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PROFESSOR: I'd like to
begin building up momentum

9
00:00:24,050 --> 00:00:26,540
by going over what we've
already done to kind of see

10
00:00:26,540 --> 00:00:28,380
how it all fits together.

11
00:00:28,380 --> 00:00:31,840
So I put the last lecture
summarised on transparencies

12
00:00:31,840 --> 00:00:32,390
here.

13
00:00:32,390 --> 00:00:33,820
And we'll go through
them quickly,

14
00:00:33,820 --> 00:00:35,560
and then we'll start
new material, which

15
00:00:35,560 --> 00:00:38,310
I'll be doing on the blackboard.

16
00:00:38,310 --> 00:00:42,790
So we've been studying
this mathematical model

17
00:00:42,790 --> 00:00:47,060
of a universe obeying
Newton's laws of gravity.

18
00:00:47,060 --> 00:00:50,070
We considered simply a
uniform distribution of mass,

19
00:00:50,070 --> 00:00:53,740
initially spherical,
and uniformly expanding,

20
00:00:53,740 --> 00:00:55,940
which means expanding
according to Hubble's law,

21
00:00:55,940 --> 00:01:00,910
with velocities proportional to
the distance from the origin.

22
00:01:00,910 --> 00:01:03,850
And Newton's laws then tell
us how that's going to evolve.

23
00:01:03,850 --> 00:01:06,510
And our job was just to
execute Newton's laws

24
00:01:06,510 --> 00:01:08,810
to calculate how
it would evolve.

25
00:01:08,810 --> 00:01:12,810
And we did that by describing
the evolution in terms

26
00:01:12,810 --> 00:01:16,850
of a function little r, which
is a function of r, i, and t.

27
00:01:16,850 --> 00:01:21,740
And that's the radius at time t
of the shell that was initially

28
00:01:21,740 --> 00:01:23,219
a radius r sub i.

29
00:01:23,219 --> 00:01:25,510
So we're trying to track
every particle in this sphere,

30
00:01:25,510 --> 00:01:27,970
not just the particles
on the surface.

31
00:01:27,970 --> 00:01:32,430
And we want to verify that it
will remain uniform in time.

32
00:01:32,430 --> 00:01:35,110
Matter will not collect
near the edge of the center.

33
00:01:35,110 --> 00:01:38,160
And we discovered it will
remain uniform if we have a 1

34
00:01:38,160 --> 00:01:40,810
over r squared force law,
but for any other force law

35
00:01:40,810 --> 00:01:43,650
it will in fact
not remain uniform.

36
00:01:43,650 --> 00:01:46,080
So we derived the
equations and found

37
00:01:46,080 --> 00:01:48,730
that it obeyed this
scaling relationship, which

38
00:01:48,730 --> 00:01:52,680
is what indicated the
maintenance of uniformity,

39
00:01:52,680 --> 00:01:55,470
wholesale scale by the
same factor, which we then

40
00:01:55,470 --> 00:01:57,490
called a of t.

41
00:01:57,490 --> 00:01:59,250
So the physical
distance of any shell

42
00:01:59,250 --> 00:02:02,950
from the origin at any time
is just equal to a of t times

43
00:02:02,950 --> 00:02:05,510
the initial distance
from the origin.

44
00:02:05,510 --> 00:02:08,720
And furthermore, we were able
to drive equations of motion

45
00:02:08,720 --> 00:02:11,840
for the scale factor.

46
00:02:11,840 --> 00:02:15,110
And it obeys the equations,
which in fact were derived

47
00:02:15,110 --> 00:02:19,640
from general relativity by
Alexander Freedman in 1922,

48
00:02:19,640 --> 00:02:22,164
and therefore they're called
the Freedman equations.

49
00:02:22,164 --> 00:02:24,080
There's a second order
equation, a double dot,

50
00:02:24,080 --> 00:02:27,210
which just tells us how
the expansion slowed down

51
00:02:27,210 --> 00:02:28,770
by Newtonian gravity.

52
00:02:28,770 --> 00:02:30,925
a double dot is negative,
so the expansion

53
00:02:30,925 --> 00:02:33,730
is being slowed by the
gravitational attraction

54
00:02:33,730 --> 00:02:37,130
of every particle in this
spherical distribution

55
00:02:37,130 --> 00:02:39,650
towards every other particle.

56
00:02:39,650 --> 00:02:42,360
And we also were able to
find a first order equation

57
00:02:42,360 --> 00:02:45,550
by integrating the
second order equation.

58
00:02:45,550 --> 00:02:47,494
And the first equation
has this form.

59
00:02:47,494 --> 00:02:49,410
It could be written a
number of different ways

60
00:02:49,410 --> 00:02:50,743
depending on how you arrange it.

61
00:02:50,743 --> 00:02:55,050
But this is the way that
I consider most common.

62
00:02:55,050 --> 00:02:59,720
And many books refer to the
second equation as the Freedman

63
00:02:59,720 --> 00:03:00,632
equation.

64
00:03:00,632 --> 00:03:03,090
Both of these equations were
derived by Alexander Freedman.

65
00:03:03,090 --> 00:03:06,310
I think it's perfect called
both Freedman equations.

66
00:03:06,310 --> 00:03:08,920
But most books do not do that.

67
00:03:08,920 --> 00:03:12,450
And in addition to finding
the equations of evolution

68
00:03:12,450 --> 00:03:16,800
for a of t, we also understood
how rho of t evolves.

69
00:03:16,800 --> 00:03:21,230
And that really is pretty
trivial to begin with.

70
00:03:21,230 --> 00:03:24,470
The Newtonian mass of this
sphere stays the same.

71
00:03:24,470 --> 00:03:28,360
It just spreads out over a
larger volume as a of t grows.

72
00:03:28,360 --> 00:03:30,220
So if the mass stays
the same and the volume

73
00:03:30,220 --> 00:03:33,700
grows as a cubed,
then the density

74
00:03:33,700 --> 00:03:36,820
has to go like one over a cubed.

75
00:03:36,820 --> 00:03:39,320
And this alone could be
written more precisely

76
00:03:39,320 --> 00:03:43,450
as an equation by saying the
easiest way to view the logic

77
00:03:43,450 --> 00:03:45,920
is that this equation
implies that a cubed times

78
00:03:45,920 --> 00:03:48,880
rho is independent of time.

79
00:03:48,880 --> 00:03:50,640
And then once you know
that a cubed times

80
00:03:50,640 --> 00:03:53,280
rho is independent of time,
you can write the equation

81
00:03:53,280 --> 00:03:55,890
in this form,
which if I multiply

82
00:03:55,890 --> 00:03:59,344
a of t cubed to the left
hand side of the equation,

83
00:03:59,344 --> 00:04:03,120
we can just say that a
of t cubed times rho is

84
00:04:03,120 --> 00:04:07,370
the same at time t as it
is at some other time t1.

85
00:04:07,370 --> 00:04:10,710
Now in lecture last time,
we wrote this equation

86
00:04:10,710 --> 00:04:16,450
where t1 was t sub i the initial
time, and a of t sub i was one.

87
00:04:16,450 --> 00:04:17,826
So we didn't
include that factor.

88
00:04:17,826 --> 00:04:19,325
So this is slightly
more general way

89
00:04:19,325 --> 00:04:20,839
of writing it than
we did last time.

90
00:04:20,839 --> 00:04:24,610
But it still has no more
content than the statement

91
00:04:24,610 --> 00:04:27,740
that rho of t falls as
1 over a cubed of t.

92
00:04:33,120 --> 00:04:37,590
We introduced a special set
of units to describe this.

93
00:04:37,590 --> 00:04:38,780
Question, yes?

94
00:04:38,780 --> 00:04:41,450
AUDIENCE: I had a question
about the Freedman equations

95
00:04:41,450 --> 00:04:42,881
and what we're
calling e squared,

96
00:04:42,881 --> 00:04:44,172
and the interpretation of that.

97
00:04:46,970 --> 00:04:48,950
PROFESSOR: What we're
calling what squared?

98
00:04:48,950 --> 00:04:49,783
AUDIENCE: h squared.

99
00:04:49,783 --> 00:04:52,246
PROFESSOR: h squared, yes.

100
00:04:52,246 --> 00:04:54,860
AUDIENCE: So in the
spherical universe,

101
00:04:54,860 --> 00:04:58,960
we showed that if e was
positive that corresponded

102
00:04:58,960 --> 00:05:02,806
to an open universe that
would expand forever.

103
00:05:02,806 --> 00:05:09,414
So doing the p set for this week
for the cylindrical universe,

104
00:05:09,414 --> 00:05:13,190
I think we found that that
universe would collapse.

105
00:05:13,190 --> 00:05:14,071
PROFESSOR: Yes.

106
00:05:14,071 --> 00:05:16,526
AUDIENCE: E was also
positive for that one.

107
00:05:16,526 --> 00:05:19,226
So I was just wondering
about the interpretation

108
00:05:19,226 --> 00:05:19,963
of [INAUDIBLE]

109
00:05:23,540 --> 00:05:25,982
PROFESSOR: Right, well I'll
be coming to that issue later.

110
00:05:25,982 --> 00:05:27,190
Let me come back to that, OK.

111
00:05:27,190 --> 00:05:29,920
Since I haven't
introduced e yet.

112
00:05:29,920 --> 00:05:31,926
And the slide is here.

113
00:05:31,926 --> 00:05:33,050
So an interesting question.

114
00:05:33,050 --> 00:05:34,120
We'll come back to that.

115
00:05:38,470 --> 00:05:41,890
So to describe this
system of equations,

116
00:05:41,890 --> 00:05:44,790
I like to introduce this
notion of a notch, which

117
00:05:44,790 --> 00:05:50,760
is a special unit used only to
measure co-moving coordinates.

118
00:05:50,760 --> 00:05:54,790
And in this case, r sub i
is our co-moving coordinate.

119
00:05:54,790 --> 00:05:56,650
As our shells
move, we label them

120
00:05:56,650 --> 00:05:58,610
all by where they
were a time t i.

121
00:05:58,610 --> 00:06:00,150
We don't change those labels.

122
00:06:00,150 --> 00:06:02,680
So those are the
co-moving coordinates,

123
00:06:02,680 --> 00:06:05,900
a coordinate system that
expands with the universe.

124
00:06:05,900 --> 00:06:08,650
So instead of
measuring r i in meters

125
00:06:08,650 --> 00:06:10,950
or any other physical
length, I like

126
00:06:10,950 --> 00:06:12,990
to measure them in a
new unit called a notch,

127
00:06:12,990 --> 00:06:15,250
just to keep things separate.

128
00:06:15,250 --> 00:06:18,270
And a notch is
defined so that a of t

129
00:06:18,270 --> 00:06:23,030
is measured in meters per
notch, and at time t i,

130
00:06:23,030 --> 00:06:24,730
one notch equals 1 meter.

131
00:06:24,730 --> 00:06:26,430
But at different
times, the relationship

132
00:06:26,430 --> 00:06:28,055
between notches and
meters is different

133
00:06:28,055 --> 00:06:32,920
because it's given by this time
dependent scale factor of t.

134
00:06:32,920 --> 00:06:35,940
If [INAUDIBLE] works to have
things depend on these units,

135
00:06:35,940 --> 00:06:38,600
one find that this new
quantity that we introduced,

136
00:06:38,600 --> 00:06:41,220
little k, which all we know
is that it's a constant,

137
00:06:41,220 --> 00:06:44,290
has the units of 1
over a notch squared.

138
00:06:44,290 --> 00:06:45,900
And that has some
relevance to us,

139
00:06:45,900 --> 00:06:48,920
because it means that we can
make little k have any value we

140
00:06:48,920 --> 00:06:51,880
want by choosing different
definitions for the notch.

141
00:06:51,880 --> 00:06:53,520
And the notch is up for grabs.

142
00:06:53,520 --> 00:06:55,640
We are just inventing
a unit to use

143
00:06:55,640 --> 00:06:58,710
to measure our co-moving
coordinate system.

144
00:06:58,710 --> 00:07:00,250
So we can always
adjust the meaning

145
00:07:00,250 --> 00:07:04,380
of a notch so that k has
whatever value we want.

146
00:07:04,380 --> 00:07:07,250
As long as we can't change its
sign by changing the units.

147
00:07:07,250 --> 00:07:09,820
And if it's zero we can't
change it by changing its units.

148
00:07:09,820 --> 00:07:11,920
As long as non-zero we
can make it any value

149
00:07:11,920 --> 00:07:16,817
we want, and in fact that is
often used in many textbooks.

150
00:07:16,817 --> 00:07:18,400
And that's I guess
what I want to talk

151
00:07:18,400 --> 00:07:23,750
about next, the conventions
that are used to define a of t.

152
00:07:23,750 --> 00:07:28,310
And for us, I'm going to treat
this notch as being arbitrary.

153
00:07:28,310 --> 00:07:33,090
We've defined the notch
originally so that a of t i

154
00:07:33,090 --> 00:07:36,250
was one meters per
notch at time t i,

155
00:07:36,250 --> 00:07:39,390
and that gave the notch more
or less a specific meaning.

156
00:07:39,390 --> 00:07:43,570
But the specific meaning
depends on what t i is.

157
00:07:43,570 --> 00:07:45,700
One can take the
same relationship

158
00:07:45,700 --> 00:07:48,325
and view it as simply
a definition of t i.

159
00:07:48,325 --> 00:07:50,450
t i is the time
at which the scale

160
00:07:50,450 --> 00:07:53,520
factor is one meter per notch.

161
00:07:53,520 --> 00:07:55,935
We take the definition of t
i and we can let the notch

162
00:07:55,935 --> 00:07:58,210
be anything we want, and
there will be some time t

163
00:07:58,210 --> 00:08:01,680
i that will still make
that statement true.

164
00:08:01,680 --> 00:08:05,280
So what I want to do basically
is to think of this equation,

165
00:08:05,280 --> 00:08:07,700
a of t i equals 1
meter per notch,

166
00:08:07,700 --> 00:08:09,920
not as a definition
of a notch, which

167
00:08:09,920 --> 00:08:12,790
I want to leave arbitrary, but
rather as a definition of t sub

168
00:08:12,790 --> 00:08:13,650
i.

169
00:08:13,650 --> 00:08:15,715
And after defining
t sub i, I want

170
00:08:15,715 --> 00:08:17,560
to just forget about t sub i.

171
00:08:17,560 --> 00:08:21,340
t sub i in fact will not
enter our equations anywhere.

172
00:08:21,340 --> 00:08:25,250
So we don't need to
remember its definition.

173
00:08:25,250 --> 00:08:27,980
I decided like the cubit.

174
00:08:27,980 --> 00:08:30,770
All of us know that the
cubit is some unit distance,

175
00:08:30,770 --> 00:08:33,900
but we don't care what it is
because we never use cubits.

176
00:08:33,900 --> 00:08:34,730
Same thing here.

177
00:08:34,730 --> 00:08:36,447
We'll just never use t sub i.

178
00:08:36,447 --> 00:08:38,030
And since we're never
going to use it.

179
00:08:38,030 --> 00:08:41,250
We don't need to remember
how it was initially defined.

180
00:08:41,250 --> 00:08:43,789
It's only of
historical interest.

181
00:08:43,789 --> 00:08:46,080
So the bottom line then is
simply that for us the notch

182
00:08:46,080 --> 00:08:49,710
is just an undefined
unit of distance

183
00:08:49,710 --> 00:08:52,490
in the co-moving
coordinate system.

184
00:08:52,490 --> 00:08:54,190
Other people use
different definitions.

185
00:08:54,190 --> 00:08:56,760
Ryden, for example
uses the definition

186
00:08:56,760 --> 00:09:01,000
where a of t sub, a of the
present time is equal to 1.

187
00:09:01,000 --> 00:09:04,220
And we would interpret that
as meaning one meter per notch

188
00:09:04,220 --> 00:09:05,106
today.

189
00:09:05,106 --> 00:09:06,730
And that's a perfectly
good definition.

190
00:09:06,730 --> 00:09:08,380
And we can use it
whenever we want,

191
00:09:08,380 --> 00:09:11,140
because our notch is
initially undefined.

192
00:09:11,140 --> 00:09:13,490
So that allows us the
freedom to define it

193
00:09:13,490 --> 00:09:17,510
in any particular problem
in any way that we want.

194
00:09:17,510 --> 00:09:19,370
Many other books take
advantage of the fact

195
00:09:19,370 --> 00:09:23,290
that this quantity k has units
of inverse notch squared,

196
00:09:23,290 --> 00:09:25,100
even though they don't say that.

197
00:09:25,100 --> 00:09:29,380
But that means you could rescale
the co-moving coordinate system

198
00:09:29,380 --> 00:09:32,360
to make k equal to
whatever value you want.

199
00:09:32,360 --> 00:09:34,750
So in many books k is always
equal to plus or minus 1

200
00:09:34,750 --> 00:09:36,920
if it's non-zero.

201
00:09:36,920 --> 00:09:39,570
The co-moving coordinate
systems is just scaled.

202
00:09:39,570 --> 00:09:43,680
We would do is we scaling of the
notch to make k have magnitude

203
00:09:43,680 --> 00:09:44,180
1.

204
00:09:47,390 --> 00:09:51,540
OK, having derived
these equations,

205
00:09:51,540 --> 00:09:54,020
the next step is
to go about asking

206
00:09:54,020 --> 00:09:57,700
what do the solutions to
the equations look like.

207
00:09:57,700 --> 00:09:59,850
And that's where things
start getting interestingly

208
00:09:59,850 --> 00:10:03,090
when we start getting some
real nontrivial results.

209
00:10:03,090 --> 00:10:05,590
The Freedman equation,
the first order one,

210
00:10:05,590 --> 00:10:08,100
could be rewritten this way.

211
00:10:08,100 --> 00:10:10,820
It's just a rewriting
of rearranging things.

212
00:10:10,820 --> 00:10:14,470
And I used here the fact
that rho times a cubed

213
00:10:14,470 --> 00:10:15,830
was a constant.

214
00:10:15,830 --> 00:10:19,040
If we took our original form
of the Freedman equation,

215
00:10:19,040 --> 00:10:22,280
this would be rho
sub i times a cubed

216
00:10:22,280 --> 00:10:24,940
of t sub i, which would be 1.

217
00:10:24,940 --> 00:10:27,300
But knowing that rho times
a cubed is a constant,

218
00:10:27,300 --> 00:10:30,180
we can let t argument
here be anything we want,

219
00:10:30,180 --> 00:10:33,420
which is what I'll do, just
to emphasize that we don't

220
00:10:33,420 --> 00:10:36,080
care anymore what t sub i was.

221
00:10:36,080 --> 00:10:37,970
It really has disappeared
from our problem.

222
00:10:37,970 --> 00:10:41,350
It was just our way
of getting started.

223
00:10:41,350 --> 00:10:43,000
So this equation holds.

224
00:10:43,000 --> 00:10:46,920
And we can use it to discuss
how the different classes

225
00:10:46,920 --> 00:10:48,760
of solutions will behave.

226
00:10:48,760 --> 00:10:51,550
And what we can see very
quickly from this equation

227
00:10:51,550 --> 00:10:53,760
is that the behavior
of the solutions

228
00:10:53,760 --> 00:10:57,350
will depend crucially
on the sign of k.

229
00:10:57,350 --> 00:10:59,716
And it's useful
here to remember,

230
00:10:59,716 --> 00:11:01,340
although we don't
need to know anything

231
00:11:01,340 --> 00:11:03,750
more than this
proportionality, that k

232
00:11:03,750 --> 00:11:07,500
is proportional to the negative
of something that we call e.

233
00:11:07,500 --> 00:11:08,920
And that the thing
that we call e

234
00:11:08,920 --> 00:11:12,567
is related to the overall
energy of this thing.

235
00:11:12,567 --> 00:11:14,400
So we'll keep that in
the back of our minds.

236
00:11:14,400 --> 00:11:16,110
But it's really
only for intuition.

237
00:11:16,110 --> 00:11:17,526
Everything that
we're going to say

238
00:11:17,526 --> 00:11:19,230
follows directly
from this equation,

239
00:11:19,230 --> 00:11:23,560
where we don't know
anything about e.

240
00:11:23,560 --> 00:11:25,380
There are three
types of solutions,

241
00:11:25,380 --> 00:11:27,670
depending on whether k is
positive, negative, or zero,

242
00:11:27,670 --> 00:11:30,530
and those are the options
for what k might be.

243
00:11:30,530 --> 00:11:32,310
It's a real number.

244
00:11:32,310 --> 00:11:34,420
So first we consider
with the solutions

245
00:11:34,420 --> 00:11:39,070
where k is less than zero, which
means e is greater than zero.

246
00:11:39,070 --> 00:11:41,520
And e being greater
than zero means

247
00:11:41,520 --> 00:11:44,270
the system has more
energy than zero.

248
00:11:44,270 --> 00:11:46,560
And in this case,
zero energy would

249
00:11:46,560 --> 00:11:50,550
correspond to having all the
particles infinitely far away.

250
00:11:50,550 --> 00:11:52,420
So the potential
energies would be zero.

251
00:11:52,420 --> 00:11:54,740
And all particles the rest,
so the kinetic energies

252
00:11:54,740 --> 00:11:56,520
would be zero.

253
00:11:56,520 --> 00:11:59,910
So in particular, zero energy
would correspond to the system

254
00:11:59,910 --> 00:12:04,520
being completely dispersed,
no longer compact.

255
00:12:04,520 --> 00:12:08,270
And in this case, our system
has more energy than that.

256
00:12:08,270 --> 00:12:10,130
And more energy
than that means it

257
00:12:10,130 --> 00:12:14,109
can blow outward without limit.

258
00:12:14,109 --> 00:12:15,900
And we see that directly
from the equation.

259
00:12:15,900 --> 00:12:18,950
If the second term has
a negative value of k,

260
00:12:18,950 --> 00:12:21,190
then the second term
itself is positive.

261
00:12:21,190 --> 00:12:23,890
And the first term is
also always positive.

262
00:12:23,890 --> 00:12:25,742
And that means
that a dot squared,

263
00:12:25,742 --> 00:12:27,450
no matter what happens
to the first term,

264
00:12:27,450 --> 00:12:29,408
is always at least bigger
than the second term,

265
00:12:29,408 --> 00:12:31,320
which is a constant.

266
00:12:31,320 --> 00:12:33,320
And if a dot is always
bigger than some constant

267
00:12:33,320 --> 00:12:35,360
means that a grows indefinitely.

268
00:12:35,360 --> 00:12:37,530
And that's called
an open universe.

269
00:12:37,530 --> 00:12:39,445
And it goes on
expanding forever.

270
00:12:42,040 --> 00:12:44,780
Second case we'll
consider is k greater

271
00:12:44,780 --> 00:12:48,580
than zero, which corresponds
to e less than zero.

272
00:12:48,580 --> 00:12:51,880
And that means since 0
corresponds to the system

273
00:12:51,880 --> 00:12:55,320
being completely dispersed,
e less than 0 means

274
00:12:55,320 --> 00:12:56,960
the system does not
have enough energy

275
00:12:56,960 --> 00:13:00,130
to ever become
completely dispersed.

276
00:13:00,130 --> 00:13:02,100
So we'll have some maximum size.

277
00:13:02,100 --> 00:13:06,170
And the maximum size follows
immediately from this equation.

278
00:13:06,170 --> 00:13:08,389
a dot squared has
to be positive.

279
00:13:08,389 --> 00:13:09,430
It can ever get negative.

280
00:13:09,430 --> 00:13:13,190
It can become zero, but
it can never get negative.

281
00:13:13,190 --> 00:13:21,960
In this case, the minus k
c squared term is negative.

282
00:13:21,960 --> 00:13:24,970
And that means that if this
term gets to be too small,

283
00:13:24,970 --> 00:13:27,550
the sum will be negative,
which is not possible.

284
00:13:27,550 --> 00:13:30,150
So this term cannot
get to be too small.

285
00:13:30,150 --> 00:13:31,775
And since a of t is
in the denominator,

286
00:13:31,775 --> 00:13:34,530
that means a of t cannot
get to be too big.

287
00:13:34,530 --> 00:13:36,730
And you can easily
derive the inequality

288
00:13:36,730 --> 00:13:39,140
that a has to obey for
the right hand side

289
00:13:39,140 --> 00:13:40,306
to always be positive.

290
00:13:40,306 --> 00:13:41,930
And in that case you
[INAUDIBLE] a max,

291
00:13:41,930 --> 00:13:44,346
which is what you would get
if you just set the right hand

292
00:13:44,346 --> 00:13:46,675
side equal to 0, given
by that expression.

293
00:13:46,675 --> 00:13:49,152
a can never get bigger than
that, because if it did,

294
00:13:49,152 --> 00:13:50,610
the right hand side
of the equation

295
00:13:50,610 --> 00:13:54,210
become negative,
which is not possible.

296
00:13:54,210 --> 00:13:56,600
So this universe will
reach a maximum size,

297
00:13:56,600 --> 00:13:58,820
which we just calculated,
and then [INAUDIBLE]

298
00:13:58,820 --> 00:14:00,640
will come back.

299
00:14:00,640 --> 00:14:03,790
So we already have a very
nontrivial result here.

300
00:14:03,790 --> 00:14:06,720
Given a description of
a universe of this type,

301
00:14:06,720 --> 00:14:08,889
we can calculate
how big it will get

302
00:14:08,889 --> 00:14:10,430
before it turns
around and collapses.

303
00:14:13,990 --> 00:14:16,410
And this of universe
ultimately undergoes

304
00:14:16,410 --> 00:14:20,180
a big crunch when it collapses,
where the word big crunch

305
00:14:20,180 --> 00:14:23,750
was made as an analogy
to the phrase big bang.

306
00:14:23,750 --> 00:14:26,890
It's called a closed universe.

307
00:14:26,890 --> 00:14:31,080
And then finally, we've now
considered the k less than zero

308
00:14:31,080 --> 00:14:32,310
and k greater than zero.

309
00:14:32,310 --> 00:14:35,102
There are many cases
when k equals zero.

310
00:14:35,102 --> 00:14:36,560
And that's called
the critical mass

311
00:14:36,560 --> 00:14:39,570
density or critical universe.

312
00:14:39,570 --> 00:14:42,480
And we can figure
out what it means

313
00:14:42,480 --> 00:14:44,370
in terms of the mass density.

314
00:14:44,370 --> 00:14:47,580
This again is our
Freedman equation.

315
00:14:47,580 --> 00:14:49,880
If k is zero, this
last term is absent.

316
00:14:49,880 --> 00:14:53,220
So we just have a relationship
between rho and h, the Hubble

317
00:14:53,220 --> 00:14:55,250
expansion rate.

318
00:14:55,250 --> 00:14:56,290
And we can solve that.

319
00:14:56,290 --> 00:14:59,190
And the value of rho which
satisfies that equation

320
00:14:59,190 --> 00:15:01,820
is called the critical density.

321
00:15:01,820 --> 00:15:03,940
As the density is equal
to the critical density,

322
00:15:03,940 --> 00:15:06,540
it means that k is zero.

323
00:15:06,540 --> 00:15:09,180
And that's called a flat case.

324
00:15:09,180 --> 00:15:12,790
We'll figure out in a
minute how it evolves.

325
00:15:12,790 --> 00:15:15,340
It's not clear if it will be
collapsed or stop or what.

326
00:15:15,340 --> 00:15:16,340
But we'll find out soon.

327
00:15:16,340 --> 00:15:18,010
It's really on the
borderline between something

328
00:15:18,010 --> 00:15:19,860
which we know expands
and something which

329
00:15:19,860 --> 00:15:22,110
we know collapses.

330
00:15:22,110 --> 00:15:23,930
And it's called a flat universe.

331
00:15:23,930 --> 00:15:26,430
The word flat suggests
geometry, and we'll

332
00:15:26,430 --> 00:15:28,090
be learning about that later.

333
00:15:28,090 --> 00:15:30,130
General relativity tells
us a little bit more

334
00:15:30,130 --> 00:15:31,240
than we learn here.

335
00:15:31,240 --> 00:15:33,010
These equations are
all exactly true

336
00:15:33,010 --> 00:15:35,090
in the context of
general relativity.

337
00:15:35,090 --> 00:15:37,290
But general relativity
also tells us

338
00:15:37,290 --> 00:15:38,800
that these equations
are connected

339
00:15:38,800 --> 00:15:41,270
to the geometry of space.

340
00:15:41,270 --> 00:15:46,940
And only for this critical mass
density is the space Euclidean.

341
00:15:46,940 --> 00:15:52,040
The word flat here is used
in the sense of Euclidean.

342
00:15:52,040 --> 00:15:55,900
So to summarize what we've said,
if the mass density is bigger

343
00:15:55,900 --> 00:15:58,350
than this critical value,
we get a closed universe,

344
00:15:58,350 --> 00:16:01,060
which reaches a maximum
size and then collapses.

345
00:16:01,060 --> 00:16:03,390
If the mass density is less
than the critical density,

346
00:16:03,390 --> 00:16:06,680
we get an open universe, which
goes on expanding forever.

347
00:16:06,680 --> 00:16:08,561
And if the mass
density is exactly

348
00:16:08,561 --> 00:16:10,810
equal to the critical density,
that's called a flat k.

349
00:16:10,810 --> 00:16:14,260
So we'll explore a little
bit more in a minute.

350
00:16:14,260 --> 00:16:16,870
It's interesting to know what
this critical density is.

351
00:16:16,870 --> 00:16:19,140
It depends on the
expansion rate.

352
00:16:19,140 --> 00:16:22,530
But the expansion rate has now
been measured quite accurately.

353
00:16:22,530 --> 00:16:25,760
So if I take the
value of 67.3, which

354
00:16:25,760 --> 00:16:29,600
is the value that comes from
the Planck satellite combining

355
00:16:29,600 --> 00:16:33,180
their results with
several other experiments,

356
00:16:33,180 --> 00:16:35,380
they get a value of 67.3.

357
00:16:35,380 --> 00:16:37,740
And when we'll put
that into this formula,

358
00:16:37,740 --> 00:16:39,950
the number one gets
is 8.4 times 10

359
00:16:39,950 --> 00:16:43,950
to the minus 30 grams
per centimeter cubed,

360
00:16:43,950 --> 00:16:49,010
which is only about 5 proton
masses per cubic meter.

361
00:16:49,010 --> 00:16:52,540
It's an unbelievably empty
universe that we live in.

362
00:16:52,540 --> 00:16:54,020
I say the universe
that we live in

363
00:16:54,020 --> 00:16:56,290
because in fact the mass
density of our universe

364
00:16:56,290 --> 00:16:58,610
is very close to
this critical value.

365
00:16:58,610 --> 00:17:04,000
It's equal to it to within about
half of a percent we now know.

366
00:17:04,000 --> 00:17:05,501
An important
definition, which we'll

367
00:17:05,501 --> 00:17:07,125
be continuing to use
through the course

368
00:17:07,125 --> 00:17:08,689
and which cosmologists
always use,

369
00:17:08,689 --> 00:17:12,670
is omega, where omega
means capital Greek omega.

370
00:17:12,670 --> 00:17:15,246
And that's just defined
to be the actual density

371
00:17:15,246 --> 00:17:17,394
of the universe,
whatever it is, divided

372
00:17:17,394 --> 00:17:18,435
by this critical density.

373
00:17:22,329 --> 00:17:25,319
OK, the one remaining thing
that we did last time,

374
00:17:25,319 --> 00:17:27,220
and we'll summarize
this and go on,

375
00:17:27,220 --> 00:17:30,080
is we figured out
what the evolution is

376
00:17:30,080 --> 00:17:32,520
for a flat universe.

377
00:17:32,520 --> 00:17:34,750
And we can do that just by
solving the differential

378
00:17:34,750 --> 00:17:38,350
equation, which is a fairly
simple differential equation.

379
00:17:38,350 --> 00:17:40,520
If we leave out the k
term, the Freedman equation

380
00:17:40,520 --> 00:17:43,440
becomes a dot over a
squared is equal to 8 pi

381
00:17:43,440 --> 00:17:47,440
g over 3 times rho And we
know how rho depends on a.

382
00:17:47,440 --> 00:17:50,397
It's proportional
to 1 over a cubed.

383
00:17:50,397 --> 00:17:52,230
So the right hand side
here is some constant

384
00:17:52,230 --> 00:17:54,890
divided by a cubed.

385
00:17:54,890 --> 00:17:57,880
And by just
rearranging things, we

386
00:17:57,880 --> 00:18:00,670
can rewrite that
as da over dt is

387
00:18:00,670 --> 00:18:04,040
equal to some constant
over a to the 1/2.

388
00:18:04,040 --> 00:18:07,480
In this slide I use this
symbol const a number of times.

389
00:18:07,480 --> 00:18:09,480
Those constants are not
all equal to each other.

390
00:18:09,480 --> 00:18:11,145
But they're all
constants, which you

391
00:18:11,145 --> 00:18:12,892
can keep track of
if you wanted to.

392
00:18:12,892 --> 00:18:14,600
But there's no need
to keep track of them

393
00:18:14,600 --> 00:18:18,240
because they have no bearing
on the answer anyway.

394
00:18:18,240 --> 00:18:23,270
So I just called
these constants const.

395
00:18:23,270 --> 00:18:25,039
So again, da over dt
equals const over a

396
00:18:25,039 --> 00:18:27,330
to the one half, which is an
easy differential equation

397
00:18:27,330 --> 00:18:29,090
to solve.

398
00:18:29,090 --> 00:18:32,780
We just multiply through by
dt and a to the one half,

399
00:18:32,780 --> 00:18:37,790
and write it in as a
to the one half times

400
00:18:37,790 --> 00:18:41,680
da equals constant times
dt, which can easily

401
00:18:41,680 --> 00:18:43,390
be integrated both
sides of the equation

402
00:18:43,390 --> 00:18:45,530
as indefinite integrals.

403
00:18:45,530 --> 00:18:48,840
And then we get 2/3
times a to the 3/2

404
00:18:48,840 --> 00:18:51,672
is equal to a constant times
t, where this constant happens

405
00:18:51,672 --> 00:18:54,130
to be the same as that constant,
for whatever that's worth,

406
00:18:54,130 --> 00:18:57,900
plus a new constant of
integration, c prime.

407
00:18:57,900 --> 00:19:00,940
Then we argue that
the value of c prime

408
00:19:00,940 --> 00:19:03,530
depends on how we
synchronize our clocks.

409
00:19:03,530 --> 00:19:09,060
If we reset our clock by
changing t by a constant

410
00:19:09,060 --> 00:19:12,180
that would change
the value of c prime.

411
00:19:12,180 --> 00:19:13,830
And since we haven't
said anything yet

412
00:19:13,830 --> 00:19:17,080
about how we're
going set our clock,

413
00:19:17,080 --> 00:19:18,590
we're perfectly
free at this point

414
00:19:18,590 --> 00:19:23,200
to just say that we're going to
set our clock so that t equals

415
00:19:23,200 --> 00:19:27,840
0 corresponds to the same
time that a is equal to zero.

416
00:19:27,840 --> 00:19:29,930
The initial singularity
of the universe

417
00:19:29,930 --> 00:19:33,700
starts from zero
size with a as zero.

418
00:19:33,700 --> 00:19:36,170
So if we do that when
a is zero, t is zero.

419
00:19:36,170 --> 00:19:38,270
That means that c prime is zero.

420
00:19:38,270 --> 00:19:40,440
So setting c prime
equal to zero is just

421
00:19:40,440 --> 00:19:43,970
a choice of how
to set our clocks.

422
00:19:43,970 --> 00:19:44,930
So we do that.

423
00:19:44,930 --> 00:19:47,390
And then we can take the
2/3 power of this equation.

424
00:19:47,390 --> 00:19:49,590
And since constants are
just constants that we don't

425
00:19:49,590 --> 00:19:51,320
care about, we
end up with a of t

426
00:19:51,320 --> 00:19:54,460
is proportional to t to the 2/3.

427
00:19:54,460 --> 00:19:57,440
And proportional is
all we need to know,

428
00:19:57,440 --> 00:19:59,210
because the constant
of proportionality

429
00:19:59,210 --> 00:20:01,175
would depend on the
definition of the notch.

430
00:20:01,175 --> 00:20:02,550
And we haven't
defined the notch.

431
00:20:02,550 --> 00:20:04,410
And we don't need
to define the notch.

432
00:20:04,410 --> 00:20:08,470
And so anything that depends on
the constant of proportionality

433
00:20:08,470 --> 00:20:11,610
will never enter
any physical answer.

434
00:20:11,610 --> 00:20:16,260
It will be relevant to
questions like how many notches

435
00:20:16,260 --> 00:20:18,410
are a certain
distance on your map.

436
00:20:18,410 --> 00:20:21,010
But for any physical
answer, we don't care.

437
00:20:21,010 --> 00:20:22,730
If we want to talk
about our map,

438
00:20:22,730 --> 00:20:25,271
we could just define a notch to
be whatever we want it to be.

439
00:20:29,440 --> 00:20:32,320
OK, that's the
end of my summary.

440
00:20:32,320 --> 00:20:34,100
Any questions about any of that?

441
00:20:34,100 --> 00:20:34,140
I'm sorry.

442
00:20:34,140 --> 00:20:35,890
I didn't come back to
answer your question

443
00:20:35,890 --> 00:20:37,240
about the cylinder.

444
00:20:37,240 --> 00:20:40,624
The cylinder problem does
end up always giving you

445
00:20:40,624 --> 00:20:42,790
a closed universe no matter
what the parameters are.

446
00:20:42,790 --> 00:20:44,800
It always collapses.

447
00:20:44,800 --> 00:20:50,870
And even though its
energy as you compute it

448
00:20:50,870 --> 00:20:54,680
would turn out to be positive,
the difference though

449
00:20:54,680 --> 00:20:58,090
is that for the case
of the cylinder,

450
00:20:58,090 --> 00:21:01,480
the potential energy
does not go to 0

451
00:21:01,480 --> 00:21:03,635
as the thing becomes
infinitely big.

452
00:21:03,635 --> 00:21:07,840
The potential energy has a
logarithmic diversion in it.

453
00:21:07,840 --> 00:21:09,789
So the zero is just
placed differently

454
00:21:09,789 --> 00:21:11,330
for the case of the
cylinder problem.

455
00:21:14,450 --> 00:21:20,950
So it ends up being closed no
matter how fast it's expanding.

456
00:21:24,070 --> 00:21:24,877
Yes?

457
00:21:24,877 --> 00:21:25,752
AUDIENCE: [INAUDIBLE]

458
00:21:33,840 --> 00:21:36,580
PROFESSOR: Well that is true.

459
00:21:36,580 --> 00:21:38,080
Certainly the
differential equations

460
00:21:38,080 --> 00:21:39,538
break down when a
is equal to zero.

461
00:21:39,538 --> 00:21:42,360
The mass density
goes to infinity.

462
00:21:42,360 --> 00:21:45,900
But we're still certainly
free to set our clocks

463
00:21:45,900 --> 00:21:49,510
so that the equations
themselves, when extrapolated

464
00:21:49,510 --> 00:21:51,920
to 0, would have the
property that a equals zero

465
00:21:51,920 --> 00:21:53,470
when t is equal to zero.

466
00:21:53,470 --> 00:21:54,910
It's certainly
correct, and I was

467
00:21:54,910 --> 00:21:56,701
going to be talking
about this in a minute,

468
00:21:56,701 --> 00:22:00,580
that you should not trust
these equations back

469
00:22:00,580 --> 00:22:01,990
to t equals zero.

470
00:22:01,990 --> 00:22:04,880
But that doesn't
stop you for choosing

471
00:22:04,880 --> 00:22:06,815
whatever you want
as your origin of t.

472
00:22:06,815 --> 00:22:08,440
And if these are the
equations we have,

473
00:22:08,440 --> 00:22:10,790
the simplest way to deal
with these equations

474
00:22:10,790 --> 00:22:14,590
is to use the zero of t when
equations say that a was zero.

475
00:22:19,067 --> 00:22:19,900
Any other questions?

476
00:22:32,530 --> 00:22:37,620
OK, in that case, we will
leave the slides for a bit

477
00:22:37,620 --> 00:22:39,800
and proceed on the blackboard.

478
00:22:56,845 --> 00:22:58,220
So so far I think
we have learned

479
00:22:58,220 --> 00:23:00,750
two varied nontrivial things
from this calculation.

480
00:23:00,750 --> 00:23:02,954
We learned how to calculate
the critical density.

481
00:23:02,954 --> 00:23:05,620
We learned how to calculate what
density the universe would have

482
00:23:05,620 --> 00:23:07,390
to have so it would re-collapse.

483
00:23:07,390 --> 00:23:11,190
And we've also learned that
if the universe is closed,

484
00:23:11,190 --> 00:23:16,985
we can calculate how large it
will get before it collapses.

485
00:23:16,985 --> 00:23:19,110
So those are two very
nontrivial results coming out

486
00:23:19,110 --> 00:23:20,850
of this Newtonian calculation.

487
00:23:20,850 --> 00:23:22,620
The next question I
want to ask is still

488
00:23:22,620 --> 00:23:25,000
about the flat universe.

489
00:23:25,000 --> 00:23:27,870
It's a fairly trivial
extension of what we have.

490
00:23:27,870 --> 00:23:29,810
Given this formula
for a of t, I would

491
00:23:29,810 --> 00:23:34,122
like to calculate the
age of a flat universe.

492
00:23:34,122 --> 00:23:36,330
If you were living in a flat
universe that was matter

493
00:23:36,330 --> 00:23:39,070
dominated like the
one we're describing,

494
00:23:39,070 --> 00:23:41,780
how would you determine
how old it was?

495
00:23:41,780 --> 00:23:45,440
And the answer is that
it's immediately related

496
00:23:45,440 --> 00:23:47,680
to the Hubble expansion
rate, and the age

497
00:23:47,680 --> 00:23:50,820
can be expressed in terms of
the Hubble expansion rate.

498
00:23:50,820 --> 00:23:59,540
To see that-- so we're
calculating the age of a matter

499
00:23:59,540 --> 00:24:01,125
dominated flat universe.

500
00:24:12,794 --> 00:24:14,210
And these age
calculations will be

501
00:24:14,210 --> 00:24:15,793
extending as we go
through the course.

502
00:24:15,793 --> 00:24:18,250
So in the end, we'll
have the full calculation

503
00:24:18,250 --> 00:24:21,180
for the real model that
we have of our universe.

504
00:24:21,180 --> 00:24:22,720
But you have to start somewhere.

505
00:24:22,720 --> 00:24:24,178
So we're starting
with just a flat,

506
00:24:24,178 --> 00:24:26,190
matter dominated universe.

507
00:24:26,190 --> 00:24:32,600
We know that a of t is equal
to some constant, which

508
00:24:32,600 --> 00:24:40,955
I will call little v
times t to the 2/3 power.

509
00:24:40,955 --> 00:24:42,580
Previously I just
used proportional to,

510
00:24:42,580 --> 00:24:44,910
but now it's just more
convenient to give a name

511
00:24:44,910 --> 00:24:46,490
to the constant of
proportionality.

512
00:24:46,490 --> 00:24:48,090
We'll never need
to know what it is.

513
00:24:48,090 --> 00:24:50,781
But v is some constant
of proportionality.

514
00:24:50,781 --> 00:24:52,780
This by the way already
tells us something else,

515
00:24:52,780 --> 00:24:54,696
which wasn't obvious
from the beginning, which

516
00:24:54,696 --> 00:24:59,140
is our flat universe does go
on expanding forever, somewhat

517
00:24:59,140 --> 00:25:00,630
like an open universe.

518
00:25:00,630 --> 00:25:06,000
An important difference
is that if you calculate

519
00:25:06,000 --> 00:25:09,420
da dt for the open
universe, that approaches

520
00:25:09,420 --> 00:25:12,220
a constant as time
goes to infinity.

521
00:25:12,220 --> 00:25:14,620
That is, the universe
keeps on expanding

522
00:25:14,620 --> 00:25:17,610
at some-- minimal rate forever.

523
00:25:17,610 --> 00:25:22,620
In this case, if you calculate
the adt, it goes to 0 at times.

524
00:25:22,620 --> 00:25:25,079
So the flat universe
expands forever,

525
00:25:25,079 --> 00:25:26,870
but at an ever, ever,
ever decreasing rate.

526
00:25:29,810 --> 00:25:33,370
We know how to
relate a of t to h.

527
00:25:33,370 --> 00:25:35,810
The Hubble expansion
rate is a dot over a,

528
00:25:35,810 --> 00:25:38,010
we learned a long time ago.

529
00:25:38,010 --> 00:25:41,170
And if we know what a
is, we know what this is.

530
00:25:41,170 --> 00:25:46,055
So this is just 2 over 3t.

531
00:25:53,930 --> 00:25:55,920
The 2/3 coming from
differentiating the 2/3.

532
00:25:55,920 --> 00:25:58,150
And that gives you a
t to the minus 1/3,

533
00:25:58,150 --> 00:25:59,840
but then you're
also dividing by a,

534
00:25:59,840 --> 00:26:03,100
which turns that t to the minus
1/3 to a t to the minus 1.

535
00:26:03,100 --> 00:26:06,570
So you get 2 over 3t
is the final answer.

536
00:26:06,570 --> 00:26:09,430
So this is the relationship
now between h and t,

537
00:26:09,430 --> 00:26:11,890
and the question we asked
is how to calculate the age.

538
00:26:11,890 --> 00:26:13,330
The age is t.

539
00:26:13,330 --> 00:26:16,960
This is all defined as where t
is equal to 0 at the big bang.

540
00:26:16,960 --> 00:26:21,770
So t really is the time
elapsed since the big bang.

541
00:26:21,770 --> 00:26:25,250
So we're left immediately
with a simple result,

542
00:26:25,250 --> 00:26:30,255
that t is equal to
2/3 times h inverse.

543
00:26:38,620 --> 00:26:41,200
Now this result immediately
makes rigorous contact

544
00:26:41,200 --> 00:26:44,260
with something that we talked
about in vague terms earlier.

545
00:26:44,260 --> 00:26:49,600
If you are so unfortunate
as to badly mis-measure h,

546
00:26:49,600 --> 00:26:52,550
you can get a pretty wild
answer for the h of your model

547
00:26:52,550 --> 00:26:53,950
universe.

548
00:26:53,950 --> 00:26:57,620
And Hubble mis-measured
h by about a factor

549
00:26:57,620 --> 00:27:00,305
of seven comparative to
present modern values.

550
00:27:00,305 --> 00:27:02,560
He got h to be too high.

551
00:27:02,560 --> 00:27:05,130
His value of h was too high
by a factor of about seven,

552
00:27:05,130 --> 00:27:07,410
and that meant that when big
bang theorists calculated

553
00:27:07,410 --> 00:27:09,280
the age of the
universe were getting

554
00:27:09,280 --> 00:27:12,250
ages that were too low
by a factor of seven.

555
00:27:12,250 --> 00:27:13,940
And in particular
that meant they

556
00:27:13,940 --> 00:27:16,790
were getting ages of the
order of 2 billion years.

557
00:27:16,790 --> 00:27:20,650
And even back in
the 1920s and '30s,

558
00:27:20,650 --> 00:27:22,640
there was sufficient
geological evidence

559
00:27:22,640 --> 00:27:25,000
that the Earth was older
than 2 billion years.

560
00:27:25,000 --> 00:27:27,010
There was also
significant understanding

561
00:27:27,010 --> 00:27:28,640
of stellar evolution,
that starts

562
00:27:28,640 --> 00:27:31,640
took longer to evolve
than 2 billion years.

563
00:27:31,640 --> 00:27:34,550
So the big bang model was
in trouble from the start,

564
00:27:34,550 --> 00:27:38,610
largely because of this
very serious mis-measurement

565
00:27:38,610 --> 00:27:40,990
in the early days of the
Hubble expansion rate.

566
00:27:51,550 --> 00:27:54,010
If we put in some
numbers, this of course

567
00:27:54,010 --> 00:27:57,820
is not an accurate model for
our universe, we now know.

568
00:27:57,820 --> 00:28:01,450
Our universe is now currently
dark energy dominated.

569
00:28:01,450 --> 00:28:04,140
But nonetheless, just
to see how this works,

570
00:28:04,140 --> 00:28:07,290
we can put in numbers.

571
00:28:07,290 --> 00:28:20,640
So h, using this Planck value
that I quoted earlier 67.3

572
00:28:20,640 --> 00:28:29,720
plus or minus 1.2 kilometers
per second per megaparsec.

573
00:28:39,570 --> 00:28:41,320
To be able to get
an answer in years,

574
00:28:41,320 --> 00:28:46,110
one has to be able to convert
this into inverse years.

575
00:28:46,110 --> 00:28:48,810
h is actually an inverse time.

576
00:28:48,810 --> 00:28:53,520
And a useful conversion
number is 1 over 10

577
00:28:53,520 --> 00:28:57,370
to the 10 years is
equal to almost 100,

578
00:28:57,370 --> 00:29:06,920
but not quite, 97.8 kilometers
per second per megaparsec,

579
00:29:06,920 --> 00:29:10,640
which allows you to convert
these Hubble expansion rate

580
00:29:10,640 --> 00:29:14,420
units into inverse years.

581
00:29:14,420 --> 00:29:21,990
And using that one finds that
the age of the universe using

582
00:29:21,990 --> 00:29:37,770
the 2/3 h inverse formula
is 9.7 plus or minus

583
00:29:37,770 --> 00:29:40,625
0.2 billion years.

584
00:29:51,810 --> 00:29:57,360
Now this number played a role
in the fairly recent history

585
00:29:57,360 --> 00:29:58,910
of cosmology.

586
00:29:58,910 --> 00:30:02,170
Before 1998, when
the dark energy

587
00:30:02,170 --> 00:30:05,430
was discovered, which kind of
settled all these questions,

588
00:30:05,430 --> 00:30:10,790
but before 1998, we thought the
universe what matter dominated.

589
00:30:10,790 --> 00:30:11,790
It might have been open.

590
00:30:11,790 --> 00:30:12,990
It didn't have to be flat.

591
00:30:12,990 --> 00:30:14,610
That was debated.

592
00:30:14,610 --> 00:30:17,330
It looked more open and flat.

593
00:30:17,330 --> 00:30:20,830
But some of us wanted to
hold out for a flat universe

594
00:30:20,830 --> 00:30:23,310
because we were
fans of inflation

595
00:30:23,310 --> 00:30:26,290
and admired inflation's
other successes,

596
00:30:26,290 --> 00:30:28,690
which we'll learn about
later in the course,

597
00:30:28,690 --> 00:30:31,470
and thought therefore that
the universe should be flat,

598
00:30:31,470 --> 00:30:34,400
and wanted to try to
reconcile all this.

599
00:30:34,400 --> 00:30:40,270
And the problem was that like
cosmology in the '20s and '30s

600
00:30:40,270 --> 00:30:42,250
when the age of the
universe that you calculate

601
00:30:42,250 --> 00:30:47,920
was too young, the same thing
was happening here before 1998,

602
00:30:47,920 --> 00:30:50,190
when we thought the universe
was matter dominated.

603
00:30:50,190 --> 00:30:54,040
This is the age that we got,
modified a little bit by having

604
00:30:54,040 --> 00:30:57,040
different values of h,
but pretty close to this.

605
00:30:57,040 --> 00:30:59,550
At the same time,
there were calculations

606
00:30:59,550 --> 00:31:01,650
about how old the
universe had to be

607
00:31:01,650 --> 00:31:04,560
to accommodate the oldest stars.

608
00:31:04,560 --> 00:31:07,760
And in the lecture notes
I quote a particular paper

609
00:31:07,760 --> 00:31:09,453
by Krauss and Chaboyer.

610
00:31:20,300 --> 00:31:24,170
Lawrence Krauss is an
MIT PhD by the way.

611
00:31:24,170 --> 00:31:29,504
And what they decided by
studying globular clusters,

612
00:31:29,504 --> 00:31:31,170
which are supposed
to contain the oldest

613
00:31:31,170 --> 00:31:39,430
stars that astronomers know
about, that the oldest stars,

614
00:31:39,430 --> 00:31:56,060
they said, had an age of
12.6 plus 3.4 minus 2.2

615
00:31:56,060 --> 00:31:56,670
billion years.

616
00:32:00,830 --> 00:32:05,340
And this is a 95%
confidence number.

617
00:32:15,110 --> 00:32:17,680
That is, instead of using
one sigma, which are often

618
00:32:17,680 --> 00:32:21,200
used to quote errors,
these are two sigma errors,

619
00:32:21,200 --> 00:32:24,580
which have probabilities
of being wrong by only 5%

620
00:32:24,580 --> 00:32:28,670
if things work properly
according to the statistics.

621
00:32:28,670 --> 00:32:32,470
So then we're going to
think about 95% limits.

622
00:32:32,470 --> 00:32:34,810
So they were willing to
take the lower limit here,

623
00:32:34,810 --> 00:32:45,750
which was 10.4.

624
00:32:45,750 --> 00:32:54,350
So they got a minimum
age of 10.4 billion years

625
00:32:54,350 --> 00:32:56,390
for the oldest stars.

626
00:32:56,390 --> 00:32:59,070
But they also argued that the
stars really could not possibly

627
00:32:59,070 --> 00:33:02,460
start to form until
about 0.8 billion years

628
00:33:02,460 --> 00:33:03,835
into the history
of the universe.

629
00:33:24,790 --> 00:33:29,190
And doing a little bit of
simple arithmetic there,

630
00:33:29,190 --> 00:33:31,420
they decided that the
minimum possible age

631
00:33:31,420 --> 00:33:34,690
for the universe at the
95% confidence level,

632
00:33:34,690 --> 00:34:07,180
would be 10.4 plus 0.8 or 11.2.

633
00:34:07,180 --> 00:34:14,679
And 11.2 is older than
9.7, and by a fair number

634
00:34:14,679 --> 00:34:17,699
of standard deviations,
although [INAUDIBLE]

635
00:34:17,699 --> 00:34:21,730
were somewhat bigger in
1998 than they are now.

636
00:34:21,730 --> 00:34:25,210
But in any case, this led to I
think what people at the time

637
00:34:25,210 --> 00:34:28,570
regarded as a tension between
the age of the universe

638
00:34:28,570 --> 00:34:32,320
question and the possibility
of having a flat universe.

639
00:34:32,320 --> 00:34:34,710
A flat universe
seemed to produce

640
00:34:34,710 --> 00:34:38,050
ages that were too young
to be consistent with what

641
00:34:38,050 --> 00:34:40,070
we knew about stars.

642
00:34:40,070 --> 00:34:43,170
Yet there was still
evidence in terms

643
00:34:43,170 --> 00:34:46,199
of the desired to make
inflationary models

644
00:34:46,199 --> 00:34:49,580
work to indicate that
the universe was flat.

645
00:34:49,580 --> 00:34:53,500
And actually it's also
true that by 1998, there

646
00:34:53,500 --> 00:34:56,942
was evidence from the
Kobe satellite measuring

647
00:34:56,942 --> 00:34:59,150
fluctuation in the cosmic
background radiation, which

648
00:34:59,150 --> 00:35:02,180
also suggested
that omega was one,

649
00:35:02,180 --> 00:35:03,950
that the universe with flat.

650
00:35:03,950 --> 00:35:07,160
So things didn't fit together
very well before 1998.

651
00:35:07,160 --> 00:35:10,250
And this was at the
crux of the argument.

652
00:35:10,250 --> 00:35:12,960
It all got settled with the
discovery of the dark energy,

653
00:35:12,960 --> 00:35:16,496
which we'll learn how to
account for in a few weeks.

654
00:35:16,496 --> 00:35:18,620
When we includes dark energy
in these calculations,

655
00:35:18,620 --> 00:35:22,810
the ages go up,
and everything does

656
00:35:22,810 --> 00:35:24,557
come into accord
with the idea now

657
00:35:24,557 --> 00:35:26,265
that the age of the
universe is estimated

658
00:35:26,265 --> 00:35:28,570
at 13.8 billion years.

659
00:35:28,570 --> 00:35:31,070
And that's consistent with the
Hubble expansion rate given

660
00:35:31,070 --> 00:35:34,030
here, as long as one
has dark energy and not

661
00:35:34,030 --> 00:35:36,110
just relativistic matter.

662
00:35:39,000 --> 00:35:40,600
OK any question about that?

663
00:35:46,621 --> 00:35:48,620
OK next thing I wanted
to say a little bit about

664
00:35:48,620 --> 00:35:51,950
is what exactly we
mean here by age.

665
00:35:51,950 --> 00:35:56,260
And the question of what we
mean by age does of course

666
00:35:56,260 --> 00:35:59,850
connect to the question of how
do we think it actually began.

667
00:35:59,850 --> 00:36:03,730
Because age means time since
the beginning presumably.

668
00:36:03,730 --> 00:36:05,730
And the answer really
is that we don't

669
00:36:05,730 --> 00:36:08,170
know how the universe began.

670
00:36:08,170 --> 00:36:12,740
The big bang is often said to be
the beginning of the universe.

671
00:36:12,740 --> 00:36:14,942
But I would argue that
we don't know that,

672
00:36:14,942 --> 00:36:16,400
and I think most
cosmologists would

673
00:36:16,400 --> 00:36:18,550
agree that we don't know that.

674
00:36:18,550 --> 00:36:21,220
As we extrapolate
backwards, we're

675
00:36:21,220 --> 00:36:24,540
using our knowledge of physics
that we measure in laboratories

676
00:36:24,540 --> 00:36:27,910
and physics that we confirm
with other astrophysical type

677
00:36:27,910 --> 00:36:31,350
observations, but nonetheless,
as we get closer and closer

678
00:36:31,350 --> 00:36:35,910
to t equals zero, the mass
density in this approximation

679
00:36:35,910 --> 00:36:39,430
grows like 1 over the
scale factor cubed, which

680
00:36:39,430 --> 00:36:42,260
means it goes up
arbitrarily large.

681
00:36:42,260 --> 00:36:46,720
Later we'll learn that when the
universe gets to be very young,

682
00:36:46,720 --> 00:36:50,600
we have to include radiation and
not just relativistic matter.

683
00:36:50,600 --> 00:36:53,152
The dark energy is actually
totally unimportant

684
00:36:53,152 --> 00:36:54,360
when we go backwards in time.

685
00:36:54,360 --> 00:36:57,900
It becomes important when
we go forwards in time.

686
00:36:57,900 --> 00:37:01,110
But when we put in radiation,
it does not solve this problem.

687
00:37:01,110 --> 00:37:04,800
The universe still
requires a mass density

688
00:37:04,800 --> 00:37:09,140
that goes to infinity as
we approach t equals zero.

689
00:37:09,140 --> 00:37:11,470
People had wondered
whether maybe that's

690
00:37:11,470 --> 00:37:13,890
an idealization associated
with our approximation

691
00:37:13,890 --> 00:37:16,720
of exact homogeneity
and isotropy

692
00:37:16,720 --> 00:37:18,940
which after all do break
down at some level.

693
00:37:18,940 --> 00:37:22,870
Maybe if we put in a slightly
inhomogeneous and slightly

694
00:37:22,870 --> 00:37:24,980
anisotropic universe
and ran it backwards,

695
00:37:24,980 --> 00:37:28,300
maybe the mass density
would not climb to infinity.

696
00:37:28,300 --> 00:37:31,700
Hawking proved that
that was not a way out.

697
00:37:31,700 --> 00:37:33,785
The universe would
become singular.

698
00:37:33,785 --> 00:37:36,160
He didn't really prove the
mass density went to infinity,

699
00:37:36,160 --> 00:37:39,020
but he proved it becomes
singular in other ways

700
00:37:39,020 --> 00:37:44,100
as t went to zero no matter
what geometry you put in.

701
00:37:44,100 --> 00:37:49,790
So the bottom line is that
classical general relativity

702
00:37:49,790 --> 00:37:53,020
does predict a
singularity of some sort

703
00:37:53,020 --> 00:37:55,740
as we extrapolate
backwards in time.

704
00:37:55,740 --> 00:38:01,610
But the important
qualification is

705
00:38:01,610 --> 00:38:04,880
that once the mass density goes
far above any mass densities

706
00:38:04,880 --> 00:38:06,750
that we've had any
experience with,

707
00:38:06,750 --> 00:38:09,500
we really don't know how
things are going to behave.

708
00:38:09,500 --> 00:38:12,230
And we don't really know how
classical general relativity

709
00:38:12,230 --> 00:38:13,790
holds in that regime.

710
00:38:13,790 --> 00:38:17,440
And in fact, we have
very strong ways

711
00:38:17,440 --> 00:38:20,170
is to believe that classical
general relativity will not

712
00:38:20,170 --> 00:38:21,850
hold in that regime.

713
00:38:21,850 --> 00:38:23,720
Because classical
general relativity

714
00:38:23,720 --> 00:38:27,010
is after all a classical
theory, a theory in which one

715
00:38:27,010 --> 00:38:28,990
talks about fields that
have definite values

716
00:38:28,990 --> 00:38:33,460
at definite times
without incorporating

717
00:38:33,460 --> 00:38:37,950
the ideas of the uncertainty
principle of quantum theory.

718
00:38:37,950 --> 00:38:42,050
So nobody in fact knows how to
build a theory in which matter

719
00:38:42,050 --> 00:38:47,020
is quantized and gravity
is not quantized.

720
00:38:47,020 --> 00:38:51,930
So all the smart money bets on
the fact that gravity is really

721
00:38:51,930 --> 00:38:54,770
a quantum theory, even though
we don't yet quite understand it

722
00:38:54,770 --> 00:38:55,785
as a quantum theory.

723
00:38:55,785 --> 00:38:57,910
And that as we go back in
time, the quantum effects

724
00:38:57,910 --> 00:39:00,200
become more and more important.

725
00:39:00,200 --> 00:39:03,440
So there's no reason to trust
classical general relativity

726
00:39:03,440 --> 00:39:06,970
as we approach t equals
zero, and therefore no reason

727
00:39:06,970 --> 00:39:10,840
to really take this
singularity seriously.

728
00:39:10,840 --> 00:39:14,550
Furthermore, we'll even see
at the end of the course

729
00:39:14,550 --> 00:39:20,130
that most inflationary
scenarios imply

730
00:39:20,130 --> 00:39:22,335
that what we call
the big bang is not

731
00:39:22,335 --> 00:39:24,770
a unique beginning
of the universe.

732
00:39:24,770 --> 00:39:27,430
But rather it now
seems pretty likely,

733
00:39:27,430 --> 00:39:30,250
although we sure don't
know, that our universe

734
00:39:30,250 --> 00:39:32,570
is part of a multiverse,
where we are just

735
00:39:32,570 --> 00:39:36,890
one universe in the multiverse,
and that the big bang, what

736
00:39:36,890 --> 00:39:40,650
we call the big bang, is really
our big bang, the beginning

737
00:39:40,650 --> 00:39:42,900
of our pocket universe.

738
00:39:42,900 --> 00:39:45,770
But before that the space
of time already existed.

739
00:39:45,770 --> 00:39:48,270
The big bang is
just a nucleation

740
00:39:48,270 --> 00:39:49,410
of a phase transition.

741
00:39:49,410 --> 00:39:51,960
It's not really a beginning.

742
00:39:51,960 --> 00:39:54,670
And that there was
other stuff that

743
00:39:54,670 --> 00:39:58,600
existed before what
we call the big bang.

744
00:39:58,600 --> 00:40:01,460
I should add though that the
inflationary scenario does not

745
00:40:01,460 --> 00:40:04,920
provide any answer whatever
to the question of how did it

746
00:40:04,920 --> 00:40:06,950
all ultimately begin.

747
00:40:06,950 --> 00:40:08,660
That's still very
much an open question.

748
00:40:08,660 --> 00:40:10,470
And it's clear that
inflation by itself

749
00:40:10,470 --> 00:40:14,460
does not even offer an
answer to that question.

750
00:40:14,460 --> 00:40:16,610
So when we talk about
the age of the universe,

751
00:40:16,610 --> 00:40:18,100
what are we talking about?

752
00:40:18,100 --> 00:40:20,750
What we're talking about is
the age, the amount of time

753
00:40:20,750 --> 00:40:22,500
that has elapsed, since
this event that we

754
00:40:22,500 --> 00:40:23,499
call the big bang.

755
00:40:23,499 --> 00:40:26,040
The big bang might not have been
the beginning of everything,

756
00:40:26,040 --> 00:40:28,590
but certainly the evidence
is overwhelmingly strong

757
00:40:28,590 --> 00:40:29,799
that it happened.

758
00:40:29,799 --> 00:40:31,340
And we could talk
about how much time

759
00:40:31,340 --> 00:40:33,260
has elapsed since it happened.

760
00:40:33,260 --> 00:40:36,550
And that's the t that we're
trying to calculate here.

761
00:40:36,550 --> 00:40:39,620
And it will be offset
by a tiny amount

762
00:40:39,620 --> 00:40:42,760
by changing the history in
the very, very early stages,

763
00:40:42,760 --> 00:40:45,690
but only by a tiny
fraction of a second.

764
00:40:45,690 --> 00:40:47,470
So the uncertainties
of quantum gravity

765
00:40:47,470 --> 00:40:52,181
are not important in calculating
the age of the universe.

766
00:40:52,181 --> 00:40:54,180
Although they are important
in interpreting what

767
00:40:54,180 --> 00:40:55,410
you mean by it.

768
00:40:55,410 --> 00:40:59,580
I think we don't really mean
the origin of space and time,

769
00:40:59,580 --> 00:41:02,030
but rather simply
the time has elapsed

770
00:41:02,030 --> 00:41:05,190
since the event
called the big bang.

771
00:41:05,190 --> 00:41:06,460
OK any questions about that?

772
00:41:09,320 --> 00:41:14,320
All right, next event
I want to talk about

773
00:41:14,320 --> 00:41:17,930
is that if the
universe as we know

774
00:41:17,930 --> 00:41:21,720
it began some 13.8
billion years ago to use

775
00:41:21,720 --> 00:41:26,550
the actual current number, that
would mean that light could

776
00:41:26,550 --> 00:41:31,210
only have traveled some finite
distance since the beginning

777
00:41:31,210 --> 00:41:34,640
of the universe as
we know it, meaning

778
00:41:34,640 --> 00:41:37,526
the universe since the big
bang, and that would mean there

779
00:41:37,526 --> 00:41:40,510
would be some maximum distance
that we could see things.

780
00:41:40,510 --> 00:41:42,844
And beyond that there
might be more things,

781
00:41:42,844 --> 00:41:45,010
but they'd be things for
which the light has not yet

782
00:41:45,010 --> 00:41:46,960
had time to reach us.

783
00:41:46,960 --> 00:41:50,380
So that's an important
concept in cosmology,

784
00:41:50,380 --> 00:41:52,720
the maximum distance
that you could see.

785
00:41:52,720 --> 00:41:54,535
It goes by the name
the horizon distance.

786
00:41:58,964 --> 00:42:00,755
If you're sailing on
the ocean, the horizon

787
00:42:00,755 --> 00:42:03,780
is the furthest
thing you can see.

788
00:42:03,780 --> 00:42:06,950
So what we want to do now
is to calculate this horizon

789
00:42:06,950 --> 00:42:10,810
distance in the model
that we now understand,

790
00:42:10,810 --> 00:42:14,490
the flat matter
dominated universe.

791
00:42:32,819 --> 00:42:34,485
And this of course
is also a calculation

792
00:42:34,485 --> 00:42:36,164
that we will be
generalizing as we

793
00:42:36,164 --> 00:42:37,580
go through the
course learning how

794
00:42:37,580 --> 00:42:40,330
to treat more and more
complicated cases and more

795
00:42:40,330 --> 00:42:43,440
and more realistic cases.

796
00:42:43,440 --> 00:42:50,000
So this horizon distance, I
should define it more exactly.

797
00:42:50,000 --> 00:42:58,800
It's the present
distance, and maybe I

798
00:42:58,800 --> 00:43:01,750
should even stick
the word proper here.

799
00:43:04,707 --> 00:43:06,790
I've been usually using
the word physical distance

800
00:43:06,790 --> 00:43:10,000
to refer to the
distance to an object

801
00:43:10,000 --> 00:43:12,450
as it would be measured
by rulers, which

802
00:43:12,450 --> 00:43:15,920
are each along the way
moving with the velocity

803
00:43:15,920 --> 00:43:18,750
of the average matter
at those locations.

804
00:43:18,750 --> 00:43:20,910
That is also called
the proper distance,

805
00:43:20,910 --> 00:43:24,680
which is [INAUDIBLE] calls it.

806
00:43:24,680 --> 00:43:26,860
And this horizon
distance is defined

807
00:43:26,860 --> 00:43:45,610
as the present proper distance
of the most distant objects

808
00:43:45,610 --> 00:43:55,160
that can be seen, limited
only by the speed of light.

809
00:44:10,859 --> 00:44:13,400
So we pretend we have telescopes
that are incredibly powerful

810
00:44:13,400 --> 00:44:16,764
and could see anything,
any light that

811
00:44:16,764 --> 00:44:17,870
could have reached us.

812
00:44:17,870 --> 00:44:20,040
But we know the light has
a finite propagation time,

813
00:44:20,040 --> 00:44:22,980
so take that into account in
talking about this horizon

814
00:44:22,980 --> 00:44:23,480
distance.

815
00:44:26,070 --> 00:44:29,730
OK so what is the horizon
distance going to be?

816
00:44:29,730 --> 00:44:35,300
Well remember that the
coordinate velocity of light

817
00:44:35,300 --> 00:44:42,440
is equal to c divided by a of t.

818
00:44:42,440 --> 00:44:45,030
I should maybe start by
saying before we get down

819
00:44:45,030 --> 00:44:48,450
to details, that you might think
naively that the answer should

820
00:44:48,450 --> 00:44:51,130
be the speed of light times
the age of the universe.

821
00:44:51,130 --> 00:44:53,370
That's how far light can travel.

822
00:44:53,370 --> 00:44:56,010
And so if the universe was
static and just appeared

823
00:44:56,010 --> 00:44:59,030
a certain time in the past,
that would be the right answer.

824
00:44:59,030 --> 00:45:00,770
I would just start
off at the beginning

825
00:45:00,770 --> 00:45:02,930
and travel at speed c.

826
00:45:02,930 --> 00:45:05,000
But it's more complicated
because the universe

827
00:45:05,000 --> 00:45:06,520
has been expanding all along.

828
00:45:06,520 --> 00:45:09,220
And it started out with
a scale factor of 0.

829
00:45:09,220 --> 00:45:10,900
And furthermore, what
we're looking for

830
00:45:10,900 --> 00:45:12,604
is present distance
in these objects,

831
00:45:12,604 --> 00:45:14,270
and the objects of
course have continued

832
00:45:14,270 --> 00:45:17,596
to move after the light that
we're now seeing has left them.

833
00:45:17,596 --> 00:45:18,970
So it's a little
more complicated

834
00:45:18,970 --> 00:45:20,590
than just c times
the speed of light.

835
00:45:20,590 --> 00:45:22,650
And we'll see what
it is by tracking it

836
00:45:22,650 --> 00:45:24,380
through very carefully.

837
00:45:24,380 --> 00:45:26,410
We'll imagine a light
beam that leaves

838
00:45:26,410 --> 00:45:29,370
from some distant object.

839
00:45:29,370 --> 00:45:35,020
And the light beam will get the
furthest if it leaves earliest.

840
00:45:35,020 --> 00:45:36,860
So we want the earliest
possible light beam

841
00:45:36,860 --> 00:45:38,950
that could have left
this distant object.

842
00:45:38,950 --> 00:45:41,241
And that would be a light
beam that left at literally t

843
00:45:41,241 --> 00:45:42,574
equals zero.

844
00:45:42,574 --> 00:45:44,240
So the light beam
leaves the object at t

845
00:45:44,240 --> 00:45:46,570
equals zero and
reaches us today.

846
00:45:46,570 --> 00:45:48,790
And we want to know how
far away is that object?

847
00:45:48,790 --> 00:45:50,690
That's the furthest
object that we could see,

848
00:45:50,690 --> 00:45:53,060
objects for which we can
only see the light that

849
00:45:53,060 --> 00:45:56,122
was emitted from the
object at t equals zero.

850
00:45:56,122 --> 00:45:58,330
So we're going to use our
co-moving coordinate system

851
00:45:58,330 --> 00:45:59,180
to trace things.

852
00:45:59,180 --> 00:46:01,940
All calculations are done
most straightforwardly

853
00:46:01,940 --> 00:46:04,190
in the co-moving
coordinate system.

854
00:46:04,190 --> 00:46:06,780
And we know that light travels
in the co-moving coordinate

855
00:46:06,780 --> 00:46:11,050
system at the rate of dx dt is
equal to c divided by a of t.

856
00:46:11,050 --> 00:46:13,580
And this really just says
that as the light passes

857
00:46:13,580 --> 00:46:16,190
any observer in this
co-moving coordinate system,

858
00:46:16,190 --> 00:46:19,750
the observer sees speed c, as
special relativity tells us

859
00:46:19,750 --> 00:46:21,340
he must.

860
00:46:21,340 --> 00:46:24,680
But we need to convert it
into notches per second

861
00:46:24,680 --> 00:46:27,180
to be able to trace it through
the co-moving coordinate

862
00:46:27,180 --> 00:46:28,210
system.

863
00:46:28,210 --> 00:46:31,150
And the relationship between
notches and meters is a of t.

864
00:46:31,150 --> 00:46:33,570
So a of t is just a
conversion factor here

865
00:46:33,570 --> 00:46:38,250
that converts the local speed
of this light pulse from meters

866
00:46:38,250 --> 00:46:41,840
per second to notches
per second, which

867
00:46:41,840 --> 00:46:46,010
is what dx dt has
to be measured in.

868
00:46:46,010 --> 00:46:50,240
So this will be the speed.

869
00:46:50,240 --> 00:46:52,860
That means that the
maximum distance that light

870
00:46:52,860 --> 00:46:55,665
will travel, still
measured in notches

871
00:46:55,665 --> 00:46:57,880
in co-moving
coordinates, will just

872
00:46:57,880 --> 00:46:59,170
be the integral of the speed.

873
00:46:59,170 --> 00:47:03,460
The integral of dx
dt is just delta x.

874
00:47:03,460 --> 00:47:09,440
So it would be the
integral of dx dt dt,

875
00:47:09,440 --> 00:47:15,390
integrating from 0 up to
t zero, the present time.

876
00:47:27,790 --> 00:47:30,870
Now this is not the final
answer that we're interested in.

877
00:47:30,870 --> 00:47:33,530
We want to know the
present physical distance

878
00:47:33,530 --> 00:47:36,020
or the present proper
distance of this object

879
00:47:36,020 --> 00:47:39,040
that's the furthest
object that we can see.

880
00:47:39,040 --> 00:47:41,220
And the way to go from
co-moving distances

881
00:47:41,220 --> 00:47:44,419
to physical businesses is to
multiply by the scale factor.

882
00:47:44,419 --> 00:47:45,960
And we are interested
in the distance

883
00:47:45,960 --> 00:47:48,820
today, so we multiply
by the scale factor

884
00:47:48,820 --> 00:47:51,650
today, the present value
of the scale factor.

885
00:47:51,650 --> 00:47:53,630
So the answers to
our problem, which

886
00:47:53,630 --> 00:47:58,625
I will call l sub p or
sub [INAUDIBLE] of t.

887
00:48:02,060 --> 00:48:05,320
I want to get the word horizon
into the subscript someplace.

888
00:48:05,320 --> 00:48:13,830
So I will call it l sub
p comma horizon, which

889
00:48:13,830 --> 00:48:18,110
means the physical distance
to the horizon at time t

890
00:48:18,110 --> 00:48:22,110
is just equal to x
max that we have here

891
00:48:22,110 --> 00:48:25,360
times the present value
of the scale factor.

892
00:48:25,360 --> 00:48:32,460
So it's a of 2
[INAUDIBLE] times x max.

893
00:48:32,460 --> 00:48:37,040
Or the final formula, just
substituting in x max,

894
00:48:37,040 --> 00:48:43,160
will be of a of t naught
times the integral from 0

895
00:48:43,160 --> 00:48:46,690
to t naught of dx dt.

896
00:48:46,690 --> 00:48:48,705
I'm going to substitute
c over a of t.

897
00:49:06,610 --> 00:49:09,460
Let me just remind
you that [INAUDIBLE]

898
00:49:09,460 --> 00:49:11,007
variable integration
should never

899
00:49:11,007 --> 00:49:13,090
have the same symbol as
the limits of integration,

900
00:49:13,090 --> 00:49:14,548
because that just
causes confusion.

901
00:49:14,548 --> 00:49:16,930
They're never really
the same thing.

902
00:49:16,930 --> 00:49:18,740
So I called the
limits t sub zero.

903
00:49:18,740 --> 00:49:22,710
So it's perfectly OK to call
the variable of integration t.

904
00:49:22,710 --> 00:49:25,880
In the notes, I call
the value of the time

905
00:49:25,880 --> 00:49:27,630
that we want to
calculate this as t,

906
00:49:27,630 --> 00:49:30,110
and then I use t prime for
the variable of integration.

907
00:49:30,110 --> 00:49:31,651
Whatever you do,
you should make sure

908
00:49:31,651 --> 00:49:33,129
that those are not the same.

909
00:49:33,129 --> 00:49:34,670
There's one variable
that corresponds

910
00:49:34,670 --> 00:49:39,140
to the variable that varies
from the initial time

911
00:49:39,140 --> 00:49:40,120
to the final time.

912
00:49:40,120 --> 00:49:41,578
And then there's
also another value

913
00:49:41,578 --> 00:49:43,690
that represents the final time.

914
00:49:43,690 --> 00:49:47,590
OK, so now all we have
to do is plug in a of t

915
00:49:47,590 --> 00:49:50,060
is a constant times t to
the 2/3 into this formula

916
00:49:50,060 --> 00:49:51,800
and we have our answer.

917
00:49:51,800 --> 00:49:54,430
Notice that it does obey
an important property.

918
00:49:54,430 --> 00:49:57,650
There's an a in the numerator
and an a in the denominator.

919
00:49:57,650 --> 00:50:08,490
And that means that when we
put in the formula for a of t,

920
00:50:08,490 --> 00:50:12,000
the constant of proportionality
b will cancel out.

921
00:50:12,000 --> 00:50:15,310
And it must, as the
constants proportionality is

922
00:50:15,310 --> 00:50:20,560
measuring the notches or notches
per seconds to the 2/3 power

923
00:50:20,560 --> 00:50:22,250
but proportional to the notch.

924
00:50:22,250 --> 00:50:24,310
And the answer can't
depend on notches

925
00:50:24,310 --> 00:50:27,790
because notches are not
really a physical unit.

926
00:50:27,790 --> 00:50:28,930
But it works.

927
00:50:28,930 --> 00:50:30,280
That's an important check.

928
00:50:30,280 --> 00:50:33,400
So-- now it's just a
matter of plugging in here,

929
00:50:33,400 --> 00:50:37,290
and maybe I'll do it explicitly.

930
00:50:37,290 --> 00:50:38,774
I'll leave out the
b's [INAUDIBLE]

931
00:50:38,774 --> 00:50:39,690
so you see the cancel.

932
00:50:39,690 --> 00:50:46,520
We have b times t zero to the
2/3 times the integral from 0

933
00:50:46,520 --> 00:50:57,430
to t zero, times c over b
times t to the 2/3 times dt.

934
00:50:57,430 --> 00:51:00,140
The b's cancel as I claimed.

935
00:51:00,140 --> 00:51:07,740
The integral of t to the minus
2/3 is 3 times t to the 1/3.

936
00:51:07,740 --> 00:51:11,400
We then subtract the t to
the 1/3 giving it the value t

937
00:51:11,400 --> 00:51:13,760
zero on the positive
side, and then

938
00:51:13,760 --> 00:51:15,240
we subtract the
value [INAUDIBLE]

939
00:51:15,240 --> 00:51:17,530
same expression at zero.

940
00:51:17,530 --> 00:51:22,090
But t to the 2/3 when
t is zero vanishes.

941
00:51:22,090 --> 00:51:26,800
So we just get t zero to
the 2/3 from the upper limit

942
00:51:26,800 --> 00:51:28,610
of integration.

943
00:51:28,610 --> 00:51:31,010
I'm sorry, to the 1/3 power.

944
00:51:31,010 --> 00:51:33,310
We integrate minus 2/3.

945
00:51:33,310 --> 00:51:34,880
We get t to the 1/3.

946
00:51:34,880 --> 00:51:37,077
The t to the 1/3
multiplies the t

947
00:51:37,077 --> 00:51:41,290
to the 2/3 giving us
a full one power of t.

948
00:51:41,290 --> 00:51:49,907
So what we're left with is
just 3 times c times t zero,

949
00:51:49,907 --> 00:51:50,990
which has the right units.

950
00:51:50,990 --> 00:51:53,290
It should have units
of physical distance.

951
00:51:53,290 --> 00:51:56,400
Speed times time
is the distance.

952
00:51:56,400 --> 00:52:00,070
And it has a surprising
factor of 3 in it.

953
00:52:00,070 --> 00:52:03,710
The naive answer would have
just been c times t zero,

954
00:52:03,710 --> 00:52:06,890
saying that the light
at time t travels

955
00:52:06,890 --> 00:52:10,520
so it travels a
distance c times t zero.

956
00:52:10,520 --> 00:52:12,890
That would be true, as I said,
in a stationary universe.

957
00:52:12,890 --> 00:52:14,860
But the universe
is not stationary.

958
00:52:14,860 --> 00:52:15,860
It's expanding.

959
00:52:15,860 --> 00:52:17,600
And the fact it's
expanding means

960
00:52:17,600 --> 00:52:20,640
that you expect this to be
bigger than c times t zero.

961
00:52:20,640 --> 00:52:23,510
It means that at earlier
times, things were closer.

962
00:52:23,510 --> 00:52:29,010
So the light can save time by
leaving early and traveling

963
00:52:29,010 --> 00:52:31,310
a good part of the
distance while distances

964
00:52:31,310 --> 00:52:33,700
are smaller than they are now.

965
00:52:33,700 --> 00:52:36,480
And it's a full factor of 3.

966
00:52:36,480 --> 00:52:41,720
So that is the horizon
distance for a flat,

967
00:52:41,720 --> 00:52:43,960
[INAUDIBLE] universe.

968
00:52:43,960 --> 00:52:45,880
We can also, since we
know how to relate t

969
00:52:45,880 --> 00:52:51,602
sub zero to the
Hubble expansion rate,

970
00:52:51,602 --> 00:52:53,060
we can express the
horizon distance

971
00:52:53,060 --> 00:52:55,460
if we want in terms of
the Hubble expansion rate

972
00:52:55,460 --> 00:52:57,520
by just doing that substitution.

973
00:52:57,520 --> 00:53:02,610
So this then becomes 2 times
c times the current Hubble

974
00:53:02,610 --> 00:53:05,650
expansion rate inverse.

975
00:53:05,650 --> 00:53:07,402
So these are both
valid expressions

976
00:53:07,402 --> 00:53:09,443
for the horizon distance
in this particular model

977
00:53:09,443 --> 00:53:10,109
of the universe.

978
00:53:15,800 --> 00:53:18,254
Any questions about the
meaning of horizon distance?

979
00:53:18,254 --> 00:53:20,170
There is actually a
subtlety about the meaning

980
00:53:20,170 --> 00:53:21,820
of horizon, which I
should talk about.

981
00:53:25,170 --> 00:53:29,270
The initial value of the scale
factor in our model is zero.

982
00:53:29,270 --> 00:53:33,490
It's t to the 2/3, and t goes to
0, the scale factor goes to 0.

983
00:53:33,490 --> 00:53:34,990
Things are of course
singular there,

984
00:53:34,990 --> 00:53:37,640
where you don't really trust
exactly what the equations are

985
00:53:37,640 --> 00:53:39,560
telling us at t equals zero.

986
00:53:39,560 --> 00:53:42,020
But that's certainly
how they behave.

987
00:53:42,020 --> 00:53:45,290
a of t goes to 0 as t goes to 0.

988
00:53:45,290 --> 00:53:48,270
That means initially everything
was on top of everything else.

989
00:53:48,270 --> 00:53:52,820
So if everything was on
top of everything else,

990
00:53:52,820 --> 00:53:54,470
why is there any
horizon distance?

991
00:53:54,470 --> 00:53:56,940
Couldn't anything have
communicated with anything

992
00:53:56,940 --> 00:53:59,905
at t equals zero, when the
distance between anything

993
00:53:59,905 --> 00:54:03,310
and anything was zero?

994
00:54:03,310 --> 00:54:06,232
The answer to that is
perhaps somewhat ambiguous.

995
00:54:06,232 --> 00:54:08,440
We of course don't really
understand the singularity.

996
00:54:08,440 --> 00:54:11,970
We don't claim to
understand the singularity.

997
00:54:11,970 --> 00:54:14,870
And therefore anything
you want to believe

998
00:54:14,870 --> 00:54:16,270
about the singularity
at t equals

999
00:54:16,270 --> 00:54:18,640
zero you are welcome to believe.

1000
00:54:18,640 --> 00:54:22,035
And nobody intelligent is
going to contradict you.

1001
00:54:22,035 --> 00:54:23,910
They might not reinforce
you, but they're not

1002
00:54:23,910 --> 00:54:27,000
going to contradict you
either, because nobody knows.

1003
00:54:27,000 --> 00:54:31,740
So it is conceivable that
everything had a chance

1004
00:54:31,740 --> 00:54:33,510
to communicate with
everything else at t

1005
00:54:33,510 --> 00:54:35,279
equals zero at the singularity.

1006
00:54:35,279 --> 00:54:37,820
And it's conceivable that when
we understand quantum gravity,

1007
00:54:37,820 --> 00:54:38,861
it may even tell us that.

1008
00:54:38,861 --> 00:54:40,730
We don't know.

1009
00:54:40,730 --> 00:54:45,920
What is still the case is
that if you strike out t

1010
00:54:45,920 --> 00:54:50,810
equals zero exactly, then
everything is well defined.

1011
00:54:50,810 --> 00:54:54,100
You can ask what happens if a
photon is sent from one object

1012
00:54:54,100 --> 00:54:56,970
to another leaving that
object at time epsilon,

1013
00:54:56,970 --> 00:54:59,370
when epsilon is slightly
later than zero.

1014
00:54:59,370 --> 00:55:01,290
And you can ask how
long does that photon

1015
00:55:01,290 --> 00:55:05,040
take to go from
object a to object b?

1016
00:55:05,040 --> 00:55:07,450
And that's really exactly
the calculation we did,

1017
00:55:07,450 --> 00:55:09,370
except instead of going
down to t equals zero,

1018
00:55:09,370 --> 00:55:11,399
you go down to t
equals epsilon, where

1019
00:55:11,399 --> 00:55:12,940
epsilon is the
earliest time that you

1020
00:55:12,940 --> 00:55:15,300
trust your classical
calculations.

1021
00:55:15,300 --> 00:55:20,320
Then you'd be asking how far
is the furthest object that we

1022
00:55:20,320 --> 00:55:23,720
could see during this classical
era, the era that starts from t

1023
00:55:23,720 --> 00:55:24,384
equals epsilon?

1024
00:55:24,384 --> 00:55:26,550
The only difference would
be to put an epsilon there

1025
00:55:26,550 --> 00:55:27,670
instead of 0.

1026
00:55:27,670 --> 00:55:30,360
And the answer-- you
can go through it--

1027
00:55:30,360 --> 00:55:34,032
differs only by some
small multiple of epsilon.

1028
00:55:34,032 --> 00:55:36,240
And if epsilon is small, it
doesn't change the answer

1029
00:55:36,240 --> 00:55:37,850
at all.

1030
00:55:37,850 --> 00:55:40,620
Physically what is
going on is that if you

1031
00:55:40,620 --> 00:55:43,910
go to very, very early times
and look at two objects a and b

1032
00:55:43,910 --> 00:55:46,520
and trace them back, the
distance between them

1033
00:55:46,520 --> 00:55:49,240
does get smaller and smaller
as epsilon goes to 0.

1034
00:55:49,240 --> 00:55:53,260
So you might think that
communication would be trivial.

1035
00:55:53,260 --> 00:55:56,870
But at the same time as the
distance is going to zero,

1036
00:55:56,870 --> 00:55:59,460
you can calculate
the velocities.

1037
00:55:59,460 --> 00:56:02,940
h remember is going
like 2 over 3t.

1038
00:56:02,940 --> 00:56:05,110
h is blowing up.

1039
00:56:05,110 --> 00:56:06,824
The velocities between
these two objects

1040
00:56:06,824 --> 00:56:08,990
a and b are going to go to
infinity at the same time

1041
00:56:08,990 --> 00:56:11,940
that the distance between
them goes to zero.

1042
00:56:11,940 --> 00:56:14,160
So even though they
will become very close,

1043
00:56:14,160 --> 00:56:16,054
if one sends a light
beam to the other,

1044
00:56:16,054 --> 00:56:17,720
the other is actually
moving away faster

1045
00:56:17,720 --> 00:56:19,840
than the light would be moving.

1046
00:56:19,840 --> 00:56:21,860
The light would
eventually catch up,

1047
00:56:21,860 --> 00:56:24,630
but the amount of time it would
take for the light to catch up

1048
00:56:24,630 --> 00:56:27,960
is exactly what this
integral is telling us.

1049
00:56:27,960 --> 00:56:30,630
So there is no
widespread communication

1050
00:56:30,630 --> 00:56:35,320
that is possible once
these equations are valid.

1051
00:56:35,320 --> 00:56:37,290
They really do say that
in the early universe,

1052
00:56:37,290 --> 00:56:41,770
things just cannot talk to other
things because the universe is

1053
00:56:41,770 --> 00:56:45,150
expanding so fast.

1054
00:56:45,150 --> 00:56:47,950
And the maximum distance
that you can see

1055
00:56:47,950 --> 00:56:51,020
is more than c times t zero,
but less than infinity.

1056
00:56:55,244 --> 00:56:55,744
Yes?

1057
00:56:55,744 --> 00:56:58,720
AUDIENCE: Why does it
matter how far away

1058
00:56:58,720 --> 00:57:01,396
the furthest galaxies
we can see are now?

1059
00:57:01,396 --> 00:57:02,770
Because we're
seeing them as they

1060
00:57:02,770 --> 00:57:06,034
were a long time ago when
they were closer to us.

1061
00:57:06,034 --> 00:57:08,700
PROFESSOR: Yea, now we certainly
are seeing them a long time ago

1062
00:57:08,700 --> 00:57:09,866
when they were closer to us.

1063
00:57:09,866 --> 00:57:10,860
That's right.

1064
00:57:10,860 --> 00:57:14,000
I would just say that this
is sort of a figure of merit.

1065
00:57:14,000 --> 00:57:18,820
If you want to describe what
you think the universe looks

1066
00:57:18,820 --> 00:57:23,300
like now, you would assume
that those galaxies that you're

1067
00:57:23,300 --> 00:57:26,430
seeing billions of light years
in the past are still there,

1068
00:57:26,430 --> 00:57:28,290
and you'd extrapolate
to the present.

1069
00:57:28,290 --> 00:57:30,637
So it's relevant to
the picture that you

1070
00:57:30,637 --> 00:57:32,720
would draw in your mind
of what the universe looks

1071
00:57:32,720 --> 00:57:36,990
like at this instant, although
that picture would be based

1072
00:57:36,990 --> 00:57:38,935
on things you haven't
actually seen.

1073
00:57:38,935 --> 00:57:39,810
AUDIENCE: [INAUDIBLE]

1074
00:57:42,810 --> 00:57:43,577
PROFESSOR: Yes?

1075
00:57:43,577 --> 00:57:45,160
AUDIENCE: Is the
fact that this number

1076
00:57:45,160 --> 00:57:48,974
is greater than c t zero
proof that the universe is

1077
00:57:48,974 --> 00:57:50,890
the actual space of the
universe is expanding?

1078
00:57:54,390 --> 00:57:57,130
PROFESSOR: I don't think
so because if you just

1079
00:57:57,130 --> 00:58:00,330
have the objects moving
and a space that you regard

1080
00:58:00,330 --> 00:58:03,370
as absolutely fixed
and then you ask

1081
00:58:03,370 --> 00:58:07,090
the present distance of that
object given that you can see

1082
00:58:07,090 --> 00:58:10,510
a light pulse that was
emitted by that object,

1083
00:58:10,510 --> 00:58:13,640
it will still be bigger than
ct, because the object continues

1084
00:58:13,640 --> 00:58:16,710
to move after it
emits the light pulse.

1085
00:58:16,710 --> 00:58:20,330
So I don't think this is
proof of anything like that.

1086
00:58:20,330 --> 00:58:24,340
I think I should add
that certainly we

1087
00:58:24,340 --> 00:58:28,010
think of this as
the space expanding.

1088
00:58:28,010 --> 00:58:29,980
That is certainly
the easiest picture.

1089
00:58:29,980 --> 00:58:34,260
But you can't have any
absolute definition

1090
00:58:34,260 --> 00:58:36,210
of what it means for
the space to expand.

1091
00:58:36,210 --> 00:58:37,585
AUDIENCE: So you're
saying it has

1092
00:58:37,585 --> 00:58:39,354
moved three times the
original distance it

1093
00:58:39,354 --> 00:58:41,490
was when it sent out an impulse?

1094
00:58:41,490 --> 00:58:44,540
PROFESSOR: No it actually was
zero distance when it sent out

1095
00:58:44,540 --> 00:58:49,929
the light pulse, because the
first object we in principle

1096
00:58:49,929 --> 00:58:52,470
could see is an object for which
the light pulse left it at t

1097
00:58:52,470 --> 00:58:55,272
equals zero when it literally
was zero distance from us.

1098
00:58:55,272 --> 00:58:57,730
And the light pulse actually
then gets further away from us

1099
00:58:57,730 --> 00:59:00,770
and comes back and
eventually reaches us.

1100
00:59:00,770 --> 00:59:02,807
We have enough information
here to trace it.

1101
00:59:02,807 --> 00:59:03,890
And that's how it behaves.

1102
00:59:10,517 --> 00:59:11,475
OK any other questions?

1103
00:59:16,120 --> 00:59:18,475
OK, next thing I
want to do, and I

1104
00:59:18,475 --> 00:59:20,350
don't know if we'll
finish this today or not.

1105
00:59:20,350 --> 00:59:22,320
But we'll get ourselves started.

1106
00:59:22,320 --> 00:59:24,900
We'd like to now, having
solved the equation

1107
00:59:24,900 --> 00:59:27,447
for the flat universe
for determining a of t,

1108
00:59:27,447 --> 00:59:29,030
we would now like
to do the same thing

1109
00:59:29,030 --> 00:59:31,177
for the open and
closed universes.

1110
00:59:31,177 --> 00:59:32,760
And we'll do the
closed universe first

1111
00:59:32,760 --> 00:59:34,590
because it's a
little bit simpler.

1112
00:59:34,590 --> 00:59:35,610
I mean they're about
equal, but we're

1113
00:59:35,610 --> 00:59:37,170
going to do the
closed universe first

1114
00:59:37,170 --> 00:59:39,272
because it's first in the
notes that I've written.

1115
00:59:42,050 --> 00:59:44,040
So we know what equation
we're trying to solve.

1116
00:59:44,040 --> 00:59:45,930
This is really now
just an exercise

1117
00:59:45,930 --> 00:59:47,800
in solving
differential equations.

1118
00:59:47,800 --> 00:59:50,610
So one has to be clever
to solve this equation.

1119
00:59:50,610 --> 00:59:52,890
So we will go
through it together

1120
00:59:52,890 --> 00:59:56,360
and see how one can be clever
and find the solution to it.

1121
00:59:56,360 --> 00:59:58,960
The equation is a
dot over a squared

1122
00:59:58,960 --> 01:00:06,290
is equal to 8 pi
over 3 g rho minus k

1123
01:00:06,290 --> 01:00:14,410
c squared over a
squared, with rho of t

1124
01:00:14,410 --> 01:00:30,530
being equal to rho i times a
cubed of t i over a cubed of t.

1125
01:00:30,530 --> 01:00:32,130
And I'm writing i here.

1126
01:00:32,130 --> 01:00:33,460
I could just as well write 1.

1127
01:00:33,460 --> 01:00:34,420
It could be any time.

1128
01:00:37,030 --> 01:00:39,535
Rho times a cubed isn't
dependent of time.

1129
01:00:39,535 --> 01:00:41,160
So you can put any
time you want there.

1130
01:00:41,160 --> 01:00:43,110
And the numerator is
just a fixed number.

1131
01:00:55,040 --> 01:00:59,242
OK, so our goal is to
solve this equation.

1132
01:00:59,242 --> 01:01:00,950
The first thing I want
to do is something

1133
01:01:00,950 --> 01:01:02,408
which is usually
a good thing to do

1134
01:01:02,408 --> 01:01:05,797
when you are given some
physical differential equation.

1135
01:01:05,797 --> 01:01:08,380
Physical differential equations
usually have constants in them

1136
01:01:08,380 --> 01:01:12,380
like g and c squared, which
have different units which

1137
01:01:12,380 --> 01:01:13,690
complicate things.

1138
01:01:13,690 --> 01:01:16,410
And they don't complicate
things in any intrinsic way.

1139
01:01:16,410 --> 01:01:20,590
They just give you extra
factors to carry around.

1140
01:01:20,590 --> 01:01:23,440
So it's usually a little
cleaner to eliminate

1141
01:01:23,440 --> 01:01:28,260
them to begin with by redefining
variables that absorb them.

1142
01:01:28,260 --> 01:01:30,510
One can do that by
defining variables

1143
01:01:30,510 --> 01:01:33,600
that have the simplest
possible units that

1144
01:01:33,600 --> 01:01:36,650
are available for
the problem at hand.

1145
01:01:36,650 --> 01:01:39,340
And in our case, we
have these complications

1146
01:01:39,340 --> 01:01:43,720
that k is measuring in
inverse notch squared,

1147
01:01:43,720 --> 01:01:46,270
and a is meters per notch.

1148
01:01:46,270 --> 01:01:51,560
We could simplify
all that by defining

1149
01:01:51,560 --> 01:01:55,340
some auxiliary quantities.

1150
01:01:55,340 --> 01:01:58,550
So in particular, I'm
going to define an a

1151
01:01:58,550 --> 01:02:08,710
with a tilde above it, which is
sort of like the scale factor,

1152
01:02:08,710 --> 01:02:11,110
but redefined by the
square root of k.

1153
01:02:15,304 --> 01:02:16,970
And this is the case
when k is positive.

1154
01:02:16,970 --> 01:02:19,094
I said we're going to do
the closed universe first.

1155
01:02:23,637 --> 01:02:25,720
I would not want to divide
by the square root of k

1156
01:02:25,720 --> 01:02:26,610
if k were negative.

1157
01:02:26,610 --> 01:02:27,610
That would be confusing.

1158
01:02:30,522 --> 01:02:32,230
Now the nice thing
about this is remember

1159
01:02:32,230 --> 01:02:34,550
k has units of
inverse notch squared.

1160
01:02:34,550 --> 01:02:36,390
a has units of meters per notch.

1161
01:02:36,390 --> 01:02:39,170
That means the
notches just go away.

1162
01:02:39,170 --> 01:02:41,995
So this has units
of length only.

1163
01:02:48,050 --> 01:02:50,969
And that was the
motivation for dividing

1164
01:02:50,969 --> 01:02:53,010
by the square root of k,
to get rid of the notch.

1165
01:02:57,810 --> 01:03:03,046
Similarly, time has units
of meters per second.

1166
01:03:03,046 --> 01:03:04,420
I'm trying to
minimize the number

1167
01:03:04,420 --> 01:03:08,210
of distinct physical quantities
that we have in our problem.

1168
01:03:08,210 --> 01:03:13,960
So I'm going to define
a t twiddle which

1169
01:03:13,960 --> 01:03:16,354
is just equal to c times t.

1170
01:03:16,354 --> 01:03:18,770
This is of course no difference
in just saying the working

1171
01:03:18,770 --> 01:03:21,360
units is c is equal to
1, which a lot of times

1172
01:03:21,360 --> 01:03:22,855
is something people say.

1173
01:03:22,855 --> 01:03:24,230
This is the same
thing except I'm

1174
01:03:24,230 --> 01:03:28,640
a little bit more
explicit about.

1175
01:03:28,640 --> 01:03:31,330
So t twiddle now is
units of length also.

1176
01:03:38,250 --> 01:03:39,890
So the idea is to
translate everything

1177
01:03:39,890 --> 01:03:41,681
so that everything is
the same units, which

1178
01:03:41,681 --> 01:03:44,190
would be meters or
whatever physical unit

1179
01:03:44,190 --> 01:03:45,910
of length you want to use.

1180
01:03:52,074 --> 01:03:53,990
OK, now I'm just going
to rewrite the Freedman

1181
01:03:53,990 --> 01:03:56,570
equation using
these substitutions.

1182
01:03:56,570 --> 01:03:59,280
And to make way for
the substitutions,

1183
01:03:59,280 --> 01:04:01,730
I'm first just going to
divide the Freedman equation

1184
01:04:01,730 --> 01:04:02,570
by kc squared.

1185
01:04:37,680 --> 01:04:38,840
This is k to the 3/2.

1186
01:04:44,630 --> 01:04:46,760
OK now when I divided
by kc squared,

1187
01:04:46,760 --> 01:04:52,139
the kc squared over a squared
there became-- I'm sorry.

1188
01:04:52,139 --> 01:04:53,930
I'm doing more than
dividing by kc squared.

1189
01:04:53,930 --> 01:04:57,300
I'm dividing by kc squared
and multiplying by a squared.

1190
01:04:57,300 --> 01:05:01,070
So I have turned
that last term into 1

1191
01:05:01,070 --> 01:05:03,970
by multiplying by its inverse.

1192
01:05:03,970 --> 01:05:05,900
On the left hand side,
the 1 over a squared

1193
01:05:05,900 --> 01:05:07,630
went away when I
multiplied by a squared.

1194
01:05:07,630 --> 01:05:11,180
So I just have a dot squared
divided by kc squared so far.

1195
01:05:11,180 --> 01:05:13,180
We'll simplify that shortly.

1196
01:05:13,180 --> 01:05:16,190
And the middle term,
I took the liberty

1197
01:05:16,190 --> 01:05:19,270
of multiplying by
the square root of k,

1198
01:05:19,270 --> 01:05:21,350
and then we have a
k to the 3/2 here.

1199
01:05:21,350 --> 01:05:23,230
If we absorb this into
that, it would just

1200
01:05:23,230 --> 01:05:25,230
be the kc squared factor
that we divided by.

1201
01:05:29,765 --> 01:05:36,620
And the a squared
has been also divided

1202
01:05:36,620 --> 01:05:40,370
into two pieces, a cubed and a.

1203
01:05:40,370 --> 01:05:42,167
So together these
two factors make up

1204
01:05:42,167 --> 01:05:44,500
the factor of a squared that
we multiplied that term by.

1205
01:05:44,500 --> 01:05:45,980
So it's the same
thing we had, just

1206
01:05:45,980 --> 01:05:48,280
multiplied by the common factor.

1207
01:05:54,050 --> 01:05:57,440
And now the nice thing
is that this is, in fact,

1208
01:05:57,440 --> 01:06:03,690
our definition of
da tilde over dt.

1209
01:06:03,690 --> 01:06:08,460
The c turns the dt into a
dt tilde, and the 1 over k

1210
01:06:08,460 --> 01:06:11,510
turns the da into da tilde.

1211
01:06:11,510 --> 01:06:14,610
So the left hand
side now is simply

1212
01:06:14,610 --> 01:06:21,260
da tilde over dt tilde squared.

1213
01:06:25,210 --> 01:06:31,075
And the right hand side,
rho times a cubed remember

1214
01:06:31,075 --> 01:06:33,490
is a constant.

1215
01:06:33,490 --> 01:06:36,100
So the only thing that depends
on time on the right hand side

1216
01:06:36,100 --> 01:06:39,290
is the a divided by mu k,
which I've isolated here.

1217
01:06:39,290 --> 01:06:40,750
Earth That's a tilde.

1218
01:06:40,750 --> 01:06:43,000
So I'm going to take all of
this, which is a constant,

1219
01:06:43,000 --> 01:06:46,590
and just give it
a name, constant.

1220
01:06:46,590 --> 01:06:53,810
So this term becomes a constant,
we'll call 2 alpha divided

1221
01:06:53,810 --> 01:06:56,520
by a tilde.

1222
01:06:56,520 --> 01:06:59,580
And then we still have minus 1.

1223
01:06:59,580 --> 01:07:02,330
And this constant,
which I'm calling alpha,

1224
01:07:02,330 --> 01:07:07,010
is just all of this factor
except for a factor of 2.

1225
01:07:07,010 --> 01:07:19,110
So it's 4 pi over 3
g rho a tilde cubed

1226
01:07:19,110 --> 01:07:22,290
divided by c squared.

1227
01:07:22,290 --> 01:07:26,400
a tilde because we had a
cubed divided by k to the 3/2,

1228
01:07:26,400 --> 01:07:29,120
and that's a tilde cubed.

1229
01:07:29,120 --> 01:07:31,610
So I've just rearranged things.

1230
01:07:31,610 --> 01:07:33,980
But now everything has
the units of length.

1231
01:07:33,980 --> 01:07:35,860
a tilde has units of length.

1232
01:07:35,860 --> 01:07:37,260
t tilde has units of length.

1233
01:07:37,260 --> 01:07:39,080
This is dimensionless,
which is good

1234
01:07:39,080 --> 01:07:41,260
because that's
also dimensionless.

1235
01:07:41,260 --> 01:07:43,390
Alpha, if you work
out all this stuff,

1236
01:07:43,390 --> 01:07:45,900
has units of length. a
tilde has units of length.

1237
01:07:45,900 --> 01:07:48,080
This is length divided
by length, dimensionless.

1238
01:07:51,212 --> 01:07:53,170
We haven't really changed
anything significant.

1239
01:07:53,170 --> 01:07:56,060
But at least as far as
keeping track of factors,

1240
01:07:56,060 --> 01:07:57,720
that equation is
the one we'll solve.

1241
01:07:57,720 --> 01:08:01,620
And the factors are now absorbed
into the constant alpha, which

1242
01:08:01,620 --> 01:08:03,514
is the only thing we
have to worry about.

1243
01:08:03,514 --> 01:08:05,680
And we don't need to worry
about that until the end.

1244
01:08:05,680 --> 01:08:06,509
Yes?

1245
01:08:06,509 --> 01:08:07,384
AUDIENCE: [INAUDIBLE]

1246
01:08:12,005 --> 01:08:14,630
PROFESSOR: As long as these two
are evaluated at the same time,

1247
01:08:14,630 --> 01:08:15,370
it doesn't matter.

1248
01:08:15,370 --> 01:08:16,245
AUDIENCE: [INAUDIBLE]

1249
01:08:22,670 --> 01:08:24,077
PROFESSOR: That's right.

1250
01:08:24,077 --> 01:08:25,910
One does need to remember
that these two are

1251
01:08:25,910 --> 01:08:29,430
to be evaluated at the
same time, whatever it is.

1252
01:08:29,430 --> 01:08:33,210
And the product does
not depend on time.

1253
01:08:37,430 --> 01:08:38,910
So I didn't write the arguments.

1254
01:08:38,910 --> 01:08:41,470
I could have, and I could have
just put in an arbitrary time.

1255
01:08:41,470 --> 01:08:43,803
But it would be the same time
for the rho and the tilde.

1256
01:08:49,350 --> 01:08:50,530
I will do that.

1257
01:08:50,530 --> 01:08:54,149
It will be rho of
some t one times

1258
01:08:54,149 --> 01:09:00,069
a tilde cubed, evaluated the
same t one for any t one.

1259
01:09:00,069 --> 01:09:01,590
This product is
independent of time.

1260
01:09:50,830 --> 01:09:52,540
OK, now that is the
kind of equation

1261
01:09:52,540 --> 01:09:55,065
where we could move things
from one side of the equation

1262
01:09:55,065 --> 01:09:59,810
to the other to reduce it
to doing ordinary intervals.

1263
01:09:59,810 --> 01:10:03,810
So I can multiply by
d tilde and divide

1264
01:10:03,810 --> 01:10:06,960
by the expression on
the right hand side

1265
01:10:06,960 --> 01:10:13,590
and get an expression
that says that dt tilde is

1266
01:10:13,590 --> 01:10:20,100
equal to da tilde divided
by the square root of 2

1267
01:10:20,100 --> 01:10:25,740
alpha over a tilde minus 1.

1268
01:10:32,020 --> 01:10:35,570
And since this has an a
tilde in the denominator

1269
01:10:35,570 --> 01:10:38,180
of the denominator, I'm
going to multiply through

1270
01:10:38,180 --> 01:10:42,860
by a tilde to
rationalize things.

1271
01:10:42,860 --> 01:10:49,670
So I'm going to rewrite
this as a tilde da tilde

1272
01:10:49,670 --> 01:11:00,160
over the square root of 2 alpha
a tilde minus a tilde squared.

1273
01:11:00,160 --> 01:11:02,280
OK this looks better
than we've had so far.

1274
01:11:08,100 --> 01:11:11,730
Now in principle, if we imagine
we can do that integral,

1275
01:11:11,730 --> 01:11:13,430
we can just
integrate both sides.

1276
01:11:13,430 --> 01:11:15,320
We get an equation
that says t tilde is

1277
01:11:15,320 --> 01:11:20,639
equal to some expression
involving the final value of a.

1278
01:11:20,639 --> 01:11:22,180
So we're going to
imagine doing that.

1279
01:11:22,180 --> 01:11:25,020
And we will actually be
able to carry it out.

1280
01:11:25,020 --> 01:11:31,170
When we integrate, there are
two ways you can proceed.

1281
01:11:31,170 --> 01:11:33,910
When I solve the
analogous problem

1282
01:11:33,910 --> 01:11:36,160
for the flat case
with the t to the 2/3

1283
01:11:36,160 --> 01:11:37,622
it was a much simpler equation.

1284
01:11:37,622 --> 01:11:39,330
But you might remember
that at one point,

1285
01:11:39,330 --> 01:11:41,140
I had an equation
where I calculated

1286
01:11:41,140 --> 01:11:43,300
the indefinite
integral of both sides,

1287
01:11:43,300 --> 01:11:45,910
and then I got a constant
of integration, which I then

1288
01:11:45,910 --> 01:11:49,050
argued should be set
equal to zero if we wanted

1289
01:11:49,050 --> 01:11:51,760
to find the zero of time
to be the zero of a.

1290
01:11:55,760 --> 01:11:58,150
The situation is really
exactly the same,

1291
01:11:58,150 --> 01:12:00,020
but to show you
an alternative way

1292
01:12:00,020 --> 01:12:02,410
of thinking about
it, this time I'm

1293
01:12:02,410 --> 01:12:05,765
going to apply definite
integrals to both sides.

1294
01:12:05,765 --> 01:12:07,890
And if you apply a definite
integral to both sides,

1295
01:12:07,890 --> 01:12:10,710
the thing to keep in mind is
that you should be integrating

1296
01:12:10,710 --> 01:12:15,150
over the same physical
interval on both sides.

1297
01:12:15,150 --> 01:12:18,190
So on the left hand
side, I'm going

1298
01:12:18,190 --> 01:12:28,580
to integrate from zero
up to some t tilde final.

1299
01:12:28,580 --> 01:12:30,330
t tilde final is just
some arbitrary time,

1300
01:12:30,330 --> 01:12:34,370
but I'm going to give it
the name t tilde final.

1301
01:12:34,370 --> 01:12:36,530
And that integral
of course we can do.

1302
01:12:36,530 --> 01:12:40,160
It's just t tilde final.

1303
01:12:40,160 --> 01:12:42,850
And that should be equal to the
integral of the right hand side

1304
01:12:42,850 --> 01:12:44,634
over the same period of time.

1305
01:12:44,634 --> 01:12:46,300
But the right hand
side is not expressed

1306
01:12:46,300 --> 01:12:47,440
as an integral over time.

1307
01:12:47,440 --> 01:12:50,350
It's expressed as an
integral over a tilde.

1308
01:12:50,350 --> 01:12:52,450
So we have to ask
ourselves, what

1309
01:12:52,450 --> 01:12:55,030
do we call the integral
over a tilde that

1310
01:12:55,030 --> 01:13:01,770
corresponds to the integral over
time from 0 to t tilde sub f.

1311
01:13:01,770 --> 01:13:06,340
And the lower
limits should match.

1312
01:13:06,340 --> 01:13:11,617
And we know what we want a tilde
to be at t tilde equals zero.

1313
01:13:11,617 --> 01:13:13,200
We're going to use
the same dimensions

1314
01:13:13,200 --> 01:13:14,530
we had in the other case.

1315
01:13:14,530 --> 01:13:16,120
We're going to define
the zero of time

1316
01:13:16,120 --> 01:13:19,530
to be the time when the
scale factor is zero.

1317
01:13:19,530 --> 01:13:22,580
So t tilde equals
zero should correspond

1318
01:13:22,580 --> 01:13:24,520
to a tilde equals zero.

1319
01:13:24,520 --> 01:13:27,050
So to get the lower
limits of integration

1320
01:13:27,050 --> 01:13:30,530
to correspond to the same
time, we just put zero her.

1321
01:13:30,530 --> 01:13:32,210
And zero here doesn't
mean time zero.

1322
01:13:32,210 --> 01:13:33,870
It means a tilde equals zero.

1323
01:13:33,870 --> 01:13:35,900
But that's what we want.

1324
01:13:35,900 --> 01:13:40,750
And for the upper limit of
integration, that should just

1325
01:13:40,750 --> 01:13:45,800
be the value of a tilde
at the time t tilde sub f.

1326
01:13:45,800 --> 01:13:52,870
So I will call
that a tilde sub f,

1327
01:13:52,870 --> 01:13:54,820
where I might make
a note on the side

1328
01:13:54,820 --> 01:14:06,200
here that a tilde sub f is
e tilde of t tilde sub f.

1329
01:14:06,200 --> 01:14:09,270
It's just the final
value of a tilde, where

1330
01:14:09,270 --> 01:14:12,125
final means anytime I choose
to stop this integration.

1331
01:14:12,125 --> 01:14:14,460
The interval is valid
over any time period.

1332
01:14:18,960 --> 01:14:20,864
So these are moments
of integration

1333
01:14:20,864 --> 01:14:22,780
where the only thing
that's new on this line--

1334
01:14:22,780 --> 01:14:33,350
now I just copy from the
line above, 2 alpha times

1335
01:14:33,350 --> 01:14:46,210
a tilde minus a tilde squared.

1336
01:15:01,590 --> 01:15:04,940
So t tilde f is equal
to that integral.

1337
01:15:04,940 --> 01:15:08,560
And our goal now would
be to do that integral.

1338
01:15:08,560 --> 01:15:10,510
And if possible-- it
won't quite be possible.

1339
01:15:10,510 --> 01:15:11,885
We can see how
close we can come.

1340
01:15:11,885 --> 01:15:14,960
But if possible we would like
to invert that relationship

1341
01:15:14,960 --> 01:15:16,890
that we get from
doing this integral,

1342
01:15:16,890 --> 01:15:21,217
to determine a tilde as
a function of t tilde.

1343
01:15:21,217 --> 01:15:22,550
That's what we would love to do.

1344
01:15:22,550 --> 01:15:24,590
It turns out that's
not quite possible.

1345
01:15:24,590 --> 01:15:27,740
But we will nonetheless be
able to obtain something called

1346
01:15:27,740 --> 01:15:29,990
a parametric solution
to this problem.

1347
01:15:29,990 --> 01:15:32,110
And we'll stop now.

1348
01:15:32,110 --> 01:15:36,090
But I will tell you next
Tuesday after the quiz

1349
01:15:36,090 --> 01:15:40,820
how we proceed here to get
a solution to this problem.