut,
so is Los Angeles,
so is the whole rest of the
surface of the Earth.
All of the points that will
eventually make the surface of
the Earth are superposed on one
another.
And yet, the whole geometry of
the Earth, of the Old Campus,
whatever, is already somehow
encoded in that point.
Because, you know,
you're taking 0 times x,
y, z for every
point.
But then, if you increase this
so that it becomes epsilon or
some non-zero number,
times x,
y, z, then you
already get the geometry of the
Old Campus and everything else
on the surface of the Earth.
So, it is wrong to think of the
Big Bang as starting at a point
and expanding into space.
That's kind of the impression
that word the Big Bang gives
you.
And you think,
naturally enough,
of an explosion,
where something at a point
explodes into empty space.
But that's not right.
All the space,
all the empty space is
contained in that point.
It's all in there.
It's just, it's all multiplied
by 0, so it all comes out to be
on top of each other in the same
place.
How are we doing?
Wonderful.
Okay, that's the essence of the
Big Bang--the whole idea of the
Big Bang--that what is happening
is that the whole coordinate
system of the Universe is
multiplied by this constant.
And that constant changes in
time.
It gets bigger.
And that fact is inferred from
the observations of these
galaxies.
That you observe galaxies,
and that they're moving away
from us.
But more than that--that
there's a linear relationship
between how far away from us
they are and how fast they're
moving.
That implies this kind of
coordinate expansion.
And if you run it in reverse,
it implies an origin to the
Universe at some specific time
in the past.
All right.
Let's have questions.
I'll tell you what.
Talk to each other.
Come up with good questions,
and remember how we do this.
When you come up with a
question--a question can be
either to explain some of this
or to expand upon it.
When you're ready with a
question, put your hand up.
I'll answer a few of them while
other people are getting their
questions together.
And then we'll answer as many
of them as we have time for in
the remainder of the class.
So, talk to each other.
Talk amongst yourselves.
Come up with a question,
any question.
All questions are good,
and see what you can do.
I'll come around and try and
answer some of these.
Yeah – so,
talk to each other by all
means.
Yes?
Student: There is a
center of the Universe,
isn't there?
Professor Charles
Bailyn: No.
Student: Well,
I mean, like,
there's got to be a point in
the Universe at which,
like, all other points--or all
other points that are furthest
away from that point are
expanding equally fast,
right?
Professor Charles
Bailyn: No.
Well, that's true of every
point in the Universe.
That's true of every point in
the Universe.
If you're sitting on the statue
of Pierson, everything at the
same distance is moving away
from you by the same amount.
That's true of every point.
Student: Oh, I see.
Professor Charles
Bailyn: Yeah.
Student: So,
wait, so, like,
is it, like,
in four dimensions,
then?
Professor Charles
Bailyn: Imagine a
three-dimensional thing
expanding into the fourth
dimension the way the surface of
a sphere expands into three.
Student: Okay, thank you.
Student: If you switched
from the expansion to getting
close together,
would it precipitate the point
at which the expansion
[inaudible]
Professor Charles
Bailyn: Oh we'll talk about
that later.
You can imagine that gravity is
going to slow the expansion
down.
But at the moment I'm just
thinking about running backwards
in time.
Student: Okay,
and at some point you got to do
expansion [inaudible]
Professor Charles
Bailyn: Yeah,
or could.
Yeah?
Student: I don't
think--so have you had like one
of those fake event horizons
where something could be
like--appear to be moving faster
than the speed of light?
Could you then like--in like a
normal event horizon you can't
pass information between them
[inaudible]
Professor Charles
Bailyn: Right.
Student: But could you,
like relay information?
Professor Charles
Bailyn: Yes,
but by the time you had done
that--that is to say,
the Universe is expanding
faster than the information
would get to you.
So, you know,
by the time you've moved your
information from here to here,
the distance from here to here
has expanded.
And so, it's not actually
getting any closer.
Okay, let's have a few of these.
Yes, go ahead.
Student: Well I
have--first, just simply,
I didn't understand why it's 0
times the coordinate,
why there's a point at all and
why there's just nothing.
Why there's not [inaudible]
Professor Charles
Bailyn: Well,
okay.
So, if A is equal to 0
here, then all events are at
0,0, 0.
So, that point of 0,0,
0 is occupied,
if you want to think of it that
way.
It's the point--the point in
question is the one at 0,0,
0.
And if you have anything in
here times 0,
that's where you're going to
end up.
And as you go backwards and run
this scale factor down to 0 as
you think about going back in
time,
everything's going to end up
closer and closer to 0.
And then, at the moment when
this equals 0,
it's all piled on top of
itself.
Does that make sense?
Yeah go ahead.
Student: Everything has
to be equidistant [Inaudible]
Professor Charles
Bailyn: No,
no, no, no, no.
Imagine-- Student:
[Inaudible]
Professor Charles
Bailyn: Oh,
in the other dimension.
Student: Yes.
Professor Charles
Bailyn: Yeah,
okay.
In the higher dimension,
that's true.
Well, okay, let me back off.
That's true if you have
constant curvature.
You could imagine something
that, you know,
looks like this,
that also expands in exactly
the same way.
And what creates the curvature
of space-time?
Gravity.
And so, we're going to end up
in a situation where we're going
to be able to determine what the
curvature of space is by seeing
how much mass there is.
So, this actually relates back
to part two of the course.
Student: So,
is that why time could move
differently?
Professor Charles
Bailyn: Huh?
Student: Time could move
differently.
Professor Charles
Bailyn: Time could move
differently, too.
Yep.
Yes?
Student: Isn't--in the
one-dimensional circle and you
said that thing's expanding to a
higher dimension--what does that
mean?
Professor Charles
Bailyn: Okay.
So, imagine you're a
one-dimensional creature.
You can only go on a line.
But your line,
unbeknownst to you,
is curved, so that it ends on
itself.
That then--the line expands.
The circle expands.
What do you think is happening?
You think suddenly all the
distances are greater,
but you can't perceive what
it's expanding into,
because that's a second
dimension, and you're
constrained to move on the line.
Similarly, if you're an ant,
or something moving on the
surface of the Earth,
and you have no understanding
that there's up and down,
and the Earth suddenly
increases in size,
you have no perception of the
dimension you're moving into.
Student: So then,
they just keep moving into
higher dimensions?
Professor Charles
Bailyn: Well,
our perception is of a
three-dimensional Universe.
So, if you say,
okay, what happens when the
scale factor gets bigger?
You can write it down
mathematically into all these
calculations,
and what you have to imagine,
if you want to have a
conceptual understanding of
this, is that there's some
fourth spatial dimension into
which this three-dimensional
space is moving into.
But we can't perceive that.
So, it's better to do it
mathematically,
because it's hard on the brain
to imagine that fourth
dimension.
And there's basically two ways
to think about this.
One is by analogy.
You think of the
one-dimensional creature and
what it thinks about the second
dimension,
or the two-dimensional
creature--what it thinks about
the third dimension.
And then, you just,
sort of, take this leap of
faith into us moving into the
fourth.
Or, you have to write things
down mathematically.
Those are the only two options
you have.
Actually picturing this is not
going to happen.
Yes?
Student: If we were to
speed up this process would we
be able to see things getting
farther away from us?
Professor Charles
Bailyn: Ah,
interesting question.
Would we, if we cranked up this
process some,
would we actually,
you know, be able to take out
our tape measure and notice that
the distance between here and
Starbucks has increased?
No.
Here's why.
We are not--we--that is to say,
you, me, our bodies,
whatever--are not participating
in the expansion of the Universe
at the moment.
Why?
Because the molecules in our
body are being held together by
other forces--chemical forces.
We are being--the Earth is held
together by gravitational forces
and those forces have stopped
the expansion of the Universe
locally,
but not globally.
And that's why you have to go
out and measure this stuff with
galaxies.
Otherwise, you'd be able to,
you know, measure it with this.
Now, of course,
then, there's the problem,
the ruler also expands,
all this kind of stuff.
But, in fact,
what happens is that
locally--local objects,
by which I mean,
our own galaxy and anything
smaller, are held together by
other forces.
Picture a balloon expanding,
so that's the two-dimensional
case, this balloon expands.
And put a bunch of leather
patches on the balloon.
So, those leather patches don't
expand, but the distance
between--because they're held
together by something else--but
the distance between them does.
Yes?
Student: So,
what causes the expansion?
Professor Charles
Bailyn: What causes the
expansion?
Oh, that's theology.
No, no, seriously,
it's an initial condition.
Something at the start,
at T equals 0,
where this A is 0.
So, you're starting with
A of--with this scale
factor of 0, but the derivative
isn't 0.
It's expanding,
A is increasing.
Why?
Why is it increasing by the
particular amount it is?
You have to think of that as
being a parameter of the
Universe.
It's one of the things about
the--sort of like the speed of
light.
Why is that the quantity it is?
Or why is the gravitational
constant the value it is?
It's one of the parameters that
governs our Universe.
Why those particular parameters?
That's not quite a science
question, and we don't know.
You can turn it around.
You can make the following
interesting point.
If it wasn't expanding at
approximately the rate it is,
we wouldn't be here to notice
it.
Because if it was expanding a
whole lot slower,
then it would never have gotten
big enough for stars to condense
out.
If it was expanding a whole lot
faster, all the atoms would be
spread out so far that,
again, stars could never form.
So, you can make this
interesting argument that while
you don't know why it has this
particular value,
it's very important that it
does, because otherwise we
wouldn't be here to observe it.
This is a form of argumentation
called the anthropic principle.
It is highly debated among
scientists whether this is a
scientific argument or not.
But it's amusing either way.
So--yes?
Student: So if
derivative A is positive,
then A should have been
0 for that, right?
Professor Charles
Bailyn: Well,
you go back before A and
you're in trouble.
So - Student: What's
that mean?
Professor Charles
Bailyn: Okay.
So, this is another one of
these un-questions.
It's usually phrased as,
well, what happened before the
Big Bang, right?
That's the equivalent question.
And, again, that's a
theological question.
It's like asking the question:
"What's going on inside an
event horizon?"
You can write down equations.
You can talk about it.
But it's un-testable by its
very nature.
And so, something– This,
by the way, is why the
Catholics like the Big Bang
Theory so much.
In fact, the mathematics,
the relativistic mathematics
that describe the Big Bang and
the expansion of the Universe,
were worked out by a man named
Lemaitre, who was a Catholic
priest, who was a Jesuit.
They love this because it gives
you a creation moment.
It kind of gives you a
scientifically verified creation
moment.
John Paul was very enthusiastic
about astrophysics.
He used to throw big
conferences in the Vatican,
give after-dinner speeches.
There's a thing called the
Vatican Astrophysical
Observatory.
They have a bunch of Jesuits.
They run a telescope in Arizona.
They do research into this.
And so, you know,
if you want to see science and
religion converge,
you want to run away from
biology as fast you can and talk
to us about cosmology.
And what I don't understand is
why the, sort of,
fundamentalist type worry so
much about biology,
where all the science is
dead-set against them.
Whereas, in the case of
astrophysics--now,
of course, it is possible,
if you take an atheistic point
of view,
to come up with all kinds of
clever ways to avoid this
creation event.
But again, it stops being
science at a certain point.
It becomes another odd kind of
theology, and we'll talk about
that, perhaps,
a little bit.
But if you want a place where
science turned out to be the
congruent to,
at least,
certain kinds of
non-fundamentalist religious
beliefs, you're way better off
in astrophysics than you are in
biology.
Yeah, go ahead.
Student: Back to the
fourth dimension,
is that a time dimension or a
space dimension?
Professor Charles
Bailyn: No,
that's a fourth spatial
dimension that I'm talking about
in this case.
Student: So,
in other words,
spatial dimension [Inaudible]
Professor Charles
Bailyn: It's expanding into
a dimension which we don't
perceive.
Or, at least,
that's one way of
conceptualizing it.
We don't get to deal with that
dimension.
So, again, it's a concept.
It's not necessarily a physical
thing.
Yes?
Student: Back to that
one point, I was just wondering,
is that like anything that will
ever [Inaudible]
Professor Charles
Bailyn: Right.
Student: [inaudible]
and anything that will ever
expand?
Professor Charles
Bailyn: Yeah,
yeah, yeah.
It has to be.
Think of all the points that
now exist in the Universe.
Multiply them by 0,
and they're all on top of each
other.
Yes?
Student: So if you were
to go really,
really fast in a straight line
in any direction,
would it eventually end at a
[Inaudible]
Professor Charles
Bailyn: So,
this is the question of what is
the curvature of the Universe.
Is it, in fact,
like this analogy,
with the circle or the surface
of the sphere.
If you go forever,
do you come back to where you
are?
That depends on what the
overall curvature of the
Universe turns out to be.
That depends on how much mass
it contains, because mass curves
space.
And so, that is a question
that's empirically answerable.
We think the answer is no.
In any case,
though, it would take you
longer, at the speed of light,
than the age of the Universe to
accomplish it.
So, even if it were positively
curved, in this sense that it
rejoined itself,
you wouldn't actually be able
to make that voyage.
Student: [Inaudible]
Professor Charles
Bailyn: No.
Student: [Inaudible]
Professor Charles
Bailyn: Well,
you might, eventually,
unless the Universe is
expanding faster than you can
move.
This goes back to the question
of can parts of it expand faster
than the speed of light,
and the answer is yes.
So, yeah.
You take off on this voyage to
the back of your own head.
Going in that direction,
you're going to end up
here--except the Universe is
expanding faster than you can
go,
and so, you don't necessarily
end up back there.
Yes?
Student: How does the
Big Crunch happen in the
expansion?
Professor Charles
Bailyn: Oh,
one of the things that we
haven't talked about is the
expansion rate constant.
Might it be that the Universe
is expanding,
but perhaps slowing down?
Why would it slow down?
It's full of mass.
Gravity slows things down.
And so, in principle,
if you've got enough mass,
the Universe will stop,
turn around,
and fall backwards.
And you can compute from--we
will compute,
next period,
how fast you have to go to make
that happen.
Or contrary,
how dense the Universe has to
be to stop it.
And so, what you expect is the
Universe is slowing down.
Because you know you don't know
how much it's slowing down,
whether it will stop the
expansion or not,
but you expect it to be slowing
down.
The punch line of this whole
part of the course,
in order to anticipate where
we're going,
is that, in fact,
we can measure the change in
the expansion rate,
and it's speeding up,
not slowing down.
This is very disturbing,
because it means there's some
kind of pervasive cosmic
anti-gravity,
which we call dark energy,
because we don't know what it
is.
And that's basically the punch
line of this whole section of
the course.
But you could imagine that
there is a change in the
expansion rate,
either positive or negative,
and that, therefore,
various different outcomes are
possible.
Yeah?
Student: Would a
hypothetical of the Big Crunch
be sort of different from Big
Bang in that the Big Bang sort
of expanded the whole geometry
of the Universe where as the Big
Crunch being motivated by
gravity would only cause the
object [inaudible]
Professor Charles
Bailyn: No,
no, no, no, no.
It turns out it--you might
imagine that,
but you'd be wrong.
It turns out that this scale
factor--you can write down a
differential equation for the
scale factor,
and that it's the scale factor
that stops, turns around and
comes back.
Now, it is the case that when
it's going out,
the particular objects the
Universe contains are different
because of the peculiar motions.
It starts out much more evenly
spread, and then gradually stuff
accumulates due to gravity,
and you get--when you're coming
back, you get a lot of
individual points,
rather than a smooth
distribution of matter.
But that has to do with
peculiar motions,
not the overall motion of the
Universe.
One more and then we got to
continue this next time.
Yes, go ahead.
Student: Before the Big
Bang, what happens when the
minus--negative number did
[Inaudible]
Professor Charles
Bailyn: Right.
So, before the Big Bang,
can you put a negative number
into that A?
This gets you into exactly the
same kind of mathematical
difficulty as moving inside an
event horizon.
You can actually write down a
metric for the whole Universe
and you change all the signs
when that happens.
And you get into a very similar
kind of problem.
So, mathematically,
you could compute a bunch of
stuff.
The physical meaning of that is
not--first of all,
the fact that that mathematics
actually works in the real world
is impossible to determine,
and the physical nature of
consequences of that are
questionable.
All right, stop.