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PROFESSOR: OK, in that
case, let's get going.
00:00:26.492 --> 00:00:28.700
I want to begin by just
summarizing where we left off
00:00:28.700 --> 00:00:31.390
last time, because we're still
to some extent in the middle
00:00:31.390 --> 00:00:33.540
of the same discussion.
00:00:33.540 --> 00:00:37.600
We were talking about
the gravitational effects
00:00:37.600 --> 00:00:40.160
of a completely
homogeneous universe that
00:00:40.160 --> 00:00:41.930
fills all of space.
00:00:41.930 --> 00:00:44.430
And you recall that
Newton had concluded
00:00:44.430 --> 00:00:46.280
that such a system
would be stable.
00:00:46.280 --> 00:00:49.870
But I was arguing that such
a system would not be stable,
00:00:49.870 --> 00:00:52.544
even given the laws of
Newtonian mechanics.
00:00:52.544 --> 00:00:54.710
And we discussed a few of
those arguments last time.
00:00:59.260 --> 00:01:03.330
We discussed, for example,
Gauss's law formulation
00:01:03.330 --> 00:01:04.959
of Newton's law of gravity.
00:01:04.959 --> 00:01:08.040
And I remind you that it's
a very simple derivation
00:01:08.040 --> 00:01:09.870
to go from Newton's
law of gravity,
00:01:09.870 --> 00:01:13.500
as Newton stated it as an
acceleration at a distance,
00:01:13.500 --> 00:01:15.150
to Gauss's law.
00:01:15.150 --> 00:01:20.810
What you do is you just show
that if the acceleration
00:01:20.810 --> 00:01:24.550
of gravity is given by Newton's
law, then for any one particle
00:01:24.550 --> 00:01:27.880
creating such a gravitational
field, Gauss's law holds.
00:01:27.880 --> 00:01:31.860
This integral is either
equal to 0 or minus 4 pi GM,
00:01:31.860 --> 00:01:35.250
depending on whether the surface
that you're integrating over
00:01:35.250 --> 00:01:39.660
encloses or does not enclose
the charge or the mass.
00:01:39.660 --> 00:01:41.340
And once you know
it for one mass,
00:01:41.340 --> 00:01:43.060
Newton tells us
that for many masses
00:01:43.060 --> 00:01:45.115
you just add the
forces as vectors.
00:01:45.115 --> 00:01:46.740
And that means that
you'll be adding up
00:01:46.740 --> 00:01:48.600
these integrals for
each of the particles.
00:01:48.600 --> 00:01:52.200
And you're lead automatically
to the expression we have here.
00:01:52.200 --> 00:01:53.730
So really, it
follows very directly
00:01:53.730 --> 00:01:55.430
from Newton's law of gravity.
00:01:55.430 --> 00:01:59.130
On the other hand, if we
apply this formulation
00:01:59.130 --> 00:02:03.350
to an infinite distribution
of mass, if Newton was right
00:02:03.350 --> 00:02:05.710
and there were no
forces, that would
00:02:05.710 --> 00:02:08.020
mean that little g, the
acceleration of gravity,
00:02:08.020 --> 00:02:09.560
would be 0 everywhere.
00:02:09.560 --> 00:02:11.620
And then, the integral
on the left would be 0.
00:02:11.620 --> 00:02:14.230
But the term on the
right is clearly not 0,
00:02:14.230 --> 00:02:17.730
if we have a volume with
some nontrivial size that
00:02:17.730 --> 00:02:19.550
includes some mass.
00:02:19.550 --> 00:02:23.590
So this formulation of
Newton's law of gravity
00:02:23.590 --> 00:02:28.440
clearly shows that an
infinite distribution of mass
00:02:28.440 --> 00:02:30.490
could not be static.
00:02:30.490 --> 00:02:33.390
Further, I showed you there's
another formulation, more
00:02:33.390 --> 00:02:35.090
modern, of Newton's
law of gravity
00:02:35.090 --> 00:02:38.141
in terms of what's called
Poisson's equation.
00:02:38.141 --> 00:02:40.640
This was really shown for the
benefit of people who know it.
00:02:40.640 --> 00:02:42.223
If you don't it,
don't worry about it.
00:02:42.223 --> 00:02:43.831
We won't need it.
00:02:43.831 --> 00:02:46.080
But it's another way of
formulating the law of gravity
00:02:46.080 --> 00:02:49.380
by introducing a
gravitational potential, phi,
00:02:49.380 --> 00:02:51.840
and writing the
acceleration of gravity
00:02:51.840 --> 00:02:53.490
as minus the gradient of phi.
00:02:53.490 --> 00:02:55.470
That just defines phi.
00:02:55.470 --> 00:02:58.450
And then, you can show that
phi obeys Poisson's equation,
00:02:58.450 --> 00:03:01.490
del squared on phi is
equal to 4 pi G times
00:03:01.490 --> 00:03:04.494
rho, where rho is
the mass density.
00:03:04.494 --> 00:03:05.910
And again, one can
see immediately
00:03:05.910 --> 00:03:09.970
that this does not allow a
static distribution of mass.
00:03:09.970 --> 00:03:11.770
If the distribution
of mass was static,
00:03:11.770 --> 00:03:14.270
that would mean
that g vector was 0.
00:03:14.270 --> 00:03:16.480
That would mean that
gradient phi was 0.
00:03:16.480 --> 00:03:18.400
That would mean that
phi was a constant.
00:03:18.400 --> 00:03:20.500
And if phi is a constant,
del squared phi is 0,
00:03:20.500 --> 00:03:25.400
and that's inconsistent
with the Poisson equation.
00:03:25.400 --> 00:03:27.090
I might further
add-- I don't think
00:03:27.090 --> 00:03:30.550
I said this last time-- that
from a modern perspective,
00:03:30.550 --> 00:03:32.590
equations like
Poisson's equation
00:03:32.590 --> 00:03:36.140
are considered more
fundamental than Newton's
00:03:36.140 --> 00:03:37.870
original statement
of the law of gravity
00:03:37.870 --> 00:03:40.300
as an action at a distance.
00:03:40.300 --> 00:03:43.390
In particular, when
one wants to generalize
00:03:43.390 --> 00:03:46.590
Newton's law, for example,
to general relativity,
00:03:46.590 --> 00:03:48.310
Einstein started with
Poisson's equation,
00:03:48.310 --> 00:03:50.750
not with the force
law at a distance.
00:03:50.750 --> 00:03:53.010
And there's nothing like
the force law at a distance
00:03:53.010 --> 00:03:55.120
and the theory of
general relativity.
00:03:55.120 --> 00:03:59.580
General relativity is formulated
in a language very similar
00:03:59.580 --> 00:04:02.030
to the language of
Poisson's equation.
00:04:02.030 --> 00:04:05.740
The key idea that
underlies this distinction
00:04:05.740 --> 00:04:08.725
is that all the laws of
physics that we know of
00:04:08.725 --> 00:04:10.970
can be expressed in a local way.
00:04:10.970 --> 00:04:12.622
Poisson's equation
is a local equation.
00:04:12.622 --> 00:04:14.080
That's just a
differential equation
00:04:14.080 --> 00:04:15.784
that holds at each
point in space
00:04:15.784 --> 00:04:17.200
and doesn't say
anything about how
00:04:17.200 --> 00:04:20.810
something at one point in space
affects something far away.
00:04:20.810 --> 00:04:23.657
That happens as a
consequence of this equation.
00:04:23.657 --> 00:04:25.990
But it happens as a consequence
of solving the equation.
00:04:25.990 --> 00:04:30.290
It's not built into the
equation to start with.
00:04:30.290 --> 00:04:32.130
Continuing.
00:04:32.130 --> 00:04:36.180
We then discussed what
goes on if one does just
00:04:36.180 --> 00:04:40.410
try to add up the forces
using Newton's law
00:04:40.410 --> 00:04:42.070
and action at a distance.
00:04:42.070 --> 00:04:44.590
And what I argued is that it's
a conditionally convergent
00:04:44.590 --> 00:04:47.840
integral, which means it's
the kind of integral which
00:04:47.840 --> 00:04:51.260
has the property
that it converges.
00:04:51.260 --> 00:04:53.480
But it can converge
to different things
00:04:53.480 --> 00:04:55.380
depending on what
order you add up
00:04:55.380 --> 00:04:57.880
the different parts
of the integral.
00:04:57.880 --> 00:05:00.400
And we considered two
possible orderings
00:05:00.400 --> 00:05:02.740
for adding up the mass.
00:05:02.740 --> 00:05:05.090
In all cases, what
we're talking about here
00:05:05.090 --> 00:05:09.100
is just a single
location, P. And you
00:05:09.100 --> 00:05:12.880
can't tell the slide
is filled with mass,
00:05:12.880 --> 00:05:15.830
but think of that
light cyan as mass.
00:05:15.830 --> 00:05:18.880
It fills the entire slide
and fills the entire universe
00:05:18.880 --> 00:05:21.554
in our toy problem here.
00:05:21.554 --> 00:05:22.970
So we're interested
in calculating
00:05:22.970 --> 00:05:25.970
the force on some
point, P, in the midst
00:05:25.970 --> 00:05:27.952
of an infinite
distribution of mass.
00:05:27.952 --> 00:05:30.160
And the only thing that
we're going to do differently
00:05:30.160 --> 00:05:31.576
in these two
calculations is we're
00:05:31.576 --> 00:05:33.980
going to add up that mass
in a different order.
00:05:33.980 --> 00:05:36.560
And if we add up
the mass ordered
00:05:36.560 --> 00:05:40.780
by concentric shells about P,
each consecutive shell clearly
00:05:40.780 --> 00:05:44.570
contributes 0 to the force
at P. And therefore, the sum
00:05:44.570 --> 00:05:49.180
in the limit, as we go out
to infinity, will still be 0.
00:05:49.180 --> 00:05:52.070
So for this case,
we get g equals 0.
00:05:52.070 --> 00:05:55.300
We do get no acceleration
at the point, P,
00:05:55.300 --> 00:05:59.880
as long as we add up all
the masses in that way.
00:05:59.880 --> 00:06:02.960
But there's nothing in
Newton's laws that tell us
00:06:02.960 --> 00:06:04.980
what order to add up the forces.
00:06:04.980 --> 00:06:08.650
Newton just tells you
that each mass creates a 1
00:06:08.650 --> 00:06:10.680
over r squared force,
and it's a vector.
00:06:10.680 --> 00:06:13.080
And Newton says to
add the vectors.
00:06:13.080 --> 00:06:15.200
Normally, addition
of vectors commutes.
00:06:15.200 --> 00:06:17.076
It doesn't matter what
order you add them in.
00:06:17.076 --> 00:06:18.700
But what we're going
to be finding here
00:06:18.700 --> 00:06:20.250
is that it does
matter what order.
00:06:20.250 --> 00:06:22.640
And therefore, the
answer is ambiguous.
00:06:22.640 --> 00:06:26.020
And to see this, we'll
consider a different ordering.
00:06:26.020 --> 00:06:27.880
Instead-- we'll still
use spherical shells,
00:06:27.880 --> 00:06:30.364
because that's the easiest
thing to think about.
00:06:30.364 --> 00:06:31.780
We could try to
do something else,
00:06:31.780 --> 00:06:34.470
but it's much harder
to use any other shape.
00:06:34.470 --> 00:06:37.260
But this time, we'll
consider spherical shells
00:06:37.260 --> 00:06:39.200
that are centered around
a different point.
00:06:39.200 --> 00:06:41.450
And we'll call the point
that the spherical shells are
00:06:41.450 --> 00:06:44.700
centered around, Q.
We're still trying
00:06:44.700 --> 00:06:47.840
to calculate the
force at the point, P,
00:06:47.840 --> 00:06:49.570
due to the infinite
mass distribution
00:06:49.570 --> 00:06:52.010
that fills space-- so we're
doing the same problem we did
00:06:52.010 --> 00:06:54.290
before-- but we're
going to add up
00:06:54.290 --> 00:06:56.320
the contributions in
a different order.
00:06:56.320 --> 00:06:59.800
And we discussed last
time that when we do that,
00:06:59.800 --> 00:07:03.350
it turns out that all the
mass inside the sphere,
00:07:03.350 --> 00:07:06.170
centered at Q out
to the radius of P,
00:07:06.170 --> 00:07:08.740
contributes to the
acceleration at P.
00:07:08.740 --> 00:07:10.760
And all mass outside
of that, could
00:07:10.760 --> 00:07:13.500
be divided into
concentric shells, where
00:07:13.500 --> 00:07:14.690
the point, P, is inside.
00:07:14.690 --> 00:07:17.470
And there's no force
inside a concentric shell.
00:07:17.470 --> 00:07:21.380
So all of the rest of the
mass contributes zilch.
00:07:21.380 --> 00:07:25.770
And the answer you get
is then simply the answer
00:07:25.770 --> 00:07:30.020
that you would have for the
force of a point mass located
00:07:30.020 --> 00:07:34.890
at Q, whose mass was the total
mass in that shaded region.
00:07:34.890 --> 00:07:36.370
And clearly it's nonzero.
00:07:36.370 --> 00:07:37.970
And furthermore,
clearly, we can--
00:07:37.970 --> 00:07:41.650
by choosing different points,
Q-- make this anything we want.
00:07:41.650 --> 00:07:44.630
We can make it bigger by putting
the point, Q, further away;
00:07:44.630 --> 00:07:53.410
and it always points in the
direction of Q. Yeah, points
00:07:53.410 --> 00:07:55.849
in the direction
of Q. So we can let
00:07:55.849 --> 00:07:58.140
it point in any direction we
want by putting the point,
00:07:58.140 --> 00:07:59.950
Q, anywhere we want.
00:07:59.950 --> 00:08:03.400
So depending on how we
add up the contributions,
00:08:03.400 --> 00:08:04.670
we can get any answer we want.
00:08:04.670 --> 00:08:07.460
And that's a
fundamental ambiguity
00:08:07.460 --> 00:08:11.170
in trying to apply Newton using
only the original statement
00:08:11.170 --> 00:08:15.640
of the law of gravity
as Newton gave it.
00:08:15.640 --> 00:08:17.230
OK.
00:08:17.230 --> 00:08:22.310
So the conclusion, here, is
that the action at a distance
00:08:22.310 --> 00:08:24.830
description is simply ambiguous.
00:08:24.830 --> 00:08:28.030
Descriptions by Gauss's
law, or Poisson's law,
00:08:28.030 --> 00:08:30.740
tell us that the system
cannot be static.
00:08:30.740 --> 00:08:33.049
And we'll soon try to
figure out exactly how we
00:08:33.049 --> 00:08:34.809
do expect it to behave.
00:08:34.809 --> 00:08:37.429
Now, I still want come back to
one argument, which was really
00:08:37.429 --> 00:08:40.590
the argument that persuaded
Newton in the first place.
00:08:40.590 --> 00:08:43.010
Newton said that if
we want to calculate
00:08:43.010 --> 00:08:45.680
the acceleration of a certain
point in this infinite mass
00:08:45.680 --> 00:08:48.810
distribution, we have
a symmetry problem.
00:08:48.810 --> 00:08:52.360
All directions from that point
looking outward look identical.
00:08:52.360 --> 00:08:56.100
If there's going to be
an acceleration acting
00:08:56.100 --> 00:08:59.950
on any point, what
could possibly
00:08:59.950 --> 00:09:05.360
determine the direction that
the acceleration will have.
00:09:05.360 --> 00:09:08.580
So that's the symmetry argument,
which is a very sticky one.
00:09:08.580 --> 00:09:10.980
It sounds very convincing,
in Newton's reasoning.
00:09:13.841 --> 00:09:16.090
There could be no acceleration,
simply because there's
00:09:16.090 --> 00:09:19.865
no preferred direction for
the acceleration to point.
00:09:19.865 --> 00:09:21.990
To convince Newton that
that's not a valid argument
00:09:21.990 --> 00:09:22.980
would probably be hard.
00:09:22.980 --> 00:09:25.229
And I don't know if we could
succeed in convincing him
00:09:25.229 --> 00:09:26.730
or not.
00:09:26.730 --> 00:09:28.450
Don't get the chance to try.
00:09:28.450 --> 00:09:29.930
But if we did have
a chance to try,
00:09:29.930 --> 00:09:33.180
what we would try
to explain to him
00:09:33.180 --> 00:09:37.850
is that the acceleration is
usually measured relative
00:09:37.850 --> 00:09:39.720
to an inertial frame.
00:09:39.720 --> 00:09:41.470
That's how Newton
always described it.
00:09:41.470 --> 00:09:46.380
And to Newton, there was a
unique inertial frame-- unique
00:09:46.380 --> 00:09:49.030
up to changes in
velocity-- determined
00:09:49.030 --> 00:09:51.170
by the frame of the fixed stars.
00:09:51.170 --> 00:09:53.280
That was the language
that Newton used.
00:09:53.280 --> 00:09:54.910
And that defined
his inertial frame.
00:09:54.910 --> 00:09:57.160
And all of his laws
of physics were
00:09:57.160 --> 00:10:00.290
claimed to hold in
this inertial frame.
00:10:00.290 --> 00:10:03.630
On the other hand, if all of
space is filled with matter
00:10:03.630 --> 00:10:06.070
and it's all going to
collapse as we're claiming,
00:10:06.070 --> 00:10:09.490
there isn't any place
to have any fixed stars.
00:10:09.490 --> 00:10:14.630
So the whole idea of an inertial
frame really disappears.
00:10:14.630 --> 00:10:17.410
There's no object, which one
can think of as being at rest
00:10:17.410 --> 00:10:19.430
or being non-accelerating
with respect
00:10:19.430 --> 00:10:24.430
to any would-be inertial frame.
00:10:24.430 --> 00:10:26.760
So in the absence of
an inertial frame,
00:10:26.760 --> 00:10:30.820
one really has to admit
that all accelerations, just
00:10:30.820 --> 00:10:33.340
like all velocities,
have to be measured
00:10:33.340 --> 00:10:34.721
as relative accelerations.
00:10:34.721 --> 00:10:36.220
We could talk about
the acceleration
00:10:36.220 --> 00:10:38.510
of one mass relative to another.
00:10:38.510 --> 00:10:41.940
But we can't talk about what
the absolute acceleration is
00:10:41.940 --> 00:10:43.930
of a given mass,
because we don't
00:10:43.930 --> 00:10:46.620
have an inertial
frame with which
00:10:46.620 --> 00:10:50.990
to compare the acceleration
of that object.
00:10:50.990 --> 00:10:53.550
So when all accelerations
are relative,
00:10:53.550 --> 00:10:55.852
then there are
more options here.
00:10:55.852 --> 00:10:58.310
And it turns out that the right
option-- the one that we'll
00:10:58.310 --> 00:11:01.540
eventually deduce--
is an option that
00:11:01.540 --> 00:11:03.730
looks similar to Hubble's law.
00:11:03.730 --> 00:11:05.970
Hubble's law is a
law about velocities.
00:11:05.970 --> 00:11:09.207
And it says that from the
point of view of any observer,
00:11:09.207 --> 00:11:11.040
all the other objects
will look like they're
00:11:11.040 --> 00:11:14.110
moving radially outward
from that observer.
00:11:14.110 --> 00:11:16.530
And in spite of the fact
that that description makes
00:11:16.530 --> 00:11:19.590
it sound, emphatically, like the
observer you're talking about
00:11:19.590 --> 00:11:23.340
is special, you can transform to
the frame of any other observer
00:11:23.340 --> 00:11:26.100
and thus seeing
exactly the same thing.
00:11:26.100 --> 00:11:29.470
So having one observer seeing
all the other velocities
00:11:29.470 --> 00:11:34.782
moving outward from him does
not violate homogeneity.
00:11:34.782 --> 00:11:36.490
It does not violate
any of the symmetries
00:11:36.490 --> 00:11:38.500
that we're trying to
incorporate in the system.
00:11:38.500 --> 00:11:39.950
And the same thing is
true for acceleration.
00:11:39.950 --> 00:11:41.180
So I'm not going to
try to show it now.
00:11:41.180 --> 00:11:43.670
We will be showing it in
the course of our upcoming
00:11:43.670 --> 00:11:45.580
calculations.
00:11:45.580 --> 00:11:48.160
But in the kind of
collapsing universe
00:11:48.160 --> 00:11:50.560
that we're going
to be describing,
00:11:50.560 --> 00:11:54.160
any observer can consider
himself or herself
00:11:54.160 --> 00:11:56.680
to be non-accelerating.
00:11:56.680 --> 00:11:59.640
And then, that observer would
see all the other particles
00:11:59.640 --> 00:12:04.460
accelerating
directly towards her.
00:12:04.460 --> 00:12:06.300
And although that
description makes
00:12:06.300 --> 00:12:09.037
it sound like the person
at the center is special,
00:12:09.037 --> 00:12:09.620
it's not true.
00:12:09.620 --> 00:12:13.600
You can transform to any
other observer's frame.
00:12:13.600 --> 00:12:16.110
And each observer
can regard himself
00:12:16.110 --> 00:12:19.700
as being non-accelerating and
would see all the other objects
00:12:19.700 --> 00:12:21.960
accelerating
radially towards him.
00:12:27.680 --> 00:12:30.120
OK.
00:12:30.120 --> 00:12:34.870
So now we are ready
to go on and try
00:12:34.870 --> 00:12:38.420
to build a mathematical
model, which will tell us
00:12:38.420 --> 00:12:44.900
how a uniform distribution
of mass will behave.
00:12:44.900 --> 00:12:48.800
Now in doing this,
we first would
00:12:48.800 --> 00:12:52.670
like to tame the
issue of infinities.
00:12:52.670 --> 00:12:57.870
And we are going to do
that by starting out
00:12:57.870 --> 00:13:01.252
with a finite sphere.
00:13:01.252 --> 00:13:02.710
And then at the
very end, we'll let
00:13:02.710 --> 00:13:06.910
the size of that
sphere go to infinity.
00:13:06.910 --> 00:13:18.300
So our goal is to build a
mathematical model of our toy
00:13:18.300 --> 00:13:19.880
universe.
00:13:19.880 --> 00:13:21.630
And what we want to
do is to incorporate
00:13:21.630 --> 00:13:25.670
the three features
that we discussed,
00:13:25.670 --> 00:13:39.660
isotropy, homogeneity,
and Hubble's law.
00:13:43.312 --> 00:13:45.520
And we're going to build
this as a mechanical system,
00:13:45.520 --> 00:13:47.353
using the laws of
mechanics as we know them.
00:13:47.353 --> 00:13:50.675
We're, in fact, going to be
using a Newtonian description.
00:13:50.675 --> 00:13:52.310
But I will assure
you that although we
00:13:52.310 --> 00:13:53.790
are using a Newtonian
description,
00:13:53.790 --> 00:13:56.737
the answer that we'll get will
in fact be the exact answer
00:13:56.737 --> 00:13:58.820
that we would have gotten
with general relativity.
00:13:58.820 --> 00:14:02.260
And we'll talk later
about why that's the case.
00:14:02.260 --> 00:14:03.760
But we're not wasting
our time doing
00:14:03.760 --> 00:14:05.240
only an approximate calculation.
00:14:05.240 --> 00:14:08.080
This actually is a
completely valid calculation,
00:14:08.080 --> 00:14:12.040
which gives us, in the end,
entirely the correct answer.
00:14:12.040 --> 00:14:16.100
So we're going to
model our universe
00:14:16.100 --> 00:14:17.630
by introducing a
coordinate system.
00:14:17.630 --> 00:14:19.130
I'm now just going
to really re-draw
00:14:19.130 --> 00:14:20.296
the picture that's up there.
00:14:20.296 --> 00:14:24.200
But if I draw it down here,
I can point to it better.
00:14:24.200 --> 00:14:29.990
We're going to imagine
starting at our toy universe
00:14:29.990 --> 00:14:34.570
as a finite sized
sphere of matter.
00:14:34.570 --> 00:14:38.320
And we're going
to let t sub i be
00:14:38.320 --> 00:14:44.320
equal to the time of
the initial picture.
00:14:51.070 --> 00:14:53.360
t sub i need not be
particularly special in any way,
00:14:53.360 --> 00:14:55.141
in terms of the life
of our universe.
00:14:55.141 --> 00:14:56.640
Once we construct
the picture, we'll
00:14:56.640 --> 00:14:59.305
be able to calculate
how it would behave--
00:14:59.305 --> 00:15:02.560
at times later than ti and
at times earlier than ti.
00:15:02.560 --> 00:15:04.700
ti is just where
we are starting.
00:15:08.220 --> 00:15:13.675
At time, ti, we will give
our sphere a maximum size.
00:15:20.572 --> 00:15:22.030
I'm calling it
maximum, because I'm
00:15:22.030 --> 00:15:23.779
thinking of this as
filled with particles.
00:15:23.779 --> 00:15:26.660
It's, therefore, the maximum
radius for any particle.
00:15:26.660 --> 00:15:29.500
It's just the radius
of the sphere.
00:15:29.500 --> 00:15:40.110
So R sub max i is just
the initial radius.
00:15:40.110 --> 00:15:43.950
An initial means
at time, t sub i.
00:15:43.950 --> 00:15:46.910
We're going to fill
the sphere with matter.
00:15:46.910 --> 00:15:48.550
And we'll think of
this matter as being
00:15:48.550 --> 00:15:52.070
a kind of a uniform fluid, or
a dust of very small particles,
00:15:52.070 --> 00:15:54.380
which we can also
think of as a fluid.
00:15:54.380 --> 00:15:56.440
And it will have a mass
density, rho sub i.
00:16:13.990 --> 00:16:14.490
OK.
00:16:14.490 --> 00:16:18.290
So it's already homogeneous and
isotropic, at least isotropic
00:16:18.290 --> 00:16:20.330
about the center.
00:16:20.330 --> 00:16:23.140
And now, we want to
incorporate Hubble's law.
00:16:23.140 --> 00:16:27.110
So we're going to start all of
this matter, in our toy system
00:16:27.110 --> 00:16:30.770
here, expanding and expanding
in precisely the pattern
00:16:30.770 --> 00:16:34.005
that Hubble's law requires--
namely, all velocities will
00:16:34.005 --> 00:16:38.140
be moving out from the center
with a magnitude proportional
00:16:38.140 --> 00:16:38.815
to the distance.
00:16:48.030 --> 00:16:54.120
So if I label a
particle by-- I'm sorry,
00:16:54.120 --> 00:16:56.570
v sub i is just for initial.
00:16:56.570 --> 00:16:58.145
For any particle--
there's no way
00:16:58.145 --> 00:17:00.700
we're indicating a particle--
at the initial time,
00:17:00.700 --> 00:17:04.460
the velocity will just
obey Hubble's law.
00:17:04.460 --> 00:17:07.079
It will be equal to some
constant, which I'll call H sub
00:17:07.079 --> 00:17:09.349
i-- the initial value
of the Hubble expansion
00:17:09.349 --> 00:17:13.740
rate-- times the
vector r, which is just
00:17:13.740 --> 00:17:15.660
a vector from the
origin to the particle.
00:17:15.660 --> 00:17:19.490
That tells us where the particle
is that we're talking about.
00:17:19.490 --> 00:17:28.990
So v sub i is the initial
velocity of any particle.
00:17:34.660 --> 00:17:44.320
H sub i is the initial
Hubble expansion rate.
00:17:46.940 --> 00:17:49.230
And r is the position
of the particle.
00:18:00.645 --> 00:18:01.145
OK.
00:18:01.145 --> 00:18:02.890
As I said, we're starting
with a finite system,
00:18:02.890 --> 00:18:04.306
which is completely
under control.
00:18:04.306 --> 00:18:07.320
We know, unambiguously, how
to calculate-- in principle
00:18:07.320 --> 00:18:09.730
at least-- how that
system will evolve,
00:18:09.730 --> 00:18:12.020
once we set up these
initial conditions.
00:18:12.020 --> 00:18:14.170
And at the end of
the calculation,
00:18:14.170 --> 00:18:18.010
we'll take the limit as
R max i goes to infinity.
00:18:18.010 --> 00:18:20.180
And that, we hope,
will capture the idea
00:18:20.180 --> 00:18:21.830
that this model
universe is going
00:18:21.830 --> 00:18:25.620
to fill the infinite space.
00:18:25.620 --> 00:18:28.166
Now, I did want to say
a little bit here--
00:18:28.166 --> 00:18:29.540
only because it's
something which
00:18:29.540 --> 00:18:33.270
has entered my
scientific life recently
00:18:33.270 --> 00:18:35.101
and in interesting ways.
00:18:35.101 --> 00:18:36.850
I would say a few
things about infinities.
00:18:45.070 --> 00:18:47.780
Now, this is an
aside, which means
00:18:47.780 --> 00:18:49.750
that if you're struggling
to understand what's
00:18:49.750 --> 00:18:51.416
going in the course,
you can ignore this
00:18:51.416 --> 00:18:53.200
and don't worry about it.
00:18:53.200 --> 00:18:59.240
But if you're interested in
thinking about these concepts,
00:18:59.240 --> 00:19:04.640
the concept of infinity has sort
of struck cosmology in the nose
00:19:04.640 --> 00:19:06.390
in the context of
the multiverse,
00:19:06.390 --> 00:19:08.890
which I spoke of a little bit
about in the overview lecture,
00:19:08.890 --> 00:19:12.802
and which we'll get back to
at the end of the course.
00:19:12.802 --> 00:19:14.760
The multiverse has forced
us to think much more
00:19:14.760 --> 00:19:17.821
about infinities than
we had previously.
00:19:17.821 --> 00:19:19.820
And in the course of that,
I learned some things
00:19:19.820 --> 00:19:22.280
about infinity that
in fact surprised me.
00:19:22.280 --> 00:19:25.770
For the most part,
in physics, we
00:19:25.770 --> 00:19:29.620
think of infinities as the
limit of finite things,
00:19:29.620 --> 00:19:31.430
as we're doing over here.
00:19:31.430 --> 00:19:34.200
So if we want to discuss the
behavior of an infinite space,
00:19:34.200 --> 00:19:36.010
we very frequently
in physics start
00:19:36.010 --> 00:19:38.890
by discussing a finite space,
where things are much easier
00:19:38.890 --> 00:19:40.290
to control mathematically.
00:19:40.290 --> 00:19:42.010
And then we take the
limit as the space
00:19:42.010 --> 00:19:44.220
gets bigger and bigger.
00:19:44.220 --> 00:19:47.550
That works for almost
everything we do in physics.
00:19:47.550 --> 00:19:50.670
And I would say that
the reason why it works
00:19:50.670 --> 00:19:53.050
is because we are
assuming fundamentally
00:19:53.050 --> 00:19:55.150
that physical
interactions are local.
00:19:55.150 --> 00:19:59.020
Things that are vastly faraway
don't affect what happens here.
00:19:59.020 --> 00:20:02.290
So as we make this
sphere larger and larger,
00:20:02.290 --> 00:20:05.670
we'll be adding matter at
larger and larger radii.
00:20:05.670 --> 00:20:07.400
That new matter
that we're adding
00:20:07.400 --> 00:20:10.210
is not going to have much
affect on what happens inside.
00:20:10.210 --> 00:20:11.710
And in fact, for
this problem, we'll
00:20:11.710 --> 00:20:14.070
soon see that the extra
matter we had on the outside
00:20:14.070 --> 00:20:16.980
has no effect whatever
what happens inside,
00:20:16.980 --> 00:20:19.120
related to this fact that
the gravitational field
00:20:19.120 --> 00:20:22.170
inside a shell of matter is 0.
00:20:22.170 --> 00:20:26.287
So that's a typical
situation and gives physicist
00:20:26.287 --> 00:20:27.870
a very strong
motivation to always try
00:20:27.870 --> 00:20:32.210
to think of infinities as
the limit of finite systems.
00:20:32.210 --> 00:20:34.480
What I want to
point out, however,
00:20:34.480 --> 00:20:36.872
is that that's not always
the right thing to do.
00:20:36.872 --> 00:20:39.080
And there are cases where
it's, in fact, emphatically
00:20:39.080 --> 00:20:41.140
the wrong thing to do.
00:20:41.140 --> 00:20:45.670
Mathematicians know about this,
but physicists tend not to.
00:20:45.670 --> 00:20:55.600
So I want to point out
that not all infinities are
00:20:55.600 --> 00:21:08.330
well-described as limits
of finite systems.
00:21:15.084 --> 00:21:16.750
I don't want to call
this into question.
00:21:16.750 --> 00:21:18.960
This, I think, is
absolutely solid.
00:21:18.960 --> 00:21:21.260
And we will continue
with it after I go off
00:21:21.260 --> 00:21:23.990
on this side discussion.
00:21:23.990 --> 00:21:27.106
But to give you an
example of a system which
00:21:27.106 --> 00:21:28.980
is infinite and not
well-described as a limit
00:21:28.980 --> 00:21:32.980
of finite systems-- this is a
mathematical example-- we could
00:21:32.980 --> 00:21:44.770
just think of the set of
positive integers, also
00:21:44.770 --> 00:21:46.140
called the natural numbers.
00:21:46.140 --> 00:21:47.970
And I'll use the letter
that's often used
00:21:47.970 --> 00:21:50.040
for it, N with an extra
line through it to make
00:21:50.040 --> 00:21:53.160
it super bold or
something like that.
00:21:53.160 --> 00:21:54.980
So that's the set of integers.
00:21:54.980 --> 00:22:00.910
And the question is,
suppose we consider
00:22:00.910 --> 00:22:03.430
trying to describe the
set of positive integers,
00:22:03.430 --> 00:22:06.210
as a limit-- a finite set.
00:22:06.210 --> 00:22:10.970
So we could think
about the limit from N
00:22:10.970 --> 00:22:17.000
goes to infinity-- as
N goes to infinity--
00:22:17.000 --> 00:22:26.005
of the natural numbers
up to N. So we're
00:22:26.005 --> 00:22:28.380
taking sets of integers and
taking bigger and bigger sets
00:22:28.380 --> 00:22:30.100
and trying to take
the limit, and asking,
00:22:30.100 --> 00:22:31.683
does that give us
the set of integers?
00:22:34.480 --> 00:22:37.150
One might think
the answer is yes.
00:22:37.150 --> 00:22:40.050
What I will claim is that
this is definitely not
00:22:40.050 --> 00:22:42.320
equal to the set of integers.
00:22:50.440 --> 00:22:52.810
And in fact, I'll claim
that the limit does not
00:22:52.810 --> 00:22:55.790
exist at all, which is
why it couldn't possibly
00:22:55.790 --> 00:22:58.210
be equal to the set integers.
00:22:58.210 --> 00:23:01.040
And to drive this
home, I just need
00:23:01.040 --> 00:23:05.555
to remind us what a limit means.
00:23:14.660 --> 00:23:16.160
And since this is
not a math course,
00:23:16.160 --> 00:23:17.510
I won't give a
rigorous definition.
00:23:17.510 --> 00:23:18.470
But I'll just give
you an example
00:23:18.470 --> 00:23:20.390
that will strike bells
and things that you
00:23:20.390 --> 00:23:22.220
learned in math classes.
00:23:22.220 --> 00:23:25.860
Suppose we want to
talk about the limit
00:23:25.860 --> 00:23:33.449
as x goes to 0 of sine x over x.
00:23:33.449 --> 00:23:34.740
But we all know how to do that.
00:23:34.740 --> 00:23:36.760
It's usually called
L'Hopital's rule or something.
00:23:36.760 --> 00:23:39.134
But you could probably just
use the definition of a limit
00:23:39.134 --> 00:23:40.550
and get it directly.
00:23:40.550 --> 00:23:43.440
That limit is 1.
00:23:43.440 --> 00:23:46.434
And what we mean
when we say that,
00:23:46.434 --> 00:23:51.650
is that as x gets
closer and closer to 0,
00:23:51.650 --> 00:23:53.830
we could evaluate this
expression for any value of x
00:23:53.830 --> 00:23:56.650
not equal to 0-- for 0,
itself, it's ambiguous.
00:23:56.650 --> 00:23:59.550
But for any value of x not equal
to 0, we can evaluate this.
00:23:59.550 --> 00:24:02.029
And as x gets closer
and closer to 0,
00:24:02.029 --> 00:24:03.570
that evaluation
gives us numbers that
00:24:03.570 --> 00:24:05.580
are closer and closer to 1.
00:24:05.580 --> 00:24:07.030
And we can get as
close to 1 as we
00:24:07.030 --> 00:24:11.482
want by choosing x's as
close to 0 as necessary.
00:24:11.482 --> 00:24:13.940
And that's usually phrased in
terms of epsilons and deltas.
00:24:13.940 --> 00:24:15.820
But I don't think
I need that here.
00:24:15.820 --> 00:24:18.650
The point is that, the limit
is simply the statement
00:24:18.650 --> 00:24:21.500
that this can be
made as close to 1
00:24:21.500 --> 00:24:25.510
as you like by choosing x as
close as possible to the limit
00:24:25.510 --> 00:24:27.710
point, 0.
00:24:27.710 --> 00:24:30.940
Now, if you imagine
applying the same concept
00:24:30.940 --> 00:24:35.600
to this set-- the set of
integers from 1 to N--
00:24:35.600 --> 00:24:37.330
the question is, as
you make N large,
00:24:37.330 --> 00:24:40.556
does it get closer to
the set of all integers?
00:24:40.556 --> 00:24:42.430
Are the numbers from 1
to 10 close to the set
00:24:42.430 --> 00:24:43.770
of all integers?
00:24:43.770 --> 00:24:44.530
No.
00:24:44.530 --> 00:24:46.597
What about 1 to a million?
00:24:46.597 --> 00:24:47.680
Still infinitely far away.
00:24:47.680 --> 00:24:48.490
1 to a billion?
00:24:48.490 --> 00:24:49.156
1 to a google?
00:24:49.156 --> 00:24:51.280
No matter what number you
pick as that upper limit,
00:24:51.280 --> 00:24:54.500
you're still infinitely far away
from the set of all integers.
00:24:54.500 --> 00:24:56.890
You're not coming close.
00:24:56.890 --> 00:24:58.610
So there's no
sense in which this
00:24:58.610 --> 00:25:01.240
converges to the
set of all integers.
00:25:01.240 --> 00:25:03.197
It's just a different animal.
00:25:03.197 --> 00:25:04.530
Now does this make a difference?
00:25:04.530 --> 00:25:06.317
Are there any questions
where it matters
00:25:06.317 --> 00:25:08.400
whether you think of the
integers as being defined
00:25:08.400 --> 00:25:11.490
in some other way,
or by this limit?
00:25:11.490 --> 00:25:14.010
Or maybe first say
how it is defined.
00:25:14.010 --> 00:25:16.589
If you ask mathematicians how
you define the set of integers,
00:25:16.589 --> 00:25:18.380
I think all of them
will tell you, well, we
00:25:18.380 --> 00:25:20.840
used the Peano axioms.
00:25:20.840 --> 00:25:23.030
And the key thing in
the Peano axioms--
00:25:23.030 --> 00:25:26.960
if you look at them-- that
controls the fact that there's
00:25:26.960 --> 00:25:31.030
an infinite number of integers,
is the successor axiom.
00:25:31.030 --> 00:25:33.800
That is, built into
these Peano axioms that
00:25:33.800 --> 00:25:36.470
describe the integers
mathematically
00:25:36.470 --> 00:25:38.880
is a statement that every
integer has a successor.
00:25:38.880 --> 00:25:40.380
And then, there are
other statements
00:25:40.380 --> 00:25:42.046
that guarantee that
the successor is not
00:25:42.046 --> 00:25:43.890
one of the ones that's
already on the list.
00:25:43.890 --> 00:25:46.700
So you always have a new
element as the successor
00:25:46.700 --> 00:25:49.320
to your highest element so far.
00:25:49.320 --> 00:25:54.020
And that guarantees from the
beginning, the set of axioms
00:25:54.020 --> 00:25:55.010
is infinite.
00:25:55.010 --> 00:25:58.160
It's not thought of as the limit
of finite sets and cannot be
00:25:58.160 --> 00:26:00.440
thought of as the
limit of finite sets.
00:26:00.440 --> 00:26:03.140
Because, no finite
set is, in any way,
00:26:03.140 --> 00:26:05.950
resembling an infinite set.
00:26:05.950 --> 00:26:07.080
So does it matter?
00:26:07.080 --> 00:26:09.810
Are there questions
where we care
00:26:09.810 --> 00:26:13.970
whether we could describe
the integers this way or not?
00:26:13.970 --> 00:26:17.120
And the only questions I
know sound kind of contrived,
00:26:17.120 --> 00:26:18.570
I'll admit.
00:26:18.570 --> 00:26:21.856
But I also point out that
in mathematics, the word,
00:26:21.856 --> 00:26:23.470
contrived doesn't cut much ice.
00:26:23.470 --> 00:26:26.364
If you discover a
contradiction in some system,
00:26:26.364 --> 00:26:28.530
nobody's going to tell you
whether you should ignore
00:26:28.530 --> 00:26:30.740
that contradiction
because it's contrived.
00:26:30.740 --> 00:26:33.620
If it's really a
contradiction, it counts.
00:26:33.620 --> 00:26:36.000
So a question where
it makes a difference,
00:26:36.000 --> 00:26:38.280
whether you think of the
integers as being defined
00:26:38.280 --> 00:26:40.670
as infinite from the start,
or whether you think of it--
00:26:40.670 --> 00:26:43.030
or try to think of it--
as a limit of this sort,
00:26:43.030 --> 00:26:47.340
is a question such as, what
fraction of the integers are
00:26:47.340 --> 00:26:50.510
so large that they cannot
be doubled and still be
00:26:50.510 --> 00:26:52.410
an integer?
00:26:52.410 --> 00:26:56.950
And notice that, if we
consider this set, for any N,
00:26:56.950 --> 00:27:00.100
no matter how large N is,
we would conclude that half
00:27:00.100 --> 00:27:03.570
of the integers are so large
that they cannot be doubled.
00:27:03.570 --> 00:27:05.720
And that would hold no
matter how large we made N.
00:27:05.720 --> 00:27:08.220
On the other hand, if we look
at the actual set of integers,
00:27:08.220 --> 00:27:09.845
we know that any
integer can be doubled
00:27:09.845 --> 00:27:12.310
and you just get
another integer.
00:27:12.310 --> 00:27:16.040
So that's an example of a
property of the integers, which
00:27:16.040 --> 00:27:19.050
you would get wrong if you
thought that you could think
00:27:19.050 --> 00:27:21.879
of the integers as
this kind of a limit.
00:27:21.879 --> 00:27:22.920
So you really just can't.
00:27:26.380 --> 00:27:28.050
Any questions about that?
00:27:28.050 --> 00:27:29.350
Subtle point, I think.
00:27:33.270 --> 00:27:34.740
AUDIENCE: How does
that relate to--
00:27:34.740 --> 00:27:36.420
PROFESSOR: To this?
00:27:36.420 --> 00:27:38.110
It's just a warning
that you should
00:27:38.110 --> 00:27:40.610
be careful about
treating infinity
00:27:40.610 --> 00:27:42.090
as limits of finite things.
00:27:42.090 --> 00:27:43.300
That's all it is.
00:27:43.300 --> 00:27:44.940
It does not relate to this.
00:27:44.940 --> 00:27:47.300
That's why I said it was
an aside when I started.
00:27:47.300 --> 00:27:51.070
Aside means something
worth knowing but not
00:27:51.070 --> 00:27:54.580
directly related to what
we're talking about.
00:27:54.580 --> 00:27:55.450
OK.
00:27:55.450 --> 00:27:59.450
So we're going to
proceed with this model.
00:27:59.450 --> 00:28:01.770
Let me mention one other
feature of the model
00:28:01.770 --> 00:28:04.430
to discuss a little bit, the
shape that we started with.
00:28:04.430 --> 00:28:06.050
We're starting with a sphere.
00:28:06.050 --> 00:28:08.904
You might ask, why a sphere?
00:28:08.904 --> 00:28:10.820
Well, a sphere is certainly
the simplest thing
00:28:10.820 --> 00:28:11.950
we could start with.
00:28:11.950 --> 00:28:15.330
And a sphere also guarantees
isotropy-- or at least
00:28:15.330 --> 00:28:16.800
isotropy about the origin.
00:28:16.800 --> 00:28:18.750
Isotropy about one point.
00:28:18.750 --> 00:28:22.640
We could, by doing
significantly more work,
00:28:22.640 --> 00:28:24.270
have started, for
example, with a cube
00:28:24.270 --> 00:28:25.896
and let the cube get
bigger and bigger.
00:28:25.896 --> 00:28:27.478
And as the cube got
bigger and bigger,
00:28:27.478 --> 00:28:28.687
it would also fill all space.
00:28:28.687 --> 00:28:30.519
And we might think that
would be another way
00:28:30.519 --> 00:28:31.780
of getting the same answer.
00:28:31.780 --> 00:28:33.040
And that would be right.
00:28:33.040 --> 00:28:35.590
If we did it with a cube,
it would be a lot more work.
00:28:35.590 --> 00:28:38.200
But we would, in fact,
get the same answer.
00:28:38.200 --> 00:28:41.315
The cube has enough
symmetry, So it, in fact,
00:28:41.315 --> 00:28:43.120
will be the same as
sphere in this case.
00:28:43.120 --> 00:28:45.286
I'm not going to try to
tell you how to calculate it
00:28:45.286 --> 00:28:46.620
with an arbitrary shape.
00:28:46.620 --> 00:28:48.120
But I will guarantee
you that a cube
00:28:48.120 --> 00:28:49.810
will get you the same answer.
00:28:49.810 --> 00:28:52.230
On the other hand, if we started
with a rectangular solid,
00:28:52.230 --> 00:28:54.646
where the three sides were
different-- or at least not all
00:28:54.646 --> 00:28:56.260
equal to each
other-- then would be
00:28:56.260 --> 00:28:58.885
starting with something which is
asymmetric in the first place.
00:28:58.885 --> 00:29:00.930
Some direction
would be privileged.
00:29:00.930 --> 00:29:03.970
And then, if we
continued-- as we're
00:29:03.970 --> 00:29:05.740
going to be doing
for our sphere--
00:29:05.740 --> 00:29:09.410
starting with this
irregular rectangular solid,
00:29:09.410 --> 00:29:11.860
we would build in an anisotropy
from the very beginning.
00:29:11.860 --> 00:29:16.520
We end up with an anisotropic
model of the universe.
00:29:16.520 --> 00:29:19.420
So since we're trying to
model the real universe, which
00:29:19.420 --> 00:29:22.480
is to a high degree
isotropic, we
00:29:22.480 --> 00:29:25.370
will start with something
which guarantees the isotropy.
00:29:25.370 --> 00:29:26.860
And the sphere does that.
00:29:26.860 --> 00:29:28.715
And it's the simplest
shape which does that.
00:29:34.851 --> 00:29:35.350
OK.
00:29:35.350 --> 00:29:39.400
Now, we're ready to start
putting in some dynamics
00:29:39.400 --> 00:29:42.200
into this model.
00:29:42.200 --> 00:29:44.930
And the dynamics that we're
going to put in for now
00:29:44.930 --> 00:29:48.960
will just be purely
Newtonian dynamics.
00:29:48.960 --> 00:29:51.550
And in fact, we'll be
modeling the matter
00:29:51.550 --> 00:29:53.770
that makes up this
sphere as just
00:29:53.770 --> 00:29:58.600
a dust of Newtonian
particles-- or if you like,
00:29:58.600 --> 00:30:02.110
a gas of Newtonian particles.
00:30:02.110 --> 00:30:04.780
These particles will be
nonrelativistic-- as implied
00:30:04.780 --> 00:30:07.060
by the word Newtonian.
00:30:07.060 --> 00:30:11.400
And that describes
our real universe
00:30:11.400 --> 00:30:13.760
for a good chunk
of its evolution
00:30:13.760 --> 00:30:15.690
but not for all of it.
00:30:15.690 --> 00:30:19.080
So let me-- before we
proceed-- say a few words
00:30:19.080 --> 00:30:25.070
about the real universe
and the kind of matter that
00:30:25.070 --> 00:30:31.180
has dominated it during
different eras of evolution.
00:30:31.180 --> 00:30:35.140
In the earliest time,
our universe, we believe,
00:30:35.140 --> 00:30:37.940
was radiation dominated.
00:30:37.940 --> 00:30:39.800
And that means
that, if you follow
00:30:39.800 --> 00:30:45.120
the evolution of our
universe backwards in time,
00:30:45.120 --> 00:30:49.100
as one goes to earlier and
earlier times, the photons that
00:30:49.100 --> 00:30:52.840
make up the cosmic background
radiation blue shift.
00:30:52.840 --> 00:30:55.090
We've learned that they red
shift as universe expands.
00:30:55.090 --> 00:30:57.550
That means if we extrapolate
backwards, they blue shift.
00:30:57.550 --> 00:31:00.330
They get more energetic
for every photon.
00:31:00.330 --> 00:31:03.450
But the number of photons
remains constant, essentially,
00:31:03.450 --> 00:31:05.090
as one goes backwards in time.
00:31:05.090 --> 00:31:07.190
They just get squeezed
into a small volume.
00:31:07.190 --> 00:31:09.580
And they become more energetic.
00:31:09.580 --> 00:31:12.910
Now, meanwhile, the particles
of ordinary matter-- and dark
00:31:12.910 --> 00:31:15.900
matter as well, the protons
and whatever the dark matter is
00:31:15.900 --> 00:31:19.740
made of-- also gets squeezed,
as you go backwards in time.
00:31:19.740 --> 00:31:22.020
But they don't become
more energetic.
00:31:22.020 --> 00:31:24.930
They remain just--
a proton remains
00:31:24.930 --> 00:31:27.510
a particle whose mass is
the mass of a proton times c
00:31:27.510 --> 00:31:29.310
squared.
00:31:29.310 --> 00:31:33.210
So as you go backwards,
the energy density
00:31:33.210 --> 00:31:35.729
in the radiation-- in the
cosmic microwave background
00:31:35.729 --> 00:31:37.770
radiation-- gets to be
larger and larger compared
00:31:37.770 --> 00:31:39.270
to the energy density
in the matter.
00:31:39.270 --> 00:31:41.210
And later, we'll learn
how to calculate this.
00:31:41.210 --> 00:31:43.940
But the two cross at
about 50,000 years.
00:31:43.940 --> 00:31:44.440
Yes.
00:31:44.440 --> 00:31:46.332
AUDIENCE: If we think
of particles as waves,
00:31:46.332 --> 00:31:50.116
then how does that make sense
that they wouldn't change?
00:31:50.116 --> 00:31:51.801
PROFESSOR: Right.
00:31:51.801 --> 00:31:53.300
If we think of
particles like waves,
00:31:53.300 --> 00:31:55.300
how does it make sense
that they don't change?
00:31:55.300 --> 00:31:56.700
The answer is, they
do a little bit.
00:31:56.700 --> 00:31:59.033
But we're going to be assuming
that these particles have
00:31:59.033 --> 00:32:00.770
negligible velocities.
00:32:00.770 --> 00:32:04.760
What happens is their momentum
actually does blue shift.
00:32:04.760 --> 00:32:08.036
But a blue shift is proportional
to the initial value.
00:32:08.036 --> 00:32:09.660
And if the initial
value is very small,
00:32:09.660 --> 00:32:11.250
even when it blue
shifts, the momentum
00:32:11.250 --> 00:32:12.333
could still be negligible.
00:32:16.390 --> 00:32:20.830
So for the real universe,
between time, 0,
00:32:20.830 --> 00:32:31.575
and about 50,000
years, the universe
00:32:31.575 --> 00:32:32.695
was radiation dominated.
00:32:37.562 --> 00:32:39.520
And we'll be talking
about that in a few weeks.
00:32:39.520 --> 00:32:42.420
But today, we're
just ignoring that.
00:32:42.420 --> 00:32:50.270
Then, from this time
of about 50,000 years
00:32:50.270 --> 00:32:56.210
until about 9 billion
years-- so a good chunk
00:32:56.210 --> 00:32:58.770
of the history of the
universe-- the universe
00:32:58.770 --> 00:33:03.870
was what is called
matter dominated.
00:33:07.950 --> 00:33:10.850
And matter means
nonrelativistic matter.
00:33:10.850 --> 00:33:14.440
That's standard
jargon in cosmology.
00:33:14.440 --> 00:33:16.410
When we talk about a
matter-dominated universe,
00:33:16.410 --> 00:33:18.659
and even though we don't use
the word nonrelativistic,
00:33:18.659 --> 00:33:22.100
that's what everybody means by
a matter dominated universe.
00:33:22.100 --> 00:33:24.300
And that's the case we're
going to be discussing,
00:33:24.300 --> 00:33:28.450
just ordinary nonrelativistic
matter filling space.
00:33:28.450 --> 00:33:31.540
And then for our real
universe, something else
00:33:31.540 --> 00:33:34.820
happened at about 9 billion
years in its history
00:33:34.820 --> 00:33:37.730
up to the present-- and
presumably in the future
00:33:37.730 --> 00:33:41.545
as well-- is the universe
became dark energy dominated.
00:33:54.100 --> 00:33:55.980
And dark energy
is the stuff that
00:33:55.980 --> 00:33:58.910
causes the universe
to accelerate.
00:33:58.910 --> 00:34:01.310
So the universe has
been accelerating
00:34:01.310 --> 00:34:07.110
since about 9 billion
years from the Big Bang.
00:34:07.110 --> 00:34:10.150
Now, I think we mentioned--
I'll mention it again quickly--
00:34:10.150 --> 00:34:13.530
there's no conversion of
ordinary matter to dark energy
00:34:13.530 --> 00:34:17.080
here, which you might guess
from this change in domination.
00:34:17.080 --> 00:34:20.560
It's just a question of how they
behave as the universe expands.
00:34:20.560 --> 00:34:25.199
Ordinary matter has a
density, which decreases as 1
00:34:25.199 --> 00:34:27.040
over the cube of
the scale factor.
00:34:27.040 --> 00:34:29.719
You just have a fixed number
of particles scattering out
00:34:29.719 --> 00:34:31.889
over a larger and larger volume.
00:34:31.889 --> 00:34:35.480
The dark energy--
for reasons that we
00:34:35.480 --> 00:34:37.520
will come to near the
end of the course,
00:34:37.520 --> 00:34:39.810
in fact, does not
change its energy
00:34:39.810 --> 00:34:42.659
density at all as
the universe expands.
00:34:42.659 --> 00:34:44.790
So what happened 9
billion years ago
00:34:44.790 --> 00:34:47.460
was simply that the
density of ordinary matter
00:34:47.460 --> 00:34:49.350
fell below the density
of dark energy.
00:34:49.350 --> 00:34:51.225
And then, the dark energy
started to dominate
00:34:51.225 --> 00:34:52.850
and the universe
started to accelerate.
00:34:56.239 --> 00:34:59.450
Today, the dark energy, by
the way, is about 60% or 70%
00:34:59.450 --> 00:35:00.850
of the total.
00:35:00.850 --> 00:35:02.410
So it doesn't
dominate completely.
00:35:02.410 --> 00:35:04.283
But it's the largest component.
00:35:07.520 --> 00:35:08.020
OK.
00:35:08.020 --> 00:35:09.810
For today's
calculation, we're going
00:35:09.810 --> 00:35:11.370
to be focusing on
this middle period
00:35:11.370 --> 00:35:13.045
and pretending it's
the whole story.
00:35:13.045 --> 00:35:15.170
And we will come back and
discuss these other eras.
00:35:15.170 --> 00:35:16.461
We're not going to ignore them.
00:35:16.461 --> 00:35:18.514
But we will not
discuss them today.
00:35:18.514 --> 00:35:19.930
So we will be
discussing this case
00:35:19.930 --> 00:35:22.090
of a matter-dominated universe.
00:35:22.090 --> 00:35:25.392
And we're going to be discussing
it using Newtonian mechanics.
00:35:25.392 --> 00:35:26.850
And in spite of
the fact that we're
00:35:26.850 --> 00:35:29.590
using Newtonian mechanics,
I will assure you--
00:35:29.590 --> 00:35:32.200
and try to give you
some arguments later--
00:35:32.200 --> 00:35:35.150
it gives exactly the same answer
that general relativity gives.
00:35:45.650 --> 00:35:46.150
OK.
00:35:46.150 --> 00:35:47.244
So how do we proceed?
00:35:47.244 --> 00:35:48.660
How do we write
down the equations
00:35:48.660 --> 00:35:52.590
that describe how this
sphere is going to evolve?
00:35:52.590 --> 00:35:55.770
We're going to be
using a shell picture.
00:35:55.770 --> 00:35:59.290
That is, we will
describe that sphere
00:35:59.290 --> 00:36:01.245
in terms of shells
that make it up.
00:36:04.240 --> 00:36:10.690
So in other words, we
will divide the matter
00:36:10.690 --> 00:36:11.690
at the initial time.
00:36:11.690 --> 00:36:14.000
And then, we'll just imagine
labeling those particles
00:36:14.000 --> 00:36:15.692
and following them.
00:36:15.692 --> 00:36:17.150
So at the initial
time, we're going
00:36:17.150 --> 00:36:20.220
to divide the matter
into concentric shells.
00:36:34.890 --> 00:36:46.120
So each shell extends from
some value of r sub i--
00:36:46.120 --> 00:36:54.930
the initial radius-- to
r sub i plus dr sub i.
00:36:54.930 --> 00:36:57.470
Our shells will be infinitesimal
in their thickness.
00:37:00.200 --> 00:37:02.300
And each shell will have
a different r sub i.
00:37:02.300 --> 00:37:04.490
And we could let all the dr sub
i's be the same, if we want.
00:37:04.490 --> 00:37:06.060
We can let each shell
be the same thickness.
00:37:06.060 --> 00:37:06.670
We can let them be different.
00:37:06.670 --> 00:37:07.419
It doesn't matter.
00:37:12.930 --> 00:37:16.710
Now a reason why we can get
away with thinking of the matter
00:37:16.710 --> 00:37:19.190
purely has being
described by shells
00:37:19.190 --> 00:37:21.510
is that we know that
we started with all
00:37:21.510 --> 00:37:23.480
of the velocities radial.
00:37:23.480 --> 00:37:25.470
That Hubble's law
that we assumed
00:37:25.470 --> 00:37:27.360
says that the velocities
were proportional
00:37:27.360 --> 00:37:30.210
to the radius vector, which
is measured from the origin.
00:37:30.210 --> 00:37:33.027
So all our initial velocities
were radial outward.
00:37:33.027 --> 00:37:34.610
And furthermore, if
we think about how
00:37:34.610 --> 00:37:37.440
the Newtonian force of
gravity will come about
00:37:37.440 --> 00:37:41.090
for this spherical object, they
will all be radial as well.
00:37:41.090 --> 00:37:43.170
So all the motion
will remain radial.
00:37:43.170 --> 00:37:46.290
There will never be any forces
acting on any of our particles
00:37:46.290 --> 00:37:49.392
that will push them in
a tangential direction.
00:37:49.392 --> 00:37:50.600
So all motion will be radial.
00:37:50.600 --> 00:37:53.570
And as long as we keep track
of the radius of each particle,
00:37:53.570 --> 00:37:55.760
its angular variables,
theta and phi--
00:37:55.760 --> 00:37:57.910
which I will never
mention again--
00:37:57.910 --> 00:37:59.710
will just be constant in time.
00:37:59.710 --> 00:38:01.210
So we will not need
to mention them.
00:38:18.190 --> 00:38:20.715
So all motion is radial,
because the v's start radial.
00:38:23.800 --> 00:38:35.309
And there are no
tangential forces,
00:38:35.309 --> 00:38:37.600
where tangential means any
direction other than radial.
00:38:37.600 --> 00:38:39.766
So we're just saying that
all the forces are radial.
00:38:39.766 --> 00:38:43.390
And that means the
motion will stay radial.
00:38:43.390 --> 00:38:50.830
To describe the motion, I want
to give a little bit more teeth
00:38:50.830 --> 00:38:52.246
to this statement
that we're going
00:38:52.246 --> 00:38:54.630
to be describing it
in terms of shells.
00:38:54.630 --> 00:38:59.080
We'll going to describe the
motion by a function, r, which
00:38:59.080 --> 00:39:03.360
will be a function of two
variables, r sub i and t.
00:39:06.400 --> 00:39:23.850
And this is just the
radius at time, t,
00:39:23.850 --> 00:39:40.450
of the shell that was at radius,
r sub i, at time, t sub i.
00:39:40.450 --> 00:39:43.120
So each shell is labeled by
where it is at time, t sub i.
00:39:43.120 --> 00:39:45.200
And once we label it,
it keeps that label.
00:39:45.200 --> 00:39:48.790
But then, we'll talk about this
function, r, which tells us
00:39:48.790 --> 00:39:51.230
where it is at any later time.
00:39:51.230 --> 00:39:54.520
And we could also think about
earlier times if we want.
00:39:54.520 --> 00:39:58.620
So r of r sub i, t is the radius
of the shell that started at r
00:39:58.620 --> 00:40:01.400
sub i, where we're looking
at the radius at time, t.
00:40:04.650 --> 00:40:07.886
Now, I should warn you that
if you look in any textbook,
00:40:07.886 --> 00:40:09.260
you will see a
simpler derivation
00:40:09.260 --> 00:40:10.890
than what I'm about to show you.
00:40:10.890 --> 00:40:13.790
So you might wonder why am
I going to so much trouble
00:40:13.790 --> 00:40:15.764
when there's an
easier way to do it.
00:40:15.764 --> 00:40:17.430
And the answer is
that the calculation I
00:40:17.430 --> 00:40:19.510
show you will, in fact,
show you more than what's
00:40:19.510 --> 00:40:23.200
in the textbooks-- and
Ryden's textbook, for example.
00:40:23.200 --> 00:40:25.120
What most textbooks
do-- including Ryden,
00:40:25.120 --> 00:40:29.310
I believe-- is to assume that
this motion will continue
00:40:29.310 --> 00:40:30.830
to obey Hubble's
law and continue
00:40:30.830 --> 00:40:33.650
to be completely
uniform in the density.
00:40:33.650 --> 00:40:35.660
And then, you could
just calculate
00:40:35.660 --> 00:40:37.500
what happens to the
outside of the sphere.
00:40:37.500 --> 00:40:38.958
And that governs
everything, if you
00:40:38.958 --> 00:40:41.150
assume that will stay uniform.
00:40:41.150 --> 00:40:43.440
We are not going to assume
that it will stay uniform.
00:40:43.440 --> 00:40:45.780
We will show that it
will stay uniform.
00:40:45.780 --> 00:40:48.352
And from my point of
view, it's a lot better
00:40:48.352 --> 00:40:50.610
to actually show something
than to just guess
00:40:50.610 --> 00:40:53.840
it and we write about,
without showing it.
00:40:53.840 --> 00:40:55.860
So that's why we're going
to do some extra work.
00:40:55.860 --> 00:40:59.550
We will actually show that the
uniformity, once you put in,
00:40:59.550 --> 00:41:03.301
is preserved under time,
under the laws of Newtonian
00:41:03.301 --> 00:41:03.800
evolution.
00:41:13.739 --> 00:41:15.030
I guess I'll leave the picture.
00:41:15.030 --> 00:41:16.280
Maybe, I'll leave all of that.
00:41:47.840 --> 00:41:48.340
OK.
00:41:48.340 --> 00:41:51.330
Now, there's an issue that's
a little bit complicated
00:41:51.330 --> 00:41:53.160
that I'll try to describe.
00:41:53.160 --> 00:41:54.660
And again, this is
a subtlety that's
00:41:54.660 --> 00:41:56.810
probably not mentioned
in the books.
00:41:56.810 --> 00:41:59.300
We have these different
shells that are evolving.
00:41:59.300 --> 00:42:01.580
And we can calculate the
force on any given shell
00:42:01.580 --> 00:42:03.740
if we know the matter
that's inside that shell.
00:42:03.740 --> 00:42:05.420
That's what it tells us.
00:42:05.420 --> 00:42:07.710
If everything is in
shells, the shells
00:42:07.710 --> 00:42:11.780
have no effect on the matter
that is inside the shell.
00:42:11.780 --> 00:42:15.780
So if we want to know the
effect on a given shell,
00:42:15.780 --> 00:42:18.670
we only need to know the matter
that's inside that shell.
00:42:18.670 --> 00:42:20.772
The matter outside has no force.
00:42:20.772 --> 00:42:22.230
So it's very
important that we know
00:42:22.230 --> 00:42:24.290
the ordering of these shells.
00:42:24.290 --> 00:42:25.710
Now, initially, we certainly do.
00:42:25.710 --> 00:42:27.980
They're just ordered
according to r sub i.
00:42:27.980 --> 00:42:29.700
But once they start
to move around,
00:42:29.700 --> 00:42:31.930
there's, in principle,
the possibility
00:42:31.930 --> 00:42:33.990
that shells could cross.
00:42:33.990 --> 00:42:36.780
And if shells crossed,
our equations of emotion
00:42:36.780 --> 00:42:39.970
would have to change, because
then different amounts of mass
00:42:39.970 --> 00:42:41.910
would be acting on
different shells.
00:42:41.910 --> 00:42:44.440
And we'd have to take
that into account.
00:42:44.440 --> 00:42:47.440
Fortunately, that turns
out not to be a problem.
00:42:47.440 --> 00:42:51.060
It just sounds like
it could be a problem.
00:42:51.060 --> 00:42:52.670
And the way we're
going to treat it
00:42:52.670 --> 00:42:56.090
is to recognize that, initially,
all these shells are moving
00:42:56.090 --> 00:42:58.480
away from each other, because
of the Hubble expansion.
00:42:58.480 --> 00:43:00.820
If Hubble's law holds,
any two particles
00:43:00.820 --> 00:43:04.090
are moving away from each
other with a relative velocity
00:43:04.090 --> 00:43:05.570
proportional to their distance.
00:43:05.570 --> 00:43:09.710
So that holds for any
two shells, as well.
00:43:09.710 --> 00:43:12.342
So if shells are
going cross, they're
00:43:12.342 --> 00:43:14.050
certainly not going
to cross immediately.
00:43:14.050 --> 00:43:15.400
There are no two shells
that are approaching
00:43:15.400 --> 00:43:16.560
each other initially.
00:43:16.560 --> 00:43:18.960
All shells are moving
apart initially.
00:43:18.960 --> 00:43:21.160
This could be turned
around by the forces--
00:43:21.160 --> 00:43:22.670
and we'll have to see.
00:43:22.670 --> 00:43:25.720
But what we can do is, we
can write down equations
00:43:25.720 --> 00:43:29.550
that we know will hold, at
least until the time where there
00:43:29.550 --> 00:43:31.542
might be some shell crossings.
00:43:31.542 --> 00:43:33.000
That is, we'll
write down equations
00:43:33.000 --> 00:43:36.580
that will hold as long as
there are no shell crossings.
00:43:36.580 --> 00:43:38.770
Then, if there was going
to be a shell crossing,
00:43:38.770 --> 00:43:39.880
the equations we
write down would
00:43:39.880 --> 00:43:41.546
have to be valid right
up until the time
00:43:41.546 --> 00:43:42.642
of that shell crossing.
00:43:42.642 --> 00:43:44.850
And, therefore, the equations
that we're writing down
00:43:44.850 --> 00:43:47.566
would have to show us that
the shells are going to cross.
00:43:47.566 --> 00:43:49.690
The shells can't just decide
to cross independently
00:43:49.690 --> 00:43:51.991
of our equations of motion.
00:43:51.991 --> 00:43:54.240
And what we'll find is that
our equations will tell us
00:43:54.240 --> 00:43:56.042
that there will be
no shell crossings.
00:43:56.042 --> 00:43:57.500
And the equations
are valid as long
00:43:57.500 --> 00:43:58.710
as there are no shell crossings.
00:43:58.710 --> 00:44:00.501
And I think if you
think about that, that's
00:44:00.501 --> 00:44:02.852
an airtight argument,
even though we're never
00:44:02.852 --> 00:44:04.810
going to write down
equations that will tell us
00:44:04.810 --> 00:44:09.220
what would happen
if shells did cross.
00:44:09.220 --> 00:44:10.260
OK.
00:44:10.260 --> 00:44:21.350
So we're going to
write equations
00:44:21.350 --> 00:44:33.200
that hold as long as there
are no shell crossings.
00:44:39.791 --> 00:44:40.290
OK.
00:44:40.290 --> 00:44:41.873
As long as there's
no shell crossings,
00:44:41.873 --> 00:44:44.702
then the total mass
inside of any shell
00:44:44.702 --> 00:44:45.660
is independent of time.
00:44:45.660 --> 00:44:49.020
It's just the shells
that are inside it.
00:44:49.020 --> 00:45:03.760
So the shell at initial radius,
r sub i, even at later times,
00:45:03.760 --> 00:45:12.550
feels the force of
the mass inside.
00:45:12.550 --> 00:45:15.400
And we can write down a
formula for that mass inside.
00:45:15.400 --> 00:45:21.220
The mass inside the shell,
whose initial radius is r sub i,
00:45:21.220 --> 00:45:25.920
is just equal to the initial
volume of that sphere, which
00:45:25.920 --> 00:45:33.450
is 4 pi over 3 r sub i
cubed, times the initial mass
00:45:33.450 --> 00:45:34.470
density, rho sub i.
00:45:36.835 --> 00:45:38.460
So that's how much
mass there is inside
00:45:38.460 --> 00:45:41.170
a given shell when
the system starts.
00:45:41.170 --> 00:45:43.910
And that will continue to be
exactly how much mass is inside
00:45:43.910 --> 00:45:45.930
that shell as the
system evolves,
00:45:45.930 --> 00:45:47.546
unless there are
shell crossings.
00:45:47.546 --> 00:45:51.440
And our goal is simply to write
down equations that are valid
00:45:51.440 --> 00:45:53.702
until there might
be a shell crossing.
00:45:53.702 --> 00:45:55.160
And then, we can
ask whether or not
00:45:55.160 --> 00:45:56.602
there will be any
shell crossings.
00:46:15.320 --> 00:46:17.240
OK, now, Newton's
law tells us how
00:46:17.240 --> 00:46:21.400
to write down the
acceleration of an arbitrary
00:46:21.400 --> 00:46:23.650
particle in this system.
00:46:23.650 --> 00:46:27.590
Newton's law tells
us the acceleration
00:46:27.590 --> 00:46:29.790
will be proportional
to negative-- I'll
00:46:29.790 --> 00:46:31.960
put a radial vector
at the end in r hat.
00:46:31.960 --> 00:46:33.800
So it's direction,
minus r hat, will
00:46:33.800 --> 00:46:36.780
be the direction
enforced on any particle.
00:46:36.780 --> 00:46:41.340
And the magnitude, it will
be Newton's constant times
00:46:41.340 --> 00:46:46.730
the mass enclosed in the
sphere of radius, r sub i,
00:46:46.730 --> 00:46:49.060
divided by the square of
the distance of that shell
00:46:49.060 --> 00:46:50.070
from the origin.
00:46:50.070 --> 00:46:53.840
And that's exactly what we
called this function, r of ri,
00:46:53.840 --> 00:46:54.760
t.
00:46:54.760 --> 00:46:58.540
It's the radius of the
shell at any given time.
00:46:58.540 --> 00:47:05.725
So it's r squared
of r sub i and t.
00:47:05.725 --> 00:47:08.100
And then as I promised you,
there's a unit vector, r hat,
00:47:08.100 --> 00:47:09.120
at the end.
00:47:09.120 --> 00:47:12.140
So the force is-- and the
acceleration, therefore,--
00:47:12.140 --> 00:47:14.940
is in the negative
r hat direction.
00:47:14.940 --> 00:47:16.847
So this holds for any
shell, whether which
00:47:16.847 --> 00:47:19.180
shell we're talking about is
indicated by this variable,
00:47:19.180 --> 00:47:20.510
r sub i.
00:47:20.510 --> 00:47:23.220
r sub i tells us the initial
position of that shell.
00:47:28.822 --> 00:47:30.530
So is everybody happy
with this equation?
00:47:30.530 --> 00:47:31.440
This is really
the crucial thing.
00:47:31.440 --> 00:47:33.148
Once you write this
down, everything else
00:47:33.148 --> 00:47:35.390
is really just chugging along.
00:47:35.390 --> 00:47:38.460
So everybody happy with that?
00:47:38.460 --> 00:47:40.720
It's just the
statement from Newton,
00:47:40.720 --> 00:47:43.890
that if all the masses
are ranged spherically
00:47:43.890 --> 00:47:50.990
symmetrically, the mass
inside any shell-- excuse me,
00:47:50.990 --> 00:47:55.380
the force due to the mass that's
on a shell that's at a larger
00:47:55.380 --> 00:47:59.010
radius than you are, produces
no acceleration for you.
00:47:59.010 --> 00:48:00.470
The only acceleration
you feel is
00:48:00.470 --> 00:48:02.990
due to the masses
at smaller radii.
00:48:02.990 --> 00:48:04.730
And that's what
this formula says.
00:48:04.730 --> 00:48:05.860
OK, good.
00:48:05.860 --> 00:48:08.160
Now, what I want to do is--
this is a vector equation--
00:48:08.160 --> 00:48:10.870
but we know that we're just
taking these spherical motions.
00:48:10.870 --> 00:48:12.370
And all we really
have to do is keep
00:48:12.370 --> 00:48:14.964
track of how r changes
with time, little r.
00:48:14.964 --> 00:48:17.380
So we can turn this into an
ordinary differential equation
00:48:17.380 --> 00:48:21.780
with no vectors for
this function, little r.
00:48:21.780 --> 00:48:24.860
The acceleration has a
magnitude, which is just,
00:48:24.860 --> 00:48:27.270
r double dot.
00:48:27.270 --> 00:48:33.871
And that is then
equal to minus--
00:48:33.871 --> 00:48:38.940
and we write again what M
of ri is from this formula.
00:48:38.940 --> 00:48:45.490
So I'm going to write this as
minus 4 pi over 3 G times r sub
00:48:45.490 --> 00:48:53.620
i cubed rho sub i-- coming
from that formula-- divided
00:48:53.620 --> 00:48:55.250
by r squared.
00:48:55.250 --> 00:48:58.070
And I'm going to stop
writing the arguments.
00:48:58.070 --> 00:49:03.170
But this r is the
function of r sub i and t.
00:49:03.170 --> 00:49:05.080
But I will stop
writing its arguments.
00:49:05.080 --> 00:49:07.220
And the two dots means
derivative with respect
00:49:07.220 --> 00:49:08.010
to time.
00:49:08.010 --> 00:49:08.510
Yes.
00:49:08.510 --> 00:49:11.408
AUDIENCE: I guess I don't
understand how come we would,
00:49:11.408 --> 00:49:14.789
calculating the mass,
we use r sub i, which
00:49:14.789 --> 00:49:18.130
is the radius at some
initial time, ti.
00:49:18.130 --> 00:49:20.060
But then, when we're
saying the distance
00:49:20.060 --> 00:49:24.310
from that mass to our point,
we use this function, r,
00:49:24.310 --> 00:49:25.852
instead of r sub i.
00:49:25.852 --> 00:49:26.560
PROFESSOR: Right.
00:49:26.560 --> 00:49:27.850
An important question.
00:49:27.850 --> 00:49:32.500
The reason is that we're
using the initial radius,
00:49:32.500 --> 00:49:34.800
but we're also using
the initial density.
00:49:34.800 --> 00:49:36.240
So this formula
certainly gives us
00:49:36.240 --> 00:49:39.804
the mass that's inside that
circle at the beginning.
00:49:39.804 --> 00:49:41.470
And then, if there's
no shell crossings,
00:49:41.470 --> 00:49:43.080
all these circles move together.
00:49:43.080 --> 00:49:44.800
The mass inside never changes.
00:49:44.800 --> 00:49:46.640
So it's the same as it was then.
00:49:46.640 --> 00:49:48.640
While, the distance from
the center of that mass
00:49:48.640 --> 00:49:50.330
does change, as
the radius changes.
00:49:50.330 --> 00:49:53.960
So the denominator changes,
the numerator does not.
00:49:57.327 --> 00:49:58.289
Yes.
00:49:58.289 --> 00:50:00.213
AUDIENCE: So, the first
equation, g, that's
00:50:00.213 --> 00:50:02.620
the gravity--
00:50:02.620 --> 00:50:05.048
PROFESSOR: g is the
acceleration of gravity.
00:50:05.048 --> 00:50:07.824
AUDIENCE: So, why does it
have-- it has units of-- no,
00:50:07.824 --> 00:50:08.324
it doesn't.
00:50:08.324 --> 00:50:08.823
Sorry.
00:50:08.823 --> 00:50:12.160
PROFESSOR: Yes, it should
have units of acceleration,
00:50:12.160 --> 00:50:16.529
as our double dot should
have units of acceleration.
00:50:16.529 --> 00:50:17.320
Hopefully, they do.
00:50:23.950 --> 00:50:24.450
OK.
00:50:24.450 --> 00:50:27.560
Now, as the system
evolves, r sub
00:50:27.560 --> 00:50:30.250
i is just a constant--
different for each shell,
00:50:30.250 --> 00:50:32.220
but a constant in time.
00:50:32.220 --> 00:50:35.270
And imagine solving this
one shell at a time.
00:50:35.270 --> 00:50:37.070
Rho sub i is, again, a constant.
00:50:37.070 --> 00:50:40.790
It's the value of the mass
density at the initial time.
00:50:40.790 --> 00:50:43.024
So it's going to keep its value.
00:50:43.024 --> 00:50:45.190
So the differential equation
involves a differential
00:50:45.190 --> 00:50:47.570
equation in which little r
changes with time, and nothing
00:50:47.570 --> 00:50:48.630
else here does.
00:50:48.630 --> 00:50:51.450
So this is just a second
order differential equation
00:50:51.450 --> 00:50:53.542
for little r.
00:50:53.542 --> 00:50:55.500
So, we're well on the
way to having a solution,
00:50:55.500 --> 00:50:57.530
because second order
differential equations
00:50:57.530 --> 00:50:59.880
are, in principle, solvable.
00:50:59.880 --> 00:51:02.560
But the one thing
that you should all
00:51:02.560 --> 00:51:05.090
remember about second order
differential equations
00:51:05.090 --> 00:51:07.650
is that in order to
have a unique solution,
00:51:07.650 --> 00:51:09.489
you need initial conditions.
00:51:09.489 --> 00:51:11.030
And if it's a second
order equation--
00:51:11.030 --> 00:51:13.910
as Newton's equations
always are-- you
00:51:13.910 --> 00:51:15.880
need to be able to specify
the initial position
00:51:15.880 --> 00:51:18.440
and the initial velocity
before the second order
00:51:18.440 --> 00:51:20.184
equation leads to
a unique answer.
00:51:20.184 --> 00:51:21.600
So that's what we
want to do next.
00:51:21.600 --> 00:51:24.090
We want to write down
the initial value
00:51:24.090 --> 00:51:26.226
of r and the initial
value of r dot.
00:51:26.226 --> 00:51:28.030
And then, we'll
have a system, which
00:51:28.030 --> 00:51:31.362
is just something we can
turn over to a mathematician.
00:51:31.362 --> 00:51:33.070
And if the mathematician
is at all smart,
00:51:33.070 --> 00:51:35.390
he'll be able to solve it.
00:51:35.390 --> 00:51:36.665
So initial conditions.
00:52:04.210 --> 00:52:06.954
So we want the initial
value of r sub i--
00:52:06.954 --> 00:52:08.370
and initial means
a time, t sub i.
00:52:08.370 --> 00:52:11.180
That's where we're setting
our initial conditions.
00:52:11.180 --> 00:52:13.020
So we need to know
what r of r sub i,
00:52:13.020 --> 00:52:15.100
t sub i is, which
is a trivial thing
00:52:15.100 --> 00:52:18.490
if you keep track of
what this notation means.
00:52:18.490 --> 00:52:21.330
What is that?
00:52:21.330 --> 00:52:22.295
r sub i, good.
00:52:28.000 --> 00:52:31.160
The meaning of this is just
the radius at time, t sub i,
00:52:31.160 --> 00:52:34.310
of the particle, whose radius
at time, t sub i, was r sub i.
00:52:34.310 --> 00:52:36.900
So to put together
all those tautologies,
00:52:36.900 --> 00:52:40.430
the answer is just,
obviously, r sub i.
00:52:40.430 --> 00:52:40.930
OK.
00:52:40.930 --> 00:52:43.875
And then we also want to be
able to solve this equation
00:52:43.875 --> 00:52:46.500
and have a unique solution--
we want the initial value of r
00:52:46.500 --> 00:52:47.790
sub i dot.
00:52:47.790 --> 00:52:51.500
And initial means,
again, at time, t sub i.
00:52:51.500 --> 00:52:56.120
And that's specified by
our Hubble expansion.
00:52:56.120 --> 00:52:59.825
Every initial velocity is
just H sub i times the radius.
00:53:03.190 --> 00:53:05.230
So if the radius
is r sub i, this
00:53:05.230 --> 00:53:08.281
is just H sub i times r sub i.
00:53:08.281 --> 00:53:10.780
That's just the Hubble velocity
that we put in to the system
00:53:10.780 --> 00:53:11.571
when we started it.
00:53:16.390 --> 00:53:16.890
OK.
00:53:16.890 --> 00:53:19.640
So now, we have a system,
which is purely mathematical.
00:53:19.640 --> 00:53:21.680
We have a second order
differential equation
00:53:21.680 --> 00:53:24.216
and initial conditions
on r and r dot.
00:53:24.216 --> 00:53:26.810
That leads to a unique solution.
00:53:26.810 --> 00:53:28.000
Now it's pure math.
00:53:28.000 --> 00:53:31.690
No more physics, at least
at this stage of the game.
00:53:31.690 --> 00:53:33.464
However, there are
interesting things
00:53:33.464 --> 00:53:34.880
mathematically
that one can notice
00:53:34.880 --> 00:53:36.920
about this system of equations.
00:53:36.920 --> 00:53:40.240
And now what we're going
to see is the magic
00:53:40.240 --> 00:53:42.490
of these equations
for preserving
00:53:42.490 --> 00:53:45.220
the uniformity of this system.
00:53:45.220 --> 00:53:47.390
It's all built into
these equations.
00:53:47.390 --> 00:53:49.390
And let's see how.
00:53:49.390 --> 00:53:51.890
The key feature that's
somewhat miraculous
00:53:51.890 --> 00:54:01.940
that these equations
have is that r
00:54:01.940 --> 00:54:11.350
sub i can be made to disappear
by a change of variables.
00:54:11.350 --> 00:54:14.690
I'll tell you what
that is in a second.
00:54:14.690 --> 00:54:27.570
It also helps if you know
how to spell "disappear."
00:54:27.570 --> 00:54:30.560
We're going to define
a new function, u.
00:54:30.560 --> 00:54:31.880
I just made up a letter there.
00:54:31.880 --> 00:54:33.130
Could have called it anything.
00:54:33.130 --> 00:54:36.470
But u of r sub i
and t is just going
00:54:36.470 --> 00:54:44.690
to be defined to be r of r sub
i and t divided by r sub i.
00:54:49.660 --> 00:54:52.232
Given any function,
r of r sub i and t,
00:54:52.232 --> 00:54:53.690
I can always define
a new function,
00:54:53.690 --> 00:54:56.179
which is just the same
function divided by r sub i.
00:55:02.420 --> 00:55:04.545
Now, let's look at what
this does to our equations.
00:55:08.090 --> 00:55:11.970
My claim is that r sub i
was going to disappear.
00:55:11.970 --> 00:55:13.470
And now, we'll see
how that happens.
00:55:38.500 --> 00:55:43.160
OK, if u is defined this way, we
can write down the differential
00:55:43.160 --> 00:55:45.490
equation that it will
obey, by writing down
00:55:45.490 --> 00:55:47.840
an equation for u double dot.
00:55:47.840 --> 00:55:54.876
And u double dot will just be r
double dot divided by r sub i.
00:55:57.520 --> 00:56:02.130
So we take the equation for r
double dot over here and divide
00:56:02.130 --> 00:56:05.690
it by r sub i .
00:56:05.690 --> 00:56:14.390
So we can write that as minus 4
pi over 3 G-- for some reason,
00:56:14.390 --> 00:56:15.995
I decided to leave
the r sub i cubed
00:56:15.995 --> 00:56:19.120
in the numerator for now--
and just put an extra r sub
00:56:19.120 --> 00:56:22.770
i in the denominator-- the one
that we divided by-- times r
00:56:22.770 --> 00:56:23.270
squared.
00:56:34.429 --> 00:56:36.470
And now, what I'm going
to do is just replace r .
00:56:36.470 --> 00:56:38.406
We're trying to write
an equation for u.
00:56:38.406 --> 00:56:41.850
r is related to u by this
equation. r is r sub i times u.
00:56:44.610 --> 00:56:54.550
So I can write this as
minus 4 pi over 3 G r sub i
00:56:54.550 --> 00:56:59.470
cubed rho sub i
over-- now we have-- u
00:56:59.470 --> 00:57:02.170
squared times r sub i cubed.
00:57:02.170 --> 00:57:03.570
I left the numerator alone.
00:57:03.570 --> 00:57:05.890
I just rewrote the
denominator by placing
00:57:05.890 --> 00:57:07.820
r by r sub i times u.
00:57:10.630 --> 00:57:12.010
And we get this formula.
00:57:12.010 --> 00:57:15.439
And now, of course, the
r sub i cubes cancel.
00:57:15.439 --> 00:57:17.730
And I think the reason why
I kept it separated this way
00:57:17.730 --> 00:57:19.870
when I wrote my notes
is at this shows very
00:57:19.870 --> 00:57:23.400
explicitly that what you have
is a cancellation between an r
00:57:23.400 --> 00:57:28.850
sub i here, which was the power
of r that appears in a volume--
00:57:28.850 --> 00:57:31.670
r sub i cubed was just
proportional to the volume
00:57:31.670 --> 00:57:35.429
of the sphere-- and r
sub i cubed down here,
00:57:35.429 --> 00:57:37.720
where one of them came from
just a change of variables,
00:57:37.720 --> 00:57:41.117
and the other was the r squared
that appeared in the force law.
00:57:41.117 --> 00:57:43.450
So this cancellation depends
crucially on the force law,
00:57:43.450 --> 00:57:45.380
being a 1 over r
squared force law.
00:57:45.380 --> 00:57:47.450
If we had 1 over some
other power of r--
00:57:47.450 --> 00:57:50.320
even if it differed by
just a little bit-- then
00:57:50.320 --> 00:57:52.499
the r sub i would not
drop out of this formula.
00:57:52.499 --> 00:57:55.040
And we'll see in a minute that
it's the dropping out of r sub
00:57:55.040 --> 00:57:58.920
i which is crucial to the
maintenance of homogeneity
00:57:58.920 --> 00:58:00.720
as the system evolves.
00:58:00.720 --> 00:58:02.890
As it evolves, Newton
tells us what happens.
00:58:02.890 --> 00:58:05.720
We don't get to make
any further choices.
00:58:05.720 --> 00:58:08.310
And if Newton is
telling us there's a 1
00:58:08.310 --> 00:58:11.365
over r squared force law,
then it remains homogeneous.
00:58:11.365 --> 00:58:13.390
But otherwise, it would not.
00:58:13.390 --> 00:58:15.870
And that's, I think, a
very interesting fact.
00:58:15.870 --> 00:58:18.140
Continuing, we've now
crossed off these r sub i's.
00:58:18.140 --> 00:58:20.030
And we get now a
simple equation,
00:58:20.030 --> 00:58:28.970
u double dot is equal to
minus 4 pi over 3 G rho sub i
00:58:28.970 --> 00:58:34.990
over u squared with no r sub
i's in the equation anymore.
00:58:34.990 --> 00:58:36.900
And that means that
this u tells us
00:58:36.900 --> 00:58:39.406
the solution for every r sub i.
00:58:39.406 --> 00:58:40.780
We don't have
different solutions
00:58:40.780 --> 00:58:42.540
for every value of
r sub i anymore.
00:58:42.540 --> 00:58:44.460
r sub i has dropped
out of the problem.
00:58:44.460 --> 00:58:47.230
We have a unique solution
independent of r sub i.
00:58:47.230 --> 00:58:51.771
And that means it
holds for all r sub i.
00:58:51.771 --> 00:58:52.270
I'm sorry.
00:58:52.270 --> 00:58:53.520
I jumped the gun.
00:58:56.912 --> 00:58:58.870
The conclusions I just
told you about are true,
00:58:58.870 --> 00:59:01.175
but part of the logic
I haven't done yet.
00:59:01.175 --> 00:59:02.800
What part of the
logic did I leave out?
00:59:05.567 --> 00:59:07.400
Initial conditions, I
heard somebody mumble.
00:59:07.400 --> 00:59:07.900
Yes.
00:59:07.900 --> 00:59:09.005
Exactly.
00:59:09.005 --> 00:59:10.380
To get the same
solution, we have
00:59:10.380 --> 00:59:12.430
to not only have a
differential equation that's
00:59:12.430 --> 00:59:15.112
independent of u sub i, but we
don't have a unique solution
00:59:15.112 --> 00:59:16.820
unless we look at the
initial conditions.
00:59:16.820 --> 00:59:19.604
We better see if the initial
conditions depend on u sub i.
00:59:19.604 --> 00:59:20.520
And they don't either.
00:59:20.520 --> 00:59:22.450
That's the beauty of it all.
00:59:22.450 --> 00:59:30.130
Starting with the r initial
condition, the initial value
00:59:30.130 --> 00:59:34.910
of u of r sub i,
t sub i, will just
00:59:34.910 --> 00:59:39.120
be equal to the initial value
of r divided by r sub i.
00:59:39.120 --> 00:59:42.414
But the initial value
of r is r sub i.
00:59:42.414 --> 00:59:46.090
So here, we get r sub i
divided by r sub i, which
00:59:46.090 --> 00:59:48.540
is 1, independent of r sub i.
00:59:48.540 --> 00:59:53.190
The initial value of u
is 1, for any r sub i.
00:59:53.190 --> 01:00:01.750
And similarly, we now want to
look at u dot of r sub i and t.
01:00:01.750 --> 01:00:05.450
And that will just be r dot
divided by r sub i, where
01:00:05.450 --> 01:00:08.092
we take the initial
value of r dot.
01:00:08.092 --> 01:00:11.015
The initial value of r
dot is Hi times r sub i.
01:00:11.015 --> 01:00:12.940
The r sub i's cancel.
01:00:12.940 --> 01:00:15.898
And u sub i dot is just H sub i.
01:00:19.180 --> 01:00:24.280
So now, we have
justified the claim
01:00:24.280 --> 01:00:27.611
that I falsely made a little
bit prematurely a minute ago.
01:00:27.611 --> 01:00:29.110
We have a system
of equations, which
01:00:29.110 --> 01:00:32.350
can be solved to give us
a solution for u, which
01:00:32.350 --> 01:00:35.870
is independent of r sub i, which
means every value of r sub i
01:00:35.870 --> 01:00:39.820
has the same equation for u.
01:00:39.820 --> 01:00:43.960
That-- if you stare
at it a little bit--
01:00:43.960 --> 01:00:45.650
gives us a physical
interpretation
01:00:45.650 --> 01:00:47.510
for this quantity, u.
01:00:53.960 --> 01:01:00.090
u, in fact, is nothing more
nor less than the scale factor
01:01:00.090 --> 01:01:02.330
that we spoke about earlier.
01:01:02.330 --> 01:01:06.127
We have found that we do have a
homogeneously expanding system.
01:01:06.127 --> 01:01:07.710
We started it
homogeneously expanding,
01:01:07.710 --> 01:01:10.043
but we didn't know until we
looked at equation of motion
01:01:10.043 --> 01:01:12.440
that it would continue
to homogeneously expand.
01:01:12.440 --> 01:01:14.380
But it does.
01:01:14.380 --> 01:01:16.840
And that means it can be
described by a scale factor.
01:01:21.730 --> 01:01:25.130
We have u of r sub i, t.
01:01:25.130 --> 01:01:27.150
In principle, it
was defined so it
01:01:27.150 --> 01:01:28.629
might have depended on r sub i.
01:01:28.629 --> 01:01:30.920
That's why I've been writing
these r sub i's all along.
01:01:30.920 --> 01:01:33.270
But now, we discovered
that the u's are completely
01:01:33.270 --> 01:01:35.270
determined by these
equations, which
01:01:35.270 --> 01:01:38.197
have no r sub i's in them--
at least none that survived.
01:01:38.197 --> 01:01:39.780
We had an r sub i
divided by r sub it,
01:01:39.780 --> 01:01:42.510
but that's not really
an r sub i, as you know.
01:01:42.510 --> 01:01:44.100
It's 1.
01:01:44.100 --> 01:01:46.377
So u is independent
of r sub i and can
01:01:46.377 --> 01:01:47.960
be thought of as
just a function of t.
01:01:50.840 --> 01:01:54.180
And we can also change
its name to a of t
01:01:54.180 --> 01:01:56.870
to make contact with the
notion of a scale factor
01:01:56.870 --> 01:02:00.010
that we discussed last time.
01:02:00.010 --> 01:02:02.600
And we can see that
the way u arises,
01:02:02.600 --> 01:02:09.570
in terms of how it describes
the motion, r of r sub i
01:02:09.570 --> 01:02:16.040
and t is just equal to--
from this equation-- r sub
01:02:16.040 --> 01:02:19.980
i times u, which
is now called a.
01:02:19.980 --> 01:02:25.550
So r, of r sub i and t, is
equal to a of t times r sub i.
01:02:28.420 --> 01:02:30.565
That's just another way of
rewriting our definition
01:02:30.565 --> 01:02:32.860
that we started with of u.
01:02:32.860 --> 01:02:34.600
So what does that mean?
01:02:34.600 --> 01:02:38.810
These r sub i's are
comoving coordinates.
01:02:38.810 --> 01:02:41.650
We've labeled each shell by
its initial position, r sub i.
01:02:41.650 --> 01:02:43.940
And we've let that shell
keep its label, r sub i,
01:02:43.940 --> 01:02:45.050
as it evolved.
01:02:45.050 --> 01:02:46.820
That's a comoving coordinate.
01:02:46.820 --> 01:02:50.220
It labels the particles
independent of where they move.
01:02:50.220 --> 01:02:57.410
And what this is telling us is
that the physical distance--
01:02:57.410 --> 01:03:08.260
from the origin
in this case-- is
01:03:08.260 --> 01:03:16.137
equal to the scale factor
times the coordinate distance.
01:03:23.909 --> 01:03:25.950
That is the coordinate
distance between the shell
01:03:25.950 --> 01:03:28.140
and the center of the
system, the origin.
01:03:31.830 --> 01:03:32.330
OK.
01:03:32.330 --> 01:03:33.480
Any questions about that?
01:03:42.840 --> 01:03:43.340
OK.
01:03:43.340 --> 01:03:45.550
It's now useful to
rewrite these equations
01:03:45.550 --> 01:03:46.580
in a few different ways.
01:04:05.980 --> 01:04:06.480
Let's see.
01:04:06.480 --> 01:04:14.200
First of all, we have written
our differential equation,
01:04:14.200 --> 01:04:17.540
up here, in terms of rho sub i.
01:04:17.540 --> 01:04:19.915
And that was very useful
because rho sub i is a constant.
01:04:19.915 --> 01:04:21.695
It doesn't change with time.
01:04:21.695 --> 01:04:24.320
Still, it's sometimes useful to
write the differential equation
01:04:24.320 --> 01:04:26.180
in terms of the temporary
value of rho, which
01:04:26.180 --> 01:04:28.346
changes with time, to see
what the relationships are
01:04:28.346 --> 01:04:30.940
between physical
quantities at a given time.
01:04:30.940 --> 01:04:33.670
And we could certainly
do that, because we
01:04:33.670 --> 01:04:37.495
know what the density
will be at any given time.
01:04:41.710 --> 01:04:45.720
For any shell, we can
calculate the density
01:04:45.720 --> 01:04:47.705
as the total mass
divided by the volume.
01:04:47.705 --> 01:04:50.080
We know this is going to stay
uniform, because everything
01:04:50.080 --> 01:04:53.250
is moving together, when
you just have motion, which
01:04:53.250 --> 01:04:56.710
is an overall scale factor
that multiplies all distances.
01:04:56.710 --> 01:04:58.540
So the density will be constant.
01:04:58.540 --> 01:05:02.880
And we can calculate the
density inside a shell
01:05:02.880 --> 01:05:05.840
by taking M of r sub i--
which we already have formulas
01:05:05.840 --> 01:05:09.590
for and is independent
of time-- and dividing it
01:05:09.590 --> 01:05:13.045
by the volume inside the
shell, which is 4 pi r cubed.
01:05:16.880 --> 01:05:21.250
And doing some
substitutions, M of r sub i
01:05:21.250 --> 01:05:26.140
is just the initial volume
times the initial density.
01:05:26.140 --> 01:05:29.350
So that is-- as we've
written before-- 4
01:05:29.350 --> 01:05:37.040
pi over 3 r sub i
cubed times rho sub i.
01:05:37.040 --> 01:05:43.740
And then, we can
write the denominator.
01:05:43.740 --> 01:05:48.726
r is equal to a times r sub i.
01:05:48.726 --> 01:05:50.350
The physical radius
is the scale factor
01:05:50.350 --> 01:05:52.340
times the coordinate radius.
01:05:52.340 --> 01:05:56.650
So here, we have a cubed
times r sub i cubed.
01:05:56.650 --> 01:05:59.660
And now, notice that
almost everything cancels.
01:05:59.660 --> 01:06:04.540
And what we're left with is just
rho sub i divided by a cubed.
01:06:14.970 --> 01:06:17.230
So that's certainly what
we would've guess, I think.
01:06:17.230 --> 01:06:20.680
The density is just
what it started but then
01:06:20.680 --> 01:06:23.640
divided by the cube
of the scale factor.
01:06:23.640 --> 01:06:25.460
And our scale factor
is defined as 1
01:06:25.460 --> 01:06:28.400
at the initial time-- the way
we've set up these conventions.
01:06:28.400 --> 01:06:31.842
So that is just the ratio
of the scale factors cubed
01:06:31.842 --> 01:06:34.210
that appears in that equation.
01:06:34.210 --> 01:06:36.410
As the universe
expands, the density
01:06:36.410 --> 01:06:40.430
falls off by 1 over
the scale factor cubed.
01:06:40.430 --> 01:06:45.960
We can also now rewrite the
equation for a double dot.
01:06:45.960 --> 01:06:48.320
a is u, so we have
the equation up there.
01:06:48.320 --> 01:06:53.440
But we could write it in terms
of the current mass density.
01:06:53.440 --> 01:06:56.080
Starting with what
we have up there,
01:06:56.080 --> 01:07:03.940
it's minus 4 pi over 3 G
rho sub i over a squared.
01:07:08.690 --> 01:07:13.600
Notice, that I can multiply
numerator and denominator by a.
01:07:16.200 --> 01:07:18.710
And then, we have rho
sub i over a cubed
01:07:18.710 --> 01:07:24.760
here, which is just the mass
density in any given time.
01:07:24.760 --> 01:07:27.180
So I can make that substitution.
01:07:27.180 --> 01:07:31.020
And I get the
meaningful equation
01:07:31.020 --> 01:07:37.930
that a double dot is equal
to minus 4 pi over 3 G
01:07:37.930 --> 01:07:41.990
rho of t times a.
01:07:47.220 --> 01:07:49.500
So this equation
gives the deceleration
01:07:49.500 --> 01:07:53.396
of our model universe in terms
of the current mass density.
01:07:53.396 --> 01:07:55.270
And notice that it does,
in fact, depend only
01:07:55.270 --> 01:07:57.650
on the current mass density.
01:07:57.650 --> 01:08:00.420
It predicts the ratio
of a double dot over a.
01:08:00.420 --> 01:08:03.310
And that, you would expect to
be what should be predicted,
01:08:03.310 --> 01:08:07.821
because remember, a is still
measured in notches per meter.
01:08:07.821 --> 01:08:09.570
So the only way the
notches can cancel out
01:08:09.570 --> 01:08:11.030
is if there is an
a on both sides.
01:08:11.030 --> 01:08:13.215
Or if you could bring
them all to one side
01:08:13.215 --> 01:08:14.910
and have a double
dot divided by a.
01:08:14.910 --> 01:08:16.375
And then, the notches go away.
01:08:16.375 --> 01:08:18.640
And you have something which
has physical units being related
01:08:18.640 --> 01:08:20.050
to something with
physical units.
01:08:29.340 --> 01:08:31.319
OK.
01:08:31.319 --> 01:08:33.810
We said at the beginning
that when we were finished,
01:08:33.810 --> 01:08:37.250
we were going to take the
limit as R max initial goes
01:08:37.250 --> 01:08:38.727
to infinity.
01:08:38.727 --> 01:08:40.310
And lots of times
when I present this,
01:08:40.310 --> 01:08:41.950
I forget to talk about that.
01:08:41.950 --> 01:08:44.870
And the reason I forget to talk
about that is, if you notice,
01:08:44.870 --> 01:08:48.310
R max sub i doesn't appear
in any of these equations.
01:08:48.310 --> 01:08:51.380
So taking the limit as R
max sub i goes to infinity
01:08:51.380 --> 01:08:53.239
doesn't actually
involve doing anything.
01:08:53.239 --> 01:08:56.580
It really just involves pointing
out that the answers we got
01:08:56.580 --> 01:08:58.600
are independent of
how big the sphere is,
01:08:58.600 --> 01:09:00.141
as long as everything
we want to talk
01:09:00.141 --> 01:09:01.899
about fits inside the sphere.
01:09:01.899 --> 01:09:03.359
Adding extra matter
on the outside
01:09:03.359 --> 01:09:04.682
doesn't change anything at all.
01:09:04.682 --> 01:09:07.140
So taking the limit as you add
an infinite amount of matter
01:09:07.140 --> 01:09:09.402
on the outside-- as long
as you imagine doing it
01:09:09.402 --> 01:09:12.399
in these spherical shells--
is a trivial matter.
01:09:12.399 --> 01:09:14.979
So the limit as R max
sub i goes to infinity
01:09:14.979 --> 01:09:17.090
is done without any work.
01:09:27.649 --> 01:09:28.149
OK.
01:09:28.149 --> 01:09:30.637
I would like to go ahead
now, and in the end,
01:09:30.637 --> 01:09:32.720
we're going to want to
think about different kinds
01:09:32.720 --> 01:09:35.720
of solutions to this equation
and what they look like.
01:09:35.720 --> 01:09:38.210
For today, I want to
take one more step, which
01:09:38.210 --> 01:09:41.630
is to rewrite this equation in
a slightly different way, which
01:09:41.630 --> 01:09:45.012
will help us to see what
the solutions look what.
01:09:45.012 --> 01:09:48.086
What I want to do is to
find a first integral
01:09:48.086 --> 01:09:48.794
of this equation.
01:10:10.060 --> 01:10:10.740
OK.
01:10:10.740 --> 01:10:13.860
To find a first integral, I'm
going to go back to the form
01:10:13.860 --> 01:10:17.827
that we had on the top
there, where everything
01:10:17.827 --> 01:10:20.404
is expressed in terms of
rho sub i rather than rho.
01:10:20.404 --> 01:10:22.320
And the advantage of
that for current purposes
01:10:22.320 --> 01:10:25.030
is that I really want to look at
the time dependence of things.
01:10:25.030 --> 01:10:26.700
And rho has its own
time dependence,
01:10:26.700 --> 01:10:29.100
which I don't want
to worry about.
01:10:29.100 --> 01:10:32.860
So if I look at the formula
in terms of rho sub i,
01:10:32.860 --> 01:10:35.520
all time dependence is explicit.
01:10:35.520 --> 01:10:38.760
So I'm going to write the
differential equation.
01:10:38.760 --> 01:10:40.470
It's the top equation
in the box there,
01:10:40.470 --> 01:10:44.580
but I'm replacing u by a,
because we renamed u as a.
01:10:44.580 --> 01:10:46.510
And I'm going to put
everything on one side.
01:10:46.510 --> 01:10:50.000
So I'm going to write
a double dot plus 4 pi
01:10:50.000 --> 01:10:59.970
over 3 G rho sub i divided
by a squared equals 0.
01:10:59.970 --> 01:11:01.610
OK, that's our
differential equation.
01:11:04.950 --> 01:11:06.950
Now, it's a second order
differential equation,
01:11:06.950 --> 01:11:10.346
like we're very much accustomed
to from Newtonian mechanics--
01:11:10.346 --> 01:11:11.970
as this is an equation
which determines
01:11:11.970 --> 01:11:13.480
a double dot, the
acceleration of a
01:11:13.480 --> 01:11:16.331
in terms of the value of a.
01:11:16.331 --> 01:11:20.740
A common thing to make use
of in Newtonian mechanics
01:11:20.740 --> 01:11:23.236
is conservation of energy.
01:11:23.236 --> 01:11:24.860
In this case, I don't
know if we should
01:11:24.860 --> 01:11:26.526
call this conservation
of energy or not.
01:11:26.526 --> 01:11:29.920
We'll talk later about
what physical significance
01:11:29.920 --> 01:11:32.310
the quantities that
we're dealing with have.
01:11:32.310 --> 01:11:35.310
But certainly, as a
mathematical technique,
01:11:35.310 --> 01:11:36.810
we can do the same
thing that would
01:11:36.810 --> 01:11:39.260
have been done if this were a
Newtonian mechanics problem,
01:11:39.260 --> 01:11:41.960
and somebody asked you to
derive the conserved energy.
01:11:41.960 --> 01:11:44.810
Now, you might have
forgotten how to do that.
01:11:44.810 --> 01:11:46.630
But I'll remind you.
01:11:46.630 --> 01:11:49.570
To get the conserved energy
that goes with this equation,
01:11:49.570 --> 01:11:52.285
you put brackets around it.
01:11:52.285 --> 01:11:53.660
You could choose
whether you want
01:11:53.660 --> 01:11:55.680
curly brackets or
square brackets or just
01:11:55.680 --> 01:11:57.300
ordinary parentheses.
01:11:57.300 --> 01:11:59.870
But then, the important
thing is that it's
01:11:59.870 --> 01:12:01.480
useful to multiply
the entire equation
01:12:01.480 --> 01:12:04.748
by an integrating factor, a dot.
01:12:04.748 --> 01:12:07.620
And once you do that,
this entire expression
01:12:07.620 --> 01:12:09.830
is a total derivative.
01:12:09.830 --> 01:12:14.520
This equation is equivalent
to dE-- for some E
01:12:14.520 --> 01:12:16.050
that I'll define
in a second-- dt
01:12:16.050 --> 01:12:33.320
equals zero, where E equals 1/2
a dot squared minus 4 pi over 3
01:12:33.320 --> 01:12:37.936
G rho sub i over a.
01:12:43.540 --> 01:12:44.630
And you can easily check.
01:12:44.630 --> 01:12:48.216
If I differentiate this, I
get exactly that equation.
01:12:48.216 --> 01:12:51.425
So they're equivalent.
01:12:51.425 --> 01:12:54.907
That is, this is
equivalent to that.
01:12:54.907 --> 01:12:56.115
So E is a conserved quantity.
01:13:06.800 --> 01:13:10.300
Now if one wants to relate this
to the energy of something,
01:13:10.300 --> 01:13:12.980
there are different
ways you can do it.
01:13:12.980 --> 01:13:23.070
One way to do it is to multiply
by an expression, which I'll
01:13:23.070 --> 01:13:25.620
write down in a second,
and to think of it
01:13:25.620 --> 01:13:33.012
as the energy of a test particle
at the surface of this sphere.
01:13:33.012 --> 01:13:34.220
I'll show you how that works.
01:13:48.614 --> 01:13:50.030
I'm going to find
something, which
01:13:50.030 --> 01:13:53.920
I'll call E sub phys--
or physical-- meaning,
01:13:53.920 --> 01:13:56.830
it's the physical energy of
this hypothetical test particle.
01:13:56.830 --> 01:13:59.630
It's not all that physical.
01:13:59.630 --> 01:14:07.170
But it will just be defined to
be m r sub i squared times E.
01:14:07.170 --> 01:14:08.990
m is the mass of
my test particle.
01:14:08.990 --> 01:14:12.025
r sub i is the radius of
that test particle expressed
01:14:12.025 --> 01:14:15.070
in terms of its initial value.
01:14:15.070 --> 01:14:21.430
And then, if we write down
what E phys looks like,
01:14:21.430 --> 01:14:31.170
absorbing these factors, it can
be written as 1/2 m times a dot
01:14:31.170 --> 01:14:50.400
r sub i squared minus GmM of
r sub i divided by a r sub i.
01:14:50.400 --> 01:14:54.490
And that's just some algebra--
absorbed these extra factors
01:14:54.490 --> 01:14:57.220
that I put into the definition.
01:14:57.220 --> 01:14:59.690
And now, if we think of
this as describing a test
01:14:59.690 --> 01:15:03.170
particle, where r sub i is
capital R sub i max-- so we're
01:15:03.170 --> 01:15:05.960
talking about the boundary
of our sphere-- then,
01:15:05.960 --> 01:15:08.640
we can identify what's
being conserved here.
01:15:08.640 --> 01:15:12.960
What's being conserved
here is 1/2 m v squared.
01:15:12.960 --> 01:15:15.640
a dot times r sub i would
just be the velocity
01:15:15.640 --> 01:15:18.520
of the particle of the
boundary of the sphere.
01:15:18.520 --> 01:15:21.810
And then, minus G times
the product of the masses
01:15:21.810 --> 01:15:25.630
divided by the distance between
the particle in the center.
01:15:25.630 --> 01:15:29.150
And that would just be the
Newtonian energy-- kinetic
01:15:29.150 --> 01:15:31.970
energy plus potential energy,
where the potential energy
01:15:31.970 --> 01:15:37.940
is negative-- of a point
particle on the boundary, where
01:15:37.940 --> 01:15:42.120
we let r sub i be
capital R sub i
01:15:42.120 --> 01:15:45.460
max-- the boundary
of the sphere.
01:15:45.460 --> 01:15:48.790
Now, if we want to apply this
to a particle inside the sphere,
01:15:48.790 --> 01:15:52.060
it's a little trickier
to get the words right.
01:15:52.060 --> 01:15:53.920
If the particle is
inside the sphere--
01:15:53.920 --> 01:15:57.170
if r sub i is not
equal to the max--
01:15:57.170 --> 01:16:00.830
this is not really the potential
energy of the particle.
01:16:00.830 --> 01:16:02.100
Can somebody tell me why not?
01:16:13.009 --> 01:16:14.800
Well, maybe the question
is a little vague.
01:16:14.800 --> 01:16:17.540
But if I did want to
calculate the potential energy
01:16:17.540 --> 01:16:24.404
of a particle inside
the sphere-- that's
01:16:24.404 --> 01:16:25.570
meant to be in the interior.
01:16:25.570 --> 01:16:27.778
You can't really tell unless
it's the actual diagram.
01:16:27.778 --> 01:16:30.620
But that's deep inside sphere.
01:16:30.620 --> 01:16:35.559
I would do it by integrating
from infinity G da,
01:16:35.559 --> 01:16:37.100
and ask how much
work do I have to do
01:16:37.100 --> 01:16:41.210
to bring in a particle from
infinity and put it there.
01:16:41.210 --> 01:16:47.570
And in doing this line
integral, I get a contribution
01:16:47.570 --> 01:16:51.470
from the mass that's
inside this dot, which
01:16:51.470 --> 01:16:53.590
is what determines
the force on that dot.
01:16:53.590 --> 01:16:57.510
But I also get a contribution
from what's outside the dot.
01:16:57.510 --> 01:16:59.620
So I don't get-- if
I wanted to calculate
01:16:59.620 --> 01:17:01.530
the actual potential
energy of that point--
01:17:01.530 --> 01:17:05.830
I don't get simply
Gm times the mass
01:17:05.830 --> 01:17:08.320
inside divided by the
distance from the center.
01:17:08.320 --> 01:17:09.935
It's more complicated
what I get.
01:17:09.935 --> 01:17:11.643
And in fact, what I
get is not conserved.
01:17:13.900 --> 01:17:17.020
Why is it not conserved?
01:17:17.020 --> 01:17:18.590
I could ask you,
but I'll tell you.
01:17:18.590 --> 01:17:19.673
We're running out of time.
01:17:19.673 --> 01:17:21.192
It's not conserved,
because if you
01:17:21.192 --> 01:17:22.900
ask for the potential
energy of something
01:17:22.900 --> 01:17:24.942
in the presence
of moving masses,
01:17:24.942 --> 01:17:26.650
there's no reason for
it to be conserved.
01:17:26.650 --> 01:17:29.400
The potential energy
for a point particle
01:17:29.400 --> 01:17:31.920
moving in the field of
static masses is conserved.
01:17:31.920 --> 01:17:34.670
That's what you've learned
in [? AO1 ?] or whatever.
01:17:34.670 --> 01:17:38.540
But if other
particles are moving,
01:17:38.540 --> 01:17:41.230
the total energy of the full
system will be conserved.
01:17:41.230 --> 01:17:44.164
But the potential energy of a
single particle-- just thought
01:17:44.164 --> 01:17:45.830
of as a particle
moving in the potential
01:17:45.830 --> 01:17:49.730
of the other particles--
will not be conserved.
01:17:49.730 --> 01:17:50.290
OK.
01:17:50.290 --> 01:17:52.540
What is conserved,
besides the energy
01:17:52.540 --> 01:17:54.720
of this test particle
on the boundary,
01:17:54.720 --> 01:17:56.607
is the total energy
of this system,
01:17:56.607 --> 01:17:57.815
which can also be calculated.
01:17:57.815 --> 01:18:00.148
And that, in fact, will be
one of your homework problems
01:18:00.148 --> 01:18:01.980
for the coming problem
set-- to calculate
01:18:01.980 --> 01:18:03.680
the total energy of that sphere.
01:18:03.680 --> 01:18:05.877
And that will be
related to this quantity
01:18:05.877 --> 01:18:07.710
with a different constant
of proportionality
01:18:07.710 --> 01:18:11.470
and will be conserved for
the obvious physical reason.
01:18:11.470 --> 01:18:14.175
For the particles inside,
what one can imagine--
01:18:14.175 --> 01:18:16.810
and I'll just say this
and then I'll stop--
01:18:16.810 --> 01:18:21.520
you can imagine that
you know that the motion
01:18:21.520 --> 01:18:25.830
of this particle is uninfluenced
by these particles outside.
01:18:25.830 --> 01:18:28.390
And therefore, you could
pretend that they're not there
01:18:28.390 --> 01:18:30.450
and think of it as an
analog problem, where
01:18:30.450 --> 01:18:34.730
the particles outside of the
radius of the particle you're
01:18:34.730 --> 01:18:39.390
focusing on simply do not
exist in this analog problem.
01:18:39.390 --> 01:18:41.977
For that analog problem, this
would be the potential energy.
01:18:41.977 --> 01:18:43.060
And it would be conserved.
01:18:43.060 --> 01:18:44.560
And you could argue
that way, that you
01:18:44.560 --> 01:18:45.820
expect this to be a constant.
01:18:45.820 --> 01:18:46.987
And you'd be correct.
01:18:46.987 --> 01:18:48.570
But it's a little
subtle to understand
01:18:48.570 --> 01:18:52.720
exactly what's conserved
and why and how to use it.
01:18:52.720 --> 01:18:55.220
OK, that's all for today.