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PROFESSOR: OK, in that
case, let's get going.

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00:00:26,492 --> 00:00:28,700
I want to begin by just
summarizing where we left off

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00:00:28,700 --> 00:00:31,390
last time, because we're still
to some extent in the middle

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00:00:31,390 --> 00:00:33,540
of the same discussion.

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00:00:33,540 --> 00:00:37,600
We were talking about
the gravitational effects

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00:00:37,600 --> 00:00:40,160
of a completely
homogeneous universe that

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00:00:40,160 --> 00:00:41,930
fills all of space.

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00:00:41,930 --> 00:00:44,430
And you recall that
Newton had concluded

16
00:00:44,430 --> 00:00:46,280
that such a system
would be stable.

17
00:00:46,280 --> 00:00:49,870
But I was arguing that such
a system would not be stable,

18
00:00:49,870 --> 00:00:52,544
even given the laws of
Newtonian mechanics.

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00:00:52,544 --> 00:00:54,710
And we discussed a few of
those arguments last time.

20
00:00:59,260 --> 00:01:03,330
We discussed, for example,
Gauss's law formulation

21
00:01:03,330 --> 00:01:04,959
of Newton's law of gravity.

22
00:01:04,959 --> 00:01:08,040
And I remind you that it's
a very simple derivation

23
00:01:08,040 --> 00:01:09,870
to go from Newton's
law of gravity,

24
00:01:09,870 --> 00:01:13,500
as Newton stated it as an
acceleration at a distance,

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00:01:13,500 --> 00:01:15,150
to Gauss's law.

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00:01:15,150 --> 00:01:20,810
What you do is you just show
that if the acceleration

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00:01:20,810 --> 00:01:24,550
of gravity is given by Newton's
law, then for any one particle

28
00:01:24,550 --> 00:01:27,880
creating such a gravitational
field, Gauss's law holds.

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00:01:27,880 --> 00:01:31,860
This integral is either
equal to 0 or minus 4 pi GM,

30
00:01:31,860 --> 00:01:35,250
depending on whether the surface
that you're integrating over

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00:01:35,250 --> 00:01:39,660
encloses or does not enclose
the charge or the mass.

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00:01:39,660 --> 00:01:41,340
And once you know
it for one mass,

33
00:01:41,340 --> 00:01:43,060
Newton tells us
that for many masses

34
00:01:43,060 --> 00:01:45,115
you just add the
forces as vectors.

35
00:01:45,115 --> 00:01:46,740
And that means that
you'll be adding up

36
00:01:46,740 --> 00:01:48,600
these integrals for
each of the particles.

37
00:01:48,600 --> 00:01:52,200
And you're lead automatically
to the expression we have here.

38
00:01:52,200 --> 00:01:53,730
So really, it
follows very directly

39
00:01:53,730 --> 00:01:55,430
from Newton's law of gravity.

40
00:01:55,430 --> 00:01:59,130
On the other hand, if we
apply this formulation

41
00:01:59,130 --> 00:02:03,350
to an infinite distribution
of mass, if Newton was right

42
00:02:03,350 --> 00:02:05,710
and there were no
forces, that would

43
00:02:05,710 --> 00:02:08,020
mean that little g, the
acceleration of gravity,

44
00:02:08,020 --> 00:02:09,560
would be 0 everywhere.

45
00:02:09,560 --> 00:02:11,620
And then, the integral
on the left would be 0.

46
00:02:11,620 --> 00:02:14,230
But the term on the
right is clearly not 0,

47
00:02:14,230 --> 00:02:17,730
if we have a volume with
some nontrivial size that

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00:02:17,730 --> 00:02:19,550
includes some mass.

49
00:02:19,550 --> 00:02:23,590
So this formulation of
Newton's law of gravity

50
00:02:23,590 --> 00:02:28,440
clearly shows that an
infinite distribution of mass

51
00:02:28,440 --> 00:02:30,490
could not be static.

52
00:02:30,490 --> 00:02:33,390
Further, I showed you there's
another formulation, more

53
00:02:33,390 --> 00:02:35,090
modern, of Newton's
law of gravity

54
00:02:35,090 --> 00:02:38,141
in terms of what's called
Poisson's equation.

55
00:02:38,141 --> 00:02:40,640
This was really shown for the
benefit of people who know it.

56
00:02:40,640 --> 00:02:42,223
If you don't it,
don't worry about it.

57
00:02:42,223 --> 00:02:43,831
We won't need it.

58
00:02:43,831 --> 00:02:46,080
But it's another way of
formulating the law of gravity

59
00:02:46,080 --> 00:02:49,380
by introducing a
gravitational potential, phi,

60
00:02:49,380 --> 00:02:51,840
and writing the
acceleration of gravity

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00:02:51,840 --> 00:02:53,490
as minus the gradient of phi.

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00:02:53,490 --> 00:02:55,470
That just defines phi.

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00:02:55,470 --> 00:02:58,450
And then, you can show that
phi obeys Poisson's equation,

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00:02:58,450 --> 00:03:01,490
del squared on phi is
equal to 4 pi G times

65
00:03:01,490 --> 00:03:04,494
rho, where rho is
the mass density.

66
00:03:04,494 --> 00:03:05,910
And again, one can
see immediately

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00:03:05,910 --> 00:03:09,970
that this does not allow a
static distribution of mass.

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00:03:09,970 --> 00:03:11,770
If the distribution
of mass was static,

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00:03:11,770 --> 00:03:14,270
that would mean
that g vector was 0.

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00:03:14,270 --> 00:03:16,480
That would mean that
gradient phi was 0.

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00:03:16,480 --> 00:03:18,400
That would mean that
phi was a constant.

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00:03:18,400 --> 00:03:20,500
And if phi is a constant,
del squared phi is 0,

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00:03:20,500 --> 00:03:25,400
and that's inconsistent
with the Poisson equation.

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00:03:25,400 --> 00:03:27,090
I might further
add-- I don't think

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00:03:27,090 --> 00:03:30,550
I said this last time-- that
from a modern perspective,

76
00:03:30,550 --> 00:03:32,590
equations like
Poisson's equation

77
00:03:32,590 --> 00:03:36,140
are considered more
fundamental than Newton's

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00:03:36,140 --> 00:03:37,870
original statement
of the law of gravity

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00:03:37,870 --> 00:03:40,300
as an action at a distance.

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00:03:40,300 --> 00:03:43,390
In particular, when
one wants to generalize

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00:03:43,390 --> 00:03:46,590
Newton's law, for example,
to general relativity,

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00:03:46,590 --> 00:03:48,310
Einstein started with
Poisson's equation,

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00:03:48,310 --> 00:03:50,750
not with the force
law at a distance.

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00:03:50,750 --> 00:03:53,010
And there's nothing like
the force law at a distance

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00:03:53,010 --> 00:03:55,120
and the theory of
general relativity.

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00:03:55,120 --> 00:03:59,580
General relativity is formulated
in a language very similar

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00:03:59,580 --> 00:04:02,030
to the language of
Poisson's equation.

88
00:04:02,030 --> 00:04:05,740
The key idea that
underlies this distinction

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00:04:05,740 --> 00:04:08,725
is that all the laws of
physics that we know of

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00:04:08,725 --> 00:04:10,970
can be expressed in a local way.

91
00:04:10,970 --> 00:04:12,622
Poisson's equation
is a local equation.

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00:04:12,622 --> 00:04:14,080
That's just a
differential equation

93
00:04:14,080 --> 00:04:15,784
that holds at each
point in space

94
00:04:15,784 --> 00:04:17,200
and doesn't say
anything about how

95
00:04:17,200 --> 00:04:20,810
something at one point in space
affects something far away.

96
00:04:20,810 --> 00:04:23,657
That happens as a
consequence of this equation.

97
00:04:23,657 --> 00:04:25,990
But it happens as a consequence
of solving the equation.

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00:04:25,990 --> 00:04:30,290
It's not built into the
equation to start with.

99
00:04:30,290 --> 00:04:32,130
Continuing.

100
00:04:32,130 --> 00:04:36,180
We then discussed what
goes on if one does just

101
00:04:36,180 --> 00:04:40,410
try to add up the forces
using Newton's law

102
00:04:40,410 --> 00:04:42,070
and action at a distance.

103
00:04:42,070 --> 00:04:44,590
And what I argued is that it's
a conditionally convergent

104
00:04:44,590 --> 00:04:47,840
integral, which means it's
the kind of integral which

105
00:04:47,840 --> 00:04:51,260
has the property
that it converges.

106
00:04:51,260 --> 00:04:53,480
But it can converge
to different things

107
00:04:53,480 --> 00:04:55,380
depending on what
order you add up

108
00:04:55,380 --> 00:04:57,880
the different parts
of the integral.

109
00:04:57,880 --> 00:05:00,400
And we considered two
possible orderings

110
00:05:00,400 --> 00:05:02,740
for adding up the mass.

111
00:05:02,740 --> 00:05:05,090
In all cases, what
we're talking about here

112
00:05:05,090 --> 00:05:09,100
is just a single
location, P. And you

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00:05:09,100 --> 00:05:12,880
can't tell the slide
is filled with mass,

114
00:05:12,880 --> 00:05:15,830
but think of that
light cyan as mass.

115
00:05:15,830 --> 00:05:18,880
It fills the entire slide
and fills the entire universe

116
00:05:18,880 --> 00:05:21,554
in our toy problem here.

117
00:05:21,554 --> 00:05:22,970
So we're interested
in calculating

118
00:05:22,970 --> 00:05:25,970
the force on some
point, P, in the midst

119
00:05:25,970 --> 00:05:27,952
of an infinite
distribution of mass.

120
00:05:27,952 --> 00:05:30,160
And the only thing that
we're going to do differently

121
00:05:30,160 --> 00:05:31,576
in these two
calculations is we're

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00:05:31,576 --> 00:05:33,980
going to add up that mass
in a different order.

123
00:05:33,980 --> 00:05:36,560
And if we add up
the mass ordered

124
00:05:36,560 --> 00:05:40,780
by concentric shells about P,
each consecutive shell clearly

125
00:05:40,780 --> 00:05:44,570
contributes 0 to the force
at P. And therefore, the sum

126
00:05:44,570 --> 00:05:49,180
in the limit, as we go out
to infinity, will still be 0.

127
00:05:49,180 --> 00:05:52,070
So for this case,
we get g equals 0.

128
00:05:52,070 --> 00:05:55,300
We do get no acceleration
at the point, P,

129
00:05:55,300 --> 00:05:59,880
as long as we add up all
the masses in that way.

130
00:05:59,880 --> 00:06:02,960
But there's nothing in
Newton's laws that tell us

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00:06:02,960 --> 00:06:04,980
what order to add up the forces.

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00:06:04,980 --> 00:06:08,650
Newton just tells you
that each mass creates a 1

133
00:06:08,650 --> 00:06:10,680
over r squared force,
and it's a vector.

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00:06:10,680 --> 00:06:13,080
And Newton says to
add the vectors.

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00:06:13,080 --> 00:06:15,200
Normally, addition
of vectors commutes.

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00:06:15,200 --> 00:06:17,076
It doesn't matter what
order you add them in.

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00:06:17,076 --> 00:06:18,700
But what we're going
to be finding here

138
00:06:18,700 --> 00:06:20,250
is that it does
matter what order.

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00:06:20,250 --> 00:06:22,640
And therefore, the
answer is ambiguous.

140
00:06:22,640 --> 00:06:26,020
And to see this, we'll
consider a different ordering.

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00:06:26,020 --> 00:06:27,880
Instead-- we'll still
use spherical shells,

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00:06:27,880 --> 00:06:30,364
because that's the easiest
thing to think about.

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00:06:30,364 --> 00:06:31,780
We could try to
do something else,

144
00:06:31,780 --> 00:06:34,470
but it's much harder
to use any other shape.

145
00:06:34,470 --> 00:06:37,260
But this time, we'll
consider spherical shells

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00:06:37,260 --> 00:06:39,200
that are centered around
a different point.

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00:06:39,200 --> 00:06:41,450
And we'll call the point
that the spherical shells are

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00:06:41,450 --> 00:06:44,700
centered around, Q.
We're still trying

149
00:06:44,700 --> 00:06:47,840
to calculate the
force at the point, P,

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00:06:47,840 --> 00:06:49,570
due to the infinite
mass distribution

151
00:06:49,570 --> 00:06:52,010
that fills space-- so we're
doing the same problem we did

152
00:06:52,010 --> 00:06:54,290
before-- but we're
going to add up

153
00:06:54,290 --> 00:06:56,320
the contributions in
a different order.

154
00:06:56,320 --> 00:06:59,800
And we discussed last
time that when we do that,

155
00:06:59,800 --> 00:07:03,350
it turns out that all the
mass inside the sphere,

156
00:07:03,350 --> 00:07:06,170
centered at Q out
to the radius of P,

157
00:07:06,170 --> 00:07:08,740
contributes to the
acceleration at P.

158
00:07:08,740 --> 00:07:10,760
And all mass outside
of that, could

159
00:07:10,760 --> 00:07:13,500
be divided into
concentric shells, where

160
00:07:13,500 --> 00:07:14,690
the point, P, is inside.

161
00:07:14,690 --> 00:07:17,470
And there's no force
inside a concentric shell.

162
00:07:17,470 --> 00:07:21,380
So all of the rest of the
mass contributes zilch.

163
00:07:21,380 --> 00:07:25,770
And the answer you get
is then simply the answer

164
00:07:25,770 --> 00:07:30,020
that you would have for the
force of a point mass located

165
00:07:30,020 --> 00:07:34,890
at Q, whose mass was the total
mass in that shaded region.

166
00:07:34,890 --> 00:07:36,370
And clearly it's nonzero.

167
00:07:36,370 --> 00:07:37,970
And furthermore,
clearly, we can--

168
00:07:37,970 --> 00:07:41,650
by choosing different points,
Q-- make this anything we want.

169
00:07:41,650 --> 00:07:44,630
We can make it bigger by putting
the point, Q, further away;

170
00:07:44,630 --> 00:07:53,410
and it always points in the
direction of Q. Yeah, points

171
00:07:53,410 --> 00:07:55,849
in the direction
of Q. So we can let

172
00:07:55,849 --> 00:07:58,140
it point in any direction we
want by putting the point,

173
00:07:58,140 --> 00:07:59,950
Q, anywhere we want.

174
00:07:59,950 --> 00:08:03,400
So depending on how we
add up the contributions,

175
00:08:03,400 --> 00:08:04,670
we can get any answer we want.

176
00:08:04,670 --> 00:08:07,460
And that's a
fundamental ambiguity

177
00:08:07,460 --> 00:08:11,170
in trying to apply Newton using
only the original statement

178
00:08:11,170 --> 00:08:15,640
of the law of gravity
as Newton gave it.

179
00:08:15,640 --> 00:08:17,230
OK.

180
00:08:17,230 --> 00:08:22,310
So the conclusion, here, is
that the action at a distance

181
00:08:22,310 --> 00:08:24,830
description is simply ambiguous.

182
00:08:24,830 --> 00:08:28,030
Descriptions by Gauss's
law, or Poisson's law,

183
00:08:28,030 --> 00:08:30,740
tell us that the system
cannot be static.

184
00:08:30,740 --> 00:08:33,049
And we'll soon try to
figure out exactly how we

185
00:08:33,049 --> 00:08:34,809
do expect it to behave.

186
00:08:34,809 --> 00:08:37,429
Now, I still want come back to
one argument, which was really

187
00:08:37,429 --> 00:08:40,590
the argument that persuaded
Newton in the first place.

188
00:08:40,590 --> 00:08:43,010
Newton said that if
we want to calculate

189
00:08:43,010 --> 00:08:45,680
the acceleration of a certain
point in this infinite mass

190
00:08:45,680 --> 00:08:48,810
distribution, we have
a symmetry problem.

191
00:08:48,810 --> 00:08:52,360
All directions from that point
looking outward look identical.

192
00:08:52,360 --> 00:08:56,100
If there's going to be
an acceleration acting

193
00:08:56,100 --> 00:08:59,950
on any point, what
could possibly

194
00:08:59,950 --> 00:09:05,360
determine the direction that
the acceleration will have.

195
00:09:05,360 --> 00:09:08,580
So that's the symmetry argument,
which is a very sticky one.

196
00:09:08,580 --> 00:09:10,980
It sounds very convincing,
in Newton's reasoning.

197
00:09:13,841 --> 00:09:16,090
There could be no acceleration,
simply because there's

198
00:09:16,090 --> 00:09:19,865
no preferred direction for
the acceleration to point.

199
00:09:19,865 --> 00:09:21,990
To convince Newton that
that's not a valid argument

200
00:09:21,990 --> 00:09:22,980
would probably be hard.

201
00:09:22,980 --> 00:09:25,229
And I don't know if we could
succeed in convincing him

202
00:09:25,229 --> 00:09:26,730
or not.

203
00:09:26,730 --> 00:09:28,450
Don't get the chance to try.

204
00:09:28,450 --> 00:09:29,930
But if we did have
a chance to try,

205
00:09:29,930 --> 00:09:33,180
what we would try
to explain to him

206
00:09:33,180 --> 00:09:37,850
is that the acceleration is
usually measured relative

207
00:09:37,850 --> 00:09:39,720
to an inertial frame.

208
00:09:39,720 --> 00:09:41,470
That's how Newton
always described it.

209
00:09:41,470 --> 00:09:46,380
And to Newton, there was a
unique inertial frame-- unique

210
00:09:46,380 --> 00:09:49,030
up to changes in
velocity-- determined

211
00:09:49,030 --> 00:09:51,170
by the frame of the fixed stars.

212
00:09:51,170 --> 00:09:53,280
That was the language
that Newton used.

213
00:09:53,280 --> 00:09:54,910
And that defined
his inertial frame.

214
00:09:54,910 --> 00:09:57,160
And all of his laws
of physics were

215
00:09:57,160 --> 00:10:00,290
claimed to hold in
this inertial frame.

216
00:10:00,290 --> 00:10:03,630
On the other hand, if all of
space is filled with matter

217
00:10:03,630 --> 00:10:06,070
and it's all going to
collapse as we're claiming,

218
00:10:06,070 --> 00:10:09,490
there isn't any place
to have any fixed stars.

219
00:10:09,490 --> 00:10:14,630
So the whole idea of an inertial
frame really disappears.

220
00:10:14,630 --> 00:10:17,410
There's no object, which one
can think of as being at rest

221
00:10:17,410 --> 00:10:19,430
or being non-accelerating
with respect

222
00:10:19,430 --> 00:10:24,430
to any would-be inertial frame.

223
00:10:24,430 --> 00:10:26,760
So in the absence of
an inertial frame,

224
00:10:26,760 --> 00:10:30,820
one really has to admit
that all accelerations, just

225
00:10:30,820 --> 00:10:33,340
like all velocities,
have to be measured

226
00:10:33,340 --> 00:10:34,721
as relative accelerations.

227
00:10:34,721 --> 00:10:36,220
We could talk about
the acceleration

228
00:10:36,220 --> 00:10:38,510
of one mass relative to another.

229
00:10:38,510 --> 00:10:41,940
But we can't talk about what
the absolute acceleration is

230
00:10:41,940 --> 00:10:43,930
of a given mass,
because we don't

231
00:10:43,930 --> 00:10:46,620
have an inertial
frame with which

232
00:10:46,620 --> 00:10:50,990
to compare the acceleration
of that object.

233
00:10:50,990 --> 00:10:53,550
So when all accelerations
are relative,

234
00:10:53,550 --> 00:10:55,852
then there are
more options here.

235
00:10:55,852 --> 00:10:58,310
And it turns out that the right
option-- the one that we'll

236
00:10:58,310 --> 00:11:01,540
eventually deduce--
is an option that

237
00:11:01,540 --> 00:11:03,730
looks similar to Hubble's law.

238
00:11:03,730 --> 00:11:05,970
Hubble's law is a
law about velocities.

239
00:11:05,970 --> 00:11:09,207
And it says that from the
point of view of any observer,

240
00:11:09,207 --> 00:11:11,040
all the other objects
will look like they're

241
00:11:11,040 --> 00:11:14,110
moving radially outward
from that observer.

242
00:11:14,110 --> 00:11:16,530
And in spite of the fact
that that description makes

243
00:11:16,530 --> 00:11:19,590
it sound, emphatically, like the
observer you're talking about

244
00:11:19,590 --> 00:11:23,340
is special, you can transform to
the frame of any other observer

245
00:11:23,340 --> 00:11:26,100
and thus seeing
exactly the same thing.

246
00:11:26,100 --> 00:11:29,470
So having one observer seeing
all the other velocities

247
00:11:29,470 --> 00:11:34,782
moving outward from him does
not violate homogeneity.

248
00:11:34,782 --> 00:11:36,490
It does not violate
any of the symmetries

249
00:11:36,490 --> 00:11:38,500
that we're trying to
incorporate in the system.

250
00:11:38,500 --> 00:11:39,950
And the same thing is
true for acceleration.

251
00:11:39,950 --> 00:11:41,180
So I'm not going to
try to show it now.

252
00:11:41,180 --> 00:11:43,670
We will be showing it in
the course of our upcoming

253
00:11:43,670 --> 00:11:45,580
calculations.

254
00:11:45,580 --> 00:11:48,160
But in the kind of
collapsing universe

255
00:11:48,160 --> 00:11:50,560
that we're going
to be describing,

256
00:11:50,560 --> 00:11:54,160
any observer can consider
himself or herself

257
00:11:54,160 --> 00:11:56,680
to be non-accelerating.

258
00:11:56,680 --> 00:11:59,640
And then, that observer would
see all the other particles

259
00:11:59,640 --> 00:12:04,460
accelerating
directly towards her.

260
00:12:04,460 --> 00:12:06,300
And although that
description makes

261
00:12:06,300 --> 00:12:09,037
it sound like the person
at the center is special,

262
00:12:09,037 --> 00:12:09,620
it's not true.

263
00:12:09,620 --> 00:12:13,600
You can transform to any
other observer's frame.

264
00:12:13,600 --> 00:12:16,110
And each observer
can regard himself

265
00:12:16,110 --> 00:12:19,700
as being non-accelerating and
would see all the other objects

266
00:12:19,700 --> 00:12:21,960
accelerating
radially towards him.

267
00:12:27,680 --> 00:12:30,120
OK.

268
00:12:30,120 --> 00:12:34,870
So now we are ready
to go on and try

269
00:12:34,870 --> 00:12:38,420
to build a mathematical
model, which will tell us

270
00:12:38,420 --> 00:12:44,900
how a uniform distribution
of mass will behave.

271
00:12:44,900 --> 00:12:48,800
Now in doing this,
we first would

272
00:12:48,800 --> 00:12:52,670
like to tame the
issue of infinities.

273
00:12:52,670 --> 00:12:57,870
And we are going to do
that by starting out

274
00:12:57,870 --> 00:13:01,252
with a finite sphere.

275
00:13:01,252 --> 00:13:02,710
And then at the
very end, we'll let

276
00:13:02,710 --> 00:13:06,910
the size of that
sphere go to infinity.

277
00:13:06,910 --> 00:13:18,300
So our goal is to build a
mathematical model of our toy

278
00:13:18,300 --> 00:13:19,880
universe.

279
00:13:19,880 --> 00:13:21,630
And what we want to
do is to incorporate

280
00:13:21,630 --> 00:13:25,670
the three features
that we discussed,

281
00:13:25,670 --> 00:13:39,660
isotropy, homogeneity,
and Hubble's law.

282
00:13:43,312 --> 00:13:45,520
And we're going to build
this as a mechanical system,

283
00:13:45,520 --> 00:13:47,353
using the laws of
mechanics as we know them.

284
00:13:47,353 --> 00:13:50,675
We're, in fact, going to be
using a Newtonian description.

285
00:13:50,675 --> 00:13:52,310
But I will assure
you that although we

286
00:13:52,310 --> 00:13:53,790
are using a Newtonian
description,

287
00:13:53,790 --> 00:13:56,737
the answer that we'll get will
in fact be the exact answer

288
00:13:56,737 --> 00:13:58,820
that we would have gotten
with general relativity.

289
00:13:58,820 --> 00:14:02,260
And we'll talk later
about why that's the case.

290
00:14:02,260 --> 00:14:03,760
But we're not wasting
our time doing

291
00:14:03,760 --> 00:14:05,240
only an approximate calculation.

292
00:14:05,240 --> 00:14:08,080
This actually is a
completely valid calculation,

293
00:14:08,080 --> 00:14:12,040
which gives us, in the end,
entirely the correct answer.

294
00:14:12,040 --> 00:14:16,100
So we're going to
model our universe

295
00:14:16,100 --> 00:14:17,630
by introducing a
coordinate system.

296
00:14:17,630 --> 00:14:19,130
I'm now just going
to really re-draw

297
00:14:19,130 --> 00:14:20,296
the picture that's up there.

298
00:14:20,296 --> 00:14:24,200
But if I draw it down here,
I can point to it better.

299
00:14:24,200 --> 00:14:29,990
We're going to imagine
starting at our toy universe

300
00:14:29,990 --> 00:14:34,570
as a finite sized
sphere of matter.

301
00:14:34,570 --> 00:14:38,320
And we're going
to let t sub i be

302
00:14:38,320 --> 00:14:44,320
equal to the time of
the initial picture.

303
00:14:51,070 --> 00:14:53,360
t sub i need not be
particularly special in any way,

304
00:14:53,360 --> 00:14:55,141
in terms of the life
of our universe.

305
00:14:55,141 --> 00:14:56,640
Once we construct
the picture, we'll

306
00:14:56,640 --> 00:14:59,305
be able to calculate
how it would behave--

307
00:14:59,305 --> 00:15:02,560
at times later than ti and
at times earlier than ti.

308
00:15:02,560 --> 00:15:04,700
ti is just where
we are starting.

309
00:15:08,220 --> 00:15:13,675
At time, ti, we will give
our sphere a maximum size.

310
00:15:20,572 --> 00:15:22,030
I'm calling it
maximum, because I'm

311
00:15:22,030 --> 00:15:23,779
thinking of this as
filled with particles.

312
00:15:23,779 --> 00:15:26,660
It's, therefore, the maximum
radius for any particle.

313
00:15:26,660 --> 00:15:29,500
It's just the radius
of the sphere.

314
00:15:29,500 --> 00:15:40,110
So R sub max i is just
the initial radius.

315
00:15:40,110 --> 00:15:43,950
An initial means
at time, t sub i.

316
00:15:43,950 --> 00:15:46,910
We're going to fill
the sphere with matter.

317
00:15:46,910 --> 00:15:48,550
And we'll think of
this matter as being

318
00:15:48,550 --> 00:15:52,070
a kind of a uniform fluid, or
a dust of very small particles,

319
00:15:52,070 --> 00:15:54,380
which we can also
think of as a fluid.

320
00:15:54,380 --> 00:15:56,440
And it will have a mass
density, rho sub i.

321
00:16:13,990 --> 00:16:14,490
OK.

322
00:16:14,490 --> 00:16:18,290
So it's already homogeneous and
isotropic, at least isotropic

323
00:16:18,290 --> 00:16:20,330
about the center.

324
00:16:20,330 --> 00:16:23,140
And now, we want to
incorporate Hubble's law.

325
00:16:23,140 --> 00:16:27,110
So we're going to start all of
this matter, in our toy system

326
00:16:27,110 --> 00:16:30,770
here, expanding and expanding
in precisely the pattern

327
00:16:30,770 --> 00:16:34,005
that Hubble's law requires--
namely, all velocities will

328
00:16:34,005 --> 00:16:38,140
be moving out from the center
with a magnitude proportional

329
00:16:38,140 --> 00:16:38,815
to the distance.

330
00:16:48,030 --> 00:16:54,120
So if I label a
particle by-- I'm sorry,

331
00:16:54,120 --> 00:16:56,570
v sub i is just for initial.

332
00:16:56,570 --> 00:16:58,145
For any particle--
there's no way

333
00:16:58,145 --> 00:17:00,700
we're indicating a particle--
at the initial time,

334
00:17:00,700 --> 00:17:04,460
the velocity will just
obey Hubble's law.

335
00:17:04,460 --> 00:17:07,079
It will be equal to some
constant, which I'll call H sub

336
00:17:07,079 --> 00:17:09,349
i-- the initial value
of the Hubble expansion

337
00:17:09,349 --> 00:17:13,740
rate-- times the
vector r, which is just

338
00:17:13,740 --> 00:17:15,660
a vector from the
origin to the particle.

339
00:17:15,660 --> 00:17:19,490
That tells us where the particle
is that we're talking about.

340
00:17:19,490 --> 00:17:28,990
So v sub i is the initial
velocity of any particle.

341
00:17:34,660 --> 00:17:44,320
H sub i is the initial
Hubble expansion rate.

342
00:17:46,940 --> 00:17:49,230
And r is the position
of the particle.

343
00:18:00,645 --> 00:18:01,145
OK.

344
00:18:01,145 --> 00:18:02,890
As I said, we're starting
with a finite system,

345
00:18:02,890 --> 00:18:04,306
which is completely
under control.

346
00:18:04,306 --> 00:18:07,320
We know, unambiguously, how
to calculate-- in principle

347
00:18:07,320 --> 00:18:09,730
at least-- how that
system will evolve,

348
00:18:09,730 --> 00:18:12,020
once we set up these
initial conditions.

349
00:18:12,020 --> 00:18:14,170
And at the end of
the calculation,

350
00:18:14,170 --> 00:18:18,010
we'll take the limit as
R max i goes to infinity.

351
00:18:18,010 --> 00:18:20,180
And that, we hope,
will capture the idea

352
00:18:20,180 --> 00:18:21,830
that this model
universe is going

353
00:18:21,830 --> 00:18:25,620
to fill the infinite space.

354
00:18:25,620 --> 00:18:28,166
Now, I did want to say
a little bit here--

355
00:18:28,166 --> 00:18:29,540
only because it's
something which

356
00:18:29,540 --> 00:18:33,270
has entered my
scientific life recently

357
00:18:33,270 --> 00:18:35,101
and in interesting ways.

358
00:18:35,101 --> 00:18:36,850
I would say a few
things about infinities.

359
00:18:45,070 --> 00:18:47,780
Now, this is an
aside, which means

360
00:18:47,780 --> 00:18:49,750
that if you're struggling
to understand what's

361
00:18:49,750 --> 00:18:51,416
going in the course,
you can ignore this

362
00:18:51,416 --> 00:18:53,200
and don't worry about it.

363
00:18:53,200 --> 00:18:59,240
But if you're interested in
thinking about these concepts,

364
00:18:59,240 --> 00:19:04,640
the concept of infinity has sort
of struck cosmology in the nose

365
00:19:04,640 --> 00:19:06,390
in the context of
the multiverse,

366
00:19:06,390 --> 00:19:08,890
which I spoke of a little bit
about in the overview lecture,

367
00:19:08,890 --> 00:19:12,802
and which we'll get back to
at the end of the course.

368
00:19:12,802 --> 00:19:14,760
The multiverse has forced
us to think much more

369
00:19:14,760 --> 00:19:17,821
about infinities than
we had previously.

370
00:19:17,821 --> 00:19:19,820
And in the course of that,
I learned some things

371
00:19:19,820 --> 00:19:22,280
about infinity that
in fact surprised me.

372
00:19:22,280 --> 00:19:25,770
For the most part,
in physics, we

373
00:19:25,770 --> 00:19:29,620
think of infinities as the
limit of finite things,

374
00:19:29,620 --> 00:19:31,430
as we're doing over here.

375
00:19:31,430 --> 00:19:34,200
So if we want to discuss the
behavior of an infinite space,

376
00:19:34,200 --> 00:19:36,010
we very frequently
in physics start

377
00:19:36,010 --> 00:19:38,890
by discussing a finite space,
where things are much easier

378
00:19:38,890 --> 00:19:40,290
to control mathematically.

379
00:19:40,290 --> 00:19:42,010
And then we take the
limit as the space

380
00:19:42,010 --> 00:19:44,220
gets bigger and bigger.

381
00:19:44,220 --> 00:19:47,550
That works for almost
everything we do in physics.

382
00:19:47,550 --> 00:19:50,670
And I would say that
the reason why it works

383
00:19:50,670 --> 00:19:53,050
is because we are
assuming fundamentally

384
00:19:53,050 --> 00:19:55,150
that physical
interactions are local.

385
00:19:55,150 --> 00:19:59,020
Things that are vastly faraway
don't affect what happens here.

386
00:19:59,020 --> 00:20:02,290
So as we make this
sphere larger and larger,

387
00:20:02,290 --> 00:20:05,670
we'll be adding matter at
larger and larger radii.

388
00:20:05,670 --> 00:20:07,400
That new matter
that we're adding

389
00:20:07,400 --> 00:20:10,210
is not going to have much
affect on what happens inside.

390
00:20:10,210 --> 00:20:11,710
And in fact, for
this problem, we'll

391
00:20:11,710 --> 00:20:14,070
soon see that the extra
matter we had on the outside

392
00:20:14,070 --> 00:20:16,980
has no effect whatever
what happens inside,

393
00:20:16,980 --> 00:20:19,120
related to this fact that
the gravitational field

394
00:20:19,120 --> 00:20:22,170
inside a shell of matter is 0.

395
00:20:22,170 --> 00:20:26,287
So that's a typical
situation and gives physicist

396
00:20:26,287 --> 00:20:27,870
a very strong
motivation to always try

397
00:20:27,870 --> 00:20:32,210
to think of infinities as
the limit of finite systems.

398
00:20:32,210 --> 00:20:34,480
What I want to
point out, however,

399
00:20:34,480 --> 00:20:36,872
is that that's not always
the right thing to do.

400
00:20:36,872 --> 00:20:39,080
And there are cases where
it's, in fact, emphatically

401
00:20:39,080 --> 00:20:41,140
the wrong thing to do.

402
00:20:41,140 --> 00:20:45,670
Mathematicians know about this,
but physicists tend not to.

403
00:20:45,670 --> 00:20:55,600
So I want to point out
that not all infinities are

404
00:20:55,600 --> 00:21:08,330
well-described as limits
of finite systems.

405
00:21:15,084 --> 00:21:16,750
I don't want to call
this into question.

406
00:21:16,750 --> 00:21:18,960
This, I think, is
absolutely solid.

407
00:21:18,960 --> 00:21:21,260
And we will continue
with it after I go off

408
00:21:21,260 --> 00:21:23,990
on this side discussion.

409
00:21:23,990 --> 00:21:27,106
But to give you an
example of a system which

410
00:21:27,106 --> 00:21:28,980
is infinite and not
well-described as a limit

411
00:21:28,980 --> 00:21:32,980
of finite systems-- this is a
mathematical example-- we could

412
00:21:32,980 --> 00:21:44,770
just think of the set of
positive integers, also

413
00:21:44,770 --> 00:21:46,140
called the natural numbers.

414
00:21:46,140 --> 00:21:47,970
And I'll use the letter
that's often used

415
00:21:47,970 --> 00:21:50,040
for it, N with an extra
line through it to make

416
00:21:50,040 --> 00:21:53,160
it super bold or
something like that.

417
00:21:53,160 --> 00:21:54,980
So that's the set of integers.

418
00:21:54,980 --> 00:22:00,910
And the question is,
suppose we consider

419
00:22:00,910 --> 00:22:03,430
trying to describe the
set of positive integers,

420
00:22:03,430 --> 00:22:06,210
as a limit-- a finite set.

421
00:22:06,210 --> 00:22:10,970
So we could think
about the limit from N

422
00:22:10,970 --> 00:22:17,000
goes to infinity-- as
N goes to infinity--

423
00:22:17,000 --> 00:22:26,005
of the natural numbers
up to N. So we're

424
00:22:26,005 --> 00:22:28,380
taking sets of integers and
taking bigger and bigger sets

425
00:22:28,380 --> 00:22:30,100
and trying to take
the limit, and asking,

426
00:22:30,100 --> 00:22:31,683
does that give us
the set of integers?

427
00:22:34,480 --> 00:22:37,150
One might think
the answer is yes.

428
00:22:37,150 --> 00:22:40,050
What I will claim is that
this is definitely not

429
00:22:40,050 --> 00:22:42,320
equal to the set of integers.

430
00:22:50,440 --> 00:22:52,810
And in fact, I'll claim
that the limit does not

431
00:22:52,810 --> 00:22:55,790
exist at all, which is
why it couldn't possibly

432
00:22:55,790 --> 00:22:58,210
be equal to the set integers.

433
00:22:58,210 --> 00:23:01,040
And to drive this
home, I just need

434
00:23:01,040 --> 00:23:05,555
to remind us what a limit means.

435
00:23:14,660 --> 00:23:16,160
And since this is
not a math course,

436
00:23:16,160 --> 00:23:17,510
I won't give a
rigorous definition.

437
00:23:17,510 --> 00:23:18,470
But I'll just give
you an example

438
00:23:18,470 --> 00:23:20,390
that will strike bells
and things that you

439
00:23:20,390 --> 00:23:22,220
learned in math classes.

440
00:23:22,220 --> 00:23:25,860
Suppose we want to
talk about the limit

441
00:23:25,860 --> 00:23:33,449
as x goes to 0 of sine x over x.

442
00:23:33,449 --> 00:23:34,740
But we all know how to do that.

443
00:23:34,740 --> 00:23:36,760
It's usually called
L'Hopital's rule or something.

444
00:23:36,760 --> 00:23:39,134
But you could probably just
use the definition of a limit

445
00:23:39,134 --> 00:23:40,550
and get it directly.

446
00:23:40,550 --> 00:23:43,440
That limit is 1.

447
00:23:43,440 --> 00:23:46,434
And what we mean
when we say that,

448
00:23:46,434 --> 00:23:51,650
is that as x gets
closer and closer to 0,

449
00:23:51,650 --> 00:23:53,830
we could evaluate this
expression for any value of x

450
00:23:53,830 --> 00:23:56,650
not equal to 0-- for 0,
itself, it's ambiguous.

451
00:23:56,650 --> 00:23:59,550
But for any value of x not equal
to 0, we can evaluate this.

452
00:23:59,550 --> 00:24:02,029
And as x gets closer
and closer to 0,

453
00:24:02,029 --> 00:24:03,570
that evaluation
gives us numbers that

454
00:24:03,570 --> 00:24:05,580
are closer and closer to 1.

455
00:24:05,580 --> 00:24:07,030
And we can get as
close to 1 as we

456
00:24:07,030 --> 00:24:11,482
want by choosing x's as
close to 0 as necessary.

457
00:24:11,482 --> 00:24:13,940
And that's usually phrased in
terms of epsilons and deltas.

458
00:24:13,940 --> 00:24:15,820
But I don't think
I need that here.

459
00:24:15,820 --> 00:24:18,650
The point is that, the limit
is simply the statement

460
00:24:18,650 --> 00:24:21,500
that this can be
made as close to 1

461
00:24:21,500 --> 00:24:25,510
as you like by choosing x as
close as possible to the limit

462
00:24:25,510 --> 00:24:27,710
point, 0.

463
00:24:27,710 --> 00:24:30,940
Now, if you imagine
applying the same concept

464
00:24:30,940 --> 00:24:35,600
to this set-- the set of
integers from 1 to N--

465
00:24:35,600 --> 00:24:37,330
the question is, as
you make N large,

466
00:24:37,330 --> 00:24:40,556
does it get closer to
the set of all integers?

467
00:24:40,556 --> 00:24:42,430
Are the numbers from 1
to 10 close to the set

468
00:24:42,430 --> 00:24:43,770
of all integers?

469
00:24:43,770 --> 00:24:44,530
No.

470
00:24:44,530 --> 00:24:46,597
What about 1 to a million?

471
00:24:46,597 --> 00:24:47,680
Still infinitely far away.

472
00:24:47,680 --> 00:24:48,490
1 to a billion?

473
00:24:48,490 --> 00:24:49,156
1 to a google?

474
00:24:49,156 --> 00:24:51,280
No matter what number you
pick as that upper limit,

475
00:24:51,280 --> 00:24:54,500
you're still infinitely far away
from the set of all integers.

476
00:24:54,500 --> 00:24:56,890
You're not coming close.

477
00:24:56,890 --> 00:24:58,610
So there's no
sense in which this

478
00:24:58,610 --> 00:25:01,240
converges to the
set of all integers.

479
00:25:01,240 --> 00:25:03,197
It's just a different animal.

480
00:25:03,197 --> 00:25:04,530
Now does this make a difference?

481
00:25:04,530 --> 00:25:06,317
Are there any questions
where it matters

482
00:25:06,317 --> 00:25:08,400
whether you think of the
integers as being defined

483
00:25:08,400 --> 00:25:11,490
in some other way,
or by this limit?

484
00:25:11,490 --> 00:25:14,010
Or maybe first say
how it is defined.

485
00:25:14,010 --> 00:25:16,589
If you ask mathematicians how
you define the set of integers,

486
00:25:16,589 --> 00:25:18,380
I think all of them
will tell you, well, we

487
00:25:18,380 --> 00:25:20,840
used the Peano axioms.

488
00:25:20,840 --> 00:25:23,030
And the key thing in
the Peano axioms--

489
00:25:23,030 --> 00:25:26,960
if you look at them-- that
controls the fact that there's

490
00:25:26,960 --> 00:25:31,030
an infinite number of integers,
is the successor axiom.

491
00:25:31,030 --> 00:25:33,800
That is, built into
these Peano axioms that

492
00:25:33,800 --> 00:25:36,470
describe the integers
mathematically

493
00:25:36,470 --> 00:25:38,880
is a statement that every
integer has a successor.

494
00:25:38,880 --> 00:25:40,380
And then, there are
other statements

495
00:25:40,380 --> 00:25:42,046
that guarantee that
the successor is not

496
00:25:42,046 --> 00:25:43,890
one of the ones that's
already on the list.

497
00:25:43,890 --> 00:25:46,700
So you always have a new
element as the successor

498
00:25:46,700 --> 00:25:49,320
to your highest element so far.

499
00:25:49,320 --> 00:25:54,020
And that guarantees from the
beginning, the set of axioms

500
00:25:54,020 --> 00:25:55,010
is infinite.

501
00:25:55,010 --> 00:25:58,160
It's not thought of as the limit
of finite sets and cannot be

502
00:25:58,160 --> 00:26:00,440
thought of as the
limit of finite sets.

503
00:26:00,440 --> 00:26:03,140
Because, no finite
set is, in any way,

504
00:26:03,140 --> 00:26:05,950
resembling an infinite set.

505
00:26:05,950 --> 00:26:07,080
So does it matter?

506
00:26:07,080 --> 00:26:09,810
Are there questions
where we care

507
00:26:09,810 --> 00:26:13,970
whether we could describe
the integers this way or not?

508
00:26:13,970 --> 00:26:17,120
And the only questions I
know sound kind of contrived,

509
00:26:17,120 --> 00:26:18,570
I'll admit.

510
00:26:18,570 --> 00:26:21,856
But I also point out that
in mathematics, the word,

511
00:26:21,856 --> 00:26:23,470
contrived doesn't cut much ice.

512
00:26:23,470 --> 00:26:26,364
If you discover a
contradiction in some system,

513
00:26:26,364 --> 00:26:28,530
nobody's going to tell you
whether you should ignore

514
00:26:28,530 --> 00:26:30,740
that contradiction
because it's contrived.

515
00:26:30,740 --> 00:26:33,620
If it's really a
contradiction, it counts.

516
00:26:33,620 --> 00:26:36,000
So a question where
it makes a difference,

517
00:26:36,000 --> 00:26:38,280
whether you think of the
integers as being defined

518
00:26:38,280 --> 00:26:40,670
as infinite from the start,
or whether you think of it--

519
00:26:40,670 --> 00:26:43,030
or try to think of it--
as a limit of this sort,

520
00:26:43,030 --> 00:26:47,340
is a question such as, what
fraction of the integers are

521
00:26:47,340 --> 00:26:50,510
so large that they cannot
be doubled and still be

522
00:26:50,510 --> 00:26:52,410
an integer?

523
00:26:52,410 --> 00:26:56,950
And notice that, if we
consider this set, for any N,

524
00:26:56,950 --> 00:27:00,100
no matter how large N is,
we would conclude that half

525
00:27:00,100 --> 00:27:03,570
of the integers are so large
that they cannot be doubled.

526
00:27:03,570 --> 00:27:05,720
And that would hold no
matter how large we made N.

527
00:27:05,720 --> 00:27:08,220
On the other hand, if we look
at the actual set of integers,

528
00:27:08,220 --> 00:27:09,845
we know that any
integer can be doubled

529
00:27:09,845 --> 00:27:12,310
and you just get
another integer.

530
00:27:12,310 --> 00:27:16,040
So that's an example of a
property of the integers, which

531
00:27:16,040 --> 00:27:19,050
you would get wrong if you
thought that you could think

532
00:27:19,050 --> 00:27:21,879
of the integers as
this kind of a limit.

533
00:27:21,879 --> 00:27:22,920
So you really just can't.

534
00:27:26,380 --> 00:27:28,050
Any questions about that?

535
00:27:28,050 --> 00:27:29,350
Subtle point, I think.

536
00:27:33,270 --> 00:27:34,740
AUDIENCE: How does
that relate to--

537
00:27:34,740 --> 00:27:36,420
PROFESSOR: To this?

538
00:27:36,420 --> 00:27:38,110
It's just a warning
that you should

539
00:27:38,110 --> 00:27:40,610
be careful about
treating infinity

540
00:27:40,610 --> 00:27:42,090
as limits of finite things.

541
00:27:42,090 --> 00:27:43,300
That's all it is.

542
00:27:43,300 --> 00:27:44,940
It does not relate to this.

543
00:27:44,940 --> 00:27:47,300
That's why I said it was
an aside when I started.

544
00:27:47,300 --> 00:27:51,070
Aside means something
worth knowing but not

545
00:27:51,070 --> 00:27:54,580
directly related to what
we're talking about.

546
00:27:54,580 --> 00:27:55,450
OK.

547
00:27:55,450 --> 00:27:59,450
So we're going to
proceed with this model.

548
00:27:59,450 --> 00:28:01,770
Let me mention one other
feature of the model

549
00:28:01,770 --> 00:28:04,430
to discuss a little bit, the
shape that we started with.

550
00:28:04,430 --> 00:28:06,050
We're starting with a sphere.

551
00:28:06,050 --> 00:28:08,904
You might ask, why a sphere?

552
00:28:08,904 --> 00:28:10,820
Well, a sphere is certainly
the simplest thing

553
00:28:10,820 --> 00:28:11,950
we could start with.

554
00:28:11,950 --> 00:28:15,330
And a sphere also guarantees
isotropy-- or at least

555
00:28:15,330 --> 00:28:16,800
isotropy about the origin.

556
00:28:16,800 --> 00:28:18,750
Isotropy about one point.

557
00:28:18,750 --> 00:28:22,640
We could, by doing
significantly more work,

558
00:28:22,640 --> 00:28:24,270
have started, for
example, with a cube

559
00:28:24,270 --> 00:28:25,896
and let the cube get
bigger and bigger.

560
00:28:25,896 --> 00:28:27,478
And as the cube got
bigger and bigger,

561
00:28:27,478 --> 00:28:28,687
it would also fill all space.

562
00:28:28,687 --> 00:28:30,519
And we might think that
would be another way

563
00:28:30,519 --> 00:28:31,780
of getting the same answer.

564
00:28:31,780 --> 00:28:33,040
And that would be right.

565
00:28:33,040 --> 00:28:35,590
If we did it with a cube,
it would be a lot more work.

566
00:28:35,590 --> 00:28:38,200
But we would, in fact,
get the same answer.

567
00:28:38,200 --> 00:28:41,315
The cube has enough
symmetry, So it, in fact,

568
00:28:41,315 --> 00:28:43,120
will be the same as
sphere in this case.

569
00:28:43,120 --> 00:28:45,286
I'm not going to try to
tell you how to calculate it

570
00:28:45,286 --> 00:28:46,620
with an arbitrary shape.

571
00:28:46,620 --> 00:28:48,120
But I will guarantee
you that a cube

572
00:28:48,120 --> 00:28:49,810
will get you the same answer.

573
00:28:49,810 --> 00:28:52,230
On the other hand, if we started
with a rectangular solid,

574
00:28:52,230 --> 00:28:54,646
where the three sides were
different-- or at least not all

575
00:28:54,646 --> 00:28:56,260
equal to each
other-- then would be

576
00:28:56,260 --> 00:28:58,885
starting with something which is
asymmetric in the first place.

577
00:28:58,885 --> 00:29:00,930
Some direction
would be privileged.

578
00:29:00,930 --> 00:29:03,970
And then, if we
continued-- as we're

579
00:29:03,970 --> 00:29:05,740
going to be doing
for our sphere--

580
00:29:05,740 --> 00:29:09,410
starting with this
irregular rectangular solid,

581
00:29:09,410 --> 00:29:11,860
we would build in an anisotropy
from the very beginning.

582
00:29:11,860 --> 00:29:16,520
We end up with an anisotropic
model of the universe.

583
00:29:16,520 --> 00:29:19,420
So since we're trying to
model the real universe, which

584
00:29:19,420 --> 00:29:22,480
is to a high degree
isotropic, we

585
00:29:22,480 --> 00:29:25,370
will start with something
which guarantees the isotropy.

586
00:29:25,370 --> 00:29:26,860
And the sphere does that.

587
00:29:26,860 --> 00:29:28,715
And it's the simplest
shape which does that.

588
00:29:34,851 --> 00:29:35,350
OK.

589
00:29:35,350 --> 00:29:39,400
Now, we're ready to start
putting in some dynamics

590
00:29:39,400 --> 00:29:42,200
into this model.

591
00:29:42,200 --> 00:29:44,930
And the dynamics that we're
going to put in for now

592
00:29:44,930 --> 00:29:48,960
will just be purely
Newtonian dynamics.

593
00:29:48,960 --> 00:29:51,550
And in fact, we'll be
modeling the matter

594
00:29:51,550 --> 00:29:53,770
that makes up this
sphere as just

595
00:29:53,770 --> 00:29:58,600
a dust of Newtonian
particles-- or if you like,

596
00:29:58,600 --> 00:30:02,110
a gas of Newtonian particles.

597
00:30:02,110 --> 00:30:04,780
These particles will be
nonrelativistic-- as implied

598
00:30:04,780 --> 00:30:07,060
by the word Newtonian.

599
00:30:07,060 --> 00:30:11,400
And that describes
our real universe

600
00:30:11,400 --> 00:30:13,760
for a good chunk
of its evolution

601
00:30:13,760 --> 00:30:15,690
but not for all of it.

602
00:30:15,690 --> 00:30:19,080
So let me-- before we
proceed-- say a few words

603
00:30:19,080 --> 00:30:25,070
about the real universe
and the kind of matter that

604
00:30:25,070 --> 00:30:31,180
has dominated it during
different eras of evolution.

605
00:30:31,180 --> 00:30:35,140
In the earliest time,
our universe, we believe,

606
00:30:35,140 --> 00:30:37,940
was radiation dominated.

607
00:30:37,940 --> 00:30:39,800
And that means
that, if you follow

608
00:30:39,800 --> 00:30:45,120
the evolution of our
universe backwards in time,

609
00:30:45,120 --> 00:30:49,100
as one goes to earlier and
earlier times, the photons that

610
00:30:49,100 --> 00:30:52,840
make up the cosmic background
radiation blue shift.

611
00:30:52,840 --> 00:30:55,090
We've learned that they red
shift as universe expands.

612
00:30:55,090 --> 00:30:57,550
That means if we extrapolate
backwards, they blue shift.

613
00:30:57,550 --> 00:31:00,330
They get more energetic
for every photon.

614
00:31:00,330 --> 00:31:03,450
But the number of photons
remains constant, essentially,

615
00:31:03,450 --> 00:31:05,090
as one goes backwards in time.

616
00:31:05,090 --> 00:31:07,190
They just get squeezed
into a small volume.

617
00:31:07,190 --> 00:31:09,580
And they become more energetic.

618
00:31:09,580 --> 00:31:12,910
Now, meanwhile, the particles
of ordinary matter-- and dark

619
00:31:12,910 --> 00:31:15,900
matter as well, the protons
and whatever the dark matter is

620
00:31:15,900 --> 00:31:19,740
made of-- also gets squeezed,
as you go backwards in time.

621
00:31:19,740 --> 00:31:22,020
But they don't become
more energetic.

622
00:31:22,020 --> 00:31:24,930
They remain just--
a proton remains

623
00:31:24,930 --> 00:31:27,510
a particle whose mass is
the mass of a proton times c

624
00:31:27,510 --> 00:31:29,310
squared.

625
00:31:29,310 --> 00:31:33,210
So as you go backwards,
the energy density

626
00:31:33,210 --> 00:31:35,729
in the radiation-- in the
cosmic microwave background

627
00:31:35,729 --> 00:31:37,770
radiation-- gets to be
larger and larger compared

628
00:31:37,770 --> 00:31:39,270
to the energy density
in the matter.

629
00:31:39,270 --> 00:31:41,210
And later, we'll learn
how to calculate this.

630
00:31:41,210 --> 00:31:43,940
But the two cross at
about 50,000 years.

631
00:31:43,940 --> 00:31:44,440
Yes.

632
00:31:44,440 --> 00:31:46,332
AUDIENCE: If we think
of particles as waves,

633
00:31:46,332 --> 00:31:50,116
then how does that make sense
that they wouldn't change?

634
00:31:50,116 --> 00:31:51,801
PROFESSOR: Right.

635
00:31:51,801 --> 00:31:53,300
If we think of
particles like waves,

636
00:31:53,300 --> 00:31:55,300
how does it make sense
that they don't change?

637
00:31:55,300 --> 00:31:56,700
The answer is, they
do a little bit.

638
00:31:56,700 --> 00:31:59,033
But we're going to be assuming
that these particles have

639
00:31:59,033 --> 00:32:00,770
negligible velocities.

640
00:32:00,770 --> 00:32:04,760
What happens is their momentum
actually does blue shift.

641
00:32:04,760 --> 00:32:08,036
But a blue shift is proportional
to the initial value.

642
00:32:08,036 --> 00:32:09,660
And if the initial
value is very small,

643
00:32:09,660 --> 00:32:11,250
even when it blue
shifts, the momentum

644
00:32:11,250 --> 00:32:12,333
could still be negligible.

645
00:32:16,390 --> 00:32:20,830
So for the real universe,
between time, 0,

646
00:32:20,830 --> 00:32:31,575
and about 50,000
years, the universe

647
00:32:31,575 --> 00:32:32,695
was radiation dominated.

648
00:32:37,562 --> 00:32:39,520
And we'll be talking
about that in a few weeks.

649
00:32:39,520 --> 00:32:42,420
But today, we're
just ignoring that.

650
00:32:42,420 --> 00:32:50,270
Then, from this time
of about 50,000 years

651
00:32:50,270 --> 00:32:56,210
until about 9 billion
years-- so a good chunk

652
00:32:56,210 --> 00:32:58,770
of the history of the
universe-- the universe

653
00:32:58,770 --> 00:33:03,870
was what is called
matter dominated.

654
00:33:07,950 --> 00:33:10,850
And matter means
nonrelativistic matter.

655
00:33:10,850 --> 00:33:14,440
That's standard
jargon in cosmology.

656
00:33:14,440 --> 00:33:16,410
When we talk about a
matter-dominated universe,

657
00:33:16,410 --> 00:33:18,659
and even though we don't use
the word nonrelativistic,

658
00:33:18,659 --> 00:33:22,100
that's what everybody means by
a matter dominated universe.

659
00:33:22,100 --> 00:33:24,300
And that's the case we're
going to be discussing,

660
00:33:24,300 --> 00:33:28,450
just ordinary nonrelativistic
matter filling space.

661
00:33:28,450 --> 00:33:31,540
And then for our real
universe, something else

662
00:33:31,540 --> 00:33:34,820
happened at about 9 billion
years in its history

663
00:33:34,820 --> 00:33:37,730
up to the present-- and
presumably in the future

664
00:33:37,730 --> 00:33:41,545
as well-- is the universe
became dark energy dominated.

665
00:33:54,100 --> 00:33:55,980
And dark energy
is the stuff that

666
00:33:55,980 --> 00:33:58,910
causes the universe
to accelerate.

667
00:33:58,910 --> 00:34:01,310
So the universe has
been accelerating

668
00:34:01,310 --> 00:34:07,110
since about 9 billion
years from the Big Bang.

669
00:34:07,110 --> 00:34:10,150
Now, I think we mentioned--
I'll mention it again quickly--

670
00:34:10,150 --> 00:34:13,530
there's no conversion of
ordinary matter to dark energy

671
00:34:13,530 --> 00:34:17,080
here, which you might guess
from this change in domination.

672
00:34:17,080 --> 00:34:20,560
It's just a question of how they
behave as the universe expands.

673
00:34:20,560 --> 00:34:25,199
Ordinary matter has a
density, which decreases as 1

674
00:34:25,199 --> 00:34:27,040
over the cube of
the scale factor.

675
00:34:27,040 --> 00:34:29,719
You just have a fixed number
of particles scattering out

676
00:34:29,719 --> 00:34:31,889
over a larger and larger volume.

677
00:34:31,889 --> 00:34:35,480
The dark energy--
for reasons that we

678
00:34:35,480 --> 00:34:37,520
will come to near the
end of the course,

679
00:34:37,520 --> 00:34:39,810
in fact, does not
change its energy

680
00:34:39,810 --> 00:34:42,659
density at all as
the universe expands.

681
00:34:42,659 --> 00:34:44,790
So what happened 9
billion years ago

682
00:34:44,790 --> 00:34:47,460
was simply that the
density of ordinary matter

683
00:34:47,460 --> 00:34:49,350
fell below the density
of dark energy.

684
00:34:49,350 --> 00:34:51,225
And then, the dark energy
started to dominate

685
00:34:51,225 --> 00:34:52,850
and the universe
started to accelerate.

686
00:34:56,239 --> 00:34:59,450
Today, the dark energy, by
the way, is about 60% or 70%

687
00:34:59,450 --> 00:35:00,850
of the total.

688
00:35:00,850 --> 00:35:02,410
So it doesn't
dominate completely.

689
00:35:02,410 --> 00:35:04,283
But it's the largest component.

690
00:35:07,520 --> 00:35:08,020
OK.

691
00:35:08,020 --> 00:35:09,810
For today's
calculation, we're going

692
00:35:09,810 --> 00:35:11,370
to be focusing on
this middle period

693
00:35:11,370 --> 00:35:13,045
and pretending it's
the whole story.

694
00:35:13,045 --> 00:35:15,170
And we will come back and
discuss these other eras.

695
00:35:15,170 --> 00:35:16,461
We're not going to ignore them.

696
00:35:16,461 --> 00:35:18,514
But we will not
discuss them today.

697
00:35:18,514 --> 00:35:19,930
So we will be
discussing this case

698
00:35:19,930 --> 00:35:22,090
of a matter-dominated universe.

699
00:35:22,090 --> 00:35:25,392
And we're going to be discussing
it using Newtonian mechanics.

700
00:35:25,392 --> 00:35:26,850
And in spite of
the fact that we're

701
00:35:26,850 --> 00:35:29,590
using Newtonian mechanics,
I will assure you--

702
00:35:29,590 --> 00:35:32,200
and try to give you
some arguments later--

703
00:35:32,200 --> 00:35:35,150
it gives exactly the same answer
that general relativity gives.

704
00:35:45,650 --> 00:35:46,150
OK.

705
00:35:46,150 --> 00:35:47,244
So how do we proceed?

706
00:35:47,244 --> 00:35:48,660
How do we write
down the equations

707
00:35:48,660 --> 00:35:52,590
that describe how this
sphere is going to evolve?

708
00:35:52,590 --> 00:35:55,770
We're going to be
using a shell picture.

709
00:35:55,770 --> 00:35:59,290
That is, we will
describe that sphere

710
00:35:59,290 --> 00:36:01,245
in terms of shells
that make it up.

711
00:36:04,240 --> 00:36:10,690
So in other words, we
will divide the matter

712
00:36:10,690 --> 00:36:11,690
at the initial time.

713
00:36:11,690 --> 00:36:14,000
And then, we'll just imagine
labeling those particles

714
00:36:14,000 --> 00:36:15,692
and following them.

715
00:36:15,692 --> 00:36:17,150
So at the initial
time, we're going

716
00:36:17,150 --> 00:36:20,220
to divide the matter
into concentric shells.

717
00:36:34,890 --> 00:36:46,120
So each shell extends from
some value of r sub i--

718
00:36:46,120 --> 00:36:54,930
the initial radius-- to
r sub i plus dr sub i.

719
00:36:54,930 --> 00:36:57,470
Our shells will be infinitesimal
in their thickness.

720
00:37:00,200 --> 00:37:02,300
And each shell will have
a different r sub i.

721
00:37:02,300 --> 00:37:04,490
And we could let all the dr sub
i's be the same, if we want.

722
00:37:04,490 --> 00:37:06,060
We can let each shell
be the same thickness.

723
00:37:06,060 --> 00:37:06,670
We can let them be different.

724
00:37:06,670 --> 00:37:07,419
It doesn't matter.

725
00:37:12,930 --> 00:37:16,710
Now a reason why we can get
away with thinking of the matter

726
00:37:16,710 --> 00:37:19,190
purely has being
described by shells

727
00:37:19,190 --> 00:37:21,510
is that we know that
we started with all

728
00:37:21,510 --> 00:37:23,480
of the velocities radial.

729
00:37:23,480 --> 00:37:25,470
That Hubble's law
that we assumed

730
00:37:25,470 --> 00:37:27,360
says that the velocities
were proportional

731
00:37:27,360 --> 00:37:30,210
to the radius vector, which
is measured from the origin.

732
00:37:30,210 --> 00:37:33,027
So all our initial velocities
were radial outward.

733
00:37:33,027 --> 00:37:34,610
And furthermore, if
we think about how

734
00:37:34,610 --> 00:37:37,440
the Newtonian force of
gravity will come about

735
00:37:37,440 --> 00:37:41,090
for this spherical object, they
will all be radial as well.

736
00:37:41,090 --> 00:37:43,170
So all the motion
will remain radial.

737
00:37:43,170 --> 00:37:46,290
There will never be any forces
acting on any of our particles

738
00:37:46,290 --> 00:37:49,392
that will push them in
a tangential direction.

739
00:37:49,392 --> 00:37:50,600
So all motion will be radial.

740
00:37:50,600 --> 00:37:53,570
And as long as we keep track
of the radius of each particle,

741
00:37:53,570 --> 00:37:55,760
its angular variables,
theta and phi--

742
00:37:55,760 --> 00:37:57,910
which I will never
mention again--

743
00:37:57,910 --> 00:37:59,710
will just be constant in time.

744
00:37:59,710 --> 00:38:01,210
So we will not need
to mention them.

745
00:38:18,190 --> 00:38:20,715
So all motion is radial,
because the v's start radial.

746
00:38:23,800 --> 00:38:35,309
And there are no
tangential forces,

747
00:38:35,309 --> 00:38:37,600
where tangential means any
direction other than radial.

748
00:38:37,600 --> 00:38:39,766
So we're just saying that
all the forces are radial.

749
00:38:39,766 --> 00:38:43,390
And that means the
motion will stay radial.

750
00:38:43,390 --> 00:38:50,830
To describe the motion, I want
to give a little bit more teeth

751
00:38:50,830 --> 00:38:52,246
to this statement
that we're going

752
00:38:52,246 --> 00:38:54,630
to be describing it
in terms of shells.

753
00:38:54,630 --> 00:38:59,080
We'll going to describe the
motion by a function, r, which

754
00:38:59,080 --> 00:39:03,360
will be a function of two
variables, r sub i and t.

755
00:39:06,400 --> 00:39:23,850
And this is just the
radius at time, t,

756
00:39:23,850 --> 00:39:40,450
of the shell that was at radius,
r sub i, at time, t sub i.

757
00:39:40,450 --> 00:39:43,120
So each shell is labeled by
where it is at time, t sub i.

758
00:39:43,120 --> 00:39:45,200
And once we label it,
it keeps that label.

759
00:39:45,200 --> 00:39:48,790
But then, we'll talk about this
function, r, which tells us

760
00:39:48,790 --> 00:39:51,230
where it is at any later time.

761
00:39:51,230 --> 00:39:54,520
And we could also think about
earlier times if we want.

762
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

763
00:39:58,620 --> 00:40:01,400
sub i, where we're looking
at the radius at time, t.

764
00:40:04,650 --> 00:40:07,886
Now, I should warn you that
if you look in any textbook,

765
00:40:07,886 --> 00:40:09,260
you will see a
simpler derivation

766
00:40:09,260 --> 00:40:10,890
than what I'm about to show you.

767
00:40:10,890 --> 00:40:13,790
So you might wonder why am
I going to so much trouble

768
00:40:13,790 --> 00:40:15,764
when there's an
easier way to do it.

769
00:40:15,764 --> 00:40:17,430
And the answer is
that the calculation I

770
00:40:17,430 --> 00:40:19,510
show you will, in fact,
show you more than what's

771
00:40:19,510 --> 00:40:23,200
in the textbooks-- and
Ryden's textbook, for example.

772
00:40:23,200 --> 00:40:25,120
What most textbooks
do-- including Ryden,

773
00:40:25,120 --> 00:40:29,310
I believe-- is to assume that
this motion will continue

774
00:40:29,310 --> 00:40:30,830
to obey Hubble's
law and continue

775
00:40:30,830 --> 00:40:33,650
to be completely
uniform in the density.

776
00:40:33,650 --> 00:40:35,660
And then, you could
just calculate

777
00:40:35,660 --> 00:40:37,500
what happens to the
outside of the sphere.

778
00:40:37,500 --> 00:40:38,958
And that governs
everything, if you

779
00:40:38,958 --> 00:40:41,150
assume that will stay uniform.

780
00:40:41,150 --> 00:40:43,440
We are not going to assume
that it will stay uniform.

781
00:40:43,440 --> 00:40:45,780
We will show that it
will stay uniform.

782
00:40:45,780 --> 00:40:48,352
And from my point of
view, it's a lot better

783
00:40:48,352 --> 00:40:50,610
to actually show something
than to just guess

784
00:40:50,610 --> 00:40:53,840
it and we write about,
without showing it.

785
00:40:53,840 --> 00:40:55,860
So that's why we're going
to do some extra work.

786
00:40:55,860 --> 00:40:59,550
We will actually show that the
uniformity, once you put in,

787
00:40:59,550 --> 00:41:03,301
is preserved under time,
under the laws of Newtonian

788
00:41:03,301 --> 00:41:03,800
evolution.

789
00:41:13,739 --> 00:41:15,030
I guess I'll leave the picture.

790
00:41:15,030 --> 00:41:16,280
Maybe, I'll leave all of that.

791
00:41:47,840 --> 00:41:48,340
OK.

792
00:41:48,340 --> 00:41:51,330
Now, there's an issue that's
a little bit complicated

793
00:41:51,330 --> 00:41:53,160
that I'll try to describe.

794
00:41:53,160 --> 00:41:54,660
And again, this is
a subtlety that's

795
00:41:54,660 --> 00:41:56,810
probably not mentioned
in the books.

796
00:41:56,810 --> 00:41:59,300
We have these different
shells that are evolving.

797
00:41:59,300 --> 00:42:01,580
And we can calculate the
force on any given shell

798
00:42:01,580 --> 00:42:03,740
if we know the matter
that's inside that shell.

799
00:42:03,740 --> 00:42:05,420
That's what it tells us.

800
00:42:05,420 --> 00:42:07,710
If everything is in
shells, the shells

801
00:42:07,710 --> 00:42:11,780
have no effect on the matter
that is inside the shell.

802
00:42:11,780 --> 00:42:15,780
So if we want to know the
effect on a given shell,

803
00:42:15,780 --> 00:42:18,670
we only need to know the matter
that's inside that shell.

804
00:42:18,670 --> 00:42:20,772
The matter outside has no force.

805
00:42:20,772 --> 00:42:22,230
So it's very
important that we know

806
00:42:22,230 --> 00:42:24,290
the ordering of these shells.

807
00:42:24,290 --> 00:42:25,710
Now, initially, we certainly do.

808
00:42:25,710 --> 00:42:27,980
They're just ordered
according to r sub i.

809
00:42:27,980 --> 00:42:29,700
But once they start
to move around,

810
00:42:29,700 --> 00:42:31,930
there's, in principle,
the possibility

811
00:42:31,930 --> 00:42:33,990
that shells could cross.

812
00:42:33,990 --> 00:42:36,780
And if shells crossed,
our equations of emotion

813
00:42:36,780 --> 00:42:39,970
would have to change, because
then different amounts of mass

814
00:42:39,970 --> 00:42:41,910
would be acting on
different shells.

815
00:42:41,910 --> 00:42:44,440
And we'd have to take
that into account.

816
00:42:44,440 --> 00:42:47,440
Fortunately, that turns
out not to be a problem.

817
00:42:47,440 --> 00:42:51,060
It just sounds like
it could be a problem.

818
00:42:51,060 --> 00:42:52,670
And the way we're
going to treat it

819
00:42:52,670 --> 00:42:56,090
is to recognize that, initially,
all these shells are moving

820
00:42:56,090 --> 00:42:58,480
away from each other, because
of the Hubble expansion.

821
00:42:58,480 --> 00:43:00,820
If Hubble's law holds,
any two particles

822
00:43:00,820 --> 00:43:04,090
are moving away from each
other with a relative velocity

823
00:43:04,090 --> 00:43:05,570
proportional to their distance.

824
00:43:05,570 --> 00:43:09,710
So that holds for any
two shells, as well.

825
00:43:09,710 --> 00:43:12,342
So if shells are
going cross, they're

826
00:43:12,342 --> 00:43:14,050
certainly not going
to cross immediately.

827
00:43:14,050 --> 00:43:15,400
There are no two shells
that are approaching

828
00:43:15,400 --> 00:43:16,560
each other initially.

829
00:43:16,560 --> 00:43:18,960
All shells are moving
apart initially.

830
00:43:18,960 --> 00:43:21,160
This could be turned
around by the forces--

831
00:43:21,160 --> 00:43:22,670
and we'll have to see.

832
00:43:22,670 --> 00:43:25,720
But what we can do is, we
can write down equations

833
00:43:25,720 --> 00:43:29,550
that we know will hold, at
least until the time where there

834
00:43:29,550 --> 00:43:31,542
might be some shell crossings.

835
00:43:31,542 --> 00:43:33,000
That is, we'll
write down equations

836
00:43:33,000 --> 00:43:36,580
that will hold as long as
there are no shell crossings.

837
00:43:36,580 --> 00:43:38,770
Then, if there was going
to be a shell crossing,

838
00:43:38,770 --> 00:43:39,880
the equations we
write down would

839
00:43:39,880 --> 00:43:41,546
have to be valid right
up until the time

840
00:43:41,546 --> 00:43:42,642
of that shell crossing.

841
00:43:42,642 --> 00:43:44,850
And, therefore, the equations
that we're writing down

842
00:43:44,850 --> 00:43:47,566
would have to show us that
the shells are going to cross.

843
00:43:47,566 --> 00:43:49,690
The shells can't just decide
to cross independently

844
00:43:49,690 --> 00:43:51,991
of our equations of motion.

845
00:43:51,991 --> 00:43:54,240
And what we'll find is that
our equations will tell us

846
00:43:54,240 --> 00:43:56,042
that there will be
no shell crossings.

847
00:43:56,042 --> 00:43:57,500
And the equations
are valid as long

848
00:43:57,500 --> 00:43:58,710
as there are no shell crossings.

849
00:43:58,710 --> 00:44:00,501
And I think if you
think about that, that's

850
00:44:00,501 --> 00:44:02,852
an airtight argument,
even though we're never

851
00:44:02,852 --> 00:44:04,810
going to write down
equations that will tell us

852
00:44:04,810 --> 00:44:09,220
what would happen
if shells did cross.

853
00:44:09,220 --> 00:44:10,260
OK.

854
00:44:10,260 --> 00:44:21,350
So we're going to
write equations

855
00:44:21,350 --> 00:44:33,200
that hold as long as there
are no shell crossings.

856
00:44:39,791 --> 00:44:40,290
OK.

857
00:44:40,290 --> 00:44:41,873
As long as there's
no shell crossings,

858
00:44:41,873 --> 00:44:44,702
then the total mass
inside of any shell

859
00:44:44,702 --> 00:44:45,660
is independent of time.

860
00:44:45,660 --> 00:44:49,020
It's just the shells
that are inside it.

861
00:44:49,020 --> 00:45:03,760
So the shell at initial radius,
r sub i, even at later times,

862
00:45:03,760 --> 00:45:12,550
feels the force of
the mass inside.

863
00:45:12,550 --> 00:45:15,400
And we can write down a
formula for that mass inside.

864
00:45:15,400 --> 00:45:21,220
The mass inside the shell,
whose initial radius is r sub i,

865
00:45:21,220 --> 00:45:25,920
is just equal to the initial
volume of that sphere, which

866
00:45:25,920 --> 00:45:33,450
is 4 pi over 3 r sub i
cubed, times the initial mass

867
00:45:33,450 --> 00:45:34,470
density, rho sub i.

868
00:45:36,835 --> 00:45:38,460
So that's how much
mass there is inside

869
00:45:38,460 --> 00:45:41,170
a given shell when
the system starts.

870
00:45:41,170 --> 00:45:43,910
And that will continue to be
exactly how much mass is inside

871
00:45:43,910 --> 00:45:45,930
that shell as the
system evolves,

872
00:45:45,930 --> 00:45:47,546
unless there are
shell crossings.

873
00:45:47,546 --> 00:45:51,440
And our goal is simply to write
down equations that are valid

874
00:45:51,440 --> 00:45:53,702
until there might
be a shell crossing.

875
00:45:53,702 --> 00:45:55,160
And then, we can
ask whether or not

876
00:45:55,160 --> 00:45:56,602
there will be any
shell crossings.

877
00:46:15,320 --> 00:46:17,240
OK, now, Newton's
law tells us how

878
00:46:17,240 --> 00:46:21,400
to write down the
acceleration of an arbitrary

879
00:46:21,400 --> 00:46:23,650
particle in this system.

880
00:46:23,650 --> 00:46:27,590
Newton's law tells
us the acceleration

881
00:46:27,590 --> 00:46:29,790
will be proportional
to negative-- I'll

882
00:46:29,790 --> 00:46:31,960
put a radial vector
at the end in r hat.

883
00:46:31,960 --> 00:46:33,800
So it's direction,
minus r hat, will

884
00:46:33,800 --> 00:46:36,780
be the direction
enforced on any particle.

885
00:46:36,780 --> 00:46:41,340
And the magnitude, it will
be Newton's constant times

886
00:46:41,340 --> 00:46:46,730
the mass enclosed in the
sphere of radius, r sub i,

887
00:46:46,730 --> 00:46:49,060
divided by the square of
the distance of that shell

888
00:46:49,060 --> 00:46:50,070
from the origin.

889
00:46:50,070 --> 00:46:53,840
And that's exactly what we
called this function, r of ri,

890
00:46:53,840 --> 00:46:54,760
t.

891
00:46:54,760 --> 00:46:58,540
It's the radius of the
shell at any given time.

892
00:46:58,540 --> 00:47:05,725
So it's r squared
of r sub i and t.

893
00:47:05,725 --> 00:47:08,100
And then as I promised you,
there's a unit vector, r hat,

894
00:47:08,100 --> 00:47:09,120
at the end.

895
00:47:09,120 --> 00:47:12,140
So the force is-- and the
acceleration, therefore,--

896
00:47:12,140 --> 00:47:14,940
is in the negative
r hat direction.

897
00:47:14,940 --> 00:47:16,847
So this holds for any
shell, whether which

898
00:47:16,847 --> 00:47:19,180
shell we're talking about is
indicated by this variable,

899
00:47:19,180 --> 00:47:20,510
r sub i.

900
00:47:20,510 --> 00:47:23,220
r sub i tells us the initial
position of that shell.

901
00:47:28,822 --> 00:47:30,530
So is everybody happy
with this equation?

902
00:47:30,530 --> 00:47:31,440
This is really
the crucial thing.

903
00:47:31,440 --> 00:47:33,148
Once you write this
down, everything else

904
00:47:33,148 --> 00:47:35,390
is really just chugging along.

905
00:47:35,390 --> 00:47:38,460
So everybody happy with that?

906
00:47:38,460 --> 00:47:40,720
It's just the
statement from Newton,

907
00:47:40,720 --> 00:47:43,890
that if all the masses
are ranged spherically

908
00:47:43,890 --> 00:47:50,990
symmetrically, the mass
inside any shell-- excuse me,

909
00:47:50,990 --> 00:47:55,380
the force due to the mass that's
on a shell that's at a larger

910
00:47:55,380 --> 00:47:59,010
radius than you are, produces
no acceleration for you.

911
00:47:59,010 --> 00:48:00,470
The only acceleration
you feel is

912
00:48:00,470 --> 00:48:02,990
due to the masses
at smaller radii.

913
00:48:02,990 --> 00:48:04,730
And that's what
this formula says.

914
00:48:04,730 --> 00:48:05,860
OK, good.

915
00:48:05,860 --> 00:48:08,160
Now, what I want to do is--
this is a vector equation--

916
00:48:08,160 --> 00:48:10,870
but we know that we're just
taking these spherical motions.

917
00:48:10,870 --> 00:48:12,370
And all we really
have to do is keep

918
00:48:12,370 --> 00:48:14,964
track of how r changes
with time, little r.

919
00:48:14,964 --> 00:48:17,380
So we can turn this into an
ordinary differential equation

920
00:48:17,380 --> 00:48:21,780
with no vectors for
this function, little r.

921
00:48:21,780 --> 00:48:24,860
The acceleration has a
magnitude, which is just,

922
00:48:24,860 --> 00:48:27,270
r double dot.

923
00:48:27,270 --> 00:48:33,871
And that is then
equal to minus--

924
00:48:33,871 --> 00:48:38,940
and we write again what M
of ri is from this formula.

925
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

926
00:48:45,490 --> 00:48:53,620
i cubed rho sub i-- coming
from that formula-- divided

927
00:48:53,620 --> 00:48:55,250
by r squared.

928
00:48:55,250 --> 00:48:58,070
And I'm going to stop
writing the arguments.

929
00:48:58,070 --> 00:49:03,170
But this r is the
function of r sub i and t.

930
00:49:03,170 --> 00:49:05,080
But I will stop
writing its arguments.

931
00:49:05,080 --> 00:49:07,220
And the two dots means
derivative with respect

932
00:49:07,220 --> 00:49:08,010
to time.

933
00:49:08,010 --> 00:49:08,510
Yes.

934
00:49:08,510 --> 00:49:11,408
AUDIENCE: I guess I don't
understand how come we would,

935
00:49:11,408 --> 00:49:14,789
calculating the mass,
we use r sub i, which

936
00:49:14,789 --> 00:49:18,130
is the radius at some
initial time, ti.

937
00:49:18,130 --> 00:49:20,060
But then, when we're
saying the distance

938
00:49:20,060 --> 00:49:24,310
from that mass to our point,
we use this function, r,

939
00:49:24,310 --> 00:49:25,852
instead of r sub i.

940
00:49:25,852 --> 00:49:26,560
PROFESSOR: Right.

941
00:49:26,560 --> 00:49:27,850
An important question.

942
00:49:27,850 --> 00:49:32,500
The reason is that we're
using the initial radius,

943
00:49:32,500 --> 00:49:34,800
but we're also using
the initial density.

944
00:49:34,800 --> 00:49:36,240
So this formula
certainly gives us

945
00:49:36,240 --> 00:49:39,804
the mass that's inside that
circle at the beginning.

946
00:49:39,804 --> 00:49:41,470
And then, if there's
no shell crossings,

947
00:49:41,470 --> 00:49:43,080
all these circles move together.

948
00:49:43,080 --> 00:49:44,800
The mass inside never changes.

949
00:49:44,800 --> 00:49:46,640
So it's the same as it was then.

950
00:49:46,640 --> 00:49:48,640
While, the distance from
the center of that mass

951
00:49:48,640 --> 00:49:50,330
does change, as
the radius changes.

952
00:49:50,330 --> 00:49:53,960
So the denominator changes,
the numerator does not.

953
00:49:57,327 --> 00:49:58,289
Yes.

954
00:49:58,289 --> 00:50:00,213
AUDIENCE: So, the first
equation, g, that's

955
00:50:00,213 --> 00:50:02,620
the gravity--

956
00:50:02,620 --> 00:50:05,048
PROFESSOR: g is the
acceleration of gravity.

957
00:50:05,048 --> 00:50:07,824
AUDIENCE: So, why does it
have-- it has units of-- no,

958
00:50:07,824 --> 00:50:08,324
it doesn't.

959
00:50:08,324 --> 00:50:08,823
Sorry.

960
00:50:08,823 --> 00:50:12,160
PROFESSOR: Yes, it should
have units of acceleration,

961
00:50:12,160 --> 00:50:16,529
as our double dot should
have units of acceleration.

962
00:50:16,529 --> 00:50:17,320
Hopefully, they do.

963
00:50:23,950 --> 00:50:24,450
OK.

964
00:50:24,450 --> 00:50:27,560
Now, as the system
evolves, r sub

965
00:50:27,560 --> 00:50:30,250
i is just a constant--
different for each shell,

966
00:50:30,250 --> 00:50:32,220
but a constant in time.

967
00:50:32,220 --> 00:50:35,270
And imagine solving this
one shell at a time.

968
00:50:35,270 --> 00:50:37,070
Rho sub i is, again, a constant.

969
00:50:37,070 --> 00:50:40,790
It's the value of the mass
density at the initial time.

970
00:50:40,790 --> 00:50:43,024
So it's going to keep its value.

971
00:50:43,024 --> 00:50:45,190
So the differential equation
involves a differential

972
00:50:45,190 --> 00:50:47,570
equation in which little r
changes with time, and nothing

973
00:50:47,570 --> 00:50:48,630
else here does.

974
00:50:48,630 --> 00:50:51,450
So this is just a second
order differential equation

975
00:50:51,450 --> 00:50:53,542
for little r.

976
00:50:53,542 --> 00:50:55,500
So, we're well on the
way to having a solution,

977
00:50:55,500 --> 00:50:57,530
because second order
differential equations

978
00:50:57,530 --> 00:50:59,880
are, in principle, solvable.

979
00:50:59,880 --> 00:51:02,560
But the one thing
that you should all

980
00:51:02,560 --> 00:51:05,090
remember about second order
differential equations

981
00:51:05,090 --> 00:51:07,650
is that in order to
have a unique solution,

982
00:51:07,650 --> 00:51:09,489
you need initial conditions.

983
00:51:09,489 --> 00:51:11,030
And if it's a second
order equation--

984
00:51:11,030 --> 00:51:13,910
as Newton's equations
always are-- you

985
00:51:13,910 --> 00:51:15,880
need to be able to specify
the initial position

986
00:51:15,880 --> 00:51:18,440
and the initial velocity
before the second order

987
00:51:18,440 --> 00:51:20,184
equation leads to
a unique answer.

988
00:51:20,184 --> 00:51:21,600
So that's what we
want to do next.

989
00:51:21,600 --> 00:51:24,090
We want to write down
the initial value

990
00:51:24,090 --> 00:51:26,226
of r and the initial
value of r dot.

991
00:51:26,226 --> 00:51:28,030
And then, we'll
have a system, which

992
00:51:28,030 --> 00:51:31,362
is just something we can
turn over to a mathematician.

993
00:51:31,362 --> 00:51:33,070
And if the mathematician
is at all smart,

994
00:51:33,070 --> 00:51:35,390
he'll be able to solve it.

995
00:51:35,390 --> 00:51:36,665
So initial conditions.

996
00:52:04,210 --> 00:52:06,954
So we want the initial
value of r sub i--

997
00:52:06,954 --> 00:52:08,370
and initial means
a time, t sub i.

998
00:52:08,370 --> 00:52:11,180
That's where we're setting
our initial conditions.

999
00:52:11,180 --> 00:52:13,020
So we need to know
what r of r sub i,

1000
00:52:13,020 --> 00:52:15,100
t sub i is, which
is a trivial thing

1001
00:52:15,100 --> 00:52:18,490
if you keep track of
what this notation means.

1002
00:52:18,490 --> 00:52:21,330
What is that?

1003
00:52:21,330 --> 00:52:22,295
r sub i, good.

1004
00:52:28,000 --> 00:52:31,160
The meaning of this is just
the radius at time, t sub i,

1005
00:52:31,160 --> 00:52:34,310
of the particle, whose radius
at time, t sub i, was r sub i.

1006
00:52:34,310 --> 00:52:36,900
So to put together
all those tautologies,

1007
00:52:36,900 --> 00:52:40,430
the answer is just,
obviously, r sub i.

1008
00:52:40,430 --> 00:52:40,930
OK.

1009
00:52:40,930 --> 00:52:43,875
And then we also want to be
able to solve this equation

1010
00:52:43,875 --> 00:52:46,500
and have a unique solution--
we want the initial value of r

1011
00:52:46,500 --> 00:52:47,790
sub i dot.

1012
00:52:47,790 --> 00:52:51,500
And initial means,
again, at time, t sub i.

1013
00:52:51,500 --> 00:52:56,120
And that's specified by
our Hubble expansion.

1014
00:52:56,120 --> 00:52:59,825
Every initial velocity is
just H sub i times the radius.

1015
00:53:03,190 --> 00:53:05,230
So if the radius
is r sub i, this

1016
00:53:05,230 --> 00:53:08,281
is just H sub i times r sub i.

1017
00:53:08,281 --> 00:53:10,780
That's just the Hubble velocity
that we put in to the system

1018
00:53:10,780 --> 00:53:11,571
when we started it.

1019
00:53:16,390 --> 00:53:16,890
OK.

1020
00:53:16,890 --> 00:53:19,640
So now, we have a system,
which is purely mathematical.

1021
00:53:19,640 --> 00:53:21,680
We have a second order
differential equation

1022
00:53:21,680 --> 00:53:24,216
and initial conditions
on r and r dot.

1023
00:53:24,216 --> 00:53:26,810
That leads to a unique solution.

1024
00:53:26,810 --> 00:53:28,000
Now it's pure math.

1025
00:53:28,000 --> 00:53:31,690
No more physics, at least
at this stage of the game.

1026
00:53:31,690 --> 00:53:33,464
However, there are
interesting things

1027
00:53:33,464 --> 00:53:34,880
mathematically
that one can notice

1028
00:53:34,880 --> 00:53:36,920
about this system of equations.

1029
00:53:36,920 --> 00:53:40,240
And now what we're going
to see is the magic

1030
00:53:40,240 --> 00:53:42,490
of these equations
for preserving

1031
00:53:42,490 --> 00:53:45,220
the uniformity of this system.

1032
00:53:45,220 --> 00:53:47,390
It's all built into
these equations.

1033
00:53:47,390 --> 00:53:49,390
And let's see how.

1034
00:53:49,390 --> 00:53:51,890
The key feature that's
somewhat miraculous

1035
00:53:51,890 --> 00:54:01,940
that these equations
have is that r

1036
00:54:01,940 --> 00:54:11,350
sub i can be made to disappear
by a change of variables.

1037
00:54:11,350 --> 00:54:14,690
I'll tell you what
that is in a second.

1038
00:54:14,690 --> 00:54:27,570
It also helps if you know
how to spell "disappear."

1039
00:54:27,570 --> 00:54:30,560
We're going to define
a new function, u.

1040
00:54:30,560 --> 00:54:31,880
I just made up a letter there.

1041
00:54:31,880 --> 00:54:33,130
Could have called it anything.

1042
00:54:33,130 --> 00:54:36,470
But u of r sub i
and t is just going

1043
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.

1044
00:54:49,660 --> 00:54:52,232
Given any function,
r of r sub i and t,

1045
00:54:52,232 --> 00:54:53,690
I can always define
a new function,

1046
00:54:53,690 --> 00:54:56,179
which is just the same
function divided by r sub i.

1047
00:55:02,420 --> 00:55:04,545
Now, let's look at what
this does to our equations.

1048
00:55:08,090 --> 00:55:11,970
My claim is that r sub i
was going to disappear.

1049
00:55:11,970 --> 00:55:13,470
And now, we'll see
how that happens.

1050
00:55:38,500 --> 00:55:43,160
OK, if u is defined this way, we
can write down the differential

1051
00:55:43,160 --> 00:55:45,490
equation that it will
obey, by writing down

1052
00:55:45,490 --> 00:55:47,840
an equation for u double dot.

1053
00:55:47,840 --> 00:55:54,876
And u double dot will just be r
double dot divided by r sub i.

1054
00:55:57,520 --> 00:56:02,130
So we take the equation for r
double dot over here and divide

1055
00:56:02,130 --> 00:56:05,690
it by r sub i .

1056
00:56:05,690 --> 00:56:14,390
So we can write that as minus 4
pi over 3 G-- for some reason,

1057
00:56:14,390 --> 00:56:15,995
I decided to leave
the r sub i cubed

1058
00:56:15,995 --> 00:56:19,120
in the numerator for now--
and just put an extra r sub

1059
00:56:19,120 --> 00:56:22,770
i in the denominator-- the one
that we divided by-- times r

1060
00:56:22,770 --> 00:56:23,270
squared.

1061
00:56:34,429 --> 00:56:36,470
And now, what I'm going
to do is just replace r .

1062
00:56:36,470 --> 00:56:38,406
We're trying to write
an equation for u.

1063
00:56:38,406 --> 00:56:41,850
r is related to u by this
equation. r is r sub i times u.

1064
00:56:44,610 --> 00:56:54,550
So I can write this as
minus 4 pi over 3 G r sub i

1065
00:56:54,550 --> 00:56:59,470
cubed rho sub i
over-- now we have-- u

1066
00:56:59,470 --> 00:57:02,170
squared times r sub i cubed.

1067
00:57:02,170 --> 00:57:03,570
I left the numerator alone.

1068
00:57:03,570 --> 00:57:05,890
I just rewrote the
denominator by placing

1069
00:57:05,890 --> 00:57:07,820
r by r sub i times u.

1070
00:57:10,630 --> 00:57:12,010
And we get this formula.

1071
00:57:12,010 --> 00:57:15,439
And now, of course, the
r sub i cubes cancel.

1072
00:57:15,439 --> 00:57:17,730
And I think the reason why
I kept it separated this way

1073
00:57:17,730 --> 00:57:19,870
when I wrote my notes
is at this shows very

1074
00:57:19,870 --> 00:57:23,400
explicitly that what you have
is a cancellation between an r

1075
00:57:23,400 --> 00:57:28,850
sub i here, which was the power
of r that appears in a volume--

1076
00:57:28,850 --> 00:57:31,670
r sub i cubed was just
proportional to the volume

1077
00:57:31,670 --> 00:57:35,429
of the sphere-- and r
sub i cubed down here,

1078
00:57:35,429 --> 00:57:37,720
where one of them came from
just a change of variables,

1079
00:57:37,720 --> 00:57:41,117
and the other was the r squared
that appeared in the force law.

1080
00:57:41,117 --> 00:57:43,450
So this cancellation depends
crucially on the force law,

1081
00:57:43,450 --> 00:57:45,380
being a 1 over r
squared force law.

1082
00:57:45,380 --> 00:57:47,450
If we had 1 over some
other power of r--

1083
00:57:47,450 --> 00:57:50,320
even if it differed by
just a little bit-- then

1084
00:57:50,320 --> 00:57:52,499
the r sub i would not
drop out of this formula.

1085
00:57:52,499 --> 00:57:55,040
And we'll see in a minute that
it's the dropping out of r sub

1086
00:57:55,040 --> 00:57:58,920
i which is crucial to the
maintenance of homogeneity

1087
00:57:58,920 --> 00:58:00,720
as the system evolves.

1088
00:58:00,720 --> 00:58:02,890
As it evolves, Newton
tells us what happens.

1089
00:58:02,890 --> 00:58:05,720
We don't get to make
any further choices.

1090
00:58:05,720 --> 00:58:08,310
And if Newton is
telling us there's a 1

1091
00:58:08,310 --> 00:58:11,365
over r squared force law,
then it remains homogeneous.

1092
00:58:11,365 --> 00:58:13,390
But otherwise, it would not.

1093
00:58:13,390 --> 00:58:15,870
And that's, I think, a
very interesting fact.

1094
00:58:15,870 --> 00:58:18,140
Continuing, we've now
crossed off these r sub i's.

1095
00:58:18,140 --> 00:58:20,030
And we get now a
simple equation,

1096
00:58:20,030 --> 00:58:28,970
u double dot is equal to
minus 4 pi over 3 G rho sub i

1097
00:58:28,970 --> 00:58:34,990
over u squared with no r sub
i's in the equation anymore.

1098
00:58:34,990 --> 00:58:36,900
And that means that
this u tells us

1099
00:58:36,900 --> 00:58:39,406
the solution for every r sub i.

1100
00:58:39,406 --> 00:58:40,780
We don't have
different solutions

1101
00:58:40,780 --> 00:58:42,540
for every value of
r sub i anymore.

1102
00:58:42,540 --> 00:58:44,460
r sub i has dropped
out of the problem.

1103
00:58:44,460 --> 00:58:47,230
We have a unique solution
independent of r sub i.

1104
00:58:47,230 --> 00:58:51,771
And that means it
holds for all r sub i.

1105
00:58:51,771 --> 00:58:52,270
I'm sorry.

1106
00:58:52,270 --> 00:58:53,520
I jumped the gun.

1107
00:58:56,912 --> 00:58:58,870
The conclusions I just
told you about are true,

1108
00:58:58,870 --> 00:59:01,175
but part of the logic
I haven't done yet.

1109
00:59:01,175 --> 00:59:02,800
What part of the
logic did I leave out?

1110
00:59:05,567 --> 00:59:07,400
Initial conditions, I
heard somebody mumble.

1111
00:59:07,400 --> 00:59:07,900
Yes.

1112
00:59:07,900 --> 00:59:09,005
Exactly.

1113
00:59:09,005 --> 00:59:10,380
To get the same
solution, we have

1114
00:59:10,380 --> 00:59:12,430
to not only have a
differential equation that's

1115
00:59:12,430 --> 00:59:15,112
independent of u sub i, but we
don't have a unique solution

1116
00:59:15,112 --> 00:59:16,820
unless we look at the
initial conditions.

1117
00:59:16,820 --> 00:59:19,604
We better see if the initial
conditions depend on u sub i.

1118
00:59:19,604 --> 00:59:20,520
And they don't either.

1119
00:59:20,520 --> 00:59:22,450
That's the beauty of it all.

1120
00:59:22,450 --> 00:59:30,130
Starting with the r initial
condition, the initial value

1121
00:59:30,130 --> 00:59:34,910
of u of r sub i,
t sub i, will just

1122
00:59:34,910 --> 00:59:39,120
be equal to the initial value
of r divided by r sub i.

1123
00:59:39,120 --> 00:59:42,414
But the initial value
of r is r sub i.

1124
00:59:42,414 --> 00:59:46,090
So here, we get r sub i
divided by r sub i, which

1125
00:59:46,090 --> 00:59:48,540
is 1, independent of r sub i.

1126
00:59:48,540 --> 00:59:53,190
The initial value of u
is 1, for any r sub i.

1127
00:59:53,190 --> 01:00:01,750
And similarly, we now want to
look at u dot of r sub i and t.

1128
01:00:01,750 --> 01:00:05,450
And that will just be r dot
divided by r sub i, where

1129
01:00:05,450 --> 01:00:08,092
we take the initial
value of r dot.

1130
01:00:08,092 --> 01:00:11,015
The initial value of r
dot is Hi times r sub i.

1131
01:00:11,015 --> 01:00:12,940
The r sub i's cancel.

1132
01:00:12,940 --> 01:00:15,898
And u sub i dot is just H sub i.

1133
01:00:19,180 --> 01:00:24,280
So now, we have
justified the claim

1134
01:00:24,280 --> 01:00:27,611
that I falsely made a little
bit prematurely a minute ago.

1135
01:00:27,611 --> 01:00:29,110
We have a system
of equations, which

1136
01:00:29,110 --> 01:00:32,350
can be solved to give us
a solution for u, which

1137
01:00:32,350 --> 01:00:35,870
is independent of r sub i, which
means every value of r sub i

1138
01:00:35,870 --> 01:00:39,820
has the same equation for u.

1139
01:00:39,820 --> 01:00:43,960
That-- if you stare
at it a little bit--

1140
01:00:43,960 --> 01:00:45,650
gives us a physical
interpretation

1141
01:00:45,650 --> 01:00:47,510
for this quantity, u.

1142
01:00:53,960 --> 01:01:00,090
u, in fact, is nothing more
nor less than the scale factor

1143
01:01:00,090 --> 01:01:02,330
that we spoke about earlier.

1144
01:01:02,330 --> 01:01:06,127
We have found that we do have a
homogeneously expanding system.

1145
01:01:06,127 --> 01:01:07,710
We started it
homogeneously expanding,

1146
01:01:07,710 --> 01:01:10,043
but we didn't know until we
looked at equation of motion

1147
01:01:10,043 --> 01:01:12,440
that it would continue
to homogeneously expand.

1148
01:01:12,440 --> 01:01:14,380
But it does.

1149
01:01:14,380 --> 01:01:16,840
And that means it can be
described by a scale factor.

1150
01:01:21,730 --> 01:01:25,130
We have u of r sub i, t.

1151
01:01:25,130 --> 01:01:27,150
In principle, it
was defined so it

1152
01:01:27,150 --> 01:01:28,629
might have depended on r sub i.

1153
01:01:28,629 --> 01:01:30,920
That's why I've been writing
these r sub i's all along.

1154
01:01:30,920 --> 01:01:33,270
But now, we discovered
that the u's are completely

1155
01:01:33,270 --> 01:01:35,270
determined by these
equations, which

1156
01:01:35,270 --> 01:01:38,197
have no r sub i's in them--
at least none that survived.

1157
01:01:38,197 --> 01:01:39,780
We had an r sub i
divided by r sub it,

1158
01:01:39,780 --> 01:01:42,510
but that's not really
an r sub i, as you know.

1159
01:01:42,510 --> 01:01:44,100
It's 1.

1160
01:01:44,100 --> 01:01:46,377
So u is independent
of r sub i and can

1161
01:01:46,377 --> 01:01:47,960
be thought of as
just a function of t.

1162
01:01:50,840 --> 01:01:54,180
And we can also change
its name to a of t

1163
01:01:54,180 --> 01:01:56,870
to make contact with the
notion of a scale factor

1164
01:01:56,870 --> 01:02:00,010
that we discussed last time.

1165
01:02:00,010 --> 01:02:02,600
And we can see that
the way u arises,

1166
01:02:02,600 --> 01:02:09,570
in terms of how it describes
the motion, r of r sub i

1167
01:02:09,570 --> 01:02:16,040
and t is just equal to--
from this equation-- r sub

1168
01:02:16,040 --> 01:02:19,980
i times u, which
is now called a.

1169
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.

1170
01:02:28,420 --> 01:02:30,565
That's just another way of
rewriting our definition

1171
01:02:30,565 --> 01:02:32,860
that we started with of u.

1172
01:02:32,860 --> 01:02:34,600
So what does that mean?

1173
01:02:34,600 --> 01:02:38,810
These r sub i's are
comoving coordinates.

1174
01:02:38,810 --> 01:02:41,650
We've labeled each shell by
its initial position, r sub i.

1175
01:02:41,650 --> 01:02:43,940
And we've let that shell
keep its label, r sub i,

1176
01:02:43,940 --> 01:02:45,050
as it evolved.

1177
01:02:45,050 --> 01:02:46,820
That's a comoving coordinate.

1178
01:02:46,820 --> 01:02:50,220
It labels the particles
independent of where they move.

1179
01:02:50,220 --> 01:02:57,410
And what this is telling us is
that the physical distance--

1180
01:02:57,410 --> 01:03:08,260
from the origin
in this case-- is

1181
01:03:08,260 --> 01:03:16,137
equal to the scale factor
times the coordinate distance.

1182
01:03:23,909 --> 01:03:25,950
That is the coordinate
distance between the shell

1183
01:03:25,950 --> 01:03:28,140
and the center of the
system, the origin.

1184
01:03:31,830 --> 01:03:32,330
OK.

1185
01:03:32,330 --> 01:03:33,480
Any questions about that?

1186
01:03:42,840 --> 01:03:43,340
OK.

1187
01:03:43,340 --> 01:03:45,550
It's now useful to
rewrite these equations

1188
01:03:45,550 --> 01:03:46,580
in a few different ways.

1189
01:04:05,980 --> 01:04:06,480
Let's see.

1190
01:04:06,480 --> 01:04:14,200
First of all, we have written
our differential equation,

1191
01:04:14,200 --> 01:04:17,540
up here, in terms of rho sub i.

1192
01:04:17,540 --> 01:04:19,915
And that was very useful
because rho sub i is a constant.

1193
01:04:19,915 --> 01:04:21,695
It doesn't change with time.

1194
01:04:21,695 --> 01:04:24,320
Still, it's sometimes useful to
write the differential equation

1195
01:04:24,320 --> 01:04:26,180
in terms of the temporary
value of rho, which

1196
01:04:26,180 --> 01:04:28,346
changes with time, to see
what the relationships are

1197
01:04:28,346 --> 01:04:30,940
between physical
quantities at a given time.

1198
01:04:30,940 --> 01:04:33,670
And we could certainly
do that, because we

1199
01:04:33,670 --> 01:04:37,495
know what the density
will be at any given time.

1200
01:04:41,710 --> 01:04:45,720
For any shell, we can
calculate the density

1201
01:04:45,720 --> 01:04:47,705
as the total mass
divided by the volume.

1202
01:04:47,705 --> 01:04:50,080
We know this is going to stay
uniform, because everything

1203
01:04:50,080 --> 01:04:53,250
is moving together, when
you just have motion, which

1204
01:04:53,250 --> 01:04:56,710
is an overall scale factor
that multiplies all distances.

1205
01:04:56,710 --> 01:04:58,540
So the density will be constant.

1206
01:04:58,540 --> 01:05:02,880
And we can calculate the
density inside a shell

1207
01:05:02,880 --> 01:05:05,840
by taking M of r sub i--
which we already have formulas

1208
01:05:05,840 --> 01:05:09,590
for and is independent
of time-- and dividing it

1209
01:05:09,590 --> 01:05:13,045
by the volume inside the
shell, which is 4 pi r cubed.

1210
01:05:16,880 --> 01:05:21,250
And doing some
substitutions, M of r sub i

1211
01:05:21,250 --> 01:05:26,140
is just the initial volume
times the initial density.

1212
01:05:26,140 --> 01:05:29,350
So that is-- as we've
written before-- 4

1213
01:05:29,350 --> 01:05:37,040
pi over 3 r sub i
cubed times rho sub i.

1214
01:05:37,040 --> 01:05:43,740
And then, we can
write the denominator.

1215
01:05:43,740 --> 01:05:48,726
r is equal to a times r sub i.

1216
01:05:48,726 --> 01:05:50,350
The physical radius
is the scale factor

1217
01:05:50,350 --> 01:05:52,340
times the coordinate radius.

1218
01:05:52,340 --> 01:05:56,650
So here, we have a cubed
times r sub i cubed.

1219
01:05:56,650 --> 01:05:59,660
And now, notice that
almost everything cancels.

1220
01:05:59,660 --> 01:06:04,540
And what we're left with is just
rho sub i divided by a cubed.

1221
01:06:14,970 --> 01:06:17,230
So that's certainly what
we would've guess, I think.

1222
01:06:17,230 --> 01:06:20,680
The density is just
what it started but then

1223
01:06:20,680 --> 01:06:23,640
divided by the cube
of the scale factor.

1224
01:06:23,640 --> 01:06:25,460
And our scale factor
is defined as 1

1225
01:06:25,460 --> 01:06:28,400
at the initial time-- the way
we've set up these conventions.

1226
01:06:28,400 --> 01:06:31,842
So that is just the ratio
of the scale factors cubed

1227
01:06:31,842 --> 01:06:34,210
that appears in that equation.

1228
01:06:34,210 --> 01:06:36,410
As the universe
expands, the density

1229
01:06:36,410 --> 01:06:40,430
falls off by 1 over
the scale factor cubed.

1230
01:06:40,430 --> 01:06:45,960
We can also now rewrite the
equation for a double dot.

1231
01:06:45,960 --> 01:06:48,320
a is u, so we have
the equation up there.

1232
01:06:48,320 --> 01:06:53,440
But we could write it in terms
of the current mass density.

1233
01:06:53,440 --> 01:06:56,080
Starting with what
we have up there,

1234
01:06:56,080 --> 01:07:03,940
it's minus 4 pi over 3 G
rho sub i over a squared.

1235
01:07:08,690 --> 01:07:13,600
Notice, that I can multiply
numerator and denominator by a.

1236
01:07:16,200 --> 01:07:18,710
And then, we have rho
sub i over a cubed

1237
01:07:18,710 --> 01:07:24,760
here, which is just the mass
density in any given time.

1238
01:07:24,760 --> 01:07:27,180
So I can make that substitution.

1239
01:07:27,180 --> 01:07:31,020
And I get the
meaningful equation

1240
01:07:31,020 --> 01:07:37,930
that a double dot is equal
to minus 4 pi over 3 G

1241
01:07:37,930 --> 01:07:41,990
rho of t times a.

1242
01:07:47,220 --> 01:07:49,500
So this equation
gives the deceleration

1243
01:07:49,500 --> 01:07:53,396
of our model universe in terms
of the current mass density.

1244
01:07:53,396 --> 01:07:55,270
And notice that it does,
in fact, depend only

1245
01:07:55,270 --> 01:07:57,650
on the current mass density.

1246
01:07:57,650 --> 01:08:00,420
It predicts the ratio
of a double dot over a.

1247
01:08:00,420 --> 01:08:03,310
And that, you would expect to
be what should be predicted,

1248
01:08:03,310 --> 01:08:07,821
because remember, a is still
measured in notches per meter.

1249
01:08:07,821 --> 01:08:09,570
So the only way the
notches can cancel out

1250
01:08:09,570 --> 01:08:11,030
is if there is an
a on both sides.

1251
01:08:11,030 --> 01:08:13,215
Or if you could bring
them all to one side

1252
01:08:13,215 --> 01:08:14,910
and have a double
dot divided by a.

1253
01:08:14,910 --> 01:08:16,375
And then, the notches go away.

1254
01:08:16,375 --> 01:08:18,640
And you have something which
has physical units being related

1255
01:08:18,640 --> 01:08:20,050
to something with
physical units.

1256
01:08:29,340 --> 01:08:31,319
OK.

1257
01:08:31,319 --> 01:08:33,810
We said at the beginning
that when we were finished,

1258
01:08:33,810 --> 01:08:37,250
we were going to take the
limit as R max initial goes

1259
01:08:37,250 --> 01:08:38,727
to infinity.

1260
01:08:38,727 --> 01:08:40,310
And lots of times
when I present this,

1261
01:08:40,310 --> 01:08:41,950
I forget to talk about that.

1262
01:08:41,950 --> 01:08:44,870
And the reason I forget to talk
about that is, if you notice,

1263
01:08:44,870 --> 01:08:48,310
R max sub i doesn't appear
in any of these equations.

1264
01:08:48,310 --> 01:08:51,380
So taking the limit as R
max sub i goes to infinity

1265
01:08:51,380 --> 01:08:53,239
doesn't actually
involve doing anything.

1266
01:08:53,239 --> 01:08:56,580
It really just involves pointing
out that the answers we got

1267
01:08:56,580 --> 01:08:58,600
are independent of
how big the sphere is,

1268
01:08:58,600 --> 01:09:00,141
as long as everything
we want to talk

1269
01:09:00,141 --> 01:09:01,899
about fits inside the sphere.

1270
01:09:01,899 --> 01:09:03,359
Adding extra matter
on the outside

1271
01:09:03,359 --> 01:09:04,682
doesn't change anything at all.

1272
01:09:04,682 --> 01:09:07,140
So taking the limit as you add
an infinite amount of matter

1273
01:09:07,140 --> 01:09:09,402
on the outside-- as long
as you imagine doing it

1274
01:09:09,402 --> 01:09:12,399
in these spherical shells--
is a trivial matter.

1275
01:09:12,399 --> 01:09:14,979
So the limit as R max
sub i goes to infinity

1276
01:09:14,979 --> 01:09:17,090
is done without any work.

1277
01:09:27,649 --> 01:09:28,149
OK.

1278
01:09:28,149 --> 01:09:30,637
I would like to go ahead
now, and in the end,

1279
01:09:30,637 --> 01:09:32,720
we're going to want to
think about different kinds

1280
01:09:32,720 --> 01:09:35,720
of solutions to this equation
and what they look like.

1281
01:09:35,720 --> 01:09:38,210
For today, I want to
take one more step, which

1282
01:09:38,210 --> 01:09:41,630
is to rewrite this equation in
a slightly different way, which

1283
01:09:41,630 --> 01:09:45,012
will help us to see what
the solutions look what.

1284
01:09:45,012 --> 01:09:48,086
What I want to do is to
find a first integral

1285
01:09:48,086 --> 01:09:48,794
of this equation.

1286
01:10:10,060 --> 01:10:10,740
OK.

1287
01:10:10,740 --> 01:10:13,860
To find a first integral, I'm
going to go back to the form

1288
01:10:13,860 --> 01:10:17,827
that we had on the top
there, where everything

1289
01:10:17,827 --> 01:10:20,404
is expressed in terms of
rho sub i rather than rho.

1290
01:10:20,404 --> 01:10:22,320
And the advantage of
that for current purposes

1291
01:10:22,320 --> 01:10:25,030
is that I really want to look at
the time dependence of things.

1292
01:10:25,030 --> 01:10:26,700
And rho has its own
time dependence,

1293
01:10:26,700 --> 01:10:29,100
which I don't want
to worry about.

1294
01:10:29,100 --> 01:10:32,860
So if I look at the formula
in terms of rho sub i,

1295
01:10:32,860 --> 01:10:35,520
all time dependence is explicit.

1296
01:10:35,520 --> 01:10:38,760
So I'm going to write the
differential equation.

1297
01:10:38,760 --> 01:10:40,470
It's the top equation
in the box there,

1298
01:10:40,470 --> 01:10:44,580
but I'm replacing u by a,
because we renamed u as a.

1299
01:10:44,580 --> 01:10:46,510
And I'm going to put
everything on one side.

1300
01:10:46,510 --> 01:10:50,000
So I'm going to write
a double dot plus 4 pi

1301
01:10:50,000 --> 01:10:59,970
over 3 G rho sub i divided
by a squared equals 0.

1302
01:10:59,970 --> 01:11:01,610
OK, that's our
differential equation.

1303
01:11:04,950 --> 01:11:06,950
Now, it's a second order
differential equation,

1304
01:11:06,950 --> 01:11:10,346
like we're very much accustomed
to from Newtonian mechanics--

1305
01:11:10,346 --> 01:11:11,970
as this is an equation
which determines

1306
01:11:11,970 --> 01:11:13,480
a double dot, the
acceleration of a

1307
01:11:13,480 --> 01:11:16,331
in terms of the value of a.

1308
01:11:16,331 --> 01:11:20,740
A common thing to make use
of in Newtonian mechanics

1309
01:11:20,740 --> 01:11:23,236
is conservation of energy.

1310
01:11:23,236 --> 01:11:24,860
In this case, I don't
know if we should

1311
01:11:24,860 --> 01:11:26,526
call this conservation
of energy or not.

1312
01:11:26,526 --> 01:11:29,920
We'll talk later about
what physical significance

1313
01:11:29,920 --> 01:11:32,310
the quantities that
we're dealing with have.

1314
01:11:32,310 --> 01:11:35,310
But certainly, as a
mathematical technique,

1315
01:11:35,310 --> 01:11:36,810
we can do the same
thing that would

1316
01:11:36,810 --> 01:11:39,260
have been done if this were a
Newtonian mechanics problem,

1317
01:11:39,260 --> 01:11:41,960
and somebody asked you to
derive the conserved energy.

1318
01:11:41,960 --> 01:11:44,810
Now, you might have
forgotten how to do that.

1319
01:11:44,810 --> 01:11:46,630
But I'll remind you.

1320
01:11:46,630 --> 01:11:49,570
To get the conserved energy
that goes with this equation,

1321
01:11:49,570 --> 01:11:52,285
you put brackets around it.

1322
01:11:52,285 --> 01:11:53,660
You could choose
whether you want

1323
01:11:53,660 --> 01:11:55,680
curly brackets or
square brackets or just

1324
01:11:55,680 --> 01:11:57,300
ordinary parentheses.

1325
01:11:57,300 --> 01:11:59,870
But then, the important
thing is that it's

1326
01:11:59,870 --> 01:12:01,480
useful to multiply
the entire equation

1327
01:12:01,480 --> 01:12:04,748
by an integrating factor, a dot.

1328
01:12:04,748 --> 01:12:07,620
And once you do that,
this entire expression

1329
01:12:07,620 --> 01:12:09,830
is a total derivative.

1330
01:12:09,830 --> 01:12:14,520
This equation is equivalent
to dE-- for some E

1331
01:12:14,520 --> 01:12:16,050
that I'll define
in a second-- dt

1332
01:12:16,050 --> 01:12:33,320
equals zero, where E equals 1/2
a dot squared minus 4 pi over 3

1333
01:12:33,320 --> 01:12:37,936
G rho sub i over a.

1334
01:12:43,540 --> 01:12:44,630
And you can easily check.

1335
01:12:44,630 --> 01:12:48,216
If I differentiate this, I
get exactly that equation.

1336
01:12:48,216 --> 01:12:51,425
So they're equivalent.

1337
01:12:51,425 --> 01:12:54,907
That is, this is
equivalent to that.

1338
01:12:54,907 --> 01:12:56,115
So E is a conserved quantity.

1339
01:13:06,800 --> 01:13:10,300
Now if one wants to relate this
to the energy of something,

1340
01:13:10,300 --> 01:13:12,980
there are different
ways you can do it.

1341
01:13:12,980 --> 01:13:23,070
One way to do it is to multiply
by an expression, which I'll

1342
01:13:23,070 --> 01:13:25,620
write down in a second,
and to think of it

1343
01:13:25,620 --> 01:13:33,012
as the energy of a test particle
at the surface of this sphere.

1344
01:13:33,012 --> 01:13:34,220
I'll show you how that works.

1345
01:13:48,614 --> 01:13:50,030
I'm going to find
something, which

1346
01:13:50,030 --> 01:13:53,920
I'll call E sub phys--
or physical-- meaning,

1347
01:13:53,920 --> 01:13:56,830
it's the physical energy of
this hypothetical test particle.

1348
01:13:56,830 --> 01:13:59,630
It's not all that physical.

1349
01:13:59,630 --> 01:14:07,170
But it will just be defined to
be m r sub i squared times E.

1350
01:14:07,170 --> 01:14:08,990
m is the mass of
my test particle.

1351
01:14:08,990 --> 01:14:12,025
r sub i is the radius of
that test particle expressed

1352
01:14:12,025 --> 01:14:15,070
in terms of its initial value.

1353
01:14:15,070 --> 01:14:21,430
And then, if we write down
what E phys looks like,

1354
01:14:21,430 --> 01:14:31,170
absorbing these factors, it can
be written as 1/2 m times a dot

1355
01:14:31,170 --> 01:14:50,400
r sub i squared minus GmM of
r sub i divided by a r sub i.

1356
01:14:50,400 --> 01:14:54,490
And that's just some algebra--
absorbed these extra factors

1357
01:14:54,490 --> 01:14:57,220
that I put into the definition.

1358
01:14:57,220 --> 01:14:59,690
And now, if we think of
this as describing a test

1359
01:14:59,690 --> 01:15:03,170
particle, where r sub i is
capital R sub i max-- so we're

1360
01:15:03,170 --> 01:15:05,960
talking about the boundary
of our sphere-- then,

1361
01:15:05,960 --> 01:15:08,640
we can identify what's
being conserved here.

1362
01:15:08,640 --> 01:15:12,960
What's being conserved
here is 1/2 m v squared.

1363
01:15:12,960 --> 01:15:15,640
a dot times r sub i would
just be the velocity

1364
01:15:15,640 --> 01:15:18,520
of the particle of the
boundary of the sphere.

1365
01:15:18,520 --> 01:15:21,810
And then, minus G times
the product of the masses

1366
01:15:21,810 --> 01:15:25,630
divided by the distance between
the particle in the center.

1367
01:15:25,630 --> 01:15:29,150
And that would just be the
Newtonian energy-- kinetic

1368
01:15:29,150 --> 01:15:31,970
energy plus potential energy,
where the potential energy

1369
01:15:31,970 --> 01:15:37,940
is negative-- of a point
particle on the boundary, where

1370
01:15:37,940 --> 01:15:42,120
we let r sub i be
capital R sub i

1371
01:15:42,120 --> 01:15:45,460
max-- the boundary
of the sphere.

1372
01:15:45,460 --> 01:15:48,790
Now, if we want to apply this
to a particle inside the sphere,

1373
01:15:48,790 --> 01:15:52,060
it's a little trickier
to get the words right.

1374
01:15:52,060 --> 01:15:53,920
If the particle is
inside the sphere--

1375
01:15:53,920 --> 01:15:57,170
if r sub i is not
equal to the max--

1376
01:15:57,170 --> 01:16:00,830
this is not really the potential
energy of the particle.

1377
01:16:00,830 --> 01:16:02,100
Can somebody tell me why not?

1378
01:16:13,009 --> 01:16:14,800
Well, maybe the question
is a little vague.

1379
01:16:14,800 --> 01:16:17,540
But if I did want to
calculate the potential energy

1380
01:16:17,540 --> 01:16:24,404
of a particle inside
the sphere-- that's

1381
01:16:24,404 --> 01:16:25,570
meant to be in the interior.

1382
01:16:25,570 --> 01:16:27,778
You can't really tell unless
it's the actual diagram.

1383
01:16:27,778 --> 01:16:30,620
But that's deep inside sphere.

1384
01:16:30,620 --> 01:16:35,559
I would do it by integrating
from infinity G da,

1385
01:16:35,559 --> 01:16:37,100
and ask how much
work do I have to do

1386
01:16:37,100 --> 01:16:41,210
to bring in a particle from
infinity and put it there.

1387
01:16:41,210 --> 01:16:47,570
And in doing this line
integral, I get a contribution

1388
01:16:47,570 --> 01:16:51,470
from the mass that's
inside this dot, which

1389
01:16:51,470 --> 01:16:53,590
is what determines
the force on that dot.

1390
01:16:53,590 --> 01:16:57,510
But I also get a contribution
from what's outside the dot.

1391
01:16:57,510 --> 01:16:59,620
So I don't get-- if
I wanted to calculate

1392
01:16:59,620 --> 01:17:01,530
the actual potential
energy of that point--

1393
01:17:01,530 --> 01:17:05,830
I don't get simply
Gm times the mass

1394
01:17:05,830 --> 01:17:08,320
inside divided by the
distance from the center.

1395
01:17:08,320 --> 01:17:09,935
It's more complicated
what I get.

1396
01:17:09,935 --> 01:17:11,643
And in fact, what I
get is not conserved.

1397
01:17:13,900 --> 01:17:17,020
Why is it not conserved?

1398
01:17:17,020 --> 01:17:18,590
I could ask you,
but I'll tell you.

1399
01:17:18,590 --> 01:17:19,673
We're running out of time.

1400
01:17:19,673 --> 01:17:21,192
It's not conserved,
because if you

1401
01:17:21,192 --> 01:17:22,900
ask for the potential
energy of something

1402
01:17:22,900 --> 01:17:24,942
in the presence
of moving masses,

1403
01:17:24,942 --> 01:17:26,650
there's no reason for
it to be conserved.

1404
01:17:26,650 --> 01:17:29,400
The potential energy
for a point particle

1405
01:17:29,400 --> 01:17:31,920
moving in the field of
static masses is conserved.

1406
01:17:31,920 --> 01:17:34,670
That's what you've learned
in [? AO1 ?] or whatever.

1407
01:17:34,670 --> 01:17:38,540
But if other
particles are moving,

1408
01:17:38,540 --> 01:17:41,230
the total energy of the full
system will be conserved.

1409
01:17:41,230 --> 01:17:44,164
But the potential energy of a
single particle-- just thought

1410
01:17:44,164 --> 01:17:45,830
of as a particle
moving in the potential

1411
01:17:45,830 --> 01:17:49,730
of the other particles--
will not be conserved.

1412
01:17:49,730 --> 01:17:50,290
OK.

1413
01:17:50,290 --> 01:17:52,540
What is conserved,
besides the energy

1414
01:17:52,540 --> 01:17:54,720
of this test particle
on the boundary,

1415
01:17:54,720 --> 01:17:56,607
is the total energy
of this system,

1416
01:17:56,607 --> 01:17:57,815
which can also be calculated.

1417
01:17:57,815 --> 01:18:00,148
And that, in fact, will be
one of your homework problems

1418
01:18:00,148 --> 01:18:01,980
for the coming problem
set-- to calculate

1419
01:18:01,980 --> 01:18:03,680
the total energy of that sphere.

1420
01:18:03,680 --> 01:18:05,877
And that will be
related to this quantity

1421
01:18:05,877 --> 01:18:07,710
with a different constant
of proportionality

1422
01:18:07,710 --> 01:18:11,470
and will be conserved for
the obvious physical reason.

1423
01:18:11,470 --> 01:18:14,175
For the particles inside,
what one can imagine--

1424
01:18:14,175 --> 01:18:16,810
and I'll just say this
and then I'll stop--

1425
01:18:16,810 --> 01:18:21,520
you can imagine that
you know that the motion

1426
01:18:21,520 --> 01:18:25,830
of this particle is uninfluenced
by these particles outside.

1427
01:18:25,830 --> 01:18:28,390
And therefore, you could
pretend that they're not there

1428
01:18:28,390 --> 01:18:30,450
and think of it as an
analog problem, where

1429
01:18:30,450 --> 01:18:34,730
the particles outside of the
radius of the particle you're

1430
01:18:34,730 --> 01:18:39,390
focusing on simply do not
exist in this analog problem.

1431
01:18:39,390 --> 01:18:41,977
For that analog problem, this
would be the potential energy.

1432
01:18:41,977 --> 01:18:43,060
And it would be conserved.

1433
01:18:43,060 --> 01:18:44,560
And you could argue
that way, that you

1434
01:18:44,560 --> 01:18:45,820
expect this to be a constant.

1435
01:18:45,820 --> 01:18:46,987
And you'd be correct.

1436
01:18:46,987 --> 01:18:48,570
But it's a little
subtle to understand

1437
01:18:48,570 --> 01:18:52,720
exactly what's conserved
and why and how to use it.

1438
01:18:52,720 --> 01:18:55,220
OK, that's all for today.