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PROFESSOR: I think
it's a good idea

9
00:00:22,150 --> 00:00:23,780
to review where we
were because we're

10
00:00:23,780 --> 00:00:25,810
kind of in the middle
of a discussion.

11
00:00:25,810 --> 00:00:27,970
We're actually on part three.

12
00:00:27,970 --> 00:00:31,510
And probably, there will
be four parts all together

13
00:00:31,510 --> 00:00:35,290
of our discussion of
homogeneous expansion.

14
00:00:35,290 --> 00:00:40,190
So I have a few slides to just
review where we were last time.

15
00:00:40,190 --> 00:00:43,110
We were building a mathematical
model of our homogeneously

16
00:00:43,110 --> 00:00:45,550
expanding universe.

17
00:00:45,550 --> 00:00:49,940
And we modeled it as
a finite-sized sphere

18
00:00:49,940 --> 00:00:51,700
where we promised
that in the end

19
00:00:51,700 --> 00:00:53,750
we will take the limit as
the size of that sphere

20
00:00:53,750 --> 00:00:57,800
goes to infinity and
fills all the space.

21
00:00:57,800 --> 00:01:01,550
But it started with some
initial maximum, R max i, i

22
00:01:01,550 --> 00:01:02,940
for initial.

23
00:01:02,940 --> 00:01:06,540
We arranged for it to have
a uniform density, rho.

24
00:01:06,540 --> 00:01:10,550
We started at a
time called t sub i.

25
00:01:10,550 --> 00:01:14,400
There were some initial
maximum radius, R max i.

26
00:01:14,400 --> 00:01:18,832
And we also set this up to
exhibit Hubble expansion.

27
00:01:18,832 --> 00:01:20,290
And we're going to
try to calculate

28
00:01:20,290 --> 00:01:21,373
how it evolves from there.

29
00:01:21,373 --> 00:01:23,440
But we're going to start
it Hubble expanding.

30
00:01:23,440 --> 00:01:25,370
And Hubble expanding
means that we're

31
00:01:25,370 --> 00:01:28,620
starting with every particle
having an initial velocity

32
00:01:28,620 --> 00:01:32,040
which is a constant, H sub i,
the initial value of the Hubble

33
00:01:32,040 --> 00:01:35,090
expansion rate,
times the vector r.

34
00:01:35,090 --> 00:01:38,320
That is, the vector that goes
from the origin to the point

35
00:01:38,320 --> 00:01:40,130
where the particle is located.

36
00:01:40,130 --> 00:01:42,320
And that corresponds
to Hubble expansion

37
00:01:42,320 --> 00:01:45,201
centered on the
center of our sphere.

38
00:01:45,201 --> 00:01:46,700
So these are our
initial conditions.

39
00:01:46,700 --> 00:01:48,400
We just put them in
by hand because we

40
00:01:48,400 --> 00:01:51,790
think they form a good model for
what our universe looks like.

41
00:01:51,790 --> 00:01:54,440
And then, the laws of
evolution should take over

42
00:01:54,440 --> 00:01:57,310
to govern what's
going to happen later.

43
00:01:57,310 --> 00:02:00,180
And since we haven't
studied general relativity,

44
00:02:00,180 --> 00:02:01,890
we'll be using
Newton's law of gravity

45
00:02:01,890 --> 00:02:04,040
to discover how it behaves.

46
00:02:04,040 --> 00:02:06,170
But I promised you
at the beginning

47
00:02:06,170 --> 00:02:08,699
that we will, in fact, get
exactly the same equation

48
00:02:08,699 --> 00:02:11,809
that we would have gotten had
we used general relativity.

49
00:02:11,809 --> 00:02:13,350
And we'll talk later
today, probably,

50
00:02:13,350 --> 00:02:15,480
about why that's the case.

51
00:02:15,480 --> 00:02:17,770
So this is the initial setup.

52
00:02:17,770 --> 00:02:19,480
Any questions about
the initial setup?

53
00:02:23,000 --> 00:02:25,000
OK, then we derived
a lot of equations.

54
00:02:25,000 --> 00:02:27,510
And everything really
follows from the statement

55
00:02:27,510 --> 00:02:31,190
at the top here, which is
understanding what Newton tells

56
00:02:31,190 --> 00:02:34,082
us about the gravitational
field of a spherical shell.

57
00:02:34,082 --> 00:02:35,790
We could always think
of our solid sphere

58
00:02:35,790 --> 00:02:37,700
as being made up of shells.

59
00:02:37,700 --> 00:02:39,440
So if we know how
a shell behaves,

60
00:02:39,440 --> 00:02:41,610
we know all we need to know.

61
00:02:41,610 --> 00:02:42,740
And Newton told us.

62
00:02:42,740 --> 00:02:46,060
He told us that inside a
spherical shell, the effect

63
00:02:46,060 --> 00:02:48,580
of gravity, which I'm
describing in terms

64
00:02:48,580 --> 00:02:52,170
of the gravitational
acceleration vector, little g,

65
00:02:52,170 --> 00:02:55,720
inside a shell, the
gravitational field is exactly

66
00:02:55,720 --> 00:02:57,134
0.

67
00:02:57,134 --> 00:02:59,550
The force is coming from all
different parts of the shell,

68
00:02:59,550 --> 00:03:01,490
pulling outward,
cancel each other.

69
00:03:01,490 --> 00:03:04,780
And the net force on any object
anywhere inside the shell

70
00:03:04,780 --> 00:03:06,580
is exactly 0.

71
00:03:06,580 --> 00:03:10,000
Outside, the entire
shell acts exactly

72
00:03:10,000 --> 00:03:12,870
as if it were a single
point mass located

73
00:03:12,870 --> 00:03:17,340
at the origin with the
same total mass, m.

74
00:03:17,340 --> 00:03:18,710
So incredibly simple.

75
00:03:18,710 --> 00:03:21,744
It's hard to believe it's
so simple, but it is.

76
00:03:21,744 --> 00:03:23,660
By the way, if you use
Gauss's law of gravity,

77
00:03:23,660 --> 00:03:25,549
it becomes very obvious
that those statements

78
00:03:25,549 --> 00:03:26,590
are the right statements.

79
00:03:26,590 --> 00:03:28,800
Newton know about
Gauss's law of gravity,

80
00:03:28,800 --> 00:03:30,790
so Newton derived
those statements

81
00:03:30,790 --> 00:03:32,540
by brute force
integration, which

82
00:03:32,540 --> 00:03:33,770
is more of a tour de force.

83
00:03:33,770 --> 00:03:38,430
But something Newton was
capable of doing it, and he did.

84
00:03:38,430 --> 00:03:40,590
To describe how the
system is going to evolve,

85
00:03:40,590 --> 00:03:43,560
moving onward, we introduce
something that's a little bit

86
00:03:43,560 --> 00:03:47,150
complicated, a function r, which
is a function of two variables,

87
00:03:47,150 --> 00:03:50,110
r sub i and t.

88
00:03:50,110 --> 00:03:53,500
And r represents
the radius at time t

89
00:03:53,500 --> 00:03:57,790
of the shell that was
initially at radius r sub i.

90
00:03:57,790 --> 00:04:01,800
And our goal here is not just
to keep track of the particle

91
00:04:01,800 --> 00:04:03,895
on the outside,
which is for example,

92
00:04:03,895 --> 00:04:06,380
what Ryden does in her textbook.

93
00:04:06,380 --> 00:04:08,400
Ryden assumes that
everything stays homogeneous.

94
00:04:08,400 --> 00:04:10,649
And then if you follow what
happens to the outer edge,

95
00:04:10,649 --> 00:04:11,950
you know everything.

96
00:04:11,950 --> 00:04:13,180
But we're not going to
make that assumption.

97
00:04:13,180 --> 00:04:15,450
We're going to conclude
that it remains homogeneous,

98
00:04:15,450 --> 00:04:16,709
but we're going to derive it.

99
00:04:16,709 --> 00:04:18,800
Which means that we need to know
the motion of every particle

100
00:04:18,800 --> 00:04:20,360
inside the sphere
to be able to tell

101
00:04:20,360 --> 00:04:22,320
if it's going to
stay homogeneous.

102
00:04:22,320 --> 00:04:26,610
And that's why we're introducing
this more general description

103
00:04:26,610 --> 00:04:29,230
where r is a
function of r sub i.

104
00:04:29,230 --> 00:04:31,440
So that function of
the extra variable

105
00:04:31,440 --> 00:04:36,500
will tell us how every particle
moves as the system evolves.

106
00:04:36,500 --> 00:04:39,194
We know that we're going to
maintain spherical symmetry

107
00:04:39,194 --> 00:04:40,860
because we start with
spherical symmetry

108
00:04:40,860 --> 00:04:43,315
and the force law respects
spherical symmetry.

109
00:04:43,315 --> 00:04:45,190
So we're building that
in from the beginning.

110
00:04:45,190 --> 00:04:48,730
We're not allowing things to
depend on angular variables

111
00:04:48,730 --> 00:04:50,430
theta or phi.

112
00:04:50,430 --> 00:04:54,560
But assuming spherical
symmetry, motion just

113
00:04:54,560 --> 00:04:57,130
is described entirely by
giving the r-coordinate

114
00:04:57,130 --> 00:04:58,600
of each particle.

115
00:04:58,600 --> 00:05:03,430
And this function r of r i
t does that-- exactly that.

116
00:05:03,430 --> 00:05:08,640
Then, we said that
at a given radius,

117
00:05:08,640 --> 00:05:10,680
this description
about shells tells us

118
00:05:10,680 --> 00:05:12,680
that the shells that
are outside that radius

119
00:05:12,680 --> 00:05:14,800
don't do anything
at a given radius.

120
00:05:14,800 --> 00:05:17,250
But the shells that are
inside act like a point mass,

121
00:05:17,250 --> 00:05:19,560
as if it was all at the origin.

122
00:05:19,560 --> 00:05:22,750
So to understand how a given
shell is going to evolve,

123
00:05:22,750 --> 00:05:24,740
all we really need to
know is the total mass

124
00:05:24,740 --> 00:05:26,120
inside that shell.

125
00:05:26,120 --> 00:05:28,580
And that's given
by M of r sub i,

126
00:05:28,580 --> 00:05:31,470
the mass inside the
shell at radius r sub i.

127
00:05:31,470 --> 00:05:33,370
And that's just the
volume of the shell

128
00:05:33,370 --> 00:05:37,180
initially times the
initial mass density.

129
00:05:37,180 --> 00:05:40,910
As these shells move, the
total mass inside a shell

130
00:05:40,910 --> 00:05:42,340
will remain exactly
the same as it

131
00:05:42,340 --> 00:05:46,330
was as long as there's
no crossings of shells.

132
00:05:46,330 --> 00:05:49,037
Now, the shell crossing
issue is hard to talk about.

133
00:05:49,037 --> 00:05:50,620
But in the end, it
just doesn't happen

134
00:05:50,620 --> 00:05:52,660
so you don't need
to worry about it.

135
00:05:52,660 --> 00:05:55,250
But the argument was
that initially we

136
00:05:55,250 --> 00:05:57,310
know the shells are moving
apart from each other

137
00:05:57,310 --> 00:05:59,880
because we built in
this Hubble expansion

138
00:05:59,880 --> 00:06:02,690
where everything is moving
away from everything else.

139
00:06:02,690 --> 00:06:05,040
So if shells are
ever going to cross,

140
00:06:05,040 --> 00:06:06,700
they're not going to
cross immediately.

141
00:06:06,700 --> 00:06:09,710
It will take some time for
these velocities to reverse,

142
00:06:09,710 --> 00:06:13,440
and the shells that were moving
apart to move together and hit.

143
00:06:13,440 --> 00:06:16,770
So there's unambiguously
at least a period of time

144
00:06:16,770 --> 00:06:18,630
where there are no
shell crossings.

145
00:06:18,630 --> 00:06:20,230
And we could write
down the equations

146
00:06:20,230 --> 00:06:22,500
that describe the motion
during this period

147
00:06:22,500 --> 00:06:25,120
where there are no
shell crossings.

148
00:06:25,120 --> 00:06:27,210
Now, if there was going
to be a shell crossing,

149
00:06:27,210 --> 00:06:28,410
the equations that
we're writing down

150
00:06:28,410 --> 00:06:30,030
would have to hold
right up until the time

151
00:06:30,030 --> 00:06:30,990
of that first crossing.

152
00:06:30,990 --> 00:06:32,660
Because as long as
there's no crossings,

153
00:06:32,660 --> 00:06:34,340
our equations are valid.

154
00:06:34,340 --> 00:06:37,570
Which means that if there was
going to be a shell crossing,

155
00:06:37,570 --> 00:06:40,040
the equations we're writing
down had better show it.

156
00:06:40,040 --> 00:06:42,480
Because the equations
that we're writing down

157
00:06:42,480 --> 00:06:44,530
have to be valid right
up until the instant

158
00:06:44,530 --> 00:06:46,570
of any possible shell crossing.

159
00:06:46,570 --> 00:06:48,780
And what we're going to
find is that the equations

160
00:06:48,780 --> 00:06:51,140
are going to lead to just
homogeneous evolution

161
00:06:51,140 --> 00:06:53,690
where there are no
shell crossings.

162
00:06:53,690 --> 00:06:55,254
And therefore, that's
the conclusion.

163
00:06:55,254 --> 00:06:57,170
If there were going to
be any shell crossings,

164
00:06:57,170 --> 00:06:58,753
these equations would
have to show it.

165
00:06:58,753 --> 00:07:01,900
They don't show it, so there
are no shell crossings.

166
00:07:01,900 --> 00:07:04,870
So it's a complicated
paragraph, but the bottom line

167
00:07:04,870 --> 00:07:07,210
is simple- we can
ignore shell crossings.

168
00:07:07,210 --> 00:07:10,050
And that means that the total
mass inside of any shell

169
00:07:10,050 --> 00:07:12,170
will remain exactly
constant with time

170
00:07:12,170 --> 00:07:13,740
given by its initial value.

171
00:07:13,740 --> 00:07:15,730
And this formula is
the initial value.

172
00:07:15,730 --> 00:07:17,230
And therefore, it
holds at all time.

173
00:07:20,180 --> 00:07:23,315
Any questions about that?

174
00:07:23,315 --> 00:07:25,190
Any questions about the
shell crossing issue?

175
00:07:28,980 --> 00:07:30,870
OK, good.

176
00:07:30,870 --> 00:07:34,000
So whoops, sorry about that.

177
00:07:34,000 --> 00:07:37,940
So M of r sub i is the mass
inside the shell at radius r

178
00:07:37,940 --> 00:07:38,920
sub i.

179
00:07:38,920 --> 00:07:41,760
And then we can write down-- now
we use Newton's law of gravity

180
00:07:41,760 --> 00:07:42,630
directly.

181
00:07:42,630 --> 00:07:45,470
We can write down
the acceleration

182
00:07:45,470 --> 00:07:48,280
of a given particle in
terms of its radius r

183
00:07:48,280 --> 00:07:50,860
and its initial radius r sub i.

184
00:07:50,860 --> 00:07:54,910
Its initial radius determines
how much mass is inside.

185
00:07:54,910 --> 00:07:57,550
M of r sub i is
independent of time.

186
00:07:57,550 --> 00:08:00,170
But the actual radius it's
at determines how far away

187
00:08:00,170 --> 00:08:03,360
it is, or describes how far
away it is from the origin.

188
00:08:03,360 --> 00:08:07,020
And that's the 1 over r squared
that appears in the force law.

189
00:08:07,020 --> 00:08:10,120
So we have the time dependent r
in the denominator and the time

190
00:08:10,120 --> 00:08:14,030
independent initial r sub i
that appears in the numerator.

191
00:08:14,030 --> 00:08:16,170
And it's all proportional
to a unit vector r

192
00:08:16,170 --> 00:08:19,830
hat pulling everything
radially inward

193
00:08:19,830 --> 00:08:21,624
because of the
minus sign in front.

194
00:08:21,624 --> 00:08:23,290
So gravity is pulling
everything inward,

195
00:08:23,290 --> 00:08:25,770
which is what we'd expect.

196
00:08:25,770 --> 00:08:28,560
So this formula is
the key formula.

197
00:08:28,560 --> 00:08:30,220
It's a vector
formula, but we know

198
00:08:30,220 --> 00:08:31,480
that all the motion is radial.

199
00:08:31,480 --> 00:08:32,938
So all we really
have to keep track

200
00:08:32,938 --> 00:08:35,659
of is the radius as
a function of time.

201
00:08:35,659 --> 00:08:37,730
So we can turn
this vector formula

202
00:08:37,730 --> 00:08:41,000
into a formula for
little r itself.

203
00:08:41,000 --> 00:08:44,390
Just the radius number,
the radial coordinate.

204
00:08:44,390 --> 00:08:49,130
And we get r double dot is minus
4 pi over 3 G r sub i cubed rho

205
00:08:49,130 --> 00:08:51,175
sub i, taking the
formula for M of ri

206
00:08:51,175 --> 00:08:53,670
from the line above
divided by r squared.

207
00:08:53,670 --> 00:08:57,210
And the r in this formula--
I didn't write the arguments,

208
00:08:57,210 --> 00:08:59,690
but it means this
function r, which

209
00:08:59,690 --> 00:09:03,060
is a function of two
arguments, r sub i and t.

210
00:09:03,060 --> 00:09:08,350
So this differential equation
now governs our entire system

211
00:09:08,350 --> 00:09:10,400
and tells us everything
we need to know

212
00:09:10,400 --> 00:09:13,791
or everything we can know
about the actual dynamics.

213
00:09:13,791 --> 00:09:15,290
But to solve a
differential equation

214
00:09:15,290 --> 00:09:17,760
of that sort, a second
order differential equation,

215
00:09:17,760 --> 00:09:19,540
we need initial conditions.

216
00:09:19,540 --> 00:09:22,300
And we already described the
initial conditions in words.

217
00:09:22,300 --> 00:09:24,860
Now we have to just figure out
what those initial conditions

218
00:09:24,860 --> 00:09:29,080
are saying about r and r dot.

219
00:09:29,080 --> 00:09:30,782
And the answer is
straightforward.

220
00:09:30,782 --> 00:09:31,740
We argued it last time.

221
00:09:31,740 --> 00:09:34,180
The initial conditions
are that r at time t

222
00:09:34,180 --> 00:09:35,540
sub i is just r sub i.

223
00:09:35,540 --> 00:09:39,110
That was really the definition
of r sub i in the first place.

224
00:09:39,110 --> 00:09:43,020
And r dot is just H
sub i times v coming

225
00:09:43,020 --> 00:09:47,080
from the formula we had
for the initial velocities.

226
00:09:47,080 --> 00:09:50,310
So these three equations,
the two initial conditions

227
00:09:50,310 --> 00:09:53,470
and the differential
equation, lead in principle

228
00:09:53,470 --> 00:09:55,870
to a mathematical solution
that's completely unique

229
00:09:55,870 --> 00:09:58,240
and determined by
those equations.

230
00:09:58,240 --> 00:10:00,120
And our goal now is
just to figure out

231
00:10:00,120 --> 00:10:03,590
what that solution looks like.

232
00:10:03,590 --> 00:10:08,460
And we discovered a
marvelous scaling property.

233
00:10:08,460 --> 00:10:11,170
That is, instead
of talking about r,

234
00:10:11,170 --> 00:10:15,580
we divided r by r sub i and
defined a new function, which

235
00:10:15,580 --> 00:10:17,325
we initially called u.

236
00:10:17,325 --> 00:10:18,950
Initially, thinking
of it as a function

237
00:10:18,950 --> 00:10:20,990
of these two variables.

238
00:10:20,990 --> 00:10:23,160
We can write down
new equations for u.

239
00:10:23,160 --> 00:10:26,330
And those equations
end up not having

240
00:10:26,330 --> 00:10:28,420
any r sub i's in them at all.

241
00:10:28,420 --> 00:10:30,360
And once we realized
that, we realized

242
00:10:30,360 --> 00:10:32,880
we don't need to call
it u of r sub i and t.

243
00:10:32,880 --> 00:10:34,390
It's really just
a function of t.

244
00:10:34,390 --> 00:10:36,452
And at that point, we
renamed it because we also

245
00:10:36,452 --> 00:10:38,660
realized that it actually
is our old friend the scale

246
00:10:38,660 --> 00:10:40,580
factor, a of t.

247
00:10:40,580 --> 00:10:44,600
So a of t is just r of r sub
i and t divided by r sub i.

248
00:10:44,600 --> 00:10:46,176
And then the reason
this is a scale

249
00:10:46,176 --> 00:10:48,800
factor is seen most clearly from
that equation, which is really

250
00:10:48,800 --> 00:10:51,390
this equation just rearranged.

251
00:10:51,390 --> 00:10:53,770
The physical distance of
a particle from the origin

252
00:10:53,770 --> 00:10:57,300
is equal to the scale
factor times r sub i

253
00:10:57,300 --> 00:11:01,190
where r sub i plays the role
of the coordinate distance.

254
00:11:01,190 --> 00:11:06,150
r sub i is a time-independent
measure indicating

255
00:11:06,150 --> 00:11:07,540
which shell you're
talking about.

256
00:11:10,950 --> 00:11:14,320
So the equations for a then
are the equations for u,

257
00:11:14,320 --> 00:11:17,010
which are the equations for
r just divided by r sub i.

258
00:11:17,010 --> 00:11:19,710
And we can write down
what those equations are.

259
00:11:19,710 --> 00:11:21,290
We have a differential
equation for a

260
00:11:21,290 --> 00:11:23,610
and two initial
conditions where r

261
00:11:23,610 --> 00:11:26,640
sub i has dropped
out all together.

262
00:11:26,640 --> 00:11:30,230
The differential equation
is given immediately

263
00:11:30,230 --> 00:11:32,440
from the one we had up
here, but we could also we

264
00:11:32,440 --> 00:11:35,280
write it in terms
of what rho of t is.

265
00:11:35,280 --> 00:11:37,030
I didn't write the
equation here because I

266
00:11:37,030 --> 00:11:38,630
guess I was running out of room.

267
00:11:38,630 --> 00:11:40,950
But we also figured out
how rho of t behaves.

268
00:11:40,950 --> 00:11:42,760
And it behaves in
the obvious way.

269
00:11:42,760 --> 00:11:44,360
As the space
expands, the density

270
00:11:44,360 --> 00:11:47,160
just goes down as the volume.

271
00:11:47,160 --> 00:11:50,660
And the volume grows like a
cubed, the cube of the scale

272
00:11:50,660 --> 00:11:54,890
factor, because volume's
proportional to radius cubed.

273
00:11:54,890 --> 00:12:00,770
So rho of t is just rho sub
i divided by a of t cubed.

274
00:12:00,770 --> 00:12:04,080
And putting that in, we
can go from that equation

275
00:12:04,080 --> 00:12:05,410
to this equation.

276
00:12:05,410 --> 00:12:07,340
And this equation makes
no reference anymore

277
00:12:07,340 --> 00:12:08,620
to the initial time.

278
00:12:08,620 --> 00:12:11,720
It's just an equation for what
deceleration you see, what

279
00:12:11,720 --> 00:12:14,210
value of a double dot
you see, as a function

280
00:12:14,210 --> 00:12:15,880
of the mass density
and a itself.

281
00:12:18,680 --> 00:12:22,010
OK, any questions about any
of those differential equation

282
00:12:22,010 --> 00:12:23,150
manipulations?

283
00:12:23,150 --> 00:12:24,180
Yes.

284
00:12:24,180 --> 00:12:27,230
AUDIENCE: In the homework, it
says that this equation is not

285
00:12:27,230 --> 00:12:28,120
entirely general.

286
00:12:28,120 --> 00:12:29,340
We can't use it in all cases.

287
00:12:29,340 --> 00:12:32,343
Whereas, the other version
that we get from the energy

288
00:12:32,343 --> 00:12:34,520
conservation is completely OK?

289
00:12:34,520 --> 00:12:36,910
PROFESSOR: That is correct, yes.

290
00:12:36,910 --> 00:12:39,137
AUDIENCE: It says it in
there, but why is that?

291
00:12:39,137 --> 00:12:40,220
PROFESSOR: Why is it true?

292
00:12:40,220 --> 00:12:43,270
Well, as long as you have only
non-relativistic matter, which

293
00:12:43,270 --> 00:12:44,680
is what we're
talking about here,

294
00:12:44,680 --> 00:12:46,900
both of these-- this
equation is golden

295
00:12:46,900 --> 00:12:49,406
and so is the equation
we're about to talk

296
00:12:49,406 --> 00:12:51,530
about the derivation of,
the conservation of energy

297
00:12:51,530 --> 00:12:52,710
equation.

298
00:12:52,710 --> 00:12:55,990
So as long as we have the
context in which we derived it,

299
00:12:55,990 --> 00:12:57,420
it's completely valid.

300
00:12:57,420 --> 00:12:59,920
But on the homework
set, we're talking

301
00:12:59,920 --> 00:13:01,019
about more general cases.

302
00:13:01,019 --> 00:13:03,060
We gave a different formula
for the scale factor,

303
00:13:03,060 --> 00:13:05,185
which corresponds to a
different situation in terms

304
00:13:05,185 --> 00:13:09,000
of the underlying materials
that are building that universe.

305
00:13:09,000 --> 00:13:13,190
And where the change
occurs is when

306
00:13:13,190 --> 00:13:16,010
one introduces a
nonzero pressure.

307
00:13:16,010 --> 00:13:17,950
This gas of particles
that we're talking about

308
00:13:17,950 --> 00:13:19,740
is just non-relativistic
particles

309
00:13:19,740 --> 00:13:21,530
moving with the
Hubble expansion.

310
00:13:21,530 --> 00:13:25,250
There's no internal velocity
which generates a pressure.

311
00:13:25,250 --> 00:13:27,570
And it's pressure that
makes a difference.

312
00:13:27,570 --> 00:13:29,639
This formula assume
zero pressure.

313
00:13:29,639 --> 00:13:31,180
We will learn later
how to correct it

314
00:13:31,180 --> 00:13:32,789
when there's a nonzero pressure.

315
00:13:32,789 --> 00:13:34,830
And the other formula
doesn't depend on pressure,

316
00:13:34,830 --> 00:13:36,747
so it's valid whether
there's pressure or not.

317
00:13:36,747 --> 00:13:39,038
But at the moment, we have
no real way of knowing that.

318
00:13:39,038 --> 00:13:40,760
We'll talk later
about why that's true.

319
00:13:44,809 --> 00:13:45,475
Other questions?

320
00:13:49,105 --> 00:13:50,040
No.

321
00:13:50,040 --> 00:13:51,920
OK, great.

322
00:13:51,920 --> 00:13:55,500
OK, one more slide here,
not too much on it.

323
00:13:55,500 --> 00:13:58,410
At the end of the last
class, we took the equation

324
00:13:58,410 --> 00:14:00,250
that I just wrote on
the previous slide.

325
00:14:00,250 --> 00:14:03,710
I just copied it to this slide,
the equation for a double dot,

326
00:14:03,710 --> 00:14:07,790
written in terms of the
initial mass density.

327
00:14:07,790 --> 00:14:12,000
And discovered that it
can be integrated once

328
00:14:12,000 --> 00:14:13,940
to produce a kind of a
conservation of energy

329
00:14:13,940 --> 00:14:15,110
equation.

330
00:14:15,110 --> 00:14:17,250
And all you do is
you start with this,

331
00:14:17,250 --> 00:14:21,100
write it by putting everything
on one side of the equation,

332
00:14:21,100 --> 00:14:25,950
a double dot plus 4 pi over 3 G
rho i over a squared equals 0.

333
00:14:25,950 --> 00:14:28,560
And then, multiply the
whole equation by a dot.

334
00:14:28,560 --> 00:14:31,220
And a dot is called
an integrating factor.

335
00:14:31,220 --> 00:14:36,600
It turns the expression
into a total derivative.

336
00:14:36,600 --> 00:14:38,625
So once you write
it this way, it

337
00:14:38,625 --> 00:14:42,560
is equivalent to dE dt
equals 0, where e is just

338
00:14:42,560 --> 00:14:44,910
defined to be this
quantity that would

339
00:14:44,910 --> 00:14:47,140
have better as a
triple equal sign.

340
00:14:47,140 --> 00:14:49,050
e is just defined
to be that quantity.

341
00:14:49,050 --> 00:14:52,100
And if you then write
down what dE dt means,

342
00:14:52,100 --> 00:14:54,670
it means exactly that.

343
00:14:54,670 --> 00:14:56,340
So it's the same equation.

344
00:14:56,340 --> 00:14:58,850
So given our second
order equation,

345
00:14:58,850 --> 00:15:02,010
we can write down a
first order equation,

346
00:15:02,010 --> 00:15:04,340
which is that E is
equal to a constant.

347
00:15:07,880 --> 00:15:12,220
And we commented last time that
the physical interpretation

348
00:15:12,220 --> 00:15:16,970
of E is-- I'm not
sure what to say.

349
00:15:16,970 --> 00:15:19,700
There are multiple physical
interpretations of E

350
00:15:19,700 --> 00:15:21,800
is probably what I want to say.

351
00:15:21,800 --> 00:15:23,720
And one physical
interpretation is

352
00:15:23,720 --> 00:15:25,600
if you multiply by
the right factors,

353
00:15:25,600 --> 00:15:29,790
it does describe the actual
energy of a test particle

354
00:15:29,790 --> 00:15:33,930
just on the boundary of our
sphere, on the outer boundary.

355
00:15:33,930 --> 00:15:37,040
It doesn't really describe
directly the total energy

356
00:15:37,040 --> 00:15:39,940
of a particle inside that
sphere because calculating

357
00:15:39,940 --> 00:15:42,312
the potential energy of a
particle inside the sphere

358
00:15:42,312 --> 00:15:43,145
is more complicated.

359
00:15:43,145 --> 00:15:45,935
And it doesn't give
you the simple answer.

360
00:15:45,935 --> 00:15:48,185
So it doesn't really describe
particles on the inside,

361
00:15:48,185 --> 00:15:50,970
except you could argue
that if you-- talking

362
00:15:50,970 --> 00:15:53,470
about a particle on the inside,
the particles outside of it

363
00:15:53,470 --> 00:15:54,070
don't matter.

364
00:15:54,070 --> 00:15:55,486
And you pretend
they're not there.

365
00:15:55,486 --> 00:15:57,280
And then it does
describe the energy.

366
00:15:57,280 --> 00:15:58,280
That is, you could
think of any particle

367
00:15:58,280 --> 00:15:59,821
as being on the
outside of the sphere

368
00:15:59,821 --> 00:16:01,570
and ignore what's outside it.

369
00:16:01,570 --> 00:16:06,150
But that's extra sentences
that you have to put in.

370
00:16:06,150 --> 00:16:08,370
On the homework, you
will also discover

371
00:16:08,370 --> 00:16:10,690
that for this finite-sized
Newtonian sphere,

372
00:16:10,690 --> 00:16:12,630
there's certainly a
well-defined Newtonian

373
00:16:12,630 --> 00:16:15,230
expression for the total
energy of the sphere.

374
00:16:15,230 --> 00:16:17,260
And that's also
proportional to this E.

375
00:16:17,260 --> 00:16:20,040
So by multiplying it
by different constant,

376
00:16:20,040 --> 00:16:22,854
you can turn it into the
total energy of the sphere.

377
00:16:22,854 --> 00:16:24,270
So it's actually
related to energy

378
00:16:24,270 --> 00:16:25,750
and it's definitely conserved.

379
00:16:25,750 --> 00:16:29,450
Those are the important
statements to takeaway.

380
00:16:29,450 --> 00:16:31,977
And that's where we
left off last time.

381
00:16:31,977 --> 00:16:33,810
And we'll pick up from
there now pretty much

382
00:16:33,810 --> 00:16:36,370
on the blackboard for
the rest of lecture.

383
00:16:36,370 --> 00:16:40,805
Are there any further
questions about these slides?

384
00:16:52,720 --> 00:16:53,390
OK.

385
00:16:53,390 --> 00:16:59,665
In that case, we will go on.

386
00:17:10,540 --> 00:17:13,890
The first thing I want to do is
to take the same conservation

387
00:17:13,890 --> 00:17:17,140
law that we have up
there and rewrite it

388
00:17:17,140 --> 00:17:19,519
in a way that's
more conventional.

389
00:17:19,519 --> 00:17:20,560
And perhaps, more useful.

390
00:17:20,560 --> 00:17:22,990
But certainly,
more conventional.

391
00:17:22,990 --> 00:17:28,920
We started with knowing that a
quantity called E is conserved.

392
00:17:28,920 --> 00:17:36,700
And it's equal to 1/2 a dot
squared minus 4 pi over 3

393
00:17:36,700 --> 00:17:41,280
G rho i over a.

394
00:17:44,160 --> 00:17:48,300
OK, then we also
know that rho of t

395
00:17:48,300 --> 00:17:54,570
is equal to rho sub i divided
by a cubed of t, which just says

396
00:17:54,570 --> 00:17:57,270
that matter thins out with
the volume which grows

397
00:17:57,270 --> 00:18:00,300
as the cube of the scale factor.

398
00:18:00,300 --> 00:18:06,260
And that can be used to
manipulate this equation.

399
00:18:11,440 --> 00:18:15,381
For reasons that will
become clearer in a minute,

400
00:18:15,381 --> 00:18:17,880
I'm just going to manipulate
this equation by multiplying it

401
00:18:17,880 --> 00:18:18,830
by 2 over a squared.

402
00:18:18,830 --> 00:18:20,580
Just because this will
get me the equation

403
00:18:20,580 --> 00:18:22,030
I'm trying to get to.

404
00:18:22,030 --> 00:18:24,551
So if I multiply the left-hand
side by 2 over a squared,

405
00:18:24,551 --> 00:18:26,342
I have to multiply the
right-hand side by 2

406
00:18:26,342 --> 00:18:28,190
over a squared.

407
00:18:28,190 --> 00:18:30,380
The 2 cancels the half.

408
00:18:30,380 --> 00:18:34,490
The first term becomes
a dot over a squared.

409
00:18:34,490 --> 00:18:37,130
And you might remember the a dot
over a is the Hubble expansion

410
00:18:37,130 --> 00:18:40,040
rate, so it has some
physical significance.

411
00:18:40,040 --> 00:18:46,790
And then, minus the 2 turns the
4 pi over 3 into 8 pi over 3.

412
00:18:46,790 --> 00:18:54,820
And the a squared multiplies
the a to make an a cubed.

413
00:18:54,820 --> 00:18:57,740
And then here, we have
rho i over a cubed, which

414
00:18:57,740 --> 00:19:00,250
in fact is just the
current value of rho.

415
00:19:00,250 --> 00:19:08,780
So we can rewrite this as a
dot over a squared minus 8 pi

416
00:19:08,780 --> 00:19:12,750
over 3 G rho.

417
00:19:12,750 --> 00:19:15,650
No a's anymore on
the right-hand side.

418
00:19:15,650 --> 00:19:17,770
Well, on a's anymore
in this term.

419
00:19:21,280 --> 00:19:26,530
OK, now the convention
that brings our notation

420
00:19:26,530 --> 00:19:32,095
into contact with the
rest of the world.

421
00:19:32,095 --> 00:19:33,470
Nobody talks about
E, by the way.

422
00:19:33,470 --> 00:19:35,666
That's just my convention.

423
00:19:35,666 --> 00:19:37,040
But to make contact
with the rest

424
00:19:37,040 --> 00:19:39,400
of the world, the rest
of the world talks

425
00:19:39,400 --> 00:19:44,070
about a number called
little k, lowercase k.

426
00:19:44,070 --> 00:19:49,200
And it connects to our notation
by being equal to minus 2E

427
00:19:49,200 --> 00:19:50,270
divided by c squared.

428
00:19:55,450 --> 00:20:04,460
And with that connection, we
can write our conservation law

429
00:20:04,460 --> 00:20:10,560
in what is the standard
way of writing it,

430
00:20:10,560 --> 00:20:11,685
at least in many textbooks.

431
00:20:25,257 --> 00:20:25,840
And that's it.

432
00:20:25,840 --> 00:20:27,600
So I put a box around it.

433
00:20:27,600 --> 00:20:30,620
And this equation was first
derived by Alexander Friedmann

434
00:20:30,620 --> 00:20:34,620
using general
relativity in 1922.

435
00:20:34,620 --> 00:20:43,390
And it is, therefore, usually
called the Friedmann equation.

436
00:20:47,040 --> 00:20:50,170
Alexander Friedmann, by the
way, was a Russian meteorologist

437
00:20:50,170 --> 00:20:51,390
by profession.

438
00:20:51,390 --> 00:20:53,320
But as a meteorologist,
he knew a lot

439
00:20:53,320 --> 00:20:54,880
about differential equations.

440
00:20:54,880 --> 00:20:58,670
And when general relativity came
out, he got interested in it

441
00:20:58,670 --> 00:21:02,365
and was the first person to
derive using general relativity

442
00:21:02,365 --> 00:21:05,390
the equations that described
an expanding universe.

443
00:21:05,390 --> 00:21:09,395
And he wrote two famous
papers-- now famous papers--

444
00:21:09,395 --> 00:21:12,260
in 1922 and 1923.

445
00:21:12,260 --> 00:21:15,580
One of them talking about
the system of equations

446
00:21:15,580 --> 00:21:17,297
where k is positive
and another talking

447
00:21:17,297 --> 00:21:18,380
about where it's negative.

448
00:21:18,380 --> 00:21:20,549
I forget which
order they were in.

449
00:21:20,549 --> 00:21:22,590
But they correspond to
open and closed universes,

450
00:21:22,590 --> 00:21:25,286
which we'll talk about
more in a few minutes.

451
00:21:25,286 --> 00:21:28,190
Now, I just should remind you
to have our equations together.

452
00:21:28,190 --> 00:21:30,760
We also had the all-important
equation for a double dot.

453
00:21:46,315 --> 00:21:51,070
And as I was just describing
an answer to a question,

454
00:21:51,070 --> 00:21:53,360
we don't know yet how
to generalize this

455
00:21:53,360 --> 00:21:56,620
to other kinds of matter besides
the non non-relativistic dust

456
00:21:56,620 --> 00:21:58,050
that we just derived them for.

457
00:21:58,050 --> 00:22:01,490
They're certainly both correct
for our non-relativistic dust.

458
00:22:01,490 --> 00:22:04,180
But when we try to generalize
them, what we'll find

459
00:22:04,180 --> 00:22:07,510
is that the top equation
will remain true exactly

460
00:22:07,510 --> 00:22:10,390
for any kind of matter, while
the bottom equation assumes

461
00:22:10,390 --> 00:22:11,515
that pressure equals 0.

462
00:22:28,540 --> 00:22:34,710
Now, the standard terminology
is to call the top equation

463
00:22:34,710 --> 00:22:35,885
the Friedmann equation.

464
00:22:35,885 --> 00:22:37,260
In fact, both of
these equations,

465
00:22:37,260 --> 00:22:39,551
with this one including the
pressure term that we don't

466
00:22:39,551 --> 00:22:42,120
have yet, appeared in
Friedmann's original papers.

467
00:22:42,120 --> 00:22:44,100
So I usually refer to
these two equations

468
00:22:44,100 --> 00:22:46,800
as the Friedmann
equations-- plural.

469
00:22:46,800 --> 00:22:50,571
But many textbooks refer to just
the top one as the Friedmann

470
00:22:50,571 --> 00:22:52,820
equation and don't give a
name to that equation, which

471
00:22:52,820 --> 00:22:54,770
is also OK if you want.

472
00:22:54,770 --> 00:22:56,429
Yes.

473
00:22:56,429 --> 00:22:58,281
AUDIENCE: Didn't we
get the top equation

474
00:22:58,281 --> 00:23:00,031
from the bottom equation?

475
00:23:00,031 --> 00:23:00,780
PROFESSOR: We did.

476
00:23:00,780 --> 00:23:03,680
That's right.

477
00:23:03,680 --> 00:23:05,840
So how does that jive?

478
00:23:05,840 --> 00:23:09,290
The answer is-- and we'll
be coming to this later.

479
00:23:09,290 --> 00:23:11,820
But the answer is that when
we got the top equation

480
00:23:11,820 --> 00:23:15,870
from the bottom equation,
we used that equation.

481
00:23:18,480 --> 00:23:20,420
And this equation
will no longer hold

482
00:23:20,420 --> 00:23:22,420
when there's a
significant pressure.

483
00:23:22,420 --> 00:23:25,969
And in fact, when we derive
it later-- I forget what order

484
00:23:25,969 --> 00:23:26,760
we'll do things in.

485
00:23:26,760 --> 00:23:28,968
But we'll make sure that
all three of these equations

486
00:23:28,968 --> 00:23:31,990
are consistent when
we include pressure.

487
00:23:31,990 --> 00:23:34,070
The reason the top
equation changes

488
00:23:34,070 --> 00:23:37,070
if you include pressure--
may not be obvious.

489
00:23:37,070 --> 00:23:39,070
But if I tell you
why it happens,

490
00:23:39,070 --> 00:23:40,414
it will become obvious.

491
00:23:40,414 --> 00:23:42,830
The top equation looks like
it's just how things thin out.

492
00:23:45,620 --> 00:23:47,010
Like a cubed.

493
00:23:47,010 --> 00:23:51,410
But rho is the
total mass density.

494
00:23:51,410 --> 00:23:53,580
And relativistically, it's
equivalent to the energy

495
00:23:53,580 --> 00:23:53,650
density.

496
00:23:53,650 --> 00:23:55,210
If you just multiply
by c squared,

497
00:23:55,210 --> 00:23:58,110
that becomes the energy
density by the E equals

498
00:23:58,110 --> 00:24:00,659
mc squared equality.

499
00:24:00,659 --> 00:24:02,450
So it's a question of
how much energy there

500
00:24:02,450 --> 00:24:05,640
is inside this sphere or box.

501
00:24:05,640 --> 00:24:08,840
And if you imagine
a box changing size,

502
00:24:08,840 --> 00:24:11,560
if it's filled with a gas
with a positive pressure,

503
00:24:11,560 --> 00:24:13,940
as that box changes
size, the pressure

504
00:24:13,940 --> 00:24:15,640
does work on the boundary.

505
00:24:15,640 --> 00:24:18,110
If you think of it as a piston.

506
00:24:18,110 --> 00:24:20,850
And if we have a positive
pressure and a gas expands,

507
00:24:20,850 --> 00:24:22,700
it loses energy.

508
00:24:22,700 --> 00:24:25,155
And relativistically, that
means it has to also lose mass.

509
00:24:27,730 --> 00:24:30,170
The total mass inside the
box does not remain constant

510
00:24:30,170 --> 00:24:32,580
as it expands, which
is the idea that we

511
00:24:32,580 --> 00:24:36,200
use when we derive that
rho i over a cubed.

512
00:24:36,200 --> 00:24:38,870
So rho i over a cubed
is the right behavior

513
00:24:38,870 --> 00:24:41,240
for the total mass
density or energy

514
00:24:41,240 --> 00:24:43,805
density for a zero pressure gas.

515
00:24:43,805 --> 00:24:46,100
But when you include
pressure and take

516
00:24:46,100 --> 00:24:50,137
into account relativity, that's
not the right formula anymore.

517
00:24:50,137 --> 00:24:53,400
AUDIENCE: The top one, it
just cancels out somehow?

518
00:24:53,400 --> 00:24:55,570
PROFESSOR: Well, the pressure
ends up canceling out,

519
00:24:55,570 --> 00:24:57,390
so that this ends
up still being true

520
00:24:57,390 --> 00:24:59,212
and this ends up
being different.

521
00:24:59,212 --> 00:25:00,670
And we'll see later
how it happens.

522
00:25:00,670 --> 00:25:05,720
I just want to indicate where
the changes are going to be.

523
00:25:05,720 --> 00:25:08,439
We'll see what the changes
are when we get there.

524
00:25:08,439 --> 00:25:10,230
AUDIENCE: What happened
to the third factor

525
00:25:10,230 --> 00:25:12,070
of a in the second equation?

526
00:25:14,060 --> 00:25:16,060
PROFESSOR: There's a
factor of a missing, sorry.

527
00:25:19,170 --> 00:25:20,800
Yes, thank you very much.

528
00:25:20,800 --> 00:25:23,136
It's important to get
the equations right.

529
00:25:23,136 --> 00:25:24,510
That wasn't
Friedmann's equation.

530
00:25:24,510 --> 00:25:25,551
That was Alan's equation.

531
00:25:28,630 --> 00:25:31,280
Now it's Friedmann's equation.

532
00:25:31,280 --> 00:25:31,970
He got it right.

533
00:25:34,880 --> 00:25:38,250
OK, any other errors or
questions to bring up?

534
00:25:43,920 --> 00:25:44,420
OK.

535
00:25:47,350 --> 00:25:51,730
Now is probably a good time to
talk about the question of why

536
00:25:51,730 --> 00:25:54,970
we were so fortunate to discover
that the Friedmann equation

537
00:25:54,970 --> 00:25:57,730
that we derived agrees exactly
with general relativity.

538
00:25:57,730 --> 00:25:59,610
There is a simple reason for it.

539
00:25:59,610 --> 00:26:02,007
I don't think it's
an accident at all.

540
00:26:02,007 --> 00:26:03,590
The reason for it,
as I understand it,

541
00:26:03,590 --> 00:26:05,409
is that we assume
from the beginning--

542
00:26:05,409 --> 00:26:07,200
and we would be assuming
this whether we're

543
00:26:07,200 --> 00:26:08,840
talking about the
Newtonian calculation

544
00:26:08,840 --> 00:26:11,920
or the corresponding general
relativity calculation.

545
00:26:11,920 --> 00:26:13,610
We assume from the
beginning that we

546
00:26:13,610 --> 00:26:16,900
are modeling a completely
homogeneous system, where

547
00:26:16,900 --> 00:26:20,000
every part of it is identical
to every other part.

548
00:26:20,000 --> 00:26:22,730
And once you assume
that homogeneity,

549
00:26:22,730 --> 00:26:26,370
it means that if you know
what happens in a little box,

550
00:26:26,370 --> 00:26:28,162
a meter by a meter
by a meter say.

551
00:26:28,162 --> 00:26:29,870
If you know what
happens in a little box,

552
00:26:29,870 --> 00:26:31,720
you know what happens
everywhere because you assume

553
00:26:31,720 --> 00:26:34,136
that what happens everywhere
is exactly the same as what's

554
00:26:34,136 --> 00:26:36,570
happening in that box.

555
00:26:36,570 --> 00:26:38,890
So that implies
that if Newton is

556
00:26:38,890 --> 00:26:40,705
right for what
happens in the box,

557
00:26:40,705 --> 00:26:42,080
Newton has to be
right for what's

558
00:26:42,080 --> 00:26:44,390
happening in the universe.

559
00:26:44,390 --> 00:26:45,990
And we do expect
Newtonian physics

560
00:26:45,990 --> 00:26:49,030
to work on small scales,
scales of a meter,

561
00:26:49,030 --> 00:26:50,150
and small velocities.

562
00:26:50,150 --> 00:26:53,600
The Hubble expansion of
a meter is negligible.

563
00:26:53,600 --> 00:26:57,380
So we expect Newton to give
us a proper description of how

564
00:26:57,380 --> 00:27:00,220
the system is behaving
on small scales.

565
00:27:00,220 --> 00:27:01,670
And the assumption
of homogeneity

566
00:27:01,670 --> 00:27:04,150
guarantees that if you
understand the small scales,

567
00:27:04,150 --> 00:27:06,840
you also understand
the large scales.

568
00:27:06,840 --> 00:27:11,070
So I think we are guaranteed
that this had better give us

569
00:27:11,070 --> 00:27:13,570
the same results as
general relativity or else

570
00:27:13,570 --> 00:27:15,320
Newtonian physics is
not the proper limit

571
00:27:15,320 --> 00:27:16,380
of general relativity.

572
00:27:16,380 --> 00:27:17,597
But it is.

573
00:27:17,597 --> 00:27:19,180
We would not accept
general relativity

574
00:27:19,180 --> 00:27:21,530
if it did not give
Newtonian physics

575
00:27:21,530 --> 00:27:23,440
for small scales
and low velocities.

576
00:27:25,960 --> 00:27:30,530
So we expect to get the
same answer as Gr and we do.

577
00:27:30,530 --> 00:27:32,240
This is exactly
what Gr would give.

578
00:27:32,240 --> 00:27:34,580
And this is exactly what Gr
would give also for the case

579
00:27:34,580 --> 00:27:39,610
where there's no pressure--
the case we're doing.

580
00:27:39,610 --> 00:27:40,970
OK, any questions about that?

581
00:27:43,916 --> 00:27:45,090
OK, next item.

582
00:27:51,214 --> 00:27:52,630
I would like to
say a couple words

583
00:27:52,630 --> 00:27:54,129
about the units in
which we're going

584
00:27:54,129 --> 00:27:55,310
to define these quantities.

585
00:27:59,430 --> 00:28:03,280
So far, in our
mathematical model,

586
00:28:03,280 --> 00:28:07,035
r and r sub i are distances.

587
00:28:09,880 --> 00:28:12,730
And therefore, they're
measured in whatever units

588
00:28:12,730 --> 00:28:15,240
you're using to
measure distances.

589
00:28:15,240 --> 00:28:17,064
And I'll pretend
we're using SI units.

590
00:28:17,064 --> 00:28:17,855
So I'll say meters.

591
00:28:21,037 --> 00:28:22,620
We could use light
years, or whatever.

592
00:28:22,620 --> 00:28:23,860
It doesn't really matter.

593
00:28:23,860 --> 00:28:26,674
But they're both measured
in ordinary distance units.

594
00:28:26,674 --> 00:28:29,090
And earlier when we talked
about scale factors and things,

595
00:28:29,090 --> 00:28:32,060
I told you that many
books do it this way.

596
00:28:32,060 --> 00:28:35,680
Think of both the
co-moving coordinates

597
00:28:35,680 --> 00:28:37,950
and the physical distances
as being measured

598
00:28:37,950 --> 00:28:40,975
in meters and the scale
factor being dimensionless.

599
00:28:40,975 --> 00:28:42,850
But I told you I don't
like to do it that way

600
00:28:42,850 --> 00:28:44,910
because I think it's
clearer to recognize

601
00:28:44,910 --> 00:28:47,320
that these co-moving
coordinates don't have

602
00:28:47,320 --> 00:28:49,800
any real relationship
to actual distances.

603
00:28:49,800 --> 00:28:53,010
At least not as time changes.

604
00:28:53,010 --> 00:28:55,080
So for me, it's better
to have a different unit

605
00:28:55,080 --> 00:28:58,080
to describe the co-moving
coordinate systems.

606
00:28:58,080 --> 00:29:00,260
So I would like to
introduce that here.

607
00:29:00,260 --> 00:29:11,550
And all I need to do really is
say that let the unit of r sub

608
00:29:11,550 --> 00:29:16,770
i be called a notch.

609
00:29:20,760 --> 00:29:22,510
Now, we already
called it a meter,

610
00:29:22,510 --> 00:29:23,900
but that doesn't really
mean that a meter is

611
00:29:23,900 --> 00:29:24,790
the same as a notch.

612
00:29:24,790 --> 00:29:27,117
Because when we
called it a meter,

613
00:29:27,117 --> 00:29:28,700
we were not really
taking into account

614
00:29:28,700 --> 00:29:33,770
the fact that it's only
a meter at time t sub i.

615
00:29:33,770 --> 00:29:38,390
So another way of
describing this definition,

616
00:29:38,390 --> 00:29:41,670
which might not sound like we're
trying to redefine the meter,

617
00:29:41,670 --> 00:29:52,327
is to say that the statement r
sub i equals 5 notches, which

618
00:29:52,327 --> 00:29:54,743
is the kind of statement I'm
going to make now because I'm

619
00:29:54,743 --> 00:29:57,410
only going to talk about r
sub i measured in notches.

620
00:29:57,410 --> 00:29:59,910
When I say r sub i
is 5 notches, that's

621
00:29:59,910 --> 00:30:13,000
equivalent to saying that the
particle labeled by r sub i

622
00:30:13,000 --> 00:30:32,800
equals 5-- 5 notches-- was
at 5 meters from the origin

623
00:30:32,800 --> 00:30:36,040
at time t sub i.

624
00:30:39,480 --> 00:30:42,954
So giving the value of a
co-moving coordinate in notches

625
00:30:42,954 --> 00:30:44,370
tells you exactly
what distance it

626
00:30:44,370 --> 00:30:48,470
was from the origin
at time t sub i.

627
00:30:48,470 --> 00:30:54,104
Now, the reason why I don't
use this to just say, well,

628
00:30:54,104 --> 00:30:56,020
why don't we call it
meters, is that we're now

629
00:30:56,020 --> 00:30:57,810
going to forget about t sub i.

630
00:30:57,810 --> 00:30:59,220
If you look at
equations we have,

631
00:30:59,220 --> 00:31:01,580
t sub i no longer
appears in any of them.

632
00:31:01,580 --> 00:31:03,690
t sub i was just our
way of getting started.

633
00:31:03,690 --> 00:31:06,310
And we could have started
at any time we wanted.

634
00:31:06,310 --> 00:31:08,786
And once we have
these equations,

635
00:31:08,786 --> 00:31:10,535
we can talk about times
earlier than t sub

636
00:31:10,535 --> 00:31:12,240
i, times later than t sub i.

637
00:31:12,240 --> 00:31:16,180
And there's nothing special
about t sub i anymore.

638
00:31:16,180 --> 00:31:19,020
But r sub i we're going to
keep as our permanent label

639
00:31:19,020 --> 00:31:21,640
for every shell, which means
for every particle there will be

640
00:31:21,640 --> 00:31:26,030
a value of r sub i attached
to that particle which will be

641
00:31:26,030 --> 00:31:28,260
maintained as the
system evolves.

642
00:31:28,260 --> 00:31:30,750
And it will be clearly
playing the role

643
00:31:30,750 --> 00:31:33,830
of what we've called the
co-moving coordinate.

644
00:31:33,830 --> 00:31:37,000
And therefore, we will
want to keep r sub i

645
00:31:37,000 --> 00:31:39,470
and we'll want to call
its unit something.

646
00:31:39,470 --> 00:31:41,630
And I'm just saying I'm
going to call them notches.

647
00:31:41,630 --> 00:31:42,850
You can call them
meters if you want,

648
00:31:42,850 --> 00:31:44,308
but I'm going to
call them notches.

649
00:31:50,090 --> 00:31:58,630
So using this language, I'm not
changing any of the equations

650
00:31:58,630 --> 00:32:00,420
that we wrote.

651
00:32:00,420 --> 00:32:09,730
So we will still have r being
equal to a of t times r sub i.

652
00:32:09,730 --> 00:32:13,250
But now, r will be
measured in meters.

653
00:32:13,250 --> 00:32:18,250
a of t I will think of
as meters per notch.

654
00:32:18,250 --> 00:32:20,320
And r sub i will me
measured in notches.

655
00:32:25,605 --> 00:32:27,980
And the scale factor a of t
will be playing the same role

656
00:32:27,980 --> 00:32:30,021
it played ever since we
introduced the word scale

657
00:32:30,021 --> 00:32:30,630
factor.

658
00:32:30,630 --> 00:32:32,671
It just means that when
the scale factor doubles,

659
00:32:32,671 --> 00:32:35,050
all distances in our
model double exactly.

660
00:32:40,970 --> 00:32:44,310
Now that we have the sort
of new system of units,

661
00:32:44,310 --> 00:32:48,450
I just want to work out
the units of an object

662
00:32:48,450 --> 00:32:49,970
where the units are
not all obvious.

663
00:32:49,970 --> 00:32:52,020
What are the units of
this thing that we've

664
00:32:52,020 --> 00:32:54,650
defined that we called k,
which is related to this thing

665
00:32:54,650 --> 00:32:57,170
that we defined that was called
E, where E was not really

666
00:32:57,170 --> 00:32:58,490
an energy?

667
00:32:58,490 --> 00:32:59,795
But we can work out what k is.

668
00:33:02,692 --> 00:33:08,070
I'm using square brackets
to mean units of.

669
00:33:08,070 --> 00:33:10,736
k can be thought of as being
defined by this equation.

670
00:33:10,736 --> 00:33:12,360
Or in any case, this
equation certainly

671
00:33:12,360 --> 00:33:14,690
must be dimensionally
consistent because we derived

672
00:33:14,690 --> 00:33:17,135
it and you didn't point out
that I made any mistakes,

673
00:33:17,135 --> 00:33:19,850
so I must not have.

674
00:33:19,850 --> 00:33:22,490
So the units of kc
squared over a squared

675
00:33:22,490 --> 00:33:24,490
should be the same as the
units of a dot squared

676
00:33:24,490 --> 00:33:26,140
over a squared.

677
00:33:26,140 --> 00:33:30,580
And if I multiply through
to write units of k--

678
00:33:30,580 --> 00:33:35,170
to relate them to a dot,
we get the units of k

679
00:33:35,170 --> 00:33:37,270
have to be the same
as the units of 1/c

680
00:33:37,270 --> 00:33:39,276
squared times a dot squared.

681
00:33:41,794 --> 00:33:43,710
The a squareds in
the denominator

682
00:33:43,710 --> 00:33:45,540
here just cancel each other.

683
00:33:45,540 --> 00:33:48,010
So the units of k has to be
the same as the units of 1/c

684
00:33:48,010 --> 00:33:50,030
squared a dot squared.

685
00:33:50,030 --> 00:33:51,890
And we know what they are.

686
00:33:51,890 --> 00:33:55,520
The units of 1/c squared
is second squared

687
00:33:55,520 --> 00:33:58,350
per meter squared.

688
00:33:58,350 --> 00:34:01,350
Meter per second
squared, but upside down.

689
00:34:01,350 --> 00:34:03,250
That's the 1/c squared factor.

690
00:34:03,250 --> 00:34:10,050
And a dot is meters per notch
per second because of the dot.

691
00:34:10,050 --> 00:34:19,360
So meters per notch would be a.

692
00:34:19,360 --> 00:34:21,790
a dot would have
an extra second.

693
00:34:21,790 --> 00:34:25,110
And the whole thing gets
squared because it's

694
00:34:25,110 --> 00:34:26,100
a dot squared here.

695
00:34:29,469 --> 00:34:33,650
And you see then that
the meter squares cancel.

696
00:34:33,650 --> 00:34:35,850
The second squareds cancel.

697
00:34:35,850 --> 00:34:40,680
And we just get the peculiar
answer 1 over a notch squared.

698
00:34:46,540 --> 00:34:49,239
Now, there's probably not a
lot of intuition behind that.

699
00:34:49,239 --> 00:34:51,159
But what it clearly
does say is that we

700
00:34:51,159 --> 00:34:54,380
can make k change its
numerical value by changing

701
00:34:54,380 --> 00:34:56,540
our value for the notch.

702
00:34:56,540 --> 00:34:57,900
And that's important to know.

703
00:34:57,900 --> 00:35:01,724
And it's a clear consequence
of what we just did.

704
00:35:01,724 --> 00:35:03,640
So now we can talk about
different conventions

705
00:35:03,640 --> 00:35:06,080
that people use for
defining the notch.

706
00:35:06,080 --> 00:35:09,580
And hence, k, which are
clearly related to each other

707
00:35:09,580 --> 00:35:10,410
we've now learned.

708
00:35:16,680 --> 00:35:29,770
So first of all, in the
construction that we just

709
00:35:29,770 --> 00:35:33,660
did-- starting out with our
sphere, and letting it grow,

710
00:35:33,660 --> 00:35:37,660
and defining the maximum sphere
as r max comma i and so on.

711
00:35:37,660 --> 00:35:44,750
In that construction, our
initial value of t-- excuse

712
00:35:44,750 --> 00:35:48,830
me, our initial value
of a, a of ti, was one.

713
00:35:48,830 --> 00:35:51,350
And the way we
first did it where

714
00:35:51,350 --> 00:35:54,170
all lengths were
measured in meters.

715
00:35:54,170 --> 00:35:58,525
And with our new definition that
r sub i is measured in notches

716
00:35:58,525 --> 00:36:01,160
so that a is meters
per notch, it

717
00:36:01,160 --> 00:36:05,360
means that in the system
we just had, a of t sub

718
00:36:05,360 --> 00:36:07,285
i was 1 meter per notch.

719
00:36:17,670 --> 00:36:20,859
And since we
already did this, we

720
00:36:20,859 --> 00:36:22,150
don't really want to change it.

721
00:36:22,150 --> 00:36:27,330
But I point out that t sub i,
if you look at our equations,

722
00:36:27,330 --> 00:36:30,840
survives in only one place,
which is in this equation.

723
00:36:30,840 --> 00:36:32,700
It has disappeared from
every other equation

724
00:36:32,700 --> 00:36:34,300
we're going to be keeping.

725
00:36:34,300 --> 00:36:37,342
So we are perfectly safe in just
forgetting about this equation.

726
00:36:37,342 --> 00:36:38,800
Or if we want to
remember about it,

727
00:36:38,800 --> 00:36:41,417
we could just say, well, yeah,
t sub i had some significance.

728
00:36:41,417 --> 00:36:43,500
It was the time at which
the scale factor happened

729
00:36:43,500 --> 00:36:45,245
to have been equal to
1 meters per notch.

730
00:36:45,245 --> 00:36:46,745
But otherwise, it
has no importance.

731
00:36:46,745 --> 00:36:48,860
There's a different time
when the scale factor was

732
00:36:48,860 --> 00:36:53,280
10 meters per notch, or
1 light year per notch.

733
00:36:53,280 --> 00:37:09,790
So since t sub i is of
no relevance whatever,

734
00:37:09,790 --> 00:37:16,440
we can safely forget
the above equation.

735
00:37:20,480 --> 00:37:22,689
Or we could think of it as
the definition of t sub i,

736
00:37:22,689 --> 00:37:24,688
where we don't care
anything else about t sub i.

737
00:37:24,688 --> 00:37:26,230
So it was just some
symbol that was

738
00:37:26,230 --> 00:37:27,604
used in some
earlier calculation.

739
00:37:31,460 --> 00:37:35,630
So we now just have a
scale factor a of t.

740
00:37:35,630 --> 00:37:38,310
And we can talk about how
we might normalize it.

741
00:37:41,469 --> 00:37:43,510
And there are basically
two important conventions

742
00:37:43,510 --> 00:37:46,240
that are in use in textbooks.

743
00:37:46,240 --> 00:37:50,130
Some textbooks, which include
Barbara Ryden's textbook

744
00:37:50,130 --> 00:37:54,790
that we're using,
define a of t to be

745
00:37:54,790 --> 00:38:01,450
equal to 1, which I will call
1 meter per not much today.

746
00:38:06,040 --> 00:38:10,080
So a of t is just defined to be
1 today as a common notation.

747
00:38:10,080 --> 00:38:12,430
It's notation that
Barbara Ryden uses.

748
00:38:12,430 --> 00:38:14,975
And that makes a notch
equal to a meter today.

749
00:38:27,990 --> 00:38:35,460
There's another
common convention,

750
00:38:35,460 --> 00:38:38,920
which is to recognize that since
k has units 1 over notches,

751
00:38:38,920 --> 00:38:41,183
we can make k any value we
want without changing any

752
00:38:41,183 --> 00:38:43,720
of the physics, just changing
our definition of the notch,

753
00:38:43,720 --> 00:38:45,350
which is up for grabs.

754
00:38:45,350 --> 00:38:47,690
Nobody has yet
defined the notch.

755
00:38:47,690 --> 00:38:49,580
We're defining it now.

756
00:38:49,580 --> 00:38:53,380
So we can choose the
definition of a notch

757
00:38:53,380 --> 00:38:56,510
to make the value of
k something simple.

758
00:38:56,510 --> 00:38:59,510
And the obvious choice for
the simplest real number

759
00:38:59,510 --> 00:39:03,170
that you could imagine
that's not 0 is 1.

760
00:39:03,170 --> 00:39:04,650
So we could choose
the definition

761
00:39:04,650 --> 00:39:06,760
of the notch to
make k equal to 1.

762
00:39:06,760 --> 00:39:08,830
Except that we can't
change the sign of k.

763
00:39:08,830 --> 00:39:13,250
The sign of k makes important
differences in this equation.

764
00:39:13,250 --> 00:39:15,904
And we don't want to make the
definition of a notch negative

765
00:39:15,904 --> 00:39:17,320
or imaginary, I
guess, is what you

766
00:39:17,320 --> 00:39:19,300
need to change the sign of k.

767
00:39:19,300 --> 00:39:21,980
So as long as
notches are real, we

768
00:39:21,980 --> 00:39:24,710
can only change positive k's
to different positive k's

769
00:39:24,710 --> 00:39:29,410
and negative k's to different
values of negative k.

770
00:39:29,410 --> 00:39:36,710
So the convention would be
that when k is not equal to 0--

771
00:39:36,710 --> 00:39:38,260
it can be 0 as a special case.

772
00:39:38,260 --> 00:39:44,740
But when k is not equal
to 0, define the notch

773
00:39:44,740 --> 00:39:48,912
so that k is equal
to plus or minus 1.

774
00:39:48,912 --> 00:39:50,370
I would say that
this convention is

775
00:39:50,370 --> 00:39:52,245
more common than that
convention of the books

776
00:39:52,245 --> 00:39:54,720
that I've read in my
lifetime, but both conventions

777
00:39:54,720 --> 00:39:55,350
are in use.

778
00:39:58,890 --> 00:40:02,330
And one sees from this
dimensional relationship

779
00:40:02,330 --> 00:40:04,410
that one can certainly
choose a notch

780
00:40:04,410 --> 00:40:08,000
to make k if it's nonzero
have any value you

781
00:40:08,000 --> 00:40:09,397
want of the same sign.

782
00:40:09,397 --> 00:40:11,480
And that means you can
always make k plus or minus

783
00:40:11,480 --> 00:40:15,320
1 if it's not 0.

784
00:40:15,320 --> 00:40:19,700
The books that use this as their
convention, generally speaking,

785
00:40:19,700 --> 00:40:23,800
leave the notch undefined
when k equals 0.

786
00:40:36,720 --> 00:40:39,310
Undefined means arbitrary.

787
00:40:39,310 --> 00:40:42,420
And there's no problem with
that because k and the notch

788
00:40:42,420 --> 00:40:45,650
never really appear in
final physical quantities.

789
00:40:45,650 --> 00:40:47,500
The notch always
was just your choice

790
00:40:47,500 --> 00:40:50,910
of how to write down your
co-moving map of what

791
00:40:50,910 --> 00:40:51,910
the universe looks like.

792
00:40:54,730 --> 00:40:57,375
OK, any questions about these
funny issues involving units?

793
00:41:04,030 --> 00:41:04,630
OK, good.

794
00:41:04,630 --> 00:41:06,860
The next thing I
want to do now is

795
00:41:06,860 --> 00:41:10,940
to start talking about
solutions to this equation.

796
00:41:10,940 --> 00:41:16,044
And I guess I'll leave the
equation there and start

797
00:41:16,044 --> 00:41:16,960
on the new blackboard.

798
00:41:41,650 --> 00:41:46,712
OK, I'm going to rewrite
the equation almost the way

799
00:41:46,712 --> 00:41:48,550
it was written on the top there.

800
00:41:48,550 --> 00:41:50,570
In fact, exactly as it
was written on top there.

801
00:41:50,570 --> 00:41:52,610
I'm going to
rewrite the equation

802
00:41:52,610 --> 00:42:06,790
as E equals 1/2 a dot squared
minus 4 pi over 3 G rho sub i

803
00:42:06,790 --> 00:42:09,430
over a.

804
00:42:09,430 --> 00:42:14,010
And the reason I'm writing it
in this way rather than any

805
00:42:14,010 --> 00:42:16,460
of the other six or seven
ways that we've written it,

806
00:42:16,460 --> 00:42:20,065
is that this way the only
time-varying thing is a itself.

807
00:42:20,065 --> 00:42:22,150
And if we want to talk
about what the differential

808
00:42:22,150 --> 00:42:24,700
equation tells us about
the time variation of a,

809
00:42:24,700 --> 00:42:27,240
it helps a lot if we're writing
an equation where a of t

810
00:42:27,240 --> 00:42:29,170
is the only thing
that varies with time.

811
00:42:29,170 --> 00:42:33,010
And this is at least
one way of doing that.

812
00:42:33,010 --> 00:42:37,930
So in particular, I used
rho sub i rather than rho.

813
00:42:37,930 --> 00:42:46,130
So the behavior of this
equation might very well

814
00:42:46,130 --> 00:42:50,280
depend on the sign of E.
And we'll see that it does.

815
00:42:50,280 --> 00:42:53,460
And if we think it might
depend on the sign of E,

816
00:42:53,460 --> 00:42:55,180
we realize from the
beginning that E

817
00:42:55,180 --> 00:42:57,230
could be positive,
negative, or 0.

818
00:42:57,230 --> 00:42:59,320
There are those three cases.

819
00:42:59,320 --> 00:43:01,070
So those are the cases
we want to look at.

820
00:43:07,880 --> 00:43:12,086
E can be positive,
negative, or 0.

821
00:43:12,086 --> 00:43:13,460
So we'll take them
one at a time.

822
00:43:16,720 --> 00:43:19,930
It will help to make things
completely obvious to rewrite

823
00:43:19,930 --> 00:43:23,450
this as an equation
for a dot squared.

824
00:43:23,450 --> 00:43:25,510
I'll multiply by 2.

825
00:43:25,510 --> 00:43:27,850
And write this as
equation a dot squared

826
00:43:27,850 --> 00:43:38,480
is equal to 2E plus 8 pi
over 3 G rho sub i over a.

827
00:43:38,480 --> 00:43:40,630
8 pi instead of 4 pi
because we multiplied by 2.

828
00:43:45,290 --> 00:43:50,030
And we notice that this term,
proportional to rho sub i

829
00:43:50,030 --> 00:43:52,316
over a, is
unambiguously positive.

830
00:43:52,316 --> 00:43:54,440
We're not going to have
any negative mass densities

831
00:43:54,440 --> 00:43:55,170
in our problem.

832
00:43:55,170 --> 00:43:56,660
There's no way that can happen.

833
00:43:56,660 --> 00:43:58,950
And a is always positive.

834
00:43:58,950 --> 00:44:00,870
So the right term is positive.

835
00:44:00,870 --> 00:44:02,550
a dot squared had
better be positive

836
00:44:02,550 --> 00:44:04,580
because it's the
square of something.

837
00:44:04,580 --> 00:44:06,310
E could, in principle,
have either sign.

838
00:44:06,310 --> 00:44:09,090
And we'll talk about both
cases, or the 0 case.

839
00:44:09,090 --> 00:44:11,250
But if we start with the
case where E is positive,

840
00:44:11,250 --> 00:44:13,560
just to consider these
three cases one at a time.

841
00:44:19,940 --> 00:44:22,750
So suppose E is greater than 0.

842
00:44:22,750 --> 00:44:26,240
And remind you that E
and k had opposite signs.

843
00:44:26,240 --> 00:44:30,584
So that would imply
that k was negative.

844
00:44:30,584 --> 00:44:33,000
So if we start by considering
the k negative case or the E

845
00:44:33,000 --> 00:44:37,370
positive case, then we see that
we have a positive number here

846
00:44:37,370 --> 00:44:39,310
and a positive number there.

847
00:44:39,310 --> 00:44:42,280
So they will add up to always
give us a positive number.

848
00:44:42,280 --> 00:44:44,320
a dot squared will
always be positive.

849
00:44:44,320 --> 00:44:48,031
And it will just mean that
a dot will be positive.

850
00:44:48,031 --> 00:44:50,280
Square root of a positive
number is a positive number.

851
00:44:50,280 --> 00:44:54,040
At least it's only the positive
square root that matters here.

852
00:44:54,040 --> 00:44:58,030
So a will just keep growing
forever in this case.

853
00:45:00,900 --> 00:45:03,350
a dot squared will
never fall below 2E.

854
00:45:03,350 --> 00:45:05,989
So it will be a lower
bound to a dot squared.

855
00:45:05,989 --> 00:45:08,280
And that means there will
always be a minimum expansion

856
00:45:08,280 --> 00:45:11,040
rate that the
universe will have.

857
00:45:11,040 --> 00:45:19,640
So in this case, a
increases forever.

858
00:45:26,332 --> 00:45:27,790
And that's called
an open universe.

859
00:45:36,284 --> 00:45:38,481
And it's one of the
three possibilities

860
00:45:38,481 --> 00:45:40,230
that we're going to
be investigating here.

861
00:46:03,910 --> 00:46:07,470
Next case is E
less than 0, which

862
00:46:07,470 --> 00:46:13,700
is the more common notation of
k means k is greater than 0.

863
00:46:16,770 --> 00:46:20,110
In this case, if you think of
E as an energy, which it really

864
00:46:20,110 --> 00:46:22,970
is, it means we have
less than 0 energy.

865
00:46:22,970 --> 00:46:25,472
Which means that we basically
have a bound system.

866
00:46:25,472 --> 00:46:27,930
And the equation tells us that
it acts like a bound system.

867
00:46:27,930 --> 00:46:29,810
We don't have to rely
on that intuition,

868
00:46:29,810 --> 00:46:32,050
but that is the right intuition.

869
00:46:32,050 --> 00:46:37,451
The equation up there tells
us that if E is negative,

870
00:46:37,451 --> 00:46:39,450
the total right-hand side
had better be positive

871
00:46:39,450 --> 00:46:41,991
because the left-hand side is
positive and the left-hand side

872
00:46:41,991 --> 00:46:43,049
cannot go negative.

873
00:46:43,049 --> 00:46:45,090
But this term is going to
get smaller and smaller

874
00:46:45,090 --> 00:46:46,990
as a increases.

875
00:46:46,990 --> 00:46:48,850
And as this term gets
smaller and smaller,

876
00:46:48,850 --> 00:46:51,670
it runs the risk of no
longer outweighing this term

877
00:46:51,670 --> 00:46:54,230
and giving possibly
a negative answer.

878
00:46:54,230 --> 00:46:58,350
And what has to happen is a
cannot get any bigger than

879
00:46:58,350 --> 00:47:00,700
the value it would have where
the right-hand side would

880
00:47:00,700 --> 00:47:02,150
vanish.

881
00:47:02,150 --> 00:47:05,440
So a continues to grow
because a dot is positive.

882
00:47:05,440 --> 00:47:08,550
This gets smaller and
smaller until this term

883
00:47:08,550 --> 00:47:10,020
equals that term in magnitude.

884
00:47:10,020 --> 00:47:13,230
And then, a dot goes to 0.

885
00:47:13,230 --> 00:47:15,090
What happens next
is not completely

886
00:47:15,090 --> 00:47:16,540
obvious from this
equation, but it

887
00:47:16,540 --> 00:47:18,740
means that we have an
expanding universe that's

888
00:47:18,740 --> 00:47:21,570
reached a maximum
size and then stopped.

889
00:47:21,570 --> 00:47:25,190
Then, what is actually
obvious is from this equation

890
00:47:25,190 --> 00:47:28,650
is that it will
start to collapse.

891
00:47:28,650 --> 00:47:31,160
So this case corresponds
to a universe that

892
00:47:31,160 --> 00:47:33,995
reaches a maximum size and then
turns around and collapses.

893
00:47:45,140 --> 00:47:52,970
So a has a maximum value.

894
00:47:52,970 --> 00:47:55,740
And we can read off
from that equation what

895
00:47:55,740 --> 00:47:58,700
it is, a max is
just the value that

896
00:47:58,700 --> 00:48:01,080
makes the right-hand
side of that equation 0,

897
00:48:01,080 --> 00:48:08,555
which is minus 4 pi G
rho sub i divided by 3E.

898
00:48:11,325 --> 00:48:13,290
And remember, E is
negative for this case,

899
00:48:13,290 --> 00:48:15,120
so this is a positive number.

900
00:48:15,120 --> 00:48:18,260
So a has some positive
maximum value.

901
00:48:18,260 --> 00:48:21,430
Reaches that value, and then
turns around and collapses.

902
00:48:21,430 --> 00:48:21,949
Yes?

903
00:48:21,949 --> 00:48:24,240
AUDIENCE: Sorry, I don't know
if you said this already,

904
00:48:24,240 --> 00:48:27,996
but since a dot squared is
equal to some quantity, when you

905
00:48:27,996 --> 00:48:31,600
solve for a dot, you can have
positive and negative solution.

906
00:48:31,600 --> 00:48:34,634
Why do we discount
the negative solution?

907
00:48:34,634 --> 00:48:36,050
PROFESSOR: OK,
very good question.

908
00:48:36,050 --> 00:48:37,466
The question if
you didn't hear it

909
00:48:37,466 --> 00:48:40,040
is, why do we discount
the negative solution

910
00:48:40,040 --> 00:48:43,380
when we have an equation
for a dot squared?

911
00:48:43,380 --> 00:48:45,770
Couldn't a dot be
positive or negative?

912
00:48:45,770 --> 00:48:48,100
And the answer is it
certainly could be either.

913
00:48:48,100 --> 00:48:50,380
And both solutions
exist as valid solutions

914
00:48:50,380 --> 00:48:52,080
to these equations.

915
00:48:52,080 --> 00:48:54,640
But we started out with
an initial condition

916
00:48:54,640 --> 00:48:55,970
that a dot was positive.

917
00:48:55,970 --> 00:48:58,780
Our initial value
of a dot was H i.

918
00:48:58,780 --> 00:49:00,910
And once it's positive,
it can't change sign

919
00:49:00,910 --> 00:49:02,130
according to that equation.

920
00:49:02,130 --> 00:49:03,680
Except by going
through 0, which is

921
00:49:03,680 --> 00:49:04,930
what we're talking about now.

922
00:49:04,930 --> 00:49:07,320
But it will only change
sign when it goes through 0.

923
00:49:16,330 --> 00:49:20,420
So it reaches a maximum
value, then it does collapse.

924
00:49:20,420 --> 00:49:22,420
And in the collapsing
phase, that same equation,

925
00:49:22,420 --> 00:49:23,900
a dot squared equals
the right-hand side,

926
00:49:23,900 --> 00:49:25,990
holds where it would be
the negative solution that

927
00:49:25,990 --> 00:49:27,281
describes the collapsing phase.

928
00:49:33,320 --> 00:49:37,480
So the verbal description
of what's happening here

929
00:49:37,480 --> 00:49:52,830
is that the universe
reaches a maximum size

930
00:49:52,830 --> 00:49:54,330
and then collapses.

931
00:49:59,000 --> 00:50:02,690
And it collapses all the way
to a equals 0 in this model.

932
00:50:02,690 --> 00:50:06,490
And that's often
called the Big Crunch

933
00:50:06,490 --> 00:50:08,145
for lack of a better word.

934
00:50:12,520 --> 00:50:15,090
The Big Crunch being
the collapsing form that

935
00:50:15,090 --> 00:50:17,370
corresponds to the
Big Bang, which

936
00:50:17,370 --> 00:50:20,750
is the instant which
all this starts.

937
00:50:26,860 --> 00:50:30,000
And this was called
an open universe.

938
00:50:30,000 --> 00:50:31,944
As you could
probably guess, this

939
00:50:31,944 --> 00:50:33,110
is called a closed universe.

940
00:51:23,480 --> 00:51:24,490
OK.

941
00:51:24,490 --> 00:51:28,830
And now finally, we want
to consider a case where

942
00:51:28,830 --> 00:51:31,350
E is not positive
and not negative.

943
00:51:31,350 --> 00:51:33,310
The case that we're
left with is E equals 0.

944
00:51:36,770 --> 00:51:39,370
And that's called
the critical case.

945
00:51:43,430 --> 00:51:48,090
So the critical value
for E is E is equal to 0.

946
00:51:48,090 --> 00:51:50,140
And that means that k
is equal to 0 as well.

947
00:51:52,725 --> 00:51:55,660
It implies k equals 0.

948
00:51:55,660 --> 00:51:58,120
And notice that this
is a special case.

949
00:51:58,120 --> 00:51:59,009
E is a real number.

950
00:51:59,009 --> 00:52:00,800
It can be positive,
negative, and 0 is just

951
00:52:00,800 --> 00:52:05,620
a particular value on the
borderline between those two.

952
00:52:05,620 --> 00:52:08,920
For the people who are in the
habit of rescaling notches

953
00:52:08,920 --> 00:52:11,950
so that k is always
plus 1, minus 1, or 0,

954
00:52:11,950 --> 00:52:16,060
it makes it sound like there are
three totally distinct cases.

955
00:52:16,060 --> 00:52:17,760
But that's only because
of the rescaling

956
00:52:17,760 --> 00:52:19,480
that those people are doing.

957
00:52:19,480 --> 00:52:22,150
If you keep track of E as your
variable, which you certainly

958
00:52:22,150 --> 00:52:25,700
can, you do see that
the flat case, E equals

959
00:52:25,700 --> 00:52:28,229
0-- the critical
case-- is really

960
00:52:28,229 --> 00:52:29,770
just the borderline
of the other two.

961
00:52:29,770 --> 00:52:32,890
And it's therefore, arbitrarily
close to both of the other two.

962
00:52:32,890 --> 00:52:36,340
It really is where they meet.

963
00:52:36,340 --> 00:52:43,800
But working out the equations,
we have in this case

964
00:52:43,800 --> 00:52:52,130
a dot over a squared is
equal to 8 pi over 3 G rho.

965
00:52:52,130 --> 00:52:55,850
And in general, it's minus
kc squared over a squared.

966
00:52:55,850 --> 00:52:59,980
But we're now considering
the case where that vanishes.

967
00:52:59,980 --> 00:53:01,630
And that means we
have a unique value

968
00:53:01,630 --> 00:53:05,180
for rho in terms
of a dot over a.

969
00:53:05,180 --> 00:53:07,830
And at this point, it's
worth reminding ourselves

970
00:53:07,830 --> 00:53:12,910
that a dot over a is just
H. So this is H squared.

971
00:53:12,910 --> 00:53:14,389
So for this critical
case, the case

972
00:53:14,389 --> 00:53:16,930
that's just on the borderline
between being open and closed--

973
00:53:16,930 --> 00:53:19,300
and we'll be calling it flat.

974
00:53:19,300 --> 00:53:21,810
For this critical case, we
have a definite relationship

975
00:53:21,810 --> 00:53:30,540
between rho and H.
So rho has to equal

976
00:53:30,540 --> 00:53:33,220
what we call the
critical density, which

977
00:53:33,220 --> 00:53:35,220
you get by just
solving that equation.

978
00:53:35,220 --> 00:53:44,020
And it's 3H squared over 8 pi G.

979
00:53:44,020 --> 00:53:47,490
And we see, therefore, that
rho being equal to rho c

980
00:53:47,490 --> 00:53:50,220
is this dividing line
between open and closed.

981
00:53:50,220 --> 00:53:53,140
And if you think back about
the signs of what we had,

982
00:53:53,140 --> 00:53:54,850
what you'll see is
that rho bigger rho

983
00:53:54,850 --> 00:53:57,660
c is what corresponds
to a closed universe.

984
00:54:02,000 --> 00:54:05,310
Rho less than rho c
is what corresponds

985
00:54:05,310 --> 00:54:06,190
to an open universe.

986
00:54:10,570 --> 00:54:13,770
And rho equals rho
c can be called

987
00:54:13,770 --> 00:54:15,470
either a critical
universe or we'll

988
00:54:15,470 --> 00:54:17,790
be calling it a flat universe.

989
00:54:17,790 --> 00:54:25,750
And the meaning of the word
"flat" will be motivated later.

990
00:54:25,750 --> 00:54:28,550
For now, these words--
open, closed, and flat--

991
00:54:28,550 --> 00:54:30,704
refer to the time
evolution of the universe.

992
00:54:30,704 --> 00:54:32,370
We'll see later that
it's also connected

993
00:54:32,370 --> 00:54:35,720
to the geometry of the universe,
but we're not there yet.

994
00:54:35,720 --> 00:54:37,920
Then the word "flat"
will make some sense.

995
00:54:37,920 --> 00:54:38,420
Yes.

996
00:54:38,420 --> 00:54:42,475
AUDIENCE: How do we know there's
not some very large entity,

997
00:54:42,475 --> 00:54:44,350
like some cluster of
black holes or something

998
00:54:44,350 --> 00:54:47,807
that renders all of this not
applicable to our universe?

999
00:54:47,807 --> 00:54:48,390
PROFESSOR: OK.

1000
00:54:48,390 --> 00:54:50,181
The question is, how
do we know there's not

1001
00:54:50,181 --> 00:54:52,370
some humongous perturbation,
some huge collection

1002
00:54:52,370 --> 00:54:54,490
of black holes that
renders this all

1003
00:54:54,490 --> 00:54:56,570
inapplicable to our universe?

1004
00:54:56,570 --> 00:54:58,970
The answer is that it
works for our universe.

1005
00:54:58,970 --> 00:55:02,130
That is, observationally
we can test these things

1006
00:55:02,130 --> 00:55:03,430
in a number of ways.

1007
00:55:03,430 --> 00:55:07,180
Tests include calculations
of the production

1008
00:55:07,180 --> 00:55:09,310
of the light chemical
elements in the Big Bang.

1009
00:55:09,310 --> 00:55:12,030
Tests include making predictions
for what the cosmic background

1010
00:55:12,030 --> 00:55:13,950
radiation should
look like in detail.

1011
00:55:13,950 --> 00:55:16,570
And those tests work
extraordinarily well.

1012
00:55:16,570 --> 00:55:19,437
So that's why we
believe the picture.

1013
00:55:19,437 --> 00:55:21,020
But you're right,
we don't have really

1014
00:55:21,020 --> 00:55:24,470
direct confirmation
of most of this.

1015
00:55:24,470 --> 00:55:27,230
And if there was some
giant conglomeration

1016
00:55:27,230 --> 00:55:30,662
of mass out there someplace, it
might not have been found yet.

1017
00:55:30,662 --> 00:55:32,370
But so far, this
picture works very well.

1018
00:55:32,370 --> 00:55:33,937
That's all I can say.

1019
00:55:33,937 --> 00:55:35,770
And there really is
quite a bit of evidence.

1020
00:55:35,770 --> 00:55:37,311
We'll maybe talk
more about it later.

1021
00:55:40,482 --> 00:55:41,315
Any other questions?

1022
00:55:45,380 --> 00:55:47,070
OK.

1023
00:55:47,070 --> 00:55:50,900
So having understood
the importance

1024
00:55:50,900 --> 00:55:53,560
of this critical
density, it might

1025
00:55:53,560 --> 00:55:56,710
be nice to know what the
value for the critical density

1026
00:55:56,710 --> 00:55:59,279
for our universe is.

1027
00:55:59,279 --> 00:56:01,820
And we can calculate it because
it just depends on the Hubble

1028
00:56:01,820 --> 00:56:03,640
expansion rate, an the
Hubble expansion rate

1029
00:56:03,640 --> 00:56:04,389
has been measured.

1030
00:56:09,280 --> 00:56:12,270
So if we try to put
in numbers, it's

1031
00:56:12,270 --> 00:56:15,430
useful to write the present
value of the Hubble expansion

1032
00:56:15,430 --> 00:56:22,510
rate, as it's often
written, as 100 times h sub

1033
00:56:22,510 --> 00:56:28,246
0 kilometers per
second per megaparsec.

1034
00:56:32,780 --> 00:56:36,070
So this defines h sub
0, little h sub 0.

1035
00:56:36,070 --> 00:56:38,590
And I think the main advantage
of using this notation

1036
00:56:38,590 --> 00:56:41,090
is that you don't have to keep
writing kilometers per second

1037
00:56:41,090 --> 00:56:43,890
per megaparsec which gets to
be a real pain to keep writing.

1038
00:56:43,890 --> 00:56:46,570
So little h sub 0 is just
a dimensionless number

1039
00:56:46,570 --> 00:56:50,040
that defines the
Hubble expansion rate.

1040
00:56:50,040 --> 00:56:52,950
And it does then allow you
to write other formulas

1041
00:56:52,950 --> 00:56:54,800
in simple ways.

1042
00:56:54,800 --> 00:56:57,880
Numerically, Newton's
constant you can look up.

1043
00:56:57,880 --> 00:57:05,380
It's 6.672 times
10 to the minus 8

1044
00:57:05,380 --> 00:57:10,580
centimeter cubed per
gram per second squared.

1045
00:57:13,541 --> 00:57:16,165
And when you put these equations
together, all you need to know

1046
00:57:16,165 --> 00:57:18,920
is G and H squared to
know what rho critical is.

1047
00:57:18,920 --> 00:57:23,830
You find that rho critical can
be written initially for any H0

1048
00:57:23,830 --> 00:57:32,110
as 1.88 h0 squared
coming from the h squared

1049
00:57:32,110 --> 00:57:42,000
in the original formula times
10 to the minus 29 grams

1050
00:57:42,000 --> 00:57:42,975
per centimeter cubed.

1051
00:57:47,890 --> 00:57:53,570
And note the whopping smallness
of that 10 to the minus 29.

1052
00:57:53,570 --> 00:57:55,840
The mass density of
our universe is, as far

1053
00:57:55,840 --> 00:57:57,922
as we know, equal to
this critical density.

1054
00:57:57,922 --> 00:57:59,630
We know it's equal to
within about a half

1055
00:57:59,630 --> 00:58:02,350
of a percent or so.

1056
00:58:02,350 --> 00:58:04,550
And h0 is near 1.

1057
00:58:04,550 --> 00:58:10,130
h0, according to
Planck, is 0.67--

1058
00:58:10,130 --> 00:58:11,690
according to the
Planck satellite

1059
00:58:11,690 --> 00:58:15,410
measurement of the
Hubble parameter

1060
00:58:15,410 --> 00:58:22,810
And if you put
that into here, you

1061
00:58:22,810 --> 00:58:28,670
get the critical density
is about 8.4 times 10

1062
00:58:28,670 --> 00:58:36,210
to the minus 30 grams
per centimeter cubed.

1063
00:58:36,210 --> 00:58:40,820
And that is equivalent to about
5 protons per cubic meter.

1064
00:58:51,222 --> 00:58:52,680
So I've written
the answer in terms

1065
00:58:52,680 --> 00:58:55,177
of grams per cubic
centimeter because

1066
00:58:55,177 --> 00:58:57,010
to me that's a very
natural unit for density

1067
00:58:57,010 --> 00:58:59,070
because it's the
density of water.

1068
00:58:59,070 --> 00:59:03,150
We're saying that the average
density of the universe is only

1069
00:59:03,150 --> 00:59:07,564
about 10 to the minus
29 quantity 8.410.

1070
00:59:07,564 --> 00:59:09,730
The average density of the
universe is only about 10

1071
00:59:09,730 --> 00:59:14,210
to the minus 29 times
the density of water.

1072
00:59:14,210 --> 00:59:16,710
So it's an unbelievably empty
universe that we're living in.

1073
00:59:16,710 --> 00:59:19,100
It's hard to believe the
universe is that empty,

1074
00:59:19,100 --> 00:59:21,990
but there are large spaces
between the galaxies

1075
00:59:21,990 --> 00:59:24,200
that we look at.

1076
00:59:24,200 --> 00:59:26,210
So the universe is
incredibly empty.

1077
00:59:26,210 --> 00:59:30,880
And in fact, this is a
vastly better vacuum.

1078
00:59:30,880 --> 00:59:33,700
An average part of the universe
is a vastly better vacuum

1079
00:59:33,700 --> 00:59:35,730
than can be made on
Earth by any machinery

1080
00:59:35,730 --> 00:59:37,660
that we have access to.

1081
00:59:37,660 --> 00:59:41,951
So the best vacuum is empty
space, just middle of nowhere.

1082
00:59:41,951 --> 00:59:43,950
And it's vastly better
than what we can actually

1083
00:59:43,950 --> 00:59:44,770
produce on Earth.

1084
00:59:53,625 --> 00:59:54,485
Yes.

1085
00:59:54,485 --> 00:59:56,425
AUDIENCE: I see you
used protons here,

1086
00:59:56,425 --> 00:59:58,880
5 protons per cubic medium.

1087
00:59:58,880 --> 01:00:02,362
So is this density corresponding
to density of baryonic matter,

1088
01:00:02,362 --> 01:00:03,320
or all types of matter?

1089
01:00:03,320 --> 01:00:05,361
PROFESSOR: This is actually
all types of matters,

1090
01:00:05,361 --> 01:00:08,290
even though I'm using my
proton as a meter stick.

1091
01:00:08,290 --> 01:00:10,910
But it is the total mass
density of the universe that's

1092
01:00:10,910 --> 01:00:12,319
very close to the
critical value.

1093
01:00:12,319 --> 01:00:13,860
And I was just about
to say something

1094
01:00:13,860 --> 01:00:16,080
about what the total
mass is made up of.

1095
01:00:43,270 --> 01:00:45,830
Cosmologists define-- this
was certainly mentioned

1096
01:00:45,830 --> 01:00:50,120
in my first lecture-- a
Greek letter capital Omega

1097
01:00:50,120 --> 01:00:53,680
to mean the actual mass
density of the universe divided

1098
01:00:53,680 --> 01:00:56,250
by this critical density.

1099
01:00:56,250 --> 01:00:59,110
So omega equals 1 in
this language corresponds

1100
01:00:59,110 --> 01:01:02,240
to a flat universe at
this critical point.

1101
01:01:02,240 --> 01:01:05,590
Omega bigger than 1 corresponds
to a closed universe.

1102
01:01:05,590 --> 01:01:07,290
And omega less
than 1 corresponds

1103
01:01:07,290 --> 01:01:09,050
to an open universe.

1104
01:01:09,050 --> 01:01:11,120
And today, we know that
omega is equal to 1

1105
01:01:11,120 --> 01:01:14,170
to an accuracy of about
a half of a percent.

1106
01:01:14,170 --> 01:01:19,220
To a very good accuracy we
know omega is very close to 1.

1107
01:01:19,220 --> 01:01:21,500
It's made up of
different contributions.

1108
01:01:21,500 --> 01:01:26,560
And these tend to vary with
time-- as the best measurements

1109
01:01:26,560 --> 01:01:29,490
tend to vary with
time by a few percent.

1110
01:01:29,490 --> 01:01:33,170
But omega matter-- and
here I mean visible

1111
01:01:33,170 --> 01:01:49,700
plus dark matter--
is roughly about 0.3.

1112
01:01:49,700 --> 01:01:52,900
And most of the universe
today, as we mentioned earlier,

1113
01:01:52,900 --> 01:01:56,060
the universe today is pretty
much dark energy-dominated.

1114
01:01:59,040 --> 01:02:15,090
So omega dark energy
is about equal to 0.70.

1115
01:02:15,090 --> 01:02:19,020
And omega total is
pretty close to 1.

1116
01:02:23,430 --> 01:02:25,680
Plus or minus about
a half of a percent.

1117
01:02:34,650 --> 01:02:36,480
So one of the
implications here is

1118
01:02:36,480 --> 01:02:39,549
that we've been assuming
in our calculation so far

1119
01:02:39,549 --> 01:02:42,090
that we're talking about nothing
but non-relativistic matter.

1120
01:02:42,090 --> 01:02:45,290
That's actually only about
30% of the actual matter

1121
01:02:45,290 --> 01:02:47,040
in the current universe.

1122
01:02:47,040 --> 01:02:48,670
So I did say this
is the beginning.

1123
01:02:48,670 --> 01:02:51,300
The current universe today
does not obey the equations

1124
01:02:51,300 --> 01:02:52,966
that we've written
down very accurately.

1125
01:02:52,966 --> 01:02:53,820
It's pretty far off.

1126
01:02:53,820 --> 01:02:55,320
But the equations
that we wrote down

1127
01:02:55,320 --> 01:02:56,778
are pretty accurate
for the history

1128
01:02:56,778 --> 01:02:59,700
of our universe from a
period of about 50,000 years

1129
01:02:59,700 --> 01:03:02,520
after the Big Bang up
to about 9 billion years

1130
01:03:02,520 --> 01:03:04,380
after the Big Bang.

1131
01:03:04,380 --> 01:03:05,208
Yes.

1132
01:03:05,208 --> 01:03:07,200
AUDIENCE: Before dark
energy was discovered,

1133
01:03:07,200 --> 01:03:09,136
did they think omega
was [INAUDIBLE]?

1134
01:03:09,136 --> 01:03:09,760
PROFESSOR: Yes.

1135
01:03:09,760 --> 01:03:11,060
At least many people did.

1136
01:03:11,060 --> 01:03:13,120
Before dark energy
was discovered,

1137
01:03:13,120 --> 01:03:15,500
there was a controversy
in the community

1138
01:03:15,500 --> 01:03:17,470
over what we thought omega was.

1139
01:03:17,470 --> 01:03:20,205
Those of us who had
faith in inflation

1140
01:03:20,205 --> 01:03:22,080
believed that omega
would be 1 because that's

1141
01:03:22,080 --> 01:03:24,780
what inflation predicts.

1142
01:03:24,780 --> 01:03:27,420
Astronomers who just had
faith in observations

1143
01:03:27,420 --> 01:03:29,500
believed that omega
was 0.2 or 0.3

1144
01:03:29,500 --> 01:03:31,619
because that's what they saw.

1145
01:03:31,619 --> 01:03:33,160
And the truth ended
up being somewhat

1146
01:03:33,160 --> 01:03:35,130
in between in the
sense that omega

1147
01:03:35,130 --> 01:03:37,600
total we now all agree
is very close to 1

1148
01:03:37,600 --> 01:03:39,052
as inflation predicts.

1149
01:03:39,052 --> 01:03:41,510
But it's still true that the
stuff that the astronomers saw

1150
01:03:41,510 --> 01:03:45,170
at this earlier time did
only add up to 0.2 or 0.3.

1151
01:03:45,170 --> 01:03:47,379
So they correctly estimated
what they were looking at

1152
01:03:47,379 --> 01:03:49,461
and they had no way of
knowing that there was also

1153
01:03:49,461 --> 01:03:51,470
this dark energy component
until it was finally

1154
01:03:51,470 --> 01:03:56,940
discovered in 1998.

1155
01:03:56,940 --> 01:03:57,919
Yes.

1156
01:03:57,919 --> 01:04:00,544
AUDIENCE: If we don't know what
dark energy is, observationally

1157
01:04:00,544 --> 01:04:02,128
how have we been
able to measure that?

1158
01:04:02,128 --> 01:04:04,002
PROFESSOR: How do we
measure it so accurately

1159
01:04:04,002 --> 01:04:05,590
if we don't know
what it is, right?

1160
01:04:05,590 --> 01:04:06,100
Right.

1161
01:04:06,100 --> 01:04:08,140
Well, the answer
is while we're not

1162
01:04:08,140 --> 01:04:09,850
sure what it is, we
actually do think

1163
01:04:09,850 --> 01:04:11,819
we know a lot of its properties.

1164
01:04:11,819 --> 01:04:13,610
And essentially, almost
all properties that

1165
01:04:13,610 --> 01:04:15,360
are relevant to cosmology.

1166
01:04:15,360 --> 01:04:18,204
We just don't know what's
sort of like inside.

1167
01:04:18,204 --> 01:04:19,870
So we know it creates
repulsive gravity.

1168
01:04:19,870 --> 01:04:23,170
We know how much repulsive
gravity it creates.

1169
01:04:23,170 --> 01:04:27,320
And we also know to
reasonable accuracy

1170
01:04:27,320 --> 01:04:29,104
how the dark energy
has been evolving

1171
01:04:29,104 --> 01:04:30,770
with time, which is
really that it's not

1172
01:04:30,770 --> 01:04:32,440
been evolving with time.

1173
01:04:32,440 --> 01:04:34,560
And that determines
what its pressure is.

1174
01:04:34,560 --> 01:04:36,630
It determines, in
fact-- to not evolve

1175
01:04:36,630 --> 01:04:39,370
with time we'll see later
requires the pressure

1176
01:04:39,370 --> 01:04:42,680
to be equal to the negative
of the energy density.

1177
01:04:42,680 --> 01:04:45,270
Pressure is related to how
energies change with time, as I

1178
01:04:45,270 --> 01:04:47,900
mentioned a few minutes
ago in a different context.

1179
01:04:47,900 --> 01:04:50,150
If you have a box that expands
and there's a pressure,

1180
01:04:50,150 --> 01:04:53,620
the pressure does dp
dv work on the box.

1181
01:04:53,620 --> 01:04:55,690
And you can tell how much
the energy in the box

1182
01:04:55,690 --> 01:04:57,876
should change for
a given pressure.

1183
01:04:57,876 --> 01:04:59,500
And we'll do this
more carefully later,

1184
01:04:59,500 --> 01:05:01,347
but to have the energy
not change at all

1185
01:05:01,347 --> 01:05:03,680
requires a pressure, which
is the negative of the energy

1186
01:05:03,680 --> 01:05:04,700
density.

1187
01:05:04,700 --> 01:05:08,220
So we know how much acceleration
the dark energy causes.

1188
01:05:08,220 --> 01:05:10,720
We know to reasonable
accuracy and we

1189
01:05:10,720 --> 01:05:12,650
assume it's true
that the pressure is

1190
01:05:12,650 --> 01:05:14,092
equal to minus the
energy density.

1191
01:05:14,092 --> 01:05:16,550
And that's all you need to know
to calculate how much of it

1192
01:05:16,550 --> 01:05:19,629
you need to account for
that much acceleration.

1193
01:05:19,629 --> 01:05:20,670
And that's how it's done.

1194
01:05:24,772 --> 01:05:25,605
Any other questions?

1195
01:05:52,091 --> 01:05:54,090
OK next thing we want to
do is to actually solve

1196
01:05:54,090 --> 01:05:56,770
this equation for
the easiest case.

1197
01:05:56,770 --> 01:06:00,270
We'll solve it in general
later, but the easiest case

1198
01:06:00,270 --> 01:06:04,760
to solve it is the case
of the critical case.

1199
01:06:12,855 --> 01:06:14,480
We only have a few
minutes, but it only

1200
01:06:14,480 --> 01:06:17,960
takes a few minutes to solve
the equation for this case.

1201
01:06:21,784 --> 01:06:22,880
It should be over here.

1202
01:06:33,649 --> 01:06:41,516
So for the critical case,
it's the case E is equal to 0.

1203
01:06:41,516 --> 01:06:42,890
And therefore, we
just have a dot

1204
01:06:42,890 --> 01:06:46,930
squared is equal to a
constant divided by a.

1205
01:06:46,930 --> 01:06:48,337
And it won't really
matter for us

1206
01:06:48,337 --> 01:06:49,670
right now what this constant is.

1207
01:06:49,670 --> 01:06:51,378
So I don't even have
to keep track of it.

1208
01:06:51,378 --> 01:06:53,724
I'll just write it
as const, C-O-N-S-T.

1209
01:06:53,724 --> 01:06:55,640
And I'll take the square
root of this equation

1210
01:06:55,640 --> 01:06:57,348
because it's easier
to know what a dot is

1211
01:06:57,348 --> 01:06:59,105
than to know what a is.

1212
01:06:59,105 --> 01:07:01,890
Easier to make use of
knowing what a dot is.

1213
01:07:01,890 --> 01:07:04,570
So I can rewrite that equation
as a dot, which I'll now

1214
01:07:04,570 --> 01:07:07,250
write as da dt, to be a little
more explicit about what we're

1215
01:07:07,250 --> 01:07:13,020
talking about, is equal to a
constant over a to the 1/2.

1216
01:07:16,000 --> 01:07:21,479
So this now is the k
equals 0 evolution.

1217
01:07:21,479 --> 01:07:23,270
So this is just the
same Friedmann equation

1218
01:07:23,270 --> 01:07:26,190
rewritten for the
special case k equals 0.

1219
01:07:26,190 --> 01:07:29,060
And now I'm just
going to perform

1220
01:07:29,060 --> 01:07:31,727
the amazingly complicated
manipulation of multiplying

1221
01:07:31,727 --> 01:07:33,560
both sides of the
equation by a to the half.

1222
01:07:36,240 --> 01:07:39,320
So we have a to the half.

1223
01:07:39,320 --> 01:07:41,810
I'm also going to
multiply by dt.

1224
01:07:41,810 --> 01:07:47,920
So a to the half da will be
equal to a constant times dt.

1225
01:07:51,020 --> 01:07:53,140
And now we can just
integrate both sides

1226
01:07:53,140 --> 01:07:55,110
as an indefinite integral.

1227
01:07:55,110 --> 01:07:58,380
And integrating both sides
as an indefinite integral,

1228
01:07:58,380 --> 01:08:03,230
the left-hand side
becomes-- go back over here.

1229
01:08:03,230 --> 01:08:13,620
2/3 a to the 3/2 is
equal to a constant times

1230
01:08:13,620 --> 01:08:20,229
t plus an arbitrary other
constant of integration.

1231
01:08:20,229 --> 01:08:21,670
This is the most
general equation,

1232
01:08:21,670 --> 01:08:23,378
which when differentiated
gives you this.

1233
01:08:28,310 --> 01:08:32,750
And now, this
equation can be solved

1234
01:08:32,750 --> 01:08:35,930
to tell us what a is
as a function of t.

1235
01:08:35,930 --> 01:08:39,370
But before I do that, I'm going
to say something about c prime

1236
01:08:39,370 --> 01:08:40,620
here.

1237
01:08:40,620 --> 01:08:44,800
c prime is allowed
by the integration.

1238
01:08:44,800 --> 01:08:48,090
But remember that when we
defined our scale of t,

1239
01:08:48,090 --> 01:08:49,840
we just started at
some arbitrary time,

1240
01:08:49,840 --> 01:08:52,830
ti, which we didn't
even specify.

1241
01:08:52,830 --> 01:08:55,010
So there's no
particular significance

1242
01:08:55,010 --> 01:08:56,970
to the origin of
time in the equations

1243
01:08:56,970 --> 01:08:59,350
that we've written so far.

1244
01:08:59,350 --> 01:09:02,540
So we're perfectly free to
shift the origin of time

1245
01:09:02,540 --> 01:09:06,109
by just redefining our clocks.

1246
01:09:06,109 --> 01:09:07,920
Cosmic time,
remember, is defined

1247
01:09:07,920 --> 01:09:10,420
in a way which makes it
uniform throughout the universe

1248
01:09:10,420 --> 01:09:12,020
by our construction.

1249
01:09:12,020 --> 01:09:15,729
But we haven't said anything yet
about how to start cosmic time.

1250
01:09:15,729 --> 01:09:18,529
But now, we have a
good way to start it.

1251
01:09:18,529 --> 01:09:20,149
In this model,
there is going to be

1252
01:09:20,149 --> 01:09:22,770
a time at which a
is going to go to 0.

1253
01:09:22,770 --> 01:09:24,700
No matter what we
choose for c prime here,

1254
01:09:24,700 --> 01:09:27,158
there will be some t which will
make the right-hand side 0.

1255
01:09:27,158 --> 01:09:28,450
And therefore, a 0.

1256
01:09:28,450 --> 01:09:30,590
And that's the instant
of the Big Bang.

1257
01:09:30,590 --> 01:09:34,295
That's when everything starts.
a never gets smaller than 0.

1258
01:09:34,295 --> 01:09:35,920
So it's very natural
to take that to be

1259
01:09:35,920 --> 01:09:38,850
defined to be the 0 of time.

1260
01:09:38,850 --> 01:09:41,899
So that's what
we're going to do.

1261
01:09:41,899 --> 01:09:55,840
So we're going to define t
equals 0 to be when a of t

1262
01:09:55,840 --> 01:09:56,450
equals 0.

1263
01:09:59,514 --> 01:10:00,930
That's just a
choice of the origin

1264
01:10:00,930 --> 01:10:02,554
of time, which you're
certainly allowed

1265
01:10:02,554 --> 01:10:06,112
to do without contradicting
anything else that we've said.

1266
01:10:06,112 --> 01:10:08,320
And that means we're just
setting c prime equal to 0.

1267
01:10:08,320 --> 01:10:09,800
So that when that's 0, that's 0.

1268
01:10:14,910 --> 01:10:17,740
So this implies
c prime equals 0.

1269
01:10:17,740 --> 01:10:19,530
And that then
implies-- we could take

1270
01:10:19,530 --> 01:10:21,424
the 2/3 power of this equation.

1271
01:10:21,424 --> 01:10:23,840
And I told you we don't really
care what that constant is.

1272
01:10:23,840 --> 01:10:24,840
So therefore, we
don't really care

1273
01:10:24,840 --> 01:10:26,360
about what that constant is.

1274
01:10:26,360 --> 01:10:30,760
What we get is that a is
equal to some constant,

1275
01:10:30,760 --> 01:10:32,810
not necessarily related
to any of the constants

1276
01:10:32,810 --> 01:10:33,657
we've said so far.

1277
01:10:33,657 --> 01:10:35,240
Although, you can
calculate how it is.

1278
01:10:35,240 --> 01:10:38,090
But some constant times
t to the 2/3 power.

1279
01:10:42,229 --> 01:10:43,770
Or equivalently,
you could just write

1280
01:10:43,770 --> 01:10:47,235
a is proportional to
t to the 2/3, which

1281
01:10:47,235 --> 01:10:48,110
has the same content.

1282
01:10:50,576 --> 01:10:52,200
Now, you might think
you'd want to know

1283
01:10:52,200 --> 01:10:53,866
what the constant of
proportionality is.

1284
01:10:53,866 --> 01:10:55,870
But remember, the constant
of proportionality

1285
01:10:55,870 --> 01:10:58,650
just depends on the
definition of the notch.

1286
01:10:58,650 --> 01:11:01,160
If you want to define the
notch so that a is equal to 1

1287
01:11:01,160 --> 01:11:04,182
today, then you would
care what the constant is.

1288
01:11:04,182 --> 01:11:06,640
If you're willing to just leave
the definition of the notch

1289
01:11:06,640 --> 01:11:10,410
arbitrary, then you don't
care what the constant is.

1290
01:11:10,410 --> 01:11:13,230
And that's the case that
I'll be doing, actually.

1291
01:11:13,230 --> 01:11:16,206
I will not define
a to be 1 today.

1292
01:11:16,206 --> 01:11:18,080
The definition of the
notch is just arbitrary

1293
01:11:18,080 --> 01:11:20,070
as far as the equations
that I'll be writing.

1294
01:11:20,070 --> 01:11:21,903
And therefore, it will
be sufficient to know

1295
01:11:21,903 --> 01:11:24,550
that a is just proportional
to t to the 2/3

1296
01:11:24,550 --> 01:11:29,440
for the flat universe case,
for the critical density case.

1297
01:11:29,440 --> 01:11:31,280
And that's where
we'll stop today.

1298
01:11:31,280 --> 01:11:33,340
And this actually pretty
really covers everything

1299
01:11:33,340 --> 01:11:36,690
through lecture notes three,
which is the same material that

1300
01:11:36,690 --> 01:11:39,170
will be covered on
the quiz next week.

1301
01:11:39,170 --> 01:11:41,090
[INAUDIBLE] does not
seem to have shown up,

1302
01:11:41,090 --> 01:11:45,420
but we'll assume that probably
the review session will

1303
01:11:45,420 --> 01:11:46,760
be next Monday night at 7:30.

1304
01:11:46,760 --> 01:11:50,750
I will check it out and
get back to you by email.