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PROFESSOR: I think
it's a good idea
00:00:22.150 --> 00:00:23.780
to review where we
were because we're
00:00:23.780 --> 00:00:25.810
kind of in the middle
of a discussion.
00:00:25.810 --> 00:00:27.970
We're actually on part three.
00:00:27.970 --> 00:00:31.510
And probably, there will
be four parts all together
00:00:31.510 --> 00:00:35.290
of our discussion of
homogeneous expansion.
00:00:35.290 --> 00:00:40.190
So I have a few slides to just
review where we were last time.
00:00:40.190 --> 00:00:43.110
We were building a mathematical
model of our homogeneously
00:00:43.110 --> 00:00:45.550
expanding universe.
00:00:45.550 --> 00:00:49.940
And we modeled it as
a finite-sized sphere
00:00:49.940 --> 00:00:51.700
where we promised
that in the end
00:00:51.700 --> 00:00:53.750
we will take the limit as
the size of that sphere
00:00:53.750 --> 00:00:57.800
goes to infinity and
fills all the space.
00:00:57.800 --> 00:01:01.550
But it started with some
initial maximum, R max i, i
00:01:01.550 --> 00:01:02.940
for initial.
00:01:02.940 --> 00:01:06.540
We arranged for it to have
a uniform density, rho.
00:01:06.540 --> 00:01:10.550
We started at a
time called t sub i.
00:01:10.550 --> 00:01:14.400
There were some initial
maximum radius, R max i.
00:01:14.400 --> 00:01:18.832
And we also set this up to
exhibit Hubble expansion.
00:01:18.832 --> 00:01:20.290
And we're going to
try to calculate
00:01:20.290 --> 00:01:21.373
how it evolves from there.
00:01:21.373 --> 00:01:23.440
But we're going to start
it Hubble expanding.
00:01:23.440 --> 00:01:25.370
And Hubble expanding
means that we're
00:01:25.370 --> 00:01:28.620
starting with every particle
having an initial velocity
00:01:28.620 --> 00:01:32.040
which is a constant, H sub i,
the initial value of the Hubble
00:01:32.040 --> 00:01:35.090
expansion rate,
times the vector r.
00:01:35.090 --> 00:01:38.320
That is, the vector that goes
from the origin to the point
00:01:38.320 --> 00:01:40.130
where the particle is located.
00:01:40.130 --> 00:01:42.320
And that corresponds
to Hubble expansion
00:01:42.320 --> 00:01:45.201
centered on the
center of our sphere.
00:01:45.201 --> 00:01:46.700
So these are our
initial conditions.
00:01:46.700 --> 00:01:48.400
We just put them in
by hand because we
00:01:48.400 --> 00:01:51.790
think they form a good model for
what our universe looks like.
00:01:51.790 --> 00:01:54.440
And then, the laws of
evolution should take over
00:01:54.440 --> 00:01:57.310
to govern what's
going to happen later.
00:01:57.310 --> 00:02:00.180
And since we haven't
studied general relativity,
00:02:00.180 --> 00:02:01.890
we'll be using
Newton's law of gravity
00:02:01.890 --> 00:02:04.040
to discover how it behaves.
00:02:04.040 --> 00:02:06.170
But I promised you
at the beginning
00:02:06.170 --> 00:02:08.699
that we will, in fact, get
exactly the same equation
00:02:08.699 --> 00:02:11.809
that we would have gotten had
we used general relativity.
00:02:11.809 --> 00:02:13.350
And we'll talk later
today, probably,
00:02:13.350 --> 00:02:15.480
about why that's the case.
00:02:15.480 --> 00:02:17.770
So this is the initial setup.
00:02:17.770 --> 00:02:19.480
Any questions about
the initial setup?
00:02:23.000 --> 00:02:25.000
OK, then we derived
a lot of equations.
00:02:25.000 --> 00:02:27.510
And everything really
follows from the statement
00:02:27.510 --> 00:02:31.190
at the top here, which is
understanding what Newton tells
00:02:31.190 --> 00:02:34.082
us about the gravitational
field of a spherical shell.
00:02:34.082 --> 00:02:35.790
We could always think
of our solid sphere
00:02:35.790 --> 00:02:37.700
as being made up of shells.
00:02:37.700 --> 00:02:39.440
So if we know how
a shell behaves,
00:02:39.440 --> 00:02:41.610
we know all we need to know.
00:02:41.610 --> 00:02:42.740
And Newton told us.
00:02:42.740 --> 00:02:46.060
He told us that inside a
spherical shell, the effect
00:02:46.060 --> 00:02:48.580
of gravity, which I'm
describing in terms
00:02:48.580 --> 00:02:52.170
of the gravitational
acceleration vector, little g,
00:02:52.170 --> 00:02:55.720
inside a shell, the
gravitational field is exactly
00:02:55.720 --> 00:02:57.134
0.
00:02:57.134 --> 00:02:59.550
The force is coming from all
different parts of the shell,
00:02:59.550 --> 00:03:01.490
pulling outward,
cancel each other.
00:03:01.490 --> 00:03:04.780
And the net force on any object
anywhere inside the shell
00:03:04.780 --> 00:03:06.580
is exactly 0.
00:03:06.580 --> 00:03:10.000
Outside, the entire
shell acts exactly
00:03:10.000 --> 00:03:12.870
as if it were a single
point mass located
00:03:12.870 --> 00:03:17.340
at the origin with the
same total mass, m.
00:03:17.340 --> 00:03:18.710
So incredibly simple.
00:03:18.710 --> 00:03:21.744
It's hard to believe it's
so simple, but it is.
00:03:21.744 --> 00:03:23.660
By the way, if you use
Gauss's law of gravity,
00:03:23.660 --> 00:03:25.549
it becomes very obvious
that those statements
00:03:25.549 --> 00:03:26.590
are the right statements.
00:03:26.590 --> 00:03:28.800
Newton know about
Gauss's law of gravity,
00:03:28.800 --> 00:03:30.790
so Newton derived
those statements
00:03:30.790 --> 00:03:32.540
by brute force
integration, which
00:03:32.540 --> 00:03:33.770
is more of a tour de force.
00:03:33.770 --> 00:03:38.430
But something Newton was
capable of doing it, and he did.
00:03:38.430 --> 00:03:40.590
To describe how the
system is going to evolve,
00:03:40.590 --> 00:03:43.560
moving onward, we introduce
something that's a little bit
00:03:43.560 --> 00:03:47.150
complicated, a function r, which
is a function of two variables,
00:03:47.150 --> 00:03:50.110
r sub i and t.
00:03:50.110 --> 00:03:53.500
And r represents
the radius at time t
00:03:53.500 --> 00:03:57.790
of the shell that was
initially at radius r sub i.
00:03:57.790 --> 00:04:01.800
And our goal here is not just
to keep track of the particle
00:04:01.800 --> 00:04:03.895
on the outside,
which is for example,
00:04:03.895 --> 00:04:06.380
what Ryden does in her textbook.
00:04:06.380 --> 00:04:08.400
Ryden assumes that
everything stays homogeneous.
00:04:08.400 --> 00:04:10.649
And then if you follow what
happens to the outer edge,
00:04:10.649 --> 00:04:11.950
you know everything.
00:04:11.950 --> 00:04:13.180
But we're not going to
make that assumption.
00:04:13.180 --> 00:04:15.450
We're going to conclude
that it remains homogeneous,
00:04:15.450 --> 00:04:16.709
but we're going to derive it.
00:04:16.709 --> 00:04:18.800
Which means that we need to know
the motion of every particle
00:04:18.800 --> 00:04:20.360
inside the sphere
to be able to tell
00:04:20.360 --> 00:04:22.320
if it's going to
stay homogeneous.
00:04:22.320 --> 00:04:26.610
And that's why we're introducing
this more general description
00:04:26.610 --> 00:04:29.230
where r is a
function of r sub i.
00:04:29.230 --> 00:04:31.440
So that function of
the extra variable
00:04:31.440 --> 00:04:36.500
will tell us how every particle
moves as the system evolves.
00:04:36.500 --> 00:04:39.194
We know that we're going to
maintain spherical symmetry
00:04:39.194 --> 00:04:40.860
because we start with
spherical symmetry
00:04:40.860 --> 00:04:43.315
and the force law respects
spherical symmetry.
00:04:43.315 --> 00:04:45.190
So we're building that
in from the beginning.
00:04:45.190 --> 00:04:48.730
We're not allowing things to
depend on angular variables
00:04:48.730 --> 00:04:50.430
theta or phi.
00:04:50.430 --> 00:04:54.560
But assuming spherical
symmetry, motion just
00:04:54.560 --> 00:04:57.130
is described entirely by
giving the r-coordinate
00:04:57.130 --> 00:04:58.600
of each particle.
00:04:58.600 --> 00:05:03.430
And this function r of r i
t does that-- exactly that.
00:05:03.430 --> 00:05:08.640
Then, we said that
at a given radius,
00:05:08.640 --> 00:05:10.680
this description
about shells tells us
00:05:10.680 --> 00:05:12.680
that the shells that
are outside that radius
00:05:12.680 --> 00:05:14.800
don't do anything
at a given radius.
00:05:14.800 --> 00:05:17.250
But the shells that are
inside act like a point mass,
00:05:17.250 --> 00:05:19.560
as if it was all at the origin.
00:05:19.560 --> 00:05:22.750
So to understand how a given
shell is going to evolve,
00:05:22.750 --> 00:05:24.740
all we really need to
know is the total mass
00:05:24.740 --> 00:05:26.120
inside that shell.
00:05:26.120 --> 00:05:28.580
And that's given
by M of r sub i,
00:05:28.580 --> 00:05:31.470
the mass inside the
shell at radius r sub i.
00:05:31.470 --> 00:05:33.370
And that's just the
volume of the shell
00:05:33.370 --> 00:05:37.180
initially times the
initial mass density.
00:05:37.180 --> 00:05:40.910
As these shells move, the
total mass inside a shell
00:05:40.910 --> 00:05:42.340
will remain exactly
the same as it
00:05:42.340 --> 00:05:46.330
was as long as there's
no crossings of shells.
00:05:46.330 --> 00:05:49.037
Now, the shell crossing
issue is hard to talk about.
00:05:49.037 --> 00:05:50.620
But in the end, it
just doesn't happen
00:05:50.620 --> 00:05:52.660
so you don't need
to worry about it.
00:05:52.660 --> 00:05:55.250
But the argument was
that initially we
00:05:55.250 --> 00:05:57.310
know the shells are moving
apart from each other
00:05:57.310 --> 00:05:59.880
because we built in
this Hubble expansion
00:05:59.880 --> 00:06:02.690
where everything is moving
away from everything else.
00:06:02.690 --> 00:06:05.040
So if shells are
ever going to cross,
00:06:05.040 --> 00:06:06.700
they're not going to
cross immediately.
00:06:06.700 --> 00:06:09.710
It will take some time for
these velocities to reverse,
00:06:09.710 --> 00:06:13.440
and the shells that were moving
apart to move together and hit.
00:06:13.440 --> 00:06:16.770
So there's unambiguously
at least a period of time
00:06:16.770 --> 00:06:18.630
where there are no
shell crossings.
00:06:18.630 --> 00:06:20.230
And we could write
down the equations
00:06:20.230 --> 00:06:22.500
that describe the motion
during this period
00:06:22.500 --> 00:06:25.120
where there are no
shell crossings.
00:06:25.120 --> 00:06:27.210
Now, if there was going
to be a shell crossing,
00:06:27.210 --> 00:06:28.410
the equations that
we're writing down
00:06:28.410 --> 00:06:30.030
would have to hold
right up until the time
00:06:30.030 --> 00:06:30.990
of that first crossing.
00:06:30.990 --> 00:06:32.660
Because as long as
there's no crossings,
00:06:32.660 --> 00:06:34.340
our equations are valid.
00:06:34.340 --> 00:06:37.570
Which means that if there was
going to be a shell crossing,
00:06:37.570 --> 00:06:40.040
the equations we're writing
down had better show it.
00:06:40.040 --> 00:06:42.480
Because the equations
that we're writing down
00:06:42.480 --> 00:06:44.530
have to be valid right
up until the instant
00:06:44.530 --> 00:06:46.570
of any possible shell crossing.
00:06:46.570 --> 00:06:48.780
And what we're going to
find is that the equations
00:06:48.780 --> 00:06:51.140
are going to lead to just
homogeneous evolution
00:06:51.140 --> 00:06:53.690
where there are no
shell crossings.
00:06:53.690 --> 00:06:55.254
And therefore, that's
the conclusion.
00:06:55.254 --> 00:06:57.170
If there were going to
be any shell crossings,
00:06:57.170 --> 00:06:58.753
these equations would
have to show it.
00:06:58.753 --> 00:07:01.900
They don't show it, so there
are no shell crossings.
00:07:01.900 --> 00:07:04.870
So it's a complicated
paragraph, but the bottom line
00:07:04.870 --> 00:07:07.210
is simple- we can
ignore shell crossings.
00:07:07.210 --> 00:07:10.050
And that means that the total
mass inside of any shell
00:07:10.050 --> 00:07:12.170
will remain exactly
constant with time
00:07:12.170 --> 00:07:13.740
given by its initial value.
00:07:13.740 --> 00:07:15.730
And this formula is
the initial value.
00:07:15.730 --> 00:07:17.230
And therefore, it
holds at all time.
00:07:20.180 --> 00:07:23.315
Any questions about that?
00:07:23.315 --> 00:07:25.190
Any questions about the
shell crossing issue?
00:07:28.980 --> 00:07:30.870
OK, good.
00:07:30.870 --> 00:07:34.000
So whoops, sorry about that.
00:07:34.000 --> 00:07:37.940
So M of r sub i is the mass
inside the shell at radius r
00:07:37.940 --> 00:07:38.920
sub i.
00:07:38.920 --> 00:07:41.760
And then we can write down-- now
we use Newton's law of gravity
00:07:41.760 --> 00:07:42.630
directly.
00:07:42.630 --> 00:07:45.470
We can write down
the acceleration
00:07:45.470 --> 00:07:48.280
of a given particle in
terms of its radius r
00:07:48.280 --> 00:07:50.860
and its initial radius r sub i.
00:07:50.860 --> 00:07:54.910
Its initial radius determines
how much mass is inside.
00:07:54.910 --> 00:07:57.550
M of r sub i is
independent of time.
00:07:57.550 --> 00:08:00.170
But the actual radius it's
at determines how far away
00:08:00.170 --> 00:08:03.360
it is, or describes how far
away it is from the origin.
00:08:03.360 --> 00:08:07.020
And that's the 1 over r squared
that appears in the force law.
00:08:07.020 --> 00:08:10.120
So we have the time dependent r
in the denominator and the time
00:08:10.120 --> 00:08:14.030
independent initial r sub i
that appears in the numerator.
00:08:14.030 --> 00:08:16.170
And it's all proportional
to a unit vector r
00:08:16.170 --> 00:08:19.830
hat pulling everything
radially inward
00:08:19.830 --> 00:08:21.624
because of the
minus sign in front.
00:08:21.624 --> 00:08:23.290
So gravity is pulling
everything inward,
00:08:23.290 --> 00:08:25.770
which is what we'd expect.
00:08:25.770 --> 00:08:28.560
So this formula is
the key formula.
00:08:28.560 --> 00:08:30.220
It's a vector
formula, but we know
00:08:30.220 --> 00:08:31.480
that all the motion is radial.
00:08:31.480 --> 00:08:32.938
So all we really
have to keep track
00:08:32.938 --> 00:08:35.659
of is the radius as
a function of time.
00:08:35.659 --> 00:08:37.730
So we can turn
this vector formula
00:08:37.730 --> 00:08:41.000
into a formula for
little r itself.
00:08:41.000 --> 00:08:44.390
Just the radius number,
the radial coordinate.
00:08:44.390 --> 00:08:49.130
And we get r double dot is minus
4 pi over 3 G r sub i cubed rho
00:08:49.130 --> 00:08:51.175
sub i, taking the
formula for M of ri
00:08:51.175 --> 00:08:53.670
from the line above
divided by r squared.
00:08:53.670 --> 00:08:57.210
And the r in this formula--
I didn't write the arguments,
00:08:57.210 --> 00:08:59.690
but it means this
function r, which
00:08:59.690 --> 00:09:03.060
is a function of two
arguments, r sub i and t.
00:09:03.060 --> 00:09:08.350
So this differential equation
now governs our entire system
00:09:08.350 --> 00:09:10.400
and tells us everything
we need to know
00:09:10.400 --> 00:09:13.791
or everything we can know
about the actual dynamics.
00:09:13.791 --> 00:09:15.290
But to solve a
differential equation
00:09:15.290 --> 00:09:17.760
of that sort, a second
order differential equation,
00:09:17.760 --> 00:09:19.540
we need initial conditions.
00:09:19.540 --> 00:09:22.300
And we already described the
initial conditions in words.
00:09:22.300 --> 00:09:24.860
Now we have to just figure out
what those initial conditions
00:09:24.860 --> 00:09:29.080
are saying about r and r dot.
00:09:29.080 --> 00:09:30.782
And the answer is
straightforward.
00:09:30.782 --> 00:09:31.740
We argued it last time.
00:09:31.740 --> 00:09:34.180
The initial conditions
are that r at time t
00:09:34.180 --> 00:09:35.540
sub i is just r sub i.
00:09:35.540 --> 00:09:39.110
That was really the definition
of r sub i in the first place.
00:09:39.110 --> 00:09:43.020
And r dot is just H
sub i times v coming
00:09:43.020 --> 00:09:47.080
from the formula we had
for the initial velocities.
00:09:47.080 --> 00:09:50.310
So these three equations,
the two initial conditions
00:09:50.310 --> 00:09:53.470
and the differential
equation, lead in principle
00:09:53.470 --> 00:09:55.870
to a mathematical solution
that's completely unique
00:09:55.870 --> 00:09:58.240
and determined by
those equations.
00:09:58.240 --> 00:10:00.120
And our goal now is
just to figure out
00:10:00.120 --> 00:10:03.590
what that solution looks like.
00:10:03.590 --> 00:10:08.460
And we discovered a
marvelous scaling property.
00:10:08.460 --> 00:10:11.170
That is, instead
of talking about r,
00:10:11.170 --> 00:10:15.580
we divided r by r sub i and
defined a new function, which
00:10:15.580 --> 00:10:17.325
we initially called u.
00:10:17.325 --> 00:10:18.950
Initially, thinking
of it as a function
00:10:18.950 --> 00:10:20.990
of these two variables.
00:10:20.990 --> 00:10:23.160
We can write down
new equations for u.
00:10:23.160 --> 00:10:26.330
And those equations
end up not having
00:10:26.330 --> 00:10:28.420
any r sub i's in them at all.
00:10:28.420 --> 00:10:30.360
And once we realized
that, we realized
00:10:30.360 --> 00:10:32.880
we don't need to call
it u of r sub i and t.
00:10:32.880 --> 00:10:34.390
It's really just
a function of t.
00:10:34.390 --> 00:10:36.452
And at that point, we
renamed it because we also
00:10:36.452 --> 00:10:38.660
realized that it actually
is our old friend the scale
00:10:38.660 --> 00:10:40.580
factor, a of t.
00:10:40.580 --> 00:10:44.600
So a of t is just r of r sub
i and t divided by r sub i.
00:10:44.600 --> 00:10:46.176
And then the reason
this is a scale
00:10:46.176 --> 00:10:48.800
factor is seen most clearly from
that equation, which is really
00:10:48.800 --> 00:10:51.390
this equation just rearranged.
00:10:51.390 --> 00:10:53.770
The physical distance of
a particle from the origin
00:10:53.770 --> 00:10:57.300
is equal to the scale
factor times r sub i
00:10:57.300 --> 00:11:01.190
where r sub i plays the role
of the coordinate distance.
00:11:01.190 --> 00:11:06.150
r sub i is a time-independent
measure indicating
00:11:06.150 --> 00:11:07.540
which shell you're
talking about.
00:11:10.950 --> 00:11:14.320
So the equations for a then
are the equations for u,
00:11:14.320 --> 00:11:17.010
which are the equations for
r just divided by r sub i.
00:11:17.010 --> 00:11:19.710
And we can write down
what those equations are.
00:11:19.710 --> 00:11:21.290
We have a differential
equation for a
00:11:21.290 --> 00:11:23.610
and two initial
conditions where r
00:11:23.610 --> 00:11:26.640
sub i has dropped
out all together.
00:11:26.640 --> 00:11:30.230
The differential equation
is given immediately
00:11:30.230 --> 00:11:32.440
from the one we had up
here, but we could also we
00:11:32.440 --> 00:11:35.280
write it in terms
of what rho of t is.
00:11:35.280 --> 00:11:37.030
I didn't write the
equation here because I
00:11:37.030 --> 00:11:38.630
guess I was running out of room.
00:11:38.630 --> 00:11:40.950
But we also figured out
how rho of t behaves.
00:11:40.950 --> 00:11:42.760
And it behaves in
the obvious way.
00:11:42.760 --> 00:11:44.360
As the space
expands, the density
00:11:44.360 --> 00:11:47.160
just goes down as the volume.
00:11:47.160 --> 00:11:50.660
And the volume grows like a
cubed, the cube of the scale
00:11:50.660 --> 00:11:54.890
factor, because volume's
proportional to radius cubed.
00:11:54.890 --> 00:12:00.770
So rho of t is just rho sub
i divided by a of t cubed.
00:12:00.770 --> 00:12:04.080
And putting that in, we
can go from that equation
00:12:04.080 --> 00:12:05.410
to this equation.
00:12:05.410 --> 00:12:07.340
And this equation makes
no reference anymore
00:12:07.340 --> 00:12:08.620
to the initial time.
00:12:08.620 --> 00:12:11.720
It's just an equation for what
deceleration you see, what
00:12:11.720 --> 00:12:14.210
value of a double dot
you see, as a function
00:12:14.210 --> 00:12:15.880
of the mass density
and a itself.
00:12:18.680 --> 00:12:22.010
OK, any questions about any
of those differential equation
00:12:22.010 --> 00:12:23.150
manipulations?
00:12:23.150 --> 00:12:24.180
Yes.
00:12:24.180 --> 00:12:27.230
AUDIENCE: In the homework, it
says that this equation is not
00:12:27.230 --> 00:12:28.120
entirely general.
00:12:28.120 --> 00:12:29.340
We can't use it in all cases.
00:12:29.340 --> 00:12:32.343
Whereas, the other version
that we get from the energy
00:12:32.343 --> 00:12:34.520
conservation is completely OK?
00:12:34.520 --> 00:12:36.910
PROFESSOR: That is correct, yes.
00:12:36.910 --> 00:12:39.137
AUDIENCE: It says it in
there, but why is that?
00:12:39.137 --> 00:12:40.220
PROFESSOR: Why is it true?
00:12:40.220 --> 00:12:43.270
Well, as long as you have only
non-relativistic matter, which
00:12:43.270 --> 00:12:44.680
is what we're
talking about here,
00:12:44.680 --> 00:12:46.900
both of these-- this
equation is golden
00:12:46.900 --> 00:12:49.406
and so is the equation
we're about to talk
00:12:49.406 --> 00:12:51.530
about the derivation of,
the conservation of energy
00:12:51.530 --> 00:12:52.710
equation.
00:12:52.710 --> 00:12:55.990
So as long as we have the
context in which we derived it,
00:12:55.990 --> 00:12:57.420
it's completely valid.
00:12:57.420 --> 00:12:59.920
But on the homework
set, we're talking
00:12:59.920 --> 00:13:01.019
about more general cases.
00:13:01.019 --> 00:13:03.060
We gave a different formula
for the scale factor,
00:13:03.060 --> 00:13:05.185
which corresponds to a
different situation in terms
00:13:05.185 --> 00:13:09.000
of the underlying materials
that are building that universe.
00:13:09.000 --> 00:13:13.190
And where the change
occurs is when
00:13:13.190 --> 00:13:16.010
one introduces a
nonzero pressure.
00:13:16.010 --> 00:13:17.950
This gas of particles
that we're talking about
00:13:17.950 --> 00:13:19.740
is just non-relativistic
particles
00:13:19.740 --> 00:13:21.530
moving with the
Hubble expansion.
00:13:21.530 --> 00:13:25.250
There's no internal velocity
which generates a pressure.
00:13:25.250 --> 00:13:27.570
And it's pressure that
makes a difference.
00:13:27.570 --> 00:13:29.639
This formula assume
zero pressure.
00:13:29.639 --> 00:13:31.180
We will learn later
how to correct it
00:13:31.180 --> 00:13:32.789
when there's a nonzero pressure.
00:13:32.789 --> 00:13:34.830
And the other formula
doesn't depend on pressure,
00:13:34.830 --> 00:13:36.747
so it's valid whether
there's pressure or not.
00:13:36.747 --> 00:13:39.038
But at the moment, we have
no real way of knowing that.
00:13:39.038 --> 00:13:40.760
We'll talk later
about why that's true.
00:13:44.809 --> 00:13:45.475
Other questions?
00:13:49.105 --> 00:13:50.040
No.
00:13:50.040 --> 00:13:51.920
OK, great.
00:13:51.920 --> 00:13:55.500
OK, one more slide here,
not too much on it.
00:13:55.500 --> 00:13:58.410
At the end of the last
class, we took the equation
00:13:58.410 --> 00:14:00.250
that I just wrote on
the previous slide.
00:14:00.250 --> 00:14:03.710
I just copied it to this slide,
the equation for a double dot,
00:14:03.710 --> 00:14:07.790
written in terms of the
initial mass density.
00:14:07.790 --> 00:14:12.000
And discovered that it
can be integrated once
00:14:12.000 --> 00:14:13.940
to produce a kind of a
conservation of energy
00:14:13.940 --> 00:14:15.110
equation.
00:14:15.110 --> 00:14:17.250
And all you do is
you start with this,
00:14:17.250 --> 00:14:21.100
write it by putting everything
on one side of the equation,
00:14:21.100 --> 00:14:25.950
a double dot plus 4 pi over 3 G
rho i over a squared equals 0.
00:14:25.950 --> 00:14:28.560
And then, multiply the
whole equation by a dot.
00:14:28.560 --> 00:14:31.220
And a dot is called
an integrating factor.
00:14:31.220 --> 00:14:36.600
It turns the expression
into a total derivative.
00:14:36.600 --> 00:14:38.625
So once you write
it this way, it
00:14:38.625 --> 00:14:42.560
is equivalent to dE dt
equals 0, where e is just
00:14:42.560 --> 00:14:44.910
defined to be this
quantity that would
00:14:44.910 --> 00:14:47.140
have better as a
triple equal sign.
00:14:47.140 --> 00:14:49.050
e is just defined
to be that quantity.
00:14:49.050 --> 00:14:52.100
And if you then write
down what dE dt means,
00:14:52.100 --> 00:14:54.670
it means exactly that.
00:14:54.670 --> 00:14:56.340
So it's the same equation.
00:14:56.340 --> 00:14:58.850
So given our second
order equation,
00:14:58.850 --> 00:15:02.010
we can write down a
first order equation,
00:15:02.010 --> 00:15:04.340
which is that E is
equal to a constant.
00:15:07.880 --> 00:15:12.220
And we commented last time that
the physical interpretation
00:15:12.220 --> 00:15:16.970
of E is-- I'm not
sure what to say.
00:15:16.970 --> 00:15:19.700
There are multiple physical
interpretations of E
00:15:19.700 --> 00:15:21.800
is probably what I want to say.
00:15:21.800 --> 00:15:23.720
And one physical
interpretation is
00:15:23.720 --> 00:15:25.600
if you multiply by
the right factors,
00:15:25.600 --> 00:15:29.790
it does describe the actual
energy of a test particle
00:15:29.790 --> 00:15:33.930
just on the boundary of our
sphere, on the outer boundary.
00:15:33.930 --> 00:15:37.040
It doesn't really describe
directly the total energy
00:15:37.040 --> 00:15:39.940
of a particle inside that
sphere because calculating
00:15:39.940 --> 00:15:42.312
the potential energy of a
particle inside the sphere
00:15:42.312 --> 00:15:43.145
is more complicated.
00:15:43.145 --> 00:15:45.935
And it doesn't give
you the simple answer.
00:15:45.935 --> 00:15:48.185
So it doesn't really describe
particles on the inside,
00:15:48.185 --> 00:15:50.970
except you could argue
that if you-- talking
00:15:50.970 --> 00:15:53.470
about a particle on the inside,
the particles outside of it
00:15:53.470 --> 00:15:54.070
don't matter.
00:15:54.070 --> 00:15:55.486
And you pretend
they're not there.
00:15:55.486 --> 00:15:57.280
And then it does
describe the energy.
00:15:57.280 --> 00:15:58.280
That is, you could
think of any particle
00:15:58.280 --> 00:15:59.821
as being on the
outside of the sphere
00:15:59.821 --> 00:16:01.570
and ignore what's outside it.
00:16:01.570 --> 00:16:06.150
But that's extra sentences
that you have to put in.
00:16:06.150 --> 00:16:08.370
On the homework, you
will also discover
00:16:08.370 --> 00:16:10.690
that for this finite-sized
Newtonian sphere,
00:16:10.690 --> 00:16:12.630
there's certainly a
well-defined Newtonian
00:16:12.630 --> 00:16:15.230
expression for the total
energy of the sphere.
00:16:15.230 --> 00:16:17.260
And that's also
proportional to this E.
00:16:17.260 --> 00:16:20.040
So by multiplying it
by different constant,
00:16:20.040 --> 00:16:22.854
you can turn it into the
total energy of the sphere.
00:16:22.854 --> 00:16:24.270
So it's actually
related to energy
00:16:24.270 --> 00:16:25.750
and it's definitely conserved.
00:16:25.750 --> 00:16:29.450
Those are the important
statements to takeaway.
00:16:29.450 --> 00:16:31.977
And that's where we
left off last time.
00:16:31.977 --> 00:16:33.810
And we'll pick up from
there now pretty much
00:16:33.810 --> 00:16:36.370
on the blackboard for
the rest of lecture.
00:16:36.370 --> 00:16:40.805
Are there any further
questions about these slides?
00:16:52.720 --> 00:16:53.390
OK.
00:16:53.390 --> 00:16:59.665
In that case, we will go on.
00:17:10.540 --> 00:17:13.890
The first thing I want to do is
to take the same conservation
00:17:13.890 --> 00:17:17.140
law that we have up
there and rewrite it
00:17:17.140 --> 00:17:19.519
in a way that's
more conventional.
00:17:19.519 --> 00:17:20.560
And perhaps, more useful.
00:17:20.560 --> 00:17:22.990
But certainly,
more conventional.
00:17:22.990 --> 00:17:28.920
We started with knowing that a
quantity called E is conserved.
00:17:28.920 --> 00:17:36.700
And it's equal to 1/2 a dot
squared minus 4 pi over 3
00:17:36.700 --> 00:17:41.280
G rho i over a.
00:17:44.160 --> 00:17:48.300
OK, then we also
know that rho of t
00:17:48.300 --> 00:17:54.570
is equal to rho sub i divided
by a cubed of t, which just says
00:17:54.570 --> 00:17:57.270
that matter thins out with
the volume which grows
00:17:57.270 --> 00:18:00.300
as the cube of the scale factor.
00:18:00.300 --> 00:18:06.260
And that can be used to
manipulate this equation.
00:18:11.440 --> 00:18:15.381
For reasons that will
become clearer in a minute,
00:18:15.381 --> 00:18:17.880
I'm just going to manipulate
this equation by multiplying it
00:18:17.880 --> 00:18:18.830
by 2 over a squared.
00:18:18.830 --> 00:18:20.580
Just because this will
get me the equation
00:18:20.580 --> 00:18:22.030
I'm trying to get to.
00:18:22.030 --> 00:18:24.551
So if I multiply the left-hand
side by 2 over a squared,
00:18:24.551 --> 00:18:26.342
I have to multiply the
right-hand side by 2
00:18:26.342 --> 00:18:28.190
over a squared.
00:18:28.190 --> 00:18:30.380
The 2 cancels the half.
00:18:30.380 --> 00:18:34.490
The first term becomes
a dot over a squared.
00:18:34.490 --> 00:18:37.130
And you might remember the a dot
over a is the Hubble expansion
00:18:37.130 --> 00:18:40.040
rate, so it has some
physical significance.
00:18:40.040 --> 00:18:46.790
And then, minus the 2 turns the
4 pi over 3 into 8 pi over 3.
00:18:46.790 --> 00:18:54.820
And the a squared multiplies
the a to make an a cubed.
00:18:54.820 --> 00:18:57.740
And then here, we have
rho i over a cubed, which
00:18:57.740 --> 00:19:00.250
in fact is just the
current value of rho.
00:19:00.250 --> 00:19:08.780
So we can rewrite this as a
dot over a squared minus 8 pi
00:19:08.780 --> 00:19:12.750
over 3 G rho.
00:19:12.750 --> 00:19:15.650
No a's anymore on
the right-hand side.
00:19:15.650 --> 00:19:17.770
Well, on a's anymore
in this term.
00:19:21.280 --> 00:19:26.530
OK, now the convention
that brings our notation
00:19:26.530 --> 00:19:32.095
into contact with the
rest of the world.
00:19:32.095 --> 00:19:33.470
Nobody talks about
E, by the way.
00:19:33.470 --> 00:19:35.666
That's just my convention.
00:19:35.666 --> 00:19:37.040
But to make contact
with the rest
00:19:37.040 --> 00:19:39.400
of the world, the rest
of the world talks
00:19:39.400 --> 00:19:44.070
about a number called
little k, lowercase k.
00:19:44.070 --> 00:19:49.200
And it connects to our notation
by being equal to minus 2E
00:19:49.200 --> 00:19:50.270
divided by c squared.
00:19:55.450 --> 00:20:04.460
And with that connection, we
can write our conservation law
00:20:04.460 --> 00:20:10.560
in what is the standard
way of writing it,
00:20:10.560 --> 00:20:11.685
at least in many textbooks.
00:20:25.257 --> 00:20:25.840
And that's it.
00:20:25.840 --> 00:20:27.600
So I put a box around it.
00:20:27.600 --> 00:20:30.620
And this equation was first
derived by Alexander Friedmann
00:20:30.620 --> 00:20:34.620
using general
relativity in 1922.
00:20:34.620 --> 00:20:43.390
And it is, therefore, usually
called the Friedmann equation.
00:20:47.040 --> 00:20:50.170
Alexander Friedmann, by the
way, was a Russian meteorologist
00:20:50.170 --> 00:20:51.390
by profession.
00:20:51.390 --> 00:20:53.320
But as a meteorologist,
he knew a lot
00:20:53.320 --> 00:20:54.880
about differential equations.
00:20:54.880 --> 00:20:58.670
And when general relativity came
out, he got interested in it
00:20:58.670 --> 00:21:02.365
and was the first person to
derive using general relativity
00:21:02.365 --> 00:21:05.390
the equations that described
an expanding universe.
00:21:05.390 --> 00:21:09.395
And he wrote two famous
papers-- now famous papers--
00:21:09.395 --> 00:21:12.260
in 1922 and 1923.
00:21:12.260 --> 00:21:15.580
One of them talking about
the system of equations
00:21:15.580 --> 00:21:17.297
where k is positive
and another talking
00:21:17.297 --> 00:21:18.380
about where it's negative.
00:21:18.380 --> 00:21:20.549
I forget which
order they were in.
00:21:20.549 --> 00:21:22.590
But they correspond to
open and closed universes,
00:21:22.590 --> 00:21:25.286
which we'll talk about
more in a few minutes.
00:21:25.286 --> 00:21:28.190
Now, I just should remind you
to have our equations together.
00:21:28.190 --> 00:21:30.760
We also had the all-important
equation for a double dot.
00:21:46.315 --> 00:21:51.070
And as I was just describing
an answer to a question,
00:21:51.070 --> 00:21:53.360
we don't know yet how
to generalize this
00:21:53.360 --> 00:21:56.620
to other kinds of matter besides
the non non-relativistic dust
00:21:56.620 --> 00:21:58.050
that we just derived them for.
00:21:58.050 --> 00:22:01.490
They're certainly both correct
for our non-relativistic dust.
00:22:01.490 --> 00:22:04.180
But when we try to generalize
them, what we'll find
00:22:04.180 --> 00:22:07.510
is that the top equation
will remain true exactly
00:22:07.510 --> 00:22:10.390
for any kind of matter, while
the bottom equation assumes
00:22:10.390 --> 00:22:11.515
that pressure equals 0.
00:22:28.540 --> 00:22:34.710
Now, the standard terminology
is to call the top equation
00:22:34.710 --> 00:22:35.885
the Friedmann equation.
00:22:35.885 --> 00:22:37.260
In fact, both of
these equations,
00:22:37.260 --> 00:22:39.551
with this one including the
pressure term that we don't
00:22:39.551 --> 00:22:42.120
have yet, appeared in
Friedmann's original papers.
00:22:42.120 --> 00:22:44.100
So I usually refer to
these two equations
00:22:44.100 --> 00:22:46.800
as the Friedmann
equations-- plural.
00:22:46.800 --> 00:22:50.571
But many textbooks refer to just
the top one as the Friedmann
00:22:50.571 --> 00:22:52.820
equation and don't give a
name to that equation, which
00:22:52.820 --> 00:22:54.770
is also OK if you want.
00:22:54.770 --> 00:22:56.429
Yes.
00:22:56.429 --> 00:22:58.281
AUDIENCE: Didn't we
get the top equation
00:22:58.281 --> 00:23:00.031
from the bottom equation?
00:23:00.031 --> 00:23:00.780
PROFESSOR: We did.
00:23:00.780 --> 00:23:03.680
That's right.
00:23:03.680 --> 00:23:05.840
So how does that jive?
00:23:05.840 --> 00:23:09.290
The answer is-- and we'll
be coming to this later.
00:23:09.290 --> 00:23:11.820
But the answer is that when
we got the top equation
00:23:11.820 --> 00:23:15.870
from the bottom equation,
we used that equation.
00:23:18.480 --> 00:23:20.420
And this equation
will no longer hold
00:23:20.420 --> 00:23:22.420
when there's a
significant pressure.
00:23:22.420 --> 00:23:25.969
And in fact, when we derive
it later-- I forget what order
00:23:25.969 --> 00:23:26.760
we'll do things in.
00:23:26.760 --> 00:23:28.968
But we'll make sure that
all three of these equations
00:23:28.968 --> 00:23:31.990
are consistent when
we include pressure.
00:23:31.990 --> 00:23:34.070
The reason the top
equation changes
00:23:34.070 --> 00:23:37.070
if you include pressure--
may not be obvious.
00:23:37.070 --> 00:23:39.070
But if I tell you
why it happens,
00:23:39.070 --> 00:23:40.414
it will become obvious.
00:23:40.414 --> 00:23:42.830
The top equation looks like
it's just how things thin out.
00:23:45.620 --> 00:23:47.010
Like a cubed.
00:23:47.010 --> 00:23:51.410
But rho is the
total mass density.
00:23:51.410 --> 00:23:53.580
And relativistically, it's
equivalent to the energy
00:23:53.580 --> 00:23:53.650
density.
00:23:53.650 --> 00:23:55.210
If you just multiply
by c squared,
00:23:55.210 --> 00:23:58.110
that becomes the energy
density by the E equals
00:23:58.110 --> 00:24:00.659
mc squared equality.
00:24:00.659 --> 00:24:02.450
So it's a question of
how much energy there
00:24:02.450 --> 00:24:05.640
is inside this sphere or box.
00:24:05.640 --> 00:24:08.840
And if you imagine
a box changing size,
00:24:08.840 --> 00:24:11.560
if it's filled with a gas
with a positive pressure,
00:24:11.560 --> 00:24:13.940
as that box changes
size, the pressure
00:24:13.940 --> 00:24:15.640
does work on the boundary.
00:24:15.640 --> 00:24:18.110
If you think of it as a piston.
00:24:18.110 --> 00:24:20.850
And if we have a positive
pressure and a gas expands,
00:24:20.850 --> 00:24:22.700
it loses energy.
00:24:22.700 --> 00:24:25.155
And relativistically, that
means it has to also lose mass.
00:24:27.730 --> 00:24:30.170
The total mass inside the
box does not remain constant
00:24:30.170 --> 00:24:32.580
as it expands, which
is the idea that we
00:24:32.580 --> 00:24:36.200
use when we derive that
rho i over a cubed.
00:24:36.200 --> 00:24:38.870
So rho i over a cubed
is the right behavior
00:24:38.870 --> 00:24:41.240
for the total mass
density or energy
00:24:41.240 --> 00:24:43.805
density for a zero pressure gas.
00:24:43.805 --> 00:24:46.100
But when you include
pressure and take
00:24:46.100 --> 00:24:50.137
into account relativity, that's
not the right formula anymore.
00:24:50.137 --> 00:24:53.400
AUDIENCE: The top one, it
just cancels out somehow?
00:24:53.400 --> 00:24:55.570
PROFESSOR: Well, the pressure
ends up canceling out,
00:24:55.570 --> 00:24:57.390
so that this ends
up still being true
00:24:57.390 --> 00:24:59.212
and this ends up
being different.
00:24:59.212 --> 00:25:00.670
And we'll see later
how it happens.
00:25:00.670 --> 00:25:05.720
I just want to indicate where
the changes are going to be.
00:25:05.720 --> 00:25:08.439
We'll see what the changes
are when we get there.
00:25:08.439 --> 00:25:10.230
AUDIENCE: What happened
to the third factor
00:25:10.230 --> 00:25:12.070
of a in the second equation?
00:25:14.060 --> 00:25:16.060
PROFESSOR: There's a
factor of a missing, sorry.
00:25:19.170 --> 00:25:20.800
Yes, thank you very much.
00:25:20.800 --> 00:25:23.136
It's important to get
the equations right.
00:25:23.136 --> 00:25:24.510
That wasn't
Friedmann's equation.
00:25:24.510 --> 00:25:25.551
That was Alan's equation.
00:25:28.630 --> 00:25:31.280
Now it's Friedmann's equation.
00:25:31.280 --> 00:25:31.970
He got it right.
00:25:34.880 --> 00:25:38.250
OK, any other errors or
questions to bring up?
00:25:43.920 --> 00:25:44.420
OK.
00:25:47.350 --> 00:25:51.730
Now is probably a good time to
talk about the question of why
00:25:51.730 --> 00:25:54.970
we were so fortunate to discover
that the Friedmann equation
00:25:54.970 --> 00:25:57.730
that we derived agrees exactly
with general relativity.
00:25:57.730 --> 00:25:59.610
There is a simple reason for it.
00:25:59.610 --> 00:26:02.007
I don't think it's
an accident at all.
00:26:02.007 --> 00:26:03.590
The reason for it,
as I understand it,
00:26:03.590 --> 00:26:05.409
is that we assume
from the beginning--
00:26:05.409 --> 00:26:07.200
and we would be assuming
this whether we're
00:26:07.200 --> 00:26:08.840
talking about the
Newtonian calculation
00:26:08.840 --> 00:26:11.920
or the corresponding general
relativity calculation.
00:26:11.920 --> 00:26:13.610
We assume from the
beginning that we
00:26:13.610 --> 00:26:16.900
are modeling a completely
homogeneous system, where
00:26:16.900 --> 00:26:20.000
every part of it is identical
to every other part.
00:26:20.000 --> 00:26:22.730
And once you assume
that homogeneity,
00:26:22.730 --> 00:26:26.370
it means that if you know
what happens in a little box,
00:26:26.370 --> 00:26:28.162
a meter by a meter
by a meter say.
00:26:28.162 --> 00:26:29.870
If you know what
happens in a little box,
00:26:29.870 --> 00:26:31.720
you know what happens
everywhere because you assume
00:26:31.720 --> 00:26:34.136
that what happens everywhere
is exactly the same as what's
00:26:34.136 --> 00:26:36.570
happening in that box.
00:26:36.570 --> 00:26:38.890
So that implies
that if Newton is
00:26:38.890 --> 00:26:40.705
right for what
happens in the box,
00:26:40.705 --> 00:26:42.080
Newton has to be
right for what's
00:26:42.080 --> 00:26:44.390
happening in the universe.
00:26:44.390 --> 00:26:45.990
And we do expect
Newtonian physics
00:26:45.990 --> 00:26:49.030
to work on small scales,
scales of a meter,
00:26:49.030 --> 00:26:50.150
and small velocities.
00:26:50.150 --> 00:26:53.600
The Hubble expansion of
a meter is negligible.
00:26:53.600 --> 00:26:57.380
So we expect Newton to give
us a proper description of how
00:26:57.380 --> 00:27:00.220
the system is behaving
on small scales.
00:27:00.220 --> 00:27:01.670
And the assumption
of homogeneity
00:27:01.670 --> 00:27:04.150
guarantees that if you
understand the small scales,
00:27:04.150 --> 00:27:06.840
you also understand
the large scales.
00:27:06.840 --> 00:27:11.070
So I think we are guaranteed
that this had better give us
00:27:11.070 --> 00:27:13.570
the same results as
general relativity or else
00:27:13.570 --> 00:27:15.320
Newtonian physics is
not the proper limit
00:27:15.320 --> 00:27:16.380
of general relativity.
00:27:16.380 --> 00:27:17.597
But it is.
00:27:17.597 --> 00:27:19.180
We would not accept
general relativity
00:27:19.180 --> 00:27:21.530
if it did not give
Newtonian physics
00:27:21.530 --> 00:27:23.440
for small scales
and low velocities.
00:27:25.960 --> 00:27:30.530
So we expect to get the
same answer as Gr and we do.
00:27:30.530 --> 00:27:32.240
This is exactly
what Gr would give.
00:27:32.240 --> 00:27:34.580
And this is exactly what Gr
would give also for the case
00:27:34.580 --> 00:27:39.610
where there's no pressure--
the case we're doing.
00:27:39.610 --> 00:27:40.970
OK, any questions about that?
00:27:43.916 --> 00:27:45.090
OK, next item.
00:27:51.214 --> 00:27:52.630
I would like to
say a couple words
00:27:52.630 --> 00:27:54.129
about the units in
which we're going
00:27:54.129 --> 00:27:55.310
to define these quantities.
00:27:59.430 --> 00:28:03.280
So far, in our
mathematical model,
00:28:03.280 --> 00:28:07.035
r and r sub i are distances.
00:28:09.880 --> 00:28:12.730
And therefore, they're
measured in whatever units
00:28:12.730 --> 00:28:15.240
you're using to
measure distances.
00:28:15.240 --> 00:28:17.064
And I'll pretend
we're using SI units.
00:28:17.064 --> 00:28:17.855
So I'll say meters.
00:28:21.037 --> 00:28:22.620
We could use light
years, or whatever.
00:28:22.620 --> 00:28:23.860
It doesn't really matter.
00:28:23.860 --> 00:28:26.674
But they're both measured
in ordinary distance units.
00:28:26.674 --> 00:28:29.090
And earlier when we talked
about scale factors and things,
00:28:29.090 --> 00:28:32.060
I told you that many
books do it this way.
00:28:32.060 --> 00:28:35.680
Think of both the
co-moving coordinates
00:28:35.680 --> 00:28:37.950
and the physical distances
as being measured
00:28:37.950 --> 00:28:40.975
in meters and the scale
factor being dimensionless.
00:28:40.975 --> 00:28:42.850
But I told you I don't
like to do it that way
00:28:42.850 --> 00:28:44.910
because I think it's
clearer to recognize
00:28:44.910 --> 00:28:47.320
that these co-moving
coordinates don't have
00:28:47.320 --> 00:28:49.800
any real relationship
to actual distances.
00:28:49.800 --> 00:28:53.010
At least not as time changes.
00:28:53.010 --> 00:28:55.080
So for me, it's better
to have a different unit
00:28:55.080 --> 00:28:58.080
to describe the co-moving
coordinate systems.
00:28:58.080 --> 00:29:00.260
So I would like to
introduce that here.
00:29:00.260 --> 00:29:11.550
And all I need to do really is
say that let the unit of r sub
00:29:11.550 --> 00:29:16.770
i be called a notch.
00:29:20.760 --> 00:29:22.510
Now, we already
called it a meter,
00:29:22.510 --> 00:29:23.900
but that doesn't really
mean that a meter is
00:29:23.900 --> 00:29:24.790
the same as a notch.
00:29:24.790 --> 00:29:27.117
Because when we
called it a meter,
00:29:27.117 --> 00:29:28.700
we were not really
taking into account
00:29:28.700 --> 00:29:33.770
the fact that it's only
a meter at time t sub i.
00:29:33.770 --> 00:29:38.390
So another way of
describing this definition,
00:29:38.390 --> 00:29:41.670
which might not sound like we're
trying to redefine the meter,
00:29:41.670 --> 00:29:52.327
is to say that the statement r
sub i equals 5 notches, which
00:29:52.327 --> 00:29:54.743
is the kind of statement I'm
going to make now because I'm
00:29:54.743 --> 00:29:57.410
only going to talk about r
sub i measured in notches.
00:29:57.410 --> 00:29:59.910
When I say r sub i
is 5 notches, that's
00:29:59.910 --> 00:30:13.000
equivalent to saying that the
particle labeled by r sub i
00:30:13.000 --> 00:30:32.800
equals 5-- 5 notches-- was
at 5 meters from the origin
00:30:32.800 --> 00:30:36.040
at time t sub i.
00:30:39.480 --> 00:30:42.954
So giving the value of a
co-moving coordinate in notches
00:30:42.954 --> 00:30:44.370
tells you exactly
what distance it
00:30:44.370 --> 00:30:48.470
was from the origin
at time t sub i.
00:30:48.470 --> 00:30:54.104
Now, the reason why I don't
use this to just say, well,
00:30:54.104 --> 00:30:56.020
why don't we call it
meters, is that we're now
00:30:56.020 --> 00:30:57.810
going to forget about t sub i.
00:30:57.810 --> 00:30:59.220
If you look at
equations we have,
00:30:59.220 --> 00:31:01.580
t sub i no longer
appears in any of them.
00:31:01.580 --> 00:31:03.690
t sub i was just our
way of getting started.
00:31:03.690 --> 00:31:06.310
And we could have started
at any time we wanted.
00:31:06.310 --> 00:31:08.786
And once we have
these equations,
00:31:08.786 --> 00:31:10.535
we can talk about times
earlier than t sub
00:31:10.535 --> 00:31:12.240
i, times later than t sub i.
00:31:12.240 --> 00:31:16.180
And there's nothing special
about t sub i anymore.
00:31:16.180 --> 00:31:19.020
But r sub i we're going to
keep as our permanent label
00:31:19.020 --> 00:31:21.640
for every shell, which means
for every particle there will be
00:31:21.640 --> 00:31:26.030
a value of r sub i attached
to that particle which will be
00:31:26.030 --> 00:31:28.260
maintained as the
system evolves.
00:31:28.260 --> 00:31:30.750
And it will be clearly
playing the role
00:31:30.750 --> 00:31:33.830
of what we've called the
co-moving coordinate.
00:31:33.830 --> 00:31:37.000
And therefore, we will
want to keep r sub i
00:31:37.000 --> 00:31:39.470
and we'll want to call
its unit something.
00:31:39.470 --> 00:31:41.630
And I'm just saying I'm
going to call them notches.
00:31:41.630 --> 00:31:42.850
You can call them
meters if you want,
00:31:42.850 --> 00:31:44.308
but I'm going to
call them notches.
00:31:50.090 --> 00:31:58.630
So using this language, I'm not
changing any of the equations
00:31:58.630 --> 00:32:00.420
that we wrote.
00:32:00.420 --> 00:32:09.730
So we will still have r being
equal to a of t times r sub i.
00:32:09.730 --> 00:32:13.250
But now, r will be
measured in meters.
00:32:13.250 --> 00:32:18.250
a of t I will think of
as meters per notch.
00:32:18.250 --> 00:32:20.320
And r sub i will me
measured in notches.
00:32:25.605 --> 00:32:27.980
And the scale factor a of t
will be playing the same role
00:32:27.980 --> 00:32:30.021
it played ever since we
introduced the word scale
00:32:30.021 --> 00:32:30.630
factor.
00:32:30.630 --> 00:32:32.671
It just means that when
the scale factor doubles,
00:32:32.671 --> 00:32:35.050
all distances in our
model double exactly.
00:32:40.970 --> 00:32:44.310
Now that we have the sort
of new system of units,
00:32:44.310 --> 00:32:48.450
I just want to work out
the units of an object
00:32:48.450 --> 00:32:49.970
where the units are
not all obvious.
00:32:49.970 --> 00:32:52.020
What are the units of
this thing that we've
00:32:52.020 --> 00:32:54.650
defined that we called k,
which is related to this thing
00:32:54.650 --> 00:32:57.170
that we defined that was called
E, where E was not really
00:32:57.170 --> 00:32:58.490
an energy?
00:32:58.490 --> 00:32:59.795
But we can work out what k is.
00:33:02.692 --> 00:33:08.070
I'm using square brackets
to mean units of.
00:33:08.070 --> 00:33:10.736
k can be thought of as being
defined by this equation.
00:33:10.736 --> 00:33:12.360
Or in any case, this
equation certainly
00:33:12.360 --> 00:33:14.690
must be dimensionally
consistent because we derived
00:33:14.690 --> 00:33:17.135
it and you didn't point out
that I made any mistakes,
00:33:17.135 --> 00:33:19.850
so I must not have.
00:33:19.850 --> 00:33:22.490
So the units of kc
squared over a squared
00:33:22.490 --> 00:33:24.490
should be the same as the
units of a dot squared
00:33:24.490 --> 00:33:26.140
over a squared.
00:33:26.140 --> 00:33:30.580
And if I multiply through
to write units of k--
00:33:30.580 --> 00:33:35.170
to relate them to a dot,
we get the units of k
00:33:35.170 --> 00:33:37.270
have to be the same
as the units of 1/c
00:33:37.270 --> 00:33:39.276
squared times a dot squared.
00:33:41.794 --> 00:33:43.710
The a squareds in
the denominator
00:33:43.710 --> 00:33:45.540
here just cancel each other.
00:33:45.540 --> 00:33:48.010
So the units of k has to be
the same as the units of 1/c
00:33:48.010 --> 00:33:50.030
squared a dot squared.
00:33:50.030 --> 00:33:51.890
And we know what they are.
00:33:51.890 --> 00:33:55.520
The units of 1/c squared
is second squared
00:33:55.520 --> 00:33:58.350
per meter squared.
00:33:58.350 --> 00:34:01.350
Meter per second
squared, but upside down.
00:34:01.350 --> 00:34:03.250
That's the 1/c squared factor.
00:34:03.250 --> 00:34:10.050
And a dot is meters per notch
per second because of the dot.
00:34:10.050 --> 00:34:19.360
So meters per notch would be a.
00:34:19.360 --> 00:34:21.790
a dot would have
an extra second.
00:34:21.790 --> 00:34:25.110
And the whole thing gets
squared because it's
00:34:25.110 --> 00:34:26.100
a dot squared here.
00:34:29.469 --> 00:34:33.650
And you see then that
the meter squares cancel.
00:34:33.650 --> 00:34:35.850
The second squareds cancel.
00:34:35.850 --> 00:34:40.680
And we just get the peculiar
answer 1 over a notch squared.
00:34:46.540 --> 00:34:49.239
Now, there's probably not a
lot of intuition behind that.
00:34:49.239 --> 00:34:51.159
But what it clearly
does say is that we
00:34:51.159 --> 00:34:54.380
can make k change its
numerical value by changing
00:34:54.380 --> 00:34:56.540
our value for the notch.
00:34:56.540 --> 00:34:57.900
And that's important to know.
00:34:57.900 --> 00:35:01.724
And it's a clear consequence
of what we just did.
00:35:01.724 --> 00:35:03.640
So now we can talk about
different conventions
00:35:03.640 --> 00:35:06.080
that people use for
defining the notch.
00:35:06.080 --> 00:35:09.580
And hence, k, which are
clearly related to each other
00:35:09.580 --> 00:35:10.410
we've now learned.
00:35:16.680 --> 00:35:29.770
So first of all, in the
construction that we just
00:35:29.770 --> 00:35:33.660
did-- starting out with our
sphere, and letting it grow,
00:35:33.660 --> 00:35:37.660
and defining the maximum sphere
as r max comma i and so on.
00:35:37.660 --> 00:35:44.750
In that construction, our
initial value of t-- excuse
00:35:44.750 --> 00:35:48.830
me, our initial value
of a, a of ti, was one.
00:35:48.830 --> 00:35:51.350
And the way we
first did it where
00:35:51.350 --> 00:35:54.170
all lengths were
measured in meters.
00:35:54.170 --> 00:35:58.525
And with our new definition that
r sub i is measured in notches
00:35:58.525 --> 00:36:01.160
so that a is meters
per notch, it
00:36:01.160 --> 00:36:05.360
means that in the system
we just had, a of t sub
00:36:05.360 --> 00:36:07.285
i was 1 meter per notch.
00:36:17.670 --> 00:36:20.859
And since we
already did this, we
00:36:20.859 --> 00:36:22.150
don't really want to change it.
00:36:22.150 --> 00:36:27.330
But I point out that t sub i,
if you look at our equations,
00:36:27.330 --> 00:36:30.840
survives in only one place,
which is in this equation.
00:36:30.840 --> 00:36:32.700
It has disappeared from
every other equation
00:36:32.700 --> 00:36:34.300
we're going to be keeping.
00:36:34.300 --> 00:36:37.342
So we are perfectly safe in just
forgetting about this equation.
00:36:37.342 --> 00:36:38.800
Or if we want to
remember about it,
00:36:38.800 --> 00:36:41.417
we could just say, well, yeah,
t sub i had some significance.
00:36:41.417 --> 00:36:43.500
It was the time at which
the scale factor happened
00:36:43.500 --> 00:36:45.245
to have been equal to
1 meters per notch.
00:36:45.245 --> 00:36:46.745
But otherwise, it
has no importance.
00:36:46.745 --> 00:36:48.860
There's a different time
when the scale factor was
00:36:48.860 --> 00:36:53.280
10 meters per notch, or
1 light year per notch.
00:36:53.280 --> 00:37:09.790
So since t sub i is of
no relevance whatever,
00:37:09.790 --> 00:37:16.440
we can safely forget
the above equation.
00:37:20.480 --> 00:37:22.689
Or we could think of it as
the definition of t sub i,
00:37:22.689 --> 00:37:24.688
where we don't care
anything else about t sub i.
00:37:24.688 --> 00:37:26.230
So it was just some
symbol that was
00:37:26.230 --> 00:37:27.604
used in some
earlier calculation.
00:37:31.460 --> 00:37:35.630
So we now just have a
scale factor a of t.
00:37:35.630 --> 00:37:38.310
And we can talk about how
we might normalize it.
00:37:41.469 --> 00:37:43.510
And there are basically
two important conventions
00:37:43.510 --> 00:37:46.240
that are in use in textbooks.
00:37:46.240 --> 00:37:50.130
Some textbooks, which include
Barbara Ryden's textbook
00:37:50.130 --> 00:37:54.790
that we're using,
define a of t to be
00:37:54.790 --> 00:38:01.450
equal to 1, which I will call
1 meter per not much today.
00:38:06.040 --> 00:38:10.080
So a of t is just defined to be
1 today as a common notation.
00:38:10.080 --> 00:38:12.430
It's notation that
Barbara Ryden uses.
00:38:12.430 --> 00:38:14.975
And that makes a notch
equal to a meter today.
00:38:27.990 --> 00:38:35.460
There's another
common convention,
00:38:35.460 --> 00:38:38.920
which is to recognize that since
k has units 1 over notches,
00:38:38.920 --> 00:38:41.183
we can make k any value we
want without changing any
00:38:41.183 --> 00:38:43.720
of the physics, just changing
our definition of the notch,
00:38:43.720 --> 00:38:45.350
which is up for grabs.
00:38:45.350 --> 00:38:47.690
Nobody has yet
defined the notch.
00:38:47.690 --> 00:38:49.580
We're defining it now.
00:38:49.580 --> 00:38:53.380
So we can choose the
definition of a notch
00:38:53.380 --> 00:38:56.510
to make the value of
k something simple.
00:38:56.510 --> 00:38:59.510
And the obvious choice for
the simplest real number
00:38:59.510 --> 00:39:03.170
that you could imagine
that's not 0 is 1.
00:39:03.170 --> 00:39:04.650
So we could choose
the definition
00:39:04.650 --> 00:39:06.760
of the notch to
make k equal to 1.
00:39:06.760 --> 00:39:08.830
Except that we can't
change the sign of k.
00:39:08.830 --> 00:39:13.250
The sign of k makes important
differences in this equation.
00:39:13.250 --> 00:39:15.904
And we don't want to make the
definition of a notch negative
00:39:15.904 --> 00:39:17.320
or imaginary, I
guess, is what you
00:39:17.320 --> 00:39:19.300
need to change the sign of k.
00:39:19.300 --> 00:39:21.980
So as long as
notches are real, we
00:39:21.980 --> 00:39:24.710
can only change positive k's
to different positive k's
00:39:24.710 --> 00:39:29.410
and negative k's to different
values of negative k.
00:39:29.410 --> 00:39:36.710
So the convention would be
that when k is not equal to 0--
00:39:36.710 --> 00:39:38.260
it can be 0 as a special case.
00:39:38.260 --> 00:39:44.740
But when k is not equal
to 0, define the notch
00:39:44.740 --> 00:39:48.912
so that k is equal
to plus or minus 1.
00:39:48.912 --> 00:39:50.370
I would say that
this convention is
00:39:50.370 --> 00:39:52.245
more common than that
convention of the books
00:39:52.245 --> 00:39:54.720
that I've read in my
lifetime, but both conventions
00:39:54.720 --> 00:39:55.350
are in use.
00:39:58.890 --> 00:40:02.330
And one sees from this
dimensional relationship
00:40:02.330 --> 00:40:04.410
that one can certainly
choose a notch
00:40:04.410 --> 00:40:08.000
to make k if it's nonzero
have any value you
00:40:08.000 --> 00:40:09.397
want of the same sign.
00:40:09.397 --> 00:40:11.480
And that means you can
always make k plus or minus
00:40:11.480 --> 00:40:15.320
1 if it's not 0.
00:40:15.320 --> 00:40:19.700
The books that use this as their
convention, generally speaking,
00:40:19.700 --> 00:40:23.800
leave the notch undefined
when k equals 0.
00:40:36.720 --> 00:40:39.310
Undefined means arbitrary.
00:40:39.310 --> 00:40:42.420
And there's no problem with
that because k and the notch
00:40:42.420 --> 00:40:45.650
never really appear in
final physical quantities.
00:40:45.650 --> 00:40:47.500
The notch always
was just your choice
00:40:47.500 --> 00:40:50.910
of how to write down your
co-moving map of what
00:40:50.910 --> 00:40:51.910
the universe looks like.
00:40:54.730 --> 00:40:57.375
OK, any questions about these
funny issues involving units?
00:41:04.030 --> 00:41:04.630
OK, good.
00:41:04.630 --> 00:41:06.860
The next thing I
want to do now is
00:41:06.860 --> 00:41:10.940
to start talking about
solutions to this equation.
00:41:10.940 --> 00:41:16.044
And I guess I'll leave the
equation there and start
00:41:16.044 --> 00:41:16.960
on the new blackboard.
00:41:41.650 --> 00:41:46.712
OK, I'm going to rewrite
the equation almost the way
00:41:46.712 --> 00:41:48.550
it was written on the top there.
00:41:48.550 --> 00:41:50.570
In fact, exactly as it
was written on top there.
00:41:50.570 --> 00:41:52.610
I'm going to
rewrite the equation
00:41:52.610 --> 00:42:06.790
as E equals 1/2 a dot squared
minus 4 pi over 3 G rho sub i
00:42:06.790 --> 00:42:09.430
over a.
00:42:09.430 --> 00:42:14.010
And the reason I'm writing it
in this way rather than any
00:42:14.010 --> 00:42:16.460
of the other six or seven
ways that we've written it,
00:42:16.460 --> 00:42:20.065
is that this way the only
time-varying thing is a itself.
00:42:20.065 --> 00:42:22.150
And if we want to talk
about what the differential
00:42:22.150 --> 00:42:24.700
equation tells us about
the time variation of a,
00:42:24.700 --> 00:42:27.240
it helps a lot if we're writing
an equation where a of t
00:42:27.240 --> 00:42:29.170
is the only thing
that varies with time.
00:42:29.170 --> 00:42:33.010
And this is at least
one way of doing that.
00:42:33.010 --> 00:42:37.930
So in particular, I used
rho sub i rather than rho.
00:42:37.930 --> 00:42:46.130
So the behavior of this
equation might very well
00:42:46.130 --> 00:42:50.280
depend on the sign of E.
And we'll see that it does.
00:42:50.280 --> 00:42:53.460
And if we think it might
depend on the sign of E,
00:42:53.460 --> 00:42:55.180
we realize from the
beginning that E
00:42:55.180 --> 00:42:57.230
could be positive,
negative, or 0.
00:42:57.230 --> 00:42:59.320
There are those three cases.
00:42:59.320 --> 00:43:01.070
So those are the cases
we want to look at.
00:43:07.880 --> 00:43:12.086
E can be positive,
negative, or 0.
00:43:12.086 --> 00:43:13.460
So we'll take them
one at a time.
00:43:16.720 --> 00:43:19.930
It will help to make things
completely obvious to rewrite
00:43:19.930 --> 00:43:23.450
this as an equation
for a dot squared.
00:43:23.450 --> 00:43:25.510
I'll multiply by 2.
00:43:25.510 --> 00:43:27.850
And write this as
equation a dot squared
00:43:27.850 --> 00:43:38.480
is equal to 2E plus 8 pi
over 3 G rho sub i over a.
00:43:38.480 --> 00:43:40.630
8 pi instead of 4 pi
because we multiplied by 2.
00:43:45.290 --> 00:43:50.030
And we notice that this term,
proportional to rho sub i
00:43:50.030 --> 00:43:52.316
over a, is
unambiguously positive.
00:43:52.316 --> 00:43:54.440
We're not going to have
any negative mass densities
00:43:54.440 --> 00:43:55.170
in our problem.
00:43:55.170 --> 00:43:56.660
There's no way that can happen.
00:43:56.660 --> 00:43:58.950
And a is always positive.
00:43:58.950 --> 00:44:00.870
So the right term is positive.
00:44:00.870 --> 00:44:02.550
a dot squared had
better be positive
00:44:02.550 --> 00:44:04.580
because it's the
square of something.
00:44:04.580 --> 00:44:06.310
E could, in principle,
have either sign.
00:44:06.310 --> 00:44:09.090
And we'll talk about both
cases, or the 0 case.
00:44:09.090 --> 00:44:11.250
But if we start with the
case where E is positive,
00:44:11.250 --> 00:44:13.560
just to consider these
three cases one at a time.
00:44:19.940 --> 00:44:22.750
So suppose E is greater than 0.
00:44:22.750 --> 00:44:26.240
And remind you that E
and k had opposite signs.
00:44:26.240 --> 00:44:30.584
So that would imply
that k was negative.
00:44:30.584 --> 00:44:33.000
So if we start by considering
the k negative case or the E
00:44:33.000 --> 00:44:37.370
positive case, then we see that
we have a positive number here
00:44:37.370 --> 00:44:39.310
and a positive number there.
00:44:39.310 --> 00:44:42.280
So they will add up to always
give us a positive number.
00:44:42.280 --> 00:44:44.320
a dot squared will
always be positive.
00:44:44.320 --> 00:44:48.031
And it will just mean that
a dot will be positive.
00:44:48.031 --> 00:44:50.280
Square root of a positive
number is a positive number.
00:44:50.280 --> 00:44:54.040
At least it's only the positive
square root that matters here.
00:44:54.040 --> 00:44:58.030
So a will just keep growing
forever in this case.
00:45:00.900 --> 00:45:03.350
a dot squared will
never fall below 2E.
00:45:03.350 --> 00:45:05.989
So it will be a lower
bound to a dot squared.
00:45:05.989 --> 00:45:08.280
And that means there will
always be a minimum expansion
00:45:08.280 --> 00:45:11.040
rate that the
universe will have.
00:45:11.040 --> 00:45:19.640
So in this case, a
increases forever.
00:45:26.332 --> 00:45:27.790
And that's called
an open universe.
00:45:36.284 --> 00:45:38.481
And it's one of the
three possibilities
00:45:38.481 --> 00:45:40.230
that we're going to
be investigating here.
00:46:03.910 --> 00:46:07.470
Next case is E
less than 0, which
00:46:07.470 --> 00:46:13.700
is the more common notation of
k means k is greater than 0.
00:46:16.770 --> 00:46:20.110
In this case, if you think of
E as an energy, which it really
00:46:20.110 --> 00:46:22.970
is, it means we have
less than 0 energy.
00:46:22.970 --> 00:46:25.472
Which means that we basically
have a bound system.
00:46:25.472 --> 00:46:27.930
And the equation tells us that
it acts like a bound system.
00:46:27.930 --> 00:46:29.810
We don't have to rely
on that intuition,
00:46:29.810 --> 00:46:32.050
but that is the right intuition.
00:46:32.050 --> 00:46:37.451
The equation up there tells
us that if E is negative,
00:46:37.451 --> 00:46:39.450
the total right-hand side
had better be positive
00:46:39.450 --> 00:46:41.991
because the left-hand side is
positive and the left-hand side
00:46:41.991 --> 00:46:43.049
cannot go negative.
00:46:43.049 --> 00:46:45.090
But this term is going to
get smaller and smaller
00:46:45.090 --> 00:46:46.990
as a increases.
00:46:46.990 --> 00:46:48.850
And as this term gets
smaller and smaller,
00:46:48.850 --> 00:46:51.670
it runs the risk of no
longer outweighing this term
00:46:51.670 --> 00:46:54.230
and giving possibly
a negative answer.
00:46:54.230 --> 00:46:58.350
And what has to happen is a
cannot get any bigger than
00:46:58.350 --> 00:47:00.700
the value it would have where
the right-hand side would
00:47:00.700 --> 00:47:02.150
vanish.
00:47:02.150 --> 00:47:05.440
So a continues to grow
because a dot is positive.
00:47:05.440 --> 00:47:08.550
This gets smaller and
smaller until this term
00:47:08.550 --> 00:47:10.020
equals that term in magnitude.
00:47:10.020 --> 00:47:13.230
And then, a dot goes to 0.
00:47:13.230 --> 00:47:15.090
What happens next
is not completely
00:47:15.090 --> 00:47:16.540
obvious from this
equation, but it
00:47:16.540 --> 00:47:18.740
means that we have an
expanding universe that's
00:47:18.740 --> 00:47:21.570
reached a maximum
size and then stopped.
00:47:21.570 --> 00:47:25.190
Then, what is actually
obvious is from this equation
00:47:25.190 --> 00:47:28.650
is that it will
start to collapse.
00:47:28.650 --> 00:47:31.160
So this case corresponds
to a universe that
00:47:31.160 --> 00:47:33.995
reaches a maximum size and then
turns around and collapses.
00:47:45.140 --> 00:47:52.970
So a has a maximum value.
00:47:52.970 --> 00:47:55.740
And we can read off
from that equation what
00:47:55.740 --> 00:47:58.700
it is, a max is
just the value that
00:47:58.700 --> 00:48:01.080
makes the right-hand
side of that equation 0,
00:48:01.080 --> 00:48:08.555
which is minus 4 pi G
rho sub i divided by 3E.
00:48:11.325 --> 00:48:13.290
And remember, E is
negative for this case,
00:48:13.290 --> 00:48:15.120
so this is a positive number.
00:48:15.120 --> 00:48:18.260
So a has some positive
maximum value.
00:48:18.260 --> 00:48:21.430
Reaches that value, and then
turns around and collapses.
00:48:21.430 --> 00:48:21.949
Yes?
00:48:21.949 --> 00:48:24.240
AUDIENCE: Sorry, I don't know
if you said this already,
00:48:24.240 --> 00:48:27.996
but since a dot squared is
equal to some quantity, when you
00:48:27.996 --> 00:48:31.600
solve for a dot, you can have
positive and negative solution.
00:48:31.600 --> 00:48:34.634
Why do we discount
the negative solution?
00:48:34.634 --> 00:48:36.050
PROFESSOR: OK,
very good question.
00:48:36.050 --> 00:48:37.466
The question if
you didn't hear it
00:48:37.466 --> 00:48:40.040
is, why do we discount
the negative solution
00:48:40.040 --> 00:48:43.380
when we have an equation
for a dot squared?
00:48:43.380 --> 00:48:45.770
Couldn't a dot be
positive or negative?
00:48:45.770 --> 00:48:48.100
And the answer is it
certainly could be either.
00:48:48.100 --> 00:48:50.380
And both solutions
exist as valid solutions
00:48:50.380 --> 00:48:52.080
to these equations.
00:48:52.080 --> 00:48:54.640
But we started out with
an initial condition
00:48:54.640 --> 00:48:55.970
that a dot was positive.
00:48:55.970 --> 00:48:58.780
Our initial value
of a dot was H i.
00:48:58.780 --> 00:49:00.910
And once it's positive,
it can't change sign
00:49:00.910 --> 00:49:02.130
according to that equation.
00:49:02.130 --> 00:49:03.680
Except by going
through 0, which is
00:49:03.680 --> 00:49:04.930
what we're talking about now.
00:49:04.930 --> 00:49:07.320
But it will only change
sign when it goes through 0.
00:49:16.330 --> 00:49:20.420
So it reaches a maximum
value, then it does collapse.
00:49:20.420 --> 00:49:22.420
And in the collapsing
phase, that same equation,
00:49:22.420 --> 00:49:23.900
a dot squared equals
the right-hand side,
00:49:23.900 --> 00:49:25.990
holds where it would be
the negative solution that
00:49:25.990 --> 00:49:27.281
describes the collapsing phase.
00:49:33.320 --> 00:49:37.480
So the verbal description
of what's happening here
00:49:37.480 --> 00:49:52.830
is that the universe
reaches a maximum size
00:49:52.830 --> 00:49:54.330
and then collapses.
00:49:59.000 --> 00:50:02.690
And it collapses all the way
to a equals 0 in this model.
00:50:02.690 --> 00:50:06.490
And that's often
called the Big Crunch
00:50:06.490 --> 00:50:08.145
for lack of a better word.
00:50:12.520 --> 00:50:15.090
The Big Crunch being
the collapsing form that
00:50:15.090 --> 00:50:17.370
corresponds to the
Big Bang, which
00:50:17.370 --> 00:50:20.750
is the instant which
all this starts.
00:50:26.860 --> 00:50:30.000
And this was called
an open universe.
00:50:30.000 --> 00:50:31.944
As you could
probably guess, this
00:50:31.944 --> 00:50:33.110
is called a closed universe.
00:51:23.480 --> 00:51:24.490
OK.
00:51:24.490 --> 00:51:28.830
And now finally, we want
to consider a case where
00:51:28.830 --> 00:51:31.350
E is not positive
and not negative.
00:51:31.350 --> 00:51:33.310
The case that we're
left with is E equals 0.
00:51:36.770 --> 00:51:39.370
And that's called
the critical case.
00:51:43.430 --> 00:51:48.090
So the critical value
for E is E is equal to 0.
00:51:48.090 --> 00:51:50.140
And that means that k
is equal to 0 as well.
00:51:52.725 --> 00:51:55.660
It implies k equals 0.
00:51:55.660 --> 00:51:58.120
And notice that this
is a special case.
00:51:58.120 --> 00:51:59.009
E is a real number.
00:51:59.009 --> 00:52:00.800
It can be positive,
negative, and 0 is just
00:52:00.800 --> 00:52:05.620
a particular value on the
borderline between those two.
00:52:05.620 --> 00:52:08.920
For the people who are in the
habit of rescaling notches
00:52:08.920 --> 00:52:11.950
so that k is always
plus 1, minus 1, or 0,
00:52:11.950 --> 00:52:16.060
it makes it sound like there are
three totally distinct cases.
00:52:16.060 --> 00:52:17.760
But that's only because
of the rescaling
00:52:17.760 --> 00:52:19.480
that those people are doing.
00:52:19.480 --> 00:52:22.150
If you keep track of E as your
variable, which you certainly
00:52:22.150 --> 00:52:25.700
can, you do see that
the flat case, E equals
00:52:25.700 --> 00:52:28.229
0-- the critical
case-- is really
00:52:28.229 --> 00:52:29.770
just the borderline
of the other two.
00:52:29.770 --> 00:52:32.890
And it's therefore, arbitrarily
close to both of the other two.
00:52:32.890 --> 00:52:36.340
It really is where they meet.
00:52:36.340 --> 00:52:43.800
But working out the equations,
we have in this case
00:52:43.800 --> 00:52:52.130
a dot over a squared is
equal to 8 pi over 3 G rho.
00:52:52.130 --> 00:52:55.850
And in general, it's minus
kc squared over a squared.
00:52:55.850 --> 00:52:59.980
But we're now considering
the case where that vanishes.
00:52:59.980 --> 00:53:01.630
And that means we
have a unique value
00:53:01.630 --> 00:53:05.180
for rho in terms
of a dot over a.
00:53:05.180 --> 00:53:07.830
And at this point, it's
worth reminding ourselves
00:53:07.830 --> 00:53:12.910
that a dot over a is just
H. So this is H squared.
00:53:12.910 --> 00:53:14.389
So for this critical
case, the case
00:53:14.389 --> 00:53:16.930
that's just on the borderline
between being open and closed--
00:53:16.930 --> 00:53:19.300
and we'll be calling it flat.
00:53:19.300 --> 00:53:21.810
For this critical case, we
have a definite relationship
00:53:21.810 --> 00:53:30.540
between rho and H.
So rho has to equal
00:53:30.540 --> 00:53:33.220
what we call the
critical density, which
00:53:33.220 --> 00:53:35.220
you get by just
solving that equation.
00:53:35.220 --> 00:53:44.020
And it's 3H squared over 8 pi G.
00:53:44.020 --> 00:53:47.490
And we see, therefore, that
rho being equal to rho c
00:53:47.490 --> 00:53:50.220
is this dividing line
between open and closed.
00:53:50.220 --> 00:53:53.140
And if you think back about
the signs of what we had,
00:53:53.140 --> 00:53:54.850
what you'll see is
that rho bigger rho
00:53:54.850 --> 00:53:57.660
c is what corresponds
to a closed universe.
00:54:02.000 --> 00:54:05.310
Rho less than rho c
is what corresponds
00:54:05.310 --> 00:54:06.190
to an open universe.
00:54:10.570 --> 00:54:13.770
And rho equals rho
c can be called
00:54:13.770 --> 00:54:15.470
either a critical
universe or we'll
00:54:15.470 --> 00:54:17.790
be calling it a flat universe.
00:54:17.790 --> 00:54:25.750
And the meaning of the word
"flat" will be motivated later.
00:54:25.750 --> 00:54:28.550
For now, these words--
open, closed, and flat--
00:54:28.550 --> 00:54:30.704
refer to the time
evolution of the universe.
00:54:30.704 --> 00:54:32.370
We'll see later that
it's also connected
00:54:32.370 --> 00:54:35.720
to the geometry of the universe,
but we're not there yet.
00:54:35.720 --> 00:54:37.920
Then the word "flat"
will make some sense.
00:54:37.920 --> 00:54:38.420
Yes.
00:54:38.420 --> 00:54:42.475
AUDIENCE: How do we know there's
not some very large entity,
00:54:42.475 --> 00:54:44.350
like some cluster of
black holes or something
00:54:44.350 --> 00:54:47.807
that renders all of this not
applicable to our universe?
00:54:47.807 --> 00:54:48.390
PROFESSOR: OK.
00:54:48.390 --> 00:54:50.181
The question is, how
do we know there's not
00:54:50.181 --> 00:54:52.370
some humongous perturbation,
some huge collection
00:54:52.370 --> 00:54:54.490
of black holes that
renders this all
00:54:54.490 --> 00:54:56.570
inapplicable to our universe?
00:54:56.570 --> 00:54:58.970
The answer is that it
works for our universe.
00:54:58.970 --> 00:55:02.130
That is, observationally
we can test these things
00:55:02.130 --> 00:55:03.430
in a number of ways.
00:55:03.430 --> 00:55:07.180
Tests include calculations
of the production
00:55:07.180 --> 00:55:09.310
of the light chemical
elements in the Big Bang.
00:55:09.310 --> 00:55:12.030
Tests include making predictions
for what the cosmic background
00:55:12.030 --> 00:55:13.950
radiation should
look like in detail.
00:55:13.950 --> 00:55:16.570
And those tests work
extraordinarily well.
00:55:16.570 --> 00:55:19.437
So that's why we
believe the picture.
00:55:19.437 --> 00:55:21.020
But you're right,
we don't have really
00:55:21.020 --> 00:55:24.470
direct confirmation
of most of this.
00:55:24.470 --> 00:55:27.230
And if there was some
giant conglomeration
00:55:27.230 --> 00:55:30.662
of mass out there someplace, it
might not have been found yet.
00:55:30.662 --> 00:55:32.370
But so far, this
picture works very well.
00:55:32.370 --> 00:55:33.937
That's all I can say.
00:55:33.937 --> 00:55:35.770
And there really is
quite a bit of evidence.
00:55:35.770 --> 00:55:37.311
We'll maybe talk
more about it later.
00:55:40.482 --> 00:55:41.315
Any other questions?
00:55:45.380 --> 00:55:47.070
OK.
00:55:47.070 --> 00:55:50.900
So having understood
the importance
00:55:50.900 --> 00:55:53.560
of this critical
density, it might
00:55:53.560 --> 00:55:56.710
be nice to know what the
value for the critical density
00:55:56.710 --> 00:55:59.279
for our universe is.
00:55:59.279 --> 00:56:01.820
And we can calculate it because
it just depends on the Hubble
00:56:01.820 --> 00:56:03.640
expansion rate, an the
Hubble expansion rate
00:56:03.640 --> 00:56:04.389
has been measured.
00:56:09.280 --> 00:56:12.270
So if we try to put
in numbers, it's
00:56:12.270 --> 00:56:15.430
useful to write the present
value of the Hubble expansion
00:56:15.430 --> 00:56:22.510
rate, as it's often
written, as 100 times h sub
00:56:22.510 --> 00:56:28.246
0 kilometers per
second per megaparsec.
00:56:32.780 --> 00:56:36.070
So this defines h sub
0, little h sub 0.
00:56:36.070 --> 00:56:38.590
And I think the main advantage
of using this notation
00:56:38.590 --> 00:56:41.090
is that you don't have to keep
writing kilometers per second
00:56:41.090 --> 00:56:43.890
per megaparsec which gets to
be a real pain to keep writing.
00:56:43.890 --> 00:56:46.570
So little h sub 0 is just
a dimensionless number
00:56:46.570 --> 00:56:50.040
that defines the
Hubble expansion rate.
00:56:50.040 --> 00:56:52.950
And it does then allow you
to write other formulas
00:56:52.950 --> 00:56:54.800
in simple ways.
00:56:54.800 --> 00:56:57.880
Numerically, Newton's
constant you can look up.
00:56:57.880 --> 00:57:05.380
It's 6.672 times
10 to the minus 8
00:57:05.380 --> 00:57:10.580
centimeter cubed per
gram per second squared.
00:57:13.541 --> 00:57:16.165
And when you put these equations
together, all you need to know
00:57:16.165 --> 00:57:18.920
is G and H squared to
know what rho critical is.
00:57:18.920 --> 00:57:23.830
You find that rho critical can
be written initially for any H0
00:57:23.830 --> 00:57:32.110
as 1.88 h0 squared
coming from the h squared
00:57:32.110 --> 00:57:42.000
in the original formula times
10 to the minus 29 grams
00:57:42.000 --> 00:57:42.975
per centimeter cubed.
00:57:47.890 --> 00:57:53.570
And note the whopping smallness
of that 10 to the minus 29.
00:57:53.570 --> 00:57:55.840
The mass density of
our universe is, as far
00:57:55.840 --> 00:57:57.922
as we know, equal to
this critical density.
00:57:57.922 --> 00:57:59.630
We know it's equal to
within about a half
00:57:59.630 --> 00:58:02.350
of a percent or so.
00:58:02.350 --> 00:58:04.550
And h0 is near 1.
00:58:04.550 --> 00:58:10.130
h0, according to
Planck, is 0.67--
00:58:10.130 --> 00:58:11.690
according to the
Planck satellite
00:58:11.690 --> 00:58:15.410
measurement of the
Hubble parameter
00:58:15.410 --> 00:58:22.810
And if you put
that into here, you
00:58:22.810 --> 00:58:28.670
get the critical density
is about 8.4 times 10
00:58:28.670 --> 00:58:36.210
to the minus 30 grams
per centimeter cubed.
00:58:36.210 --> 00:58:40.820
And that is equivalent to about
5 protons per cubic meter.
00:58:51.222 --> 00:58:52.680
So I've written
the answer in terms
00:58:52.680 --> 00:58:55.177
of grams per cubic
centimeter because
00:58:55.177 --> 00:58:57.010
to me that's a very
natural unit for density
00:58:57.010 --> 00:58:59.070
because it's the
density of water.
00:58:59.070 --> 00:59:03.150
We're saying that the average
density of the universe is only
00:59:03.150 --> 00:59:07.564
about 10 to the minus
29 quantity 8.410.
00:59:07.564 --> 00:59:09.730
The average density of the
universe is only about 10
00:59:09.730 --> 00:59:14.210
to the minus 29 times
the density of water.
00:59:14.210 --> 00:59:16.710
So it's an unbelievably empty
universe that we're living in.
00:59:16.710 --> 00:59:19.100
It's hard to believe the
universe is that empty,
00:59:19.100 --> 00:59:21.990
but there are large spaces
between the galaxies
00:59:21.990 --> 00:59:24.200
that we look at.
00:59:24.200 --> 00:59:26.210
So the universe is
incredibly empty.
00:59:26.210 --> 00:59:30.880
And in fact, this is a
vastly better vacuum.
00:59:30.880 --> 00:59:33.700
An average part of the universe
is a vastly better vacuum
00:59:33.700 --> 00:59:35.730
than can be made on
Earth by any machinery
00:59:35.730 --> 00:59:37.660
that we have access to.
00:59:37.660 --> 00:59:41.951
So the best vacuum is empty
space, just middle of nowhere.
00:59:41.951 --> 00:59:43.950
And it's vastly better
than what we can actually
00:59:43.950 --> 00:59:44.770
produce on Earth.
00:59:53.625 --> 00:59:54.485
Yes.
00:59:54.485 --> 00:59:56.425
AUDIENCE: I see you
used protons here,
00:59:56.425 --> 00:59:58.880
5 protons per cubic medium.
00:59:58.880 --> 01:00:02.362
So is this density corresponding
to density of baryonic matter,
01:00:02.362 --> 01:00:03.320
or all types of matter?
01:00:03.320 --> 01:00:05.361
PROFESSOR: This is actually
all types of matters,
01:00:05.361 --> 01:00:08.290
even though I'm using my
proton as a meter stick.
01:00:08.290 --> 01:00:10.910
But it is the total mass
density of the universe that's
01:00:10.910 --> 01:00:12.319
very close to the
critical value.
01:00:12.319 --> 01:00:13.860
And I was just about
to say something
01:00:13.860 --> 01:00:16.080
about what the total
mass is made up of.
01:00:43.270 --> 01:00:45.830
Cosmologists define-- this
was certainly mentioned
01:00:45.830 --> 01:00:50.120
in my first lecture-- a
Greek letter capital Omega
01:00:50.120 --> 01:00:53.680
to mean the actual mass
density of the universe divided
01:00:53.680 --> 01:00:56.250
by this critical density.
01:00:56.250 --> 01:00:59.110
So omega equals 1 in
this language corresponds
01:00:59.110 --> 01:01:02.240
to a flat universe at
this critical point.
01:01:02.240 --> 01:01:05.590
Omega bigger than 1 corresponds
to a closed universe.
01:01:05.590 --> 01:01:07.290
And omega less
than 1 corresponds
01:01:07.290 --> 01:01:09.050
to an open universe.
01:01:09.050 --> 01:01:11.120
And today, we know that
omega is equal to 1
01:01:11.120 --> 01:01:14.170
to an accuracy of about
a half of a percent.
01:01:14.170 --> 01:01:19.220
To a very good accuracy we
know omega is very close to 1.
01:01:19.220 --> 01:01:21.500
It's made up of
different contributions.
01:01:21.500 --> 01:01:26.560
And these tend to vary with
time-- as the best measurements
01:01:26.560 --> 01:01:29.490
tend to vary with
time by a few percent.
01:01:29.490 --> 01:01:33.170
But omega matter-- and
here I mean visible
01:01:33.170 --> 01:01:49.700
plus dark matter--
is roughly about 0.3.
01:01:49.700 --> 01:01:52.900
And most of the universe
today, as we mentioned earlier,
01:01:52.900 --> 01:01:56.060
the universe today is pretty
much dark energy-dominated.
01:01:59.040 --> 01:02:15.090
So omega dark energy
is about equal to 0.70.
01:02:15.090 --> 01:02:19.020
And omega total is
pretty close to 1.
01:02:23.430 --> 01:02:25.680
Plus or minus about
a half of a percent.
01:02:34.650 --> 01:02:36.480
So one of the
implications here is
01:02:36.480 --> 01:02:39.549
that we've been assuming
in our calculation so far
01:02:39.549 --> 01:02:42.090
that we're talking about nothing
but non-relativistic matter.
01:02:42.090 --> 01:02:45.290
That's actually only about
30% of the actual matter
01:02:45.290 --> 01:02:47.040
in the current universe.
01:02:47.040 --> 01:02:48.670
So I did say this
is the beginning.
01:02:48.670 --> 01:02:51.300
The current universe today
does not obey the equations
01:02:51.300 --> 01:02:52.966
that we've written
down very accurately.
01:02:52.966 --> 01:02:53.820
It's pretty far off.
01:02:53.820 --> 01:02:55.320
But the equations
that we wrote down
01:02:55.320 --> 01:02:56.778
are pretty accurate
for the history
01:02:56.778 --> 01:02:59.700
of our universe from a
period of about 50,000 years
01:02:59.700 --> 01:03:02.520
after the Big Bang up
to about 9 billion years
01:03:02.520 --> 01:03:04.380
after the Big Bang.
01:03:04.380 --> 01:03:05.208
Yes.
01:03:05.208 --> 01:03:07.200
AUDIENCE: Before dark
energy was discovered,
01:03:07.200 --> 01:03:09.136
did they think omega
was [INAUDIBLE]?
01:03:09.136 --> 01:03:09.760
PROFESSOR: Yes.
01:03:09.760 --> 01:03:11.060
At least many people did.
01:03:11.060 --> 01:03:13.120
Before dark energy
was discovered,
01:03:13.120 --> 01:03:15.500
there was a controversy
in the community
01:03:15.500 --> 01:03:17.470
over what we thought omega was.
01:03:17.470 --> 01:03:20.205
Those of us who had
faith in inflation
01:03:20.205 --> 01:03:22.080
believed that omega
would be 1 because that's
01:03:22.080 --> 01:03:24.780
what inflation predicts.
01:03:24.780 --> 01:03:27.420
Astronomers who just had
faith in observations
01:03:27.420 --> 01:03:29.500
believed that omega
was 0.2 or 0.3
01:03:29.500 --> 01:03:31.619
because that's what they saw.
01:03:31.619 --> 01:03:33.160
And the truth ended
up being somewhat
01:03:33.160 --> 01:03:35.130
in between in the
sense that omega
01:03:35.130 --> 01:03:37.600
total we now all agree
is very close to 1
01:03:37.600 --> 01:03:39.052
as inflation predicts.
01:03:39.052 --> 01:03:41.510
But it's still true that the
stuff that the astronomers saw
01:03:41.510 --> 01:03:45.170
at this earlier time did
only add up to 0.2 or 0.3.
01:03:45.170 --> 01:03:47.379
So they correctly estimated
what they were looking at
01:03:47.379 --> 01:03:49.461
and they had no way of
knowing that there was also
01:03:49.461 --> 01:03:51.470
this dark energy component
until it was finally
01:03:51.470 --> 01:03:56.940
discovered in 1998.
01:03:56.940 --> 01:03:57.919
Yes.
01:03:57.919 --> 01:04:00.544
AUDIENCE: If we don't know what
dark energy is, observationally
01:04:00.544 --> 01:04:02.128
how have we been
able to measure that?
01:04:02.128 --> 01:04:04.002
PROFESSOR: How do we
measure it so accurately
01:04:04.002 --> 01:04:05.590
if we don't know
what it is, right?
01:04:05.590 --> 01:04:06.100
Right.
01:04:06.100 --> 01:04:08.140
Well, the answer
is while we're not
01:04:08.140 --> 01:04:09.850
sure what it is, we
actually do think
01:04:09.850 --> 01:04:11.819
we know a lot of its properties.
01:04:11.819 --> 01:04:13.610
And essentially, almost
all properties that
01:04:13.610 --> 01:04:15.360
are relevant to cosmology.
01:04:15.360 --> 01:04:18.204
We just don't know what's
sort of like inside.
01:04:18.204 --> 01:04:19.870
So we know it creates
repulsive gravity.
01:04:19.870 --> 01:04:23.170
We know how much repulsive
gravity it creates.
01:04:23.170 --> 01:04:27.320
And we also know to
reasonable accuracy
01:04:27.320 --> 01:04:29.104
how the dark energy
has been evolving
01:04:29.104 --> 01:04:30.770
with time, which is
really that it's not
01:04:30.770 --> 01:04:32.440
been evolving with time.
01:04:32.440 --> 01:04:34.560
And that determines
what its pressure is.
01:04:34.560 --> 01:04:36.630
It determines, in
fact-- to not evolve
01:04:36.630 --> 01:04:39.370
with time we'll see later
requires the pressure
01:04:39.370 --> 01:04:42.680
to be equal to the negative
of the energy density.
01:04:42.680 --> 01:04:45.270
Pressure is related to how
energies change with time, as I
01:04:45.270 --> 01:04:47.900
mentioned a few minutes
ago in a different context.
01:04:47.900 --> 01:04:50.150
If you have a box that expands
and there's a pressure,
01:04:50.150 --> 01:04:53.620
the pressure does dp
dv work on the box.
01:04:53.620 --> 01:04:55.690
And you can tell how much
the energy in the box
01:04:55.690 --> 01:04:57.876
should change for
a given pressure.
01:04:57.876 --> 01:04:59.500
And we'll do this
more carefully later,
01:04:59.500 --> 01:05:01.347
but to have the energy
not change at all
01:05:01.347 --> 01:05:03.680
requires a pressure, which
is the negative of the energy
01:05:03.680 --> 01:05:04.700
density.
01:05:04.700 --> 01:05:08.220
So we know how much acceleration
the dark energy causes.
01:05:08.220 --> 01:05:10.720
We know to reasonable
accuracy and we
01:05:10.720 --> 01:05:12.650
assume it's true
that the pressure is
01:05:12.650 --> 01:05:14.092
equal to minus the
energy density.
01:05:14.092 --> 01:05:16.550
And that's all you need to know
to calculate how much of it
01:05:16.550 --> 01:05:19.629
you need to account for
that much acceleration.
01:05:19.629 --> 01:05:20.670
And that's how it's done.
01:05:24.772 --> 01:05:25.605
Any other questions?
01:05:52.091 --> 01:05:54.090
OK next thing we want to
do is to actually solve
01:05:54.090 --> 01:05:56.770
this equation for
the easiest case.
01:05:56.770 --> 01:06:00.270
We'll solve it in general
later, but the easiest case
01:06:00.270 --> 01:06:04.760
to solve it is the case
of the critical case.
01:06:12.855 --> 01:06:14.480
We only have a few
minutes, but it only
01:06:14.480 --> 01:06:17.960
takes a few minutes to solve
the equation for this case.
01:06:21.784 --> 01:06:22.880
It should be over here.
01:06:33.649 --> 01:06:41.516
So for the critical case,
it's the case E is equal to 0.
01:06:41.516 --> 01:06:42.890
And therefore, we
just have a dot
01:06:42.890 --> 01:06:46.930
squared is equal to a
constant divided by a.
01:06:46.930 --> 01:06:48.337
And it won't really
matter for us
01:06:48.337 --> 01:06:49.670
right now what this constant is.
01:06:49.670 --> 01:06:51.378
So I don't even have
to keep track of it.
01:06:51.378 --> 01:06:53.724
I'll just write it
as const, C-O-N-S-T.
01:06:53.724 --> 01:06:55.640
And I'll take the square
root of this equation
01:06:55.640 --> 01:06:57.348
because it's easier
to know what a dot is
01:06:57.348 --> 01:06:59.105
than to know what a is.
01:06:59.105 --> 01:07:01.890
Easier to make use of
knowing what a dot is.
01:07:01.890 --> 01:07:04.570
So I can rewrite that equation
as a dot, which I'll now
01:07:04.570 --> 01:07:07.250
write as da dt, to be a little
more explicit about what we're
01:07:07.250 --> 01:07:13.020
talking about, is equal to a
constant over a to the 1/2.
01:07:16.000 --> 01:07:21.479
So this now is the k
equals 0 evolution.
01:07:21.479 --> 01:07:23.270
So this is just the
same Friedmann equation
01:07:23.270 --> 01:07:26.190
rewritten for the
special case k equals 0.
01:07:26.190 --> 01:07:29.060
And now I'm just
going to perform
01:07:29.060 --> 01:07:31.727
the amazingly complicated
manipulation of multiplying
01:07:31.727 --> 01:07:33.560
both sides of the
equation by a to the half.
01:07:36.240 --> 01:07:39.320
So we have a to the half.
01:07:39.320 --> 01:07:41.810
I'm also going to
multiply by dt.
01:07:41.810 --> 01:07:47.920
So a to the half da will be
equal to a constant times dt.
01:07:51.020 --> 01:07:53.140
And now we can just
integrate both sides
01:07:53.140 --> 01:07:55.110
as an indefinite integral.
01:07:55.110 --> 01:07:58.380
And integrating both sides
as an indefinite integral,
01:07:58.380 --> 01:08:03.230
the left-hand side
becomes-- go back over here.
01:08:03.230 --> 01:08:13.620
2/3 a to the 3/2 is
equal to a constant times
01:08:13.620 --> 01:08:20.229
t plus an arbitrary other
constant of integration.
01:08:20.229 --> 01:08:21.670
This is the most
general equation,
01:08:21.670 --> 01:08:23.378
which when differentiated
gives you this.
01:08:28.310 --> 01:08:32.750
And now, this
equation can be solved
01:08:32.750 --> 01:08:35.930
to tell us what a is
as a function of t.
01:08:35.930 --> 01:08:39.370
But before I do that, I'm going
to say something about c prime
01:08:39.370 --> 01:08:40.620
here.
01:08:40.620 --> 01:08:44.800
c prime is allowed
by the integration.
01:08:44.800 --> 01:08:48.090
But remember that when we
defined our scale of t,
01:08:48.090 --> 01:08:49.840
we just started at
some arbitrary time,
01:08:49.840 --> 01:08:52.830
ti, which we didn't
even specify.
01:08:52.830 --> 01:08:55.010
So there's no
particular significance
01:08:55.010 --> 01:08:56.970
to the origin of
time in the equations
01:08:56.970 --> 01:08:59.350
that we've written so far.
01:08:59.350 --> 01:09:02.540
So we're perfectly free to
shift the origin of time
01:09:02.540 --> 01:09:06.109
by just redefining our clocks.
01:09:06.109 --> 01:09:07.920
Cosmic time,
remember, is defined
01:09:07.920 --> 01:09:10.420
in a way which makes it
uniform throughout the universe
01:09:10.420 --> 01:09:12.020
by our construction.
01:09:12.020 --> 01:09:15.729
But we haven't said anything yet
about how to start cosmic time.
01:09:15.729 --> 01:09:18.529
But now, we have a
good way to start it.
01:09:18.529 --> 01:09:20.149
In this model,
there is going to be
01:09:20.149 --> 01:09:22.770
a time at which a
is going to go to 0.
01:09:22.770 --> 01:09:24.700
No matter what we
choose for c prime here,
01:09:24.700 --> 01:09:27.158
there will be some t which will
make the right-hand side 0.
01:09:27.158 --> 01:09:28.450
And therefore, a 0.
01:09:28.450 --> 01:09:30.590
And that's the instant
of the Big Bang.
01:09:30.590 --> 01:09:34.295
That's when everything starts.
a never gets smaller than 0.
01:09:34.295 --> 01:09:35.920
So it's very natural
to take that to be
01:09:35.920 --> 01:09:38.850
defined to be the 0 of time.
01:09:38.850 --> 01:09:41.899
So that's what
we're going to do.
01:09:41.899 --> 01:09:55.840
So we're going to define t
equals 0 to be when a of t
01:09:55.840 --> 01:09:56.450
equals 0.
01:09:59.514 --> 01:10:00.930
That's just a
choice of the origin
01:10:00.930 --> 01:10:02.554
of time, which you're
certainly allowed
01:10:02.554 --> 01:10:06.112
to do without contradicting
anything else that we've said.
01:10:06.112 --> 01:10:08.320
And that means we're just
setting c prime equal to 0.
01:10:08.320 --> 01:10:09.800
So that when that's 0, that's 0.
01:10:14.910 --> 01:10:17.740
So this implies
c prime equals 0.
01:10:17.740 --> 01:10:19.530
And that then
implies-- we could take
01:10:19.530 --> 01:10:21.424
the 2/3 power of this equation.
01:10:21.424 --> 01:10:23.840
And I told you we don't really
care what that constant is.
01:10:23.840 --> 01:10:24.840
So therefore, we
don't really care
01:10:24.840 --> 01:10:26.360
about what that constant is.
01:10:26.360 --> 01:10:30.760
What we get is that a is
equal to some constant,
01:10:30.760 --> 01:10:32.810
not necessarily related
to any of the constants
01:10:32.810 --> 01:10:33.657
we've said so far.
01:10:33.657 --> 01:10:35.240
Although, you can
calculate how it is.
01:10:35.240 --> 01:10:38.090
But some constant times
t to the 2/3 power.
01:10:42.229 --> 01:10:43.770
Or equivalently,
you could just write
01:10:43.770 --> 01:10:47.235
a is proportional to
t to the 2/3, which
01:10:47.235 --> 01:10:48.110
has the same content.
01:10:50.576 --> 01:10:52.200
Now, you might think
you'd want to know
01:10:52.200 --> 01:10:53.866
what the constant of
proportionality is.
01:10:53.866 --> 01:10:55.870
But remember, the constant
of proportionality
01:10:55.870 --> 01:10:58.650
just depends on the
definition of the notch.
01:10:58.650 --> 01:11:01.160
If you want to define the
notch so that a is equal to 1
01:11:01.160 --> 01:11:04.182
today, then you would
care what the constant is.
01:11:04.182 --> 01:11:06.640
If you're willing to just leave
the definition of the notch
01:11:06.640 --> 01:11:10.410
arbitrary, then you don't
care what the constant is.
01:11:10.410 --> 01:11:13.230
And that's the case that
I'll be doing, actually.
01:11:13.230 --> 01:11:16.206
I will not define
a to be 1 today.
01:11:16.206 --> 01:11:18.080
The definition of the
notch is just arbitrary
01:11:18.080 --> 01:11:20.070
as far as the equations
that I'll be writing.
01:11:20.070 --> 01:11:21.903
And therefore, it will
be sufficient to know
01:11:21.903 --> 01:11:24.550
that a is just proportional
to t to the 2/3
01:11:24.550 --> 01:11:29.440
for the flat universe case,
for the critical density case.
01:11:29.440 --> 01:11:31.280
And that's where
we'll stop today.
01:11:31.280 --> 01:11:33.340
And this actually pretty
really covers everything
01:11:33.340 --> 01:11:36.690
through lecture notes three,
which is the same material that
01:11:36.690 --> 01:11:39.170
will be covered on
the quiz next week.
01:11:39.170 --> 01:11:41.090
[INAUDIBLE] does not
seem to have shown up,
01:11:41.090 --> 01:11:45.420
but we'll assume that probably
the review session will
01:11:45.420 --> 01:11:46.760
be next Monday night at 7:30.
01:11:46.760 --> 01:11:50.750
I will check it out and
get back to you by email.