1
00:00:00,080 --> 00:00:01,670
The following
content is provided

2
00:00:01,670 --> 00:00:03,820
under a Creative
Commons license.

3
00:00:03,820 --> 00:00:06,550
Your support will help MIT
OpenCourseWare continue

4
00:00:06,550 --> 00:00:10,160
to offer high quality
educational resources for free.

5
00:00:10,160 --> 00:00:12,700
To make a donation or to
view additional materials

6
00:00:12,700 --> 00:00:16,620
from hundreds of MIT courses,
visit MIT OpenCourseWare

7
00:00:16,620 --> 00:00:17,327
at ocw.mit.edu.

8
00:00:24,820 --> 00:00:25,880
PROFESSOR: OK.

9
00:00:25,880 --> 00:00:28,370
I think it's time
for us to start.

10
00:00:28,370 --> 00:00:32,590
Last time we talked about the
Doppler shift and a little bit

11
00:00:32,590 --> 00:00:34,350
of special relativity.

12
00:00:34,350 --> 00:00:38,480
Today we'll be going on to talk
more about cosmological topics.

13
00:00:38,480 --> 00:00:40,190
We'll be talking
about kinematically

14
00:00:40,190 --> 00:00:43,240
how one describes a
homogeneously-expanding

15
00:00:43,240 --> 00:00:44,950
universe like the
one that we think

16
00:00:44,950 --> 00:00:48,510
we're living to a very
good approximation.

17
00:00:48,510 --> 00:00:51,670
In that case, let's get started.

18
00:00:51,670 --> 00:00:53,920
What I want to do today
is talk about some

19
00:00:53,920 --> 00:00:59,720
of the basic descriptive
properties of the universe

20
00:00:59,720 --> 00:01:02,100
as we will describe it.

21
00:01:02,100 --> 00:01:05,360
The universe is, of course,
a very complicated place.

22
00:01:05,360 --> 00:01:07,065
It includes you and
me, for example,

23
00:01:07,065 --> 00:01:09,590
and we're pretty
complicated structures.

24
00:01:09,590 --> 00:01:12,630
But cosmology is not really
the study of all that.

25
00:01:12,630 --> 00:01:15,890
Cosmology is the study of
the universe in the large,

26
00:01:15,890 --> 00:01:19,260
and we'll begin by discussing
the universe on its largest

27
00:01:19,260 --> 00:01:22,450
scales in which you view
approximated by a very

28
00:01:22,450 --> 00:01:26,570
simple model, which
we'll be learning about.

29
00:01:26,570 --> 00:01:29,645
So in particular on
very large scales,

30
00:01:29,645 --> 00:01:33,270
the universe is pretty well
described by threes properties,

31
00:01:33,270 --> 00:01:36,360
which we will talk
about one by one.

32
00:01:36,360 --> 00:01:44,490
The first is isotropy,
and that just

33
00:01:44,490 --> 00:01:46,230
comes from some
Greek root, which

34
00:01:46,230 --> 00:01:47,735
means the same in
all directions.

35
00:02:03,664 --> 00:02:05,830
Now, of course, as we look
around say the room here,

36
00:02:05,830 --> 00:02:07,640
the room doesn't look the
same in all directions.

37
00:02:07,640 --> 00:02:09,009
The front of the room looks
different from the back

38
00:02:09,009 --> 00:02:10,190
of the room.

39
00:02:10,190 --> 00:02:12,080
And looking towards
Mass Ave looks

40
00:02:12,080 --> 00:02:14,550
different from looking
towards the river,

41
00:02:14,550 --> 00:02:16,940
and looking further
out into space,

42
00:02:16,940 --> 00:02:19,370
looking towards the
Virgo cluster, which

43
00:02:19,370 --> 00:02:21,842
is the center of our
local super cluster,

44
00:02:21,842 --> 00:02:23,300
looks rather
different from looking

45
00:02:23,300 --> 00:02:25,870
in the opposite direction.

46
00:02:25,870 --> 00:02:28,970
But when one gets out
to looking at things

47
00:02:28,970 --> 00:02:32,600
on the very large scale
where in this case very large

48
00:02:32,600 --> 00:03:00,390
means on the scale of a few
hundred million light years,

49
00:03:00,390 --> 00:03:03,610
things begin to
look very isotropic.

50
00:03:03,610 --> 00:03:06,140
That is no matter what
direction you look,

51
00:03:06,140 --> 00:03:09,550
as long as you're averaging
over these very large scales,

52
00:03:09,550 --> 00:03:13,100
you find that you see
pretty much the same thing.

53
00:03:13,100 --> 00:03:17,730
This becomes most
emphatic when one looks

54
00:03:17,730 --> 00:03:20,730
at the cosmic background
radiation, which is really

55
00:03:20,730 --> 00:03:23,660
the furthest object
that we can look at.

56
00:03:23,660 --> 00:03:29,540
It's radiation that was admitted
shortly after the Big Bang.

57
00:03:29,540 --> 00:03:33,720
The history of the cosmic
background radiation

58
00:03:33,720 --> 00:03:36,170
in a nutshell is worth
keeping in mind here.

59
00:03:40,650 --> 00:03:44,685
I'll refer to it as the CMB for
cosmic microwave background.

60
00:04:00,090 --> 00:04:02,970
And in a nutshell, the
things to keep in mind

61
00:04:02,970 --> 00:04:06,020
in thinking about
this history is

62
00:04:06,020 --> 00:04:17,620
that until about 400,000
years after the beginning,

63
00:04:17,620 --> 00:04:23,200
the universe was a
plasma, or maybe I

64
00:04:23,200 --> 00:04:25,200
should say more accurately
that the universe was

65
00:04:25,200 --> 00:04:26,290
filled with plasma.

66
00:04:36,790 --> 00:04:41,690
And within a plasma, photons
essentially go nowhere.

67
00:04:41,690 --> 00:04:43,870
They're constantly moving
at the speed of light,

68
00:04:43,870 --> 00:04:46,490
but they have a very large
cross section for scattering off

69
00:04:46,490 --> 00:04:50,530
of the free electrons
that fill the plasma.

70
00:04:50,530 --> 00:04:53,050
And that means that the
photons are constantly

71
00:04:53,050 --> 00:04:56,660
changing directions and the net
progress in any one direction

72
00:04:56,660 --> 00:04:57,490
is negligible.

73
00:05:04,750 --> 00:05:10,754
So the photons are
frozen with the matter,

74
00:05:10,754 --> 00:05:19,660
I'll say frozen inside
the matter, which

75
00:05:19,660 --> 00:05:23,480
means that the net velocity
relative to this plasma

76
00:05:23,480 --> 00:05:25,550
is essentially zero.

77
00:05:25,550 --> 00:05:27,230
But according to
our calculations,

78
00:05:27,230 --> 00:05:29,720
and we'll learn later how
to do these calculations,

79
00:05:29,720 --> 00:05:32,900
at about 400,000 years
after the Big Bang,

80
00:05:32,900 --> 00:05:36,470
the universe cooled enough
so that it neutralized

81
00:05:36,470 --> 00:05:39,680
and then it became a neutral
gas like the air in this room.

82
00:05:39,680 --> 00:05:41,055
And the air in
this room you know

83
00:05:41,055 --> 00:05:43,370
this is very
transparent to photons,

84
00:05:43,370 --> 00:05:45,700
and that means that light
travels from my face

85
00:05:45,700 --> 00:05:47,895
to your eyes on straight
lines and allows

86
00:05:47,895 --> 00:05:51,100
you to see an image of what my
face looks like and vice versa,

87
00:05:51,100 --> 00:05:54,020
by the way.

88
00:05:54,020 --> 00:05:56,810
And it's a little dicey
to extrapolate something

89
00:05:56,810 --> 00:05:59,419
from the room to the universe.

90
00:05:59,419 --> 00:06:01,210
The orders of magnitude
are very different.

91
00:06:01,210 --> 00:06:02,793
But in this case,
the physics actually

92
00:06:02,793 --> 00:06:04,440
ends up being exactly the same.

93
00:06:04,440 --> 00:06:08,190
Once the universe becomes
filled with a neutral gas,

94
00:06:08,190 --> 00:06:11,260
it really does become
transparent to the photons

95
00:06:11,260 --> 00:06:13,810
of the cosmic
microwave background.

96
00:06:13,810 --> 00:06:17,150
So these photons have for
the most part been travelling

97
00:06:17,150 --> 00:06:20,670
on perfectly-straight lines
since 400,000 years after

98
00:06:20,670 --> 00:06:22,360
the Big Bang.

99
00:06:22,360 --> 00:06:25,530
And that means that
when we see them today,

100
00:06:25,530 --> 00:06:28,620
we are essentially
seeing an image of what

101
00:06:28,620 --> 00:06:32,262
the universe looks like at
400,000 years after the Big

102
00:06:32,262 --> 00:06:32,762
Bang.

103
00:06:36,538 --> 00:06:57,120
So at 400,000 years,
gas neutralized

104
00:06:57,120 --> 00:07:01,970
and became transparent.

105
00:07:16,360 --> 00:07:21,090
This by the way has a
name, which is universally

106
00:07:21,090 --> 00:07:22,924
what is called in
cosmology, nobody actually

107
00:07:22,924 --> 00:07:24,798
understands why it's
called this, by the way,

108
00:07:24,798 --> 00:07:26,116
but the name is recombination.

109
00:07:35,980 --> 00:07:38,130
And the mystery is what
the re is doing there

110
00:07:38,130 --> 00:07:40,550
because as far as we
know, the gas is combining

111
00:07:40,550 --> 00:07:43,130
for the first time in
the history of universe,

112
00:07:43,130 --> 00:07:45,720
but that's otherwise
what everybody calls it.

113
00:07:45,720 --> 00:07:48,290
I did actually once
ask Jim Peebles who

114
00:07:48,290 --> 00:07:51,754
might be the person who first
called it this why it was

115
00:07:51,754 --> 00:07:53,295
called this, and he
told me that this

116
00:07:53,295 --> 00:07:54,990
is what the plasma
physicists called it,

117
00:07:54,990 --> 00:07:57,565
so it was natural to just
pick up the same word

118
00:07:57,565 --> 00:07:59,840
when he was doing
cosmology, so maybe

119
00:07:59,840 --> 00:08:01,256
that's how the word originated.

120
00:08:01,256 --> 00:08:02,880
But coming from the
point of cosmology,

121
00:08:02,880 --> 00:08:06,010
it is a misnomer in
that for the theory

122
00:08:06,010 --> 00:08:08,380
that we're discussing
the prefix re here

123
00:08:08,380 --> 00:08:10,140
has absolutely no
business being there.

124
00:08:13,481 --> 00:08:15,730
So what do we see when we
look at the cosmic microwave

125
00:08:15,730 --> 00:08:16,540
background?

126
00:08:16,540 --> 00:08:20,900
We see that it is
unbelievably isotropic.

127
00:08:20,900 --> 00:08:28,865
What we find is that
there are deviations

128
00:08:28,865 --> 00:08:30,365
in the temperature
of the radiation.

129
00:08:30,365 --> 00:08:33,900
The intensity is measured
as an effective temperature.

130
00:08:33,900 --> 00:08:36,670
There are deviations in the
temperature of the radiation

131
00:08:36,670 --> 00:08:43,520
of a fractional amount of
about 10 to the minus 3,

132
00:08:43,520 --> 00:08:45,430
which is a very small
number, but it's

133
00:08:45,430 --> 00:08:48,260
even stronger than that.

134
00:08:48,260 --> 00:08:51,370
This deviation of one
part in 10 to the 3

135
00:08:51,370 --> 00:08:53,565
has a particular
angular pattern,

136
00:08:53,565 --> 00:08:55,480
and it's not the
angular pattern that you

137
00:08:55,480 --> 00:08:57,770
would expect if the
source system were moving

138
00:08:57,770 --> 00:09:00,330
through the cosmic
microwave background,

139
00:09:00,330 --> 00:09:03,180
and that's how we interpret
this 10 to the minus 3 effect.

140
00:09:09,090 --> 00:09:19,026
Motion of solar system
through the CMB.

141
00:09:24,305 --> 00:09:33,760
And after removing the
effect of the motion.

142
00:09:37,814 --> 00:09:39,230
Now actually when
we move it, it's

143
00:09:39,230 --> 00:09:41,580
not like we have an independent
way of measuring it.

144
00:09:41,580 --> 00:09:44,180
We don't really, not
to enough accuracy.

145
00:09:44,180 --> 00:09:47,920
So we're really just fitting
it to the data and removing it.

146
00:09:47,920 --> 00:09:49,480
But when we do the
split to the data,

147
00:09:49,480 --> 00:09:51,210
it's a three-parameter
fit, that is,

148
00:09:51,210 --> 00:09:53,235
we have three components
of a velocity to fit.

149
00:09:53,235 --> 00:09:55,320
We have a whole angular
pattern on the sky,

150
00:09:55,320 --> 00:09:57,754
and we only have three
numbers to play with.

151
00:09:57,754 --> 00:09:59,670
So it's strongly constrained
even though we're

152
00:09:59,670 --> 00:10:01,980
using the data
itself to determine

153
00:10:01,980 --> 00:10:04,240
what we think our
velocity is relative

154
00:10:04,240 --> 00:10:06,590
to the cosmic
microwave background.

155
00:10:06,590 --> 00:10:09,290
And after removing
it, then what we find

156
00:10:09,290 --> 00:10:13,620
is that the residual
deviations, delta t over t,

157
00:10:13,620 --> 00:10:16,985
are only at the level of
about 10 to the minus 5,

158
00:10:16,985 --> 00:10:22,170
1 part in 100,000, which
is really unbelievably

159
00:10:22,170 --> 00:10:25,370
isotropic, unbelievably uniform.

160
00:10:25,370 --> 00:10:29,140
One time I decided
to think about how

161
00:10:29,140 --> 00:10:32,505
round that is, how much
the same in all directions

162
00:10:32,505 --> 00:10:35,890
it is by asking
myself the question,

163
00:10:35,890 --> 00:10:38,920
is it possible to
grind a marble that

164
00:10:38,920 --> 00:10:43,187
would be spherical to an
accuracy of 10 to the minus 5.

165
00:10:43,187 --> 00:10:44,770
And you can think
about that yourself.

166
00:10:44,770 --> 00:10:47,610
The answer I came up with was
that yes it is, but it really

167
00:10:47,610 --> 00:10:49,340
strains the limits
of our technology.

168
00:10:49,340 --> 00:10:51,410
It correspond to sort
of the best technology

169
00:10:51,410 --> 00:10:54,470
we have for building
highly-precise lenses

170
00:10:54,470 --> 00:10:59,890
basically fractions of
a wavelength of light.

171
00:10:59,890 --> 00:11:02,320
So to round to 1
part in 10 to the 5

172
00:11:02,320 --> 00:11:06,657
is really being unbelievable
round, unbelievably isotropic.

173
00:11:06,657 --> 00:11:08,240
And that's the way
the universe looks.

174
00:11:11,150 --> 00:11:16,175
Next item in our description
of the universe is homogeneity.

175
00:11:29,800 --> 00:11:32,860
Homogeneity is harder
to test with precision

176
00:11:32,860 --> 00:11:35,410
because it means
looking out into space

177
00:11:35,410 --> 00:11:38,210
and trying to see, for example,
if the density of galaxies

178
00:11:38,210 --> 00:11:40,782
is uniform as a
function of distance.

179
00:11:40,782 --> 00:11:42,990
We always talked about as
a function of angle, that's

180
00:11:42,990 --> 00:11:46,683
isotropy, and it's
very uniform where

181
00:11:46,683 --> 00:11:48,510
one could make very
precise statements

182
00:11:48,510 --> 00:11:51,020
about the cosmic
microwave background.

183
00:11:51,020 --> 00:11:53,120
But to talk about
homogeneity, one

184
00:11:53,120 --> 00:11:56,150
has to be able to talk about how
the galaxy distribution varies

185
00:11:56,150 --> 00:11:58,920
with distance, and
distances are very

186
00:11:58,920 --> 00:12:02,060
hard to measure cosmologically.

187
00:12:02,060 --> 00:12:04,230
So as far as we could
tell, the universe

188
00:12:04,230 --> 00:12:07,920
is perfectly compatible with
being homogeneous, again,

189
00:12:07,920 --> 00:12:11,630
on length scales of a few
hundred million light years,

190
00:12:11,630 --> 00:12:15,550
but it's hard to make any
very precise statement.

191
00:12:15,550 --> 00:12:18,290
There is, of course,
relationships

192
00:12:18,290 --> 00:12:21,860
between isotropy
and homogeneity.

193
00:12:21,860 --> 00:12:23,735
Homogeneity, by the way
I didn't define that.

194
00:12:23,735 --> 00:12:25,109
I assumed you know
what it meant,

195
00:12:25,109 --> 00:12:27,110
but I should
definitely define it.

196
00:12:27,110 --> 00:12:29,280
Isotropy means the
same in all directions.

197
00:12:29,280 --> 00:12:31,060
Homogeneity means the
same at all places.

198
00:12:37,942 --> 00:12:41,870
So sometimes these are just put
together and called uniformity

199
00:12:41,870 --> 00:12:44,590
because they are very
similar concepts.

200
00:12:44,590 --> 00:12:46,980
They are, however, distinct
concepts logically,

201
00:12:46,980 --> 00:12:50,720
and it is worth spending a
little time understanding

202
00:12:50,720 --> 00:12:53,760
how they connect to each
other, in particular

203
00:12:53,760 --> 00:12:56,290
how you can have one without
the other is the best way

204
00:12:56,290 --> 00:13:00,470
to understand what
they individually mean.

205
00:13:00,470 --> 00:13:04,220
So suppose, for example,
we had a universe

206
00:13:04,220 --> 00:13:08,390
that was homogeneous
but not isotropic.

207
00:13:08,390 --> 00:13:10,590
Is that possible,
and if so, what

208
00:13:10,590 --> 00:13:12,630
would be an example
of a feature that

209
00:13:12,630 --> 00:13:15,810
would be described that way?

210
00:13:15,810 --> 00:13:17,810
Let me throw it out to you.

211
00:13:17,810 --> 00:13:21,750
We want to be homogeneous,
but not isotropic.

212
00:13:21,750 --> 00:13:22,250
Yes.

213
00:13:22,250 --> 00:13:25,363
AUDIENCE: It would be parallel
universes like a cylinder

214
00:13:25,363 --> 00:13:27,998
pointing in a z
direction, and I mean,

215
00:13:27,998 --> 00:13:29,914
matter is all homogeneous
with a cylinder

216
00:13:29,914 --> 00:13:32,310
but there is preferred
directions for isotropic.

217
00:13:32,310 --> 00:13:33,893
PROFESSOR: A preferred
direction fixed

218
00:13:33,893 --> 00:13:35,580
by the direction
of the periodicity?

219
00:13:35,580 --> 00:13:36,470
That is an example.

220
00:13:36,470 --> 00:13:37,259
That's right.

221
00:13:37,259 --> 00:13:37,800
That's right.

222
00:13:40,660 --> 00:13:45,640
Let me ask if there are other
examples people could think of.

223
00:13:45,640 --> 00:13:46,140
Yes.

224
00:13:46,140 --> 00:13:48,080
AUDIENCE: There are
galaxies everywhere

225
00:13:48,080 --> 00:13:50,343
with constant density,
but they're all

226
00:13:50,343 --> 00:13:51,970
aligned in a
particular direction.

227
00:13:51,970 --> 00:13:52,720
PROFESSOR: That's right.

228
00:13:52,720 --> 00:13:53,261
That's right.

229
00:13:53,261 --> 00:13:55,850
Galaxies have a
shape, in particular

230
00:13:55,850 --> 00:13:57,269
they have an angular momentum.

231
00:13:57,269 --> 00:13:58,810
The angular momentum
could be a line,

232
00:13:58,810 --> 00:14:02,300
and that would be an
example of a universe that

233
00:14:02,300 --> 00:14:05,350
would be homogeneous
but not isotropic.

234
00:14:05,350 --> 00:14:05,870
Very good.

235
00:14:05,870 --> 00:14:06,660
Very good.

236
00:14:06,660 --> 00:14:09,310
Another example that I'll just
throw out, which I think maybe

237
00:14:09,310 --> 00:14:11,590
is simple to think about
is the universe is filled

238
00:14:11,590 --> 00:14:14,330
with this cosmic microwave
background radiation.

239
00:14:14,330 --> 00:14:16,440
suppose all the photons
going in the z direction

240
00:14:16,440 --> 00:14:17,670
were more energetic
than the ones

241
00:14:17,670 --> 00:14:18,961
going in the x and y direction.

242
00:14:18,961 --> 00:14:21,229
That would be a
possible situation

243
00:14:21,229 --> 00:14:22,770
that could be
completely homogeneous,

244
00:14:22,770 --> 00:14:25,230
but would be an example of
something that would not

245
00:14:25,230 --> 00:14:26,445
be isotropic.

246
00:14:26,445 --> 00:14:28,445
So there are many examples
you can come up with.

247
00:14:28,445 --> 00:14:30,632
I'm very glad you came
up with the ones you did.

248
00:14:30,632 --> 00:14:32,330
That's great.

249
00:14:32,330 --> 00:14:34,794
Going the other way it's harder.

250
00:14:34,794 --> 00:14:36,710
Suppose we try to think
of the universe that's

251
00:14:36,710 --> 00:14:38,686
isotropic but not homogeneous.

252
00:14:41,310 --> 00:14:43,560
Isotropic, by the way, does
depend on the observer, so

253
00:14:43,560 --> 00:14:48,320
let's first talk about
isotropic relative to us.

254
00:14:48,320 --> 00:14:50,440
I was just going to say
imagining a universe that

255
00:14:50,440 --> 00:14:52,070
would be isotropic
relevant to us,

256
00:14:52,070 --> 00:14:53,369
but would not be homogeneous.

257
00:14:53,369 --> 00:14:53,868
Yes.

258
00:14:53,868 --> 00:14:56,520
AUDIENCE: Could it be like
if we lived in some shell.

259
00:14:56,520 --> 00:14:57,520
PROFESSOR: That's right.

260
00:14:57,520 --> 00:14:58,085
A shell structure.

261
00:14:58,085 --> 00:14:59,676
AUDIENCE: In all direction,
the shell would be there.

262
00:14:59,676 --> 00:15:00,900
PROFESSOR: That's right.

263
00:15:00,900 --> 00:15:01,500
That's right.

264
00:15:01,500 --> 00:15:03,416
I think I'll even draw
that on the blackboard.

265
00:15:06,250 --> 00:15:18,223
Example of isotropy
without homogeneity.

266
00:15:22,620 --> 00:15:28,290
So we would be
here, and the matter

267
00:15:28,290 --> 00:15:31,390
could be distributed in a
perfectly spherically-symmetric

268
00:15:31,390 --> 00:15:34,320
distribution with
us at the center.

269
00:15:34,320 --> 00:15:36,095
And that would be an
example of something

270
00:15:36,095 --> 00:15:41,050
that would be isotopic to
us but not homogeneous.

271
00:15:41,050 --> 00:15:42,450
Now, things like
that, of course,

272
00:15:42,450 --> 00:15:45,140
are considered weird because
we don't think of ourselves

273
00:15:45,140 --> 00:15:48,080
as living in any special
place in the universe,

274
00:15:48,080 --> 00:15:50,790
and that's basically what
the Copernican Revolution was

275
00:15:50,790 --> 00:15:51,850
all about.

276
00:15:51,850 --> 00:15:55,810
And the Copernican Revolution
is sunken very deeply

277
00:15:55,810 --> 00:15:58,390
into the psychology
of scientists.

278
00:15:58,390 --> 00:16:02,730
So I think scientists would
be very loathed to imagine

279
00:16:02,730 --> 00:16:04,680
the universe that
look like this,

280
00:16:04,680 --> 00:16:09,480
but it does help to understand
what these words mean.

281
00:16:09,480 --> 00:16:14,370
If a universe is going to be
isotropic to all observers,

282
00:16:14,370 --> 00:16:17,400
then it does have
to be homogeneous,

283
00:16:17,400 --> 00:16:18,912
and that's part
of the reason why

284
00:16:18,912 --> 00:16:21,120
we're pretty confident that
our universe is basically

285
00:16:21,120 --> 00:16:24,380
homogeneous, because we just
decided that's isotropic to us,

286
00:16:24,380 --> 00:16:27,890
and we decide we're
not special then it

287
00:16:27,890 --> 00:16:29,959
has to be isotropic to
everybody and then it

288
00:16:29,959 --> 00:16:30,875
has to be homogeneous.

289
00:16:37,262 --> 00:16:58,350
If the universe is
isotropic to all observers,

290
00:16:58,350 --> 00:17:10,596
it is homogeneous

291
00:17:10,596 --> 00:17:11,970
Now, a thought
which I will leave

292
00:17:11,970 --> 00:17:14,980
for you to think about between
now and the next lecture

293
00:17:14,980 --> 00:17:18,190
is whether or not really
knowing that a universe is

294
00:17:18,190 --> 00:17:21,027
isotropic with respect
to two observers

295
00:17:21,027 --> 00:17:22,735
is enough to prove
that it's homogeneous.

296
00:17:25,032 --> 00:17:26,865
That turned out to be
a more subtle question

297
00:17:26,865 --> 00:17:28,339
than it might sound.

298
00:17:28,339 --> 00:17:30,300
I don't know if it
sounds subtle or not.

299
00:17:30,300 --> 00:17:33,820
I should maybe just tell you
basically what the answer is

300
00:17:33,820 --> 00:17:36,760
and then you can try to think if
you can understand the answer.

301
00:17:36,760 --> 00:17:41,890
In the Euclidean space, isotropy
about two distinct observers

302
00:17:41,890 --> 00:17:44,096
is enough to make
it homogeneous,

303
00:17:44,096 --> 00:17:46,617
which is kind of
what you visualize.

304
00:17:46,617 --> 00:17:48,200
But if you can allow
yourself to think

305
00:17:48,200 --> 00:17:49,679
about non-Euclidean
spaces, and I

306
00:17:49,679 --> 00:17:51,720
know we haven't talked
about non-Euclidean spaces

307
00:17:51,720 --> 00:17:55,950
yet so you might not have in the
way of tools to think about it.

308
00:17:55,950 --> 00:17:58,994
But think, for example, about
surfaces in three dimensions.

309
00:17:58,994 --> 00:18:00,410
Surfaces are very
good examples of

310
00:18:00,410 --> 00:18:03,310
non-Euclidean
two-dimensional geometries.

311
00:18:03,310 --> 00:18:07,580
And see if you can invent a
two-dimensional geometry that

312
00:18:07,580 --> 00:18:10,330
would be isotropic
about two points,

313
00:18:10,330 --> 00:18:11,800
but would not be homogeneous.

314
00:18:11,800 --> 00:18:16,670
So that's your thought
assignment for next time,

315
00:18:16,670 --> 00:18:19,596
not to be handed in just to
be talked about in the lecture

316
00:18:19,596 --> 00:18:22,230
next time.

317
00:18:22,230 --> 00:18:28,610
So isotropy and homogeneity are
two of the key properties that

318
00:18:28,610 --> 00:18:34,180
define the simplicity of our
universe on very large scales.

319
00:18:34,180 --> 00:18:36,220
The next thing I
want to talk about

320
00:18:36,220 --> 00:18:40,390
is the expansion of the
universe, which is basically

321
00:18:40,390 --> 00:18:43,240
characterized by Hubble's law.

322
00:18:56,990 --> 00:18:58,490
Last time I think
I said I was going

323
00:18:58,490 --> 00:18:59,948
to call it the
Lemaitre-Hubble law.

324
00:18:59,948 --> 00:19:01,950
I decided I'll probably
call it Hubble's law.

325
00:19:01,950 --> 00:19:03,370
Now, Hubble, I
think, really does

326
00:19:03,370 --> 00:19:05,840
deserve credit for
demonstrating observations

327
00:19:05,840 --> 00:19:08,164
that the law is true,
and that's really

328
00:19:08,164 --> 00:19:10,080
what he is getting credit
for and that was not

329
00:19:10,080 --> 00:19:11,530
believed until he discovered it.

330
00:19:11,530 --> 00:19:14,540
So it really did have
a tremendous effect

331
00:19:14,540 --> 00:19:17,362
on the course of cosmology.

332
00:19:17,362 --> 00:19:21,140
So Hubble's law says that
on average all galaxies

333
00:19:21,140 --> 00:19:25,640
are receding from us
with a velocity which

334
00:19:25,640 --> 00:19:29,320
is equal to a constant, H,
called the Hubble constant--

335
00:19:29,320 --> 00:19:31,770
Hubble called it K,
by the way, capital

336
00:19:31,770 --> 00:19:37,070
K-- times the distance
to the Galaxy, r.

337
00:19:37,070 --> 00:19:40,110
And so it's not true
exactly for our universe,

338
00:19:40,110 --> 00:19:43,460
but it's true in
some average sense,

339
00:19:43,460 --> 00:19:46,300
just as isotropy
and homogeneity are,

340
00:19:46,300 --> 00:19:48,090
we're only true on
an average sense.

341
00:19:52,960 --> 00:19:56,170
I want to tell you about the
units in which it's measured

342
00:19:56,170 --> 00:19:58,910
and that leads me to the parsec.

343
00:19:58,910 --> 00:20:00,320
Let me write this on the board.

344
00:20:00,320 --> 00:20:06,260
But astronomers always
measure the Hubble constant

345
00:20:06,260 --> 00:20:08,810
or I will sometimes call
it the Hubble expansion

346
00:20:08,810 --> 00:20:22,483
rate in kilometers per
second per megaparsec.

347
00:20:27,320 --> 00:20:30,660
And it's a relationship between
a velocity and distance,

348
00:20:30,660 --> 00:20:32,330
so kilometers per
second is velocity

349
00:20:32,330 --> 00:20:35,530
and velocity per
megaparsec is the velocity

350
00:20:35,530 --> 00:20:38,007
per distance, which
is what it should be.

351
00:20:38,007 --> 00:20:39,465
Notice, however,
that I wrote that.

352
00:20:39,465 --> 00:20:42,760
A kilometer and a megaparsec
are both units of distance.

353
00:20:42,760 --> 00:20:45,400
So they actually just
have some fixed ratio.

354
00:20:45,400 --> 00:20:47,510
So in the end, this
Hubble constant really

355
00:20:47,510 --> 00:20:50,535
is just an inverse
time, and obviously,

356
00:20:50,535 --> 00:20:52,740
if you multiply an inverse
time times the distance

357
00:20:52,740 --> 00:20:55,900
you get a distance per
time, which is the velocity,

358
00:20:55,900 --> 00:20:56,940
so that works.

359
00:20:56,940 --> 00:21:01,660
But it's very seldom quoted
as simply an inverse time,

360
00:21:01,660 --> 00:21:05,340
instead it's quoted by the units
that astronomers like to use.

361
00:21:05,340 --> 00:21:09,390
They measure velocities
as a normal person would

362
00:21:09,390 --> 00:21:11,510
in kilometers per
second, but they

363
00:21:11,510 --> 00:21:13,956
measure distances
in megaparsecs,

364
00:21:13,956 --> 00:21:16,980
where a megaparsec
is a million parsecs,

365
00:21:16,980 --> 00:21:19,095
and a parsec is defined
by that diagram.

366
00:21:21,960 --> 00:21:25,380
The base of this triangle
is one astronomical unit,

367
00:21:25,380 --> 00:21:29,340
the mean distance between
the Earth the sun.

368
00:21:29,340 --> 00:21:35,540
And the distance at
which the angles attended

369
00:21:35,540 --> 00:21:38,820
by one astronomical unit
is one second of arc

370
00:21:38,820 --> 00:21:43,960
is what's called a parsec
and abbreviated pc.

371
00:21:43,960 --> 00:21:46,940
And a parsec is about
three light years.

372
00:21:46,940 --> 00:21:48,840
I'll write these
things on the board.

373
00:21:52,020 --> 00:22:04,990
One parsec equals
3.2616 light years,

374
00:22:04,990 --> 00:22:08,970
and a megaparsec is
a million of those.

375
00:22:08,970 --> 00:22:13,600
Another useful number to
keep in mind for converting,

376
00:22:13,600 --> 00:22:17,500
if you want to think
of H as inverse years,

377
00:22:17,500 --> 00:22:33,290
then the useful equality is
that 1 over 10 to the 10 years

378
00:22:33,290 --> 00:22:41,560
is equal to 97.8, and it's
suitable to remember this

379
00:22:41,560 --> 00:22:45,560
as being 100-- you can look up
the exact number when you need

380
00:22:45,560 --> 00:22:54,510
it-- and these funny things
kilometers per second

381
00:22:54,510 --> 00:22:55,625
per megaparsec.

382
00:23:04,100 --> 00:23:06,500
So what is the value
of Hubble's constant?

383
00:23:09,390 --> 00:23:17,260
It actually has a
very interesting and

384
00:23:17,260 --> 00:23:19,750
historically-significant
history.

385
00:23:19,750 --> 00:23:24,180
It was first measured in
this paper by George Lemaitre

386
00:23:24,180 --> 00:23:27,620
and in 1927, published
only in French

387
00:23:27,620 --> 00:23:31,700
and ignored by the rest of
world, at the time at least.

388
00:23:31,700 --> 00:23:33,850
It got discovered later.

389
00:23:33,850 --> 00:23:36,460
And Lemaitre was
not an astronomer.

390
00:23:36,460 --> 00:23:38,470
He was a theoretical
cosmologist.

391
00:23:38,470 --> 00:23:43,150
I mentioned a few times I
think he had a PhD from MIT

392
00:23:43,150 --> 00:23:47,740
in theoretical cosmology
in physics, in principle.

393
00:23:47,740 --> 00:23:52,290
And the value that he got based
on looking at other people's

394
00:23:52,290 --> 00:24:14,500
data, in 1927, had the
value of-- I guess actually,

395
00:24:14,500 --> 00:24:15,910
I'll give you the range.

396
00:24:15,910 --> 00:24:18,140
He gave two different
methods of calculating it.

397
00:24:18,140 --> 00:24:20,260
We've got two slightly
different answers.

398
00:24:20,260 --> 00:24:28,192
So we had 575 to 625 of these
[INAUDIBLE] units kilometers

399
00:24:28,192 --> 00:24:29,275
per second per megaparsec.

400
00:24:37,210 --> 00:24:43,120
And two years later
in his famous paper

401
00:24:43,120 --> 00:24:53,302
"Hubble," got the
value of 500 kilometers

402
00:24:53,302 --> 00:24:54,385
per second per megaparsec.

403
00:24:57,620 --> 00:24:59,986
I have a picture of Hubble too.

404
00:24:59,986 --> 00:25:00,486
Yes.

405
00:25:00,486 --> 00:25:04,101
AUDIENCE: That last in
the board right there

406
00:25:04,101 --> 00:25:11,537
where you have 1 over 10 to
the 10 amperes, is that H?

407
00:25:11,537 --> 00:25:13,370
PROFESSOR: That's just
an equality of units.

408
00:25:13,370 --> 00:25:14,670
AUDIENCE: Quality of units.

409
00:25:14,670 --> 00:25:16,900
PROFESSOR: That's just
the unit equality.

410
00:25:16,900 --> 00:25:20,930
It's relevant to H, because
H is measured in those units.

411
00:25:20,930 --> 00:25:25,060
But it really is just
an equality of units.

412
00:25:25,060 --> 00:25:28,110
1 over 10 to the 10th years
has units of inverse time,

413
00:25:28,110 --> 00:25:30,820
and kilometers per
second per megaparsec

414
00:25:30,820 --> 00:25:33,850
has units of inverse time
also because kilometers

415
00:25:33,850 --> 00:25:36,940
is distance and megaparsec
is inverse distance.

416
00:25:36,940 --> 00:25:39,860
So both sides have the same
units and the same dimensions,

417
00:25:39,860 --> 00:25:42,500
I should say, and it's
just two different ways

418
00:25:42,500 --> 00:25:44,620
of measuring the same
thing, inverse times.

419
00:25:48,750 --> 00:25:51,560
So in 1929, Hubble
published his famous paper

420
00:25:51,560 --> 00:25:53,750
which he got the value
of 500, and there's

421
00:25:53,750 --> 00:25:56,317
an important difference really
between the papers by Lemaitre

422
00:25:56,317 --> 00:25:56,820
and Hubble.

423
00:25:56,820 --> 00:25:59,410
First of all, Hubble was
using largely his own data.

424
00:25:59,410 --> 00:26:01,430
Lemaitre was using other
people's data mostly

425
00:26:01,430 --> 00:26:03,210
Hubble's actually.

426
00:26:03,210 --> 00:26:06,510
And furthermore,
Hubble made the claim

427
00:26:06,510 --> 00:26:12,010
that the data justified
the relationship

428
00:26:12,010 --> 00:26:14,940
that v is equal to
a constant times r.

429
00:26:14,940 --> 00:26:17,150
Lemaitre knew that
relation theoretically

430
00:26:17,150 --> 00:26:19,170
for a uniformly-expanding
universe,

431
00:26:19,170 --> 00:26:22,000
which we'll be
talking about shortly.

432
00:26:22,000 --> 00:26:26,710
But he did not claim to be
able to get it from the data.

433
00:26:26,710 --> 00:26:29,260
The data he had he decided
was not strong enough

434
00:26:29,260 --> 00:26:31,840
to reach that
conclusion, but he was

435
00:26:31,840 --> 00:26:36,550
still able to get a value for H
by taking the average velocity

436
00:26:36,550 --> 00:26:42,470
dividing it by the average
distance and got a number.

437
00:26:42,470 --> 00:26:44,290
I think I have
Hubble's data next.

438
00:26:44,290 --> 00:26:46,860
Yeah, here's Hubble's data.

439
00:26:46,860 --> 00:26:51,160
The data obviously
was not very good.

440
00:26:51,160 --> 00:26:54,580
It only goes up to about 1,000.

441
00:26:54,580 --> 00:26:57,130
One curiosity of this
graph that you might notice

442
00:26:57,130 --> 00:27:00,940
is that the vertical axis
is a velocity meaning

443
00:27:00,940 --> 00:27:03,390
it should be measured in
kilometers per second,

444
00:27:03,390 --> 00:27:08,210
but nonetheless Hubble
wrote it as kilometers.

445
00:27:08,210 --> 00:27:11,164
Not getting his units right,
so minus 10 or something

446
00:27:11,164 --> 00:27:15,880
like that on the graded sheet.

447
00:27:15,880 --> 00:27:17,646
But somehow it did
not stop the paper

448
00:27:17,646 --> 00:27:20,020
from getting published in the
proceedings of the National

449
00:27:20,020 --> 00:27:23,930
Academy of Sciences
and had become,

450
00:27:23,930 --> 00:27:27,420
of course, a
monumentally-famous paper.

451
00:27:27,420 --> 00:27:30,240
But you can see that
the data is scattered,

452
00:27:30,240 --> 00:27:34,230
and it has those nice
lines drawn through which

453
00:27:34,230 --> 00:27:36,640
guide your eye, but if you
imagine taking away the lines,

454
00:27:36,640 --> 00:27:39,350
it's not that clear
on the data itself

455
00:27:39,350 --> 00:27:41,790
that it really is a
linear relationship.

456
00:27:41,790 --> 00:27:44,139
But it's suggested,
at least, and Hubble

457
00:27:44,139 --> 00:27:46,180
thought it was pretty
convincing and later Hubble

458
00:27:46,180 --> 00:27:48,845
gathered more data
for this project,

459
00:27:48,845 --> 00:27:50,720
and it did become quite
convincing that there

460
00:27:50,720 --> 00:27:53,040
is a linear relationship,
and today there's

461
00:27:53,040 --> 00:27:56,930
no doubt that there is a linear
relationship between velocity

462
00:27:56,930 --> 00:27:57,929
and distance.

463
00:27:57,929 --> 00:27:59,720
At very large distances
there's deviations,

464
00:27:59,720 --> 00:28:01,450
which we can
understand and we'll

465
00:28:01,450 --> 00:28:06,100
be talking about later,
but basically, at least

466
00:28:06,100 --> 00:28:10,821
for moderate distances, one
has this linear relationship.

467
00:28:10,821 --> 00:28:13,070
I should mention that the
velocity of the solar system

468
00:28:13,070 --> 00:28:16,660
through the CMB is also the
velocity of the solar system

469
00:28:16,660 --> 00:28:18,840
through this pattern
of Hubble expansion,

470
00:28:18,840 --> 00:28:21,560
and both Hubble and Lemaitre
had to make estimates

471
00:28:21,560 --> 00:28:23,230
of the velocity of
the solar system

472
00:28:23,230 --> 00:28:25,705
relative to these galaxies
and subtract that out

473
00:28:25,705 --> 00:28:27,950
to get things that
resemble a straight line.

474
00:28:30,950 --> 00:28:33,900
Lemaitre estimated the
velocity of our solar system

475
00:28:33,900 --> 00:28:39,290
as 300 kilometers per second,
and Hubble estimated it

476
00:28:39,290 --> 00:28:41,204
as 280 kilometers per second.

477
00:28:41,204 --> 00:28:43,120
So it was a relevant
feature because remember,

478
00:28:43,120 --> 00:28:45,298
the maximum velocity there
is only 1,000 kilometers

479
00:28:45,298 --> 00:28:48,126
per second, so the correction
that he's putting in

480
00:28:48,126 --> 00:28:51,350
is about a third of the
maximum velocity seen.

481
00:28:51,350 --> 00:28:55,744
So it's a very important and not
that it was easy to determine.

482
00:28:55,744 --> 00:28:57,712
AUDIENCE: What were
they using to determine

483
00:28:57,712 --> 00:28:58,696
the [INAUDIBLE] CMB?

484
00:28:58,696 --> 00:29:00,370
PROFESSOR: I think
they were just

485
00:29:00,370 --> 00:29:02,520
looking for what
they could assume

486
00:29:02,520 --> 00:29:04,640
that would make the
average expansion

487
00:29:04,640 --> 00:29:06,936
in all directions
about the same.

488
00:29:06,936 --> 00:29:08,606
To be honest, I'm
not sure about that.

489
00:29:08,606 --> 00:29:10,980
But that's the only thing I
can see that they would have,

490
00:29:10,980 --> 00:29:12,890
so I think that must be
what they were using.

491
00:29:18,780 --> 00:29:22,466
Now since these
ancient times, there

492
00:29:22,466 --> 00:29:24,840
have been many measurements
of the Hubble expansion rate,

493
00:29:24,840 --> 00:29:27,540
and they changed a great deal.

494
00:29:27,540 --> 00:29:32,910
So in the '40s
through '60s, there

495
00:29:32,910 --> 00:29:36,290
was a whole series of
measurements dominated

496
00:29:36,290 --> 00:29:43,130
by people like Walter
Baade and Allan Sandage.

497
00:29:52,764 --> 00:29:54,180
And generally
speaking, the values

498
00:29:54,180 --> 00:29:58,052
came down steadily
from the high values

499
00:29:58,052 --> 00:29:59,760
that were measured by
Hubble and Lemaitre

500
00:29:59,760 --> 00:30:02,320
in the very early days.

501
00:30:02,320 --> 00:30:05,790
When I was a graduate
student, if you asked anybody

502
00:30:05,790 --> 00:30:09,050
what the Hubble constant was,
you always got the same answer.

503
00:30:09,050 --> 00:30:11,640
It was somewhere
between 50 and 100,

504
00:30:11,640 --> 00:30:13,730
still uncertain
by a factor of 2,

505
00:30:13,730 --> 00:30:18,200
but much lower by a factor
of 5 or 10 from the values

506
00:30:18,200 --> 00:30:23,860
that Hubble was talk
about, and was still

507
00:30:23,860 --> 00:30:28,100
a major source of uncertainty
in talking about cosmology.

508
00:30:28,100 --> 00:30:38,880
Values started to become
more precise around 2001.

509
00:30:38,880 --> 00:30:40,854
So in 2001, there was
the Hubble Key Project

510
00:30:40,854 --> 00:30:42,020
that released these results.

511
00:30:45,060 --> 00:30:47,550
The word Hubble here refers
to the Hubble satellite,

512
00:30:47,550 --> 00:30:51,140
which was named after
Hubble-- Hubble, Edwin.

513
00:30:51,140 --> 00:30:56,160
And they were able to use
the Hubble telescope to see

514
00:30:56,160 --> 00:30:59,140
Cepheid variables and galaxies,
that was significantly further

515
00:30:59,140 --> 00:31:01,800
than Cepheid variables
can ever be seen before

516
00:31:01,800 --> 00:31:04,225
and thereby make a much better
calibration of the distance

517
00:31:04,225 --> 00:31:05,440
scale.

518
00:31:05,440 --> 00:31:08,100
As you'll learn about
when you do your reading,

519
00:31:08,100 --> 00:31:10,370
Cepheid variables are
crucial to determining

520
00:31:10,370 --> 00:31:12,950
the cosmological distance scale.

521
00:31:12,950 --> 00:31:20,200
So the value that they
got was much more precise

522
00:31:20,200 --> 00:31:24,380
than anything previous, 72 plus
or minus 8 of these [? quad ?]

523
00:31:24,380 --> 00:31:26,260
units kilometers per
second per megaparsec.

524
00:31:28,810 --> 00:31:31,240
Meanwhile things were
still controversial.

525
00:31:31,240 --> 00:31:34,090
I should have added that when
people said it was 50 to 100

526
00:31:34,090 --> 00:31:37,920
when I was a graduate student,
it wasn't that people really

527
00:31:37,920 --> 00:31:39,910
understood the error
bars to be that large.

528
00:31:39,910 --> 00:31:42,510
The real situation is that there
were a group of astronomers

529
00:31:42,510 --> 00:31:43,992
that claimed
adamantly it was 50,

530
00:31:43,992 --> 00:31:45,950
and there were other
groups of astronomers that

531
00:31:45,950 --> 00:31:47,980
claimed adamantly
that it was 100.

532
00:31:47,980 --> 00:31:50,530
Anyway On person is shouting
in your ear saying it's 100,

533
00:31:50,530 --> 00:31:52,113
another person is
shouting in your ear

534
00:31:52,113 --> 00:31:55,740
saying it's 50 the conclusion
is that it's 50 to 100,

535
00:31:55,740 --> 00:31:59,120
and that's the situation when
I was a graduate student.

536
00:31:59,120 --> 00:32:05,730
So this was a somewhat high
value relative to the argument.

537
00:32:05,730 --> 00:32:08,240
The people who are
arguing on the low side

538
00:32:08,240 --> 00:32:11,360
were still in business at this
time and still in fact also

539
00:32:11,360 --> 00:32:14,190
using Hubble telescope data.

540
00:32:14,190 --> 00:32:17,700
So Tammann and
Sandage, the same year

541
00:32:17,700 --> 00:32:22,090
using the same instrument--
let me put the year here,

542
00:32:22,090 --> 00:32:27,450
and it's 2001-- Tammann
and Sandage were estimating

543
00:32:27,450 --> 00:32:31,240
60 plus or minus they
said less than 10%.

544
00:32:34,880 --> 00:32:36,380
so these didn't quite mesh.

545
00:32:55,660 --> 00:33:09,312
Coming to more modern
times, in 2003, WMAP,

546
00:33:09,312 --> 00:33:21,130
the satellite called the
Wilkinson Microwave Anisotropy

547
00:33:21,130 --> 00:33:26,890
Probe, a satellite
dedicate to measuring

548
00:33:26,890 --> 00:33:30,140
these minute variations of the
cosmic microwave background

549
00:33:30,140 --> 00:33:32,755
at the level of 1
part in 100,000,

550
00:33:32,755 --> 00:33:34,290
it turns out that
those measurements

551
00:33:34,290 --> 00:33:36,700
are estimated at the
Hubble expansion rate

552
00:33:36,700 --> 00:33:39,950
by fitting the data to
a theoretical model.

553
00:33:39,950 --> 00:33:51,440
And their initial number
was 72 plus or minus 5.

554
00:33:54,490 --> 00:33:57,590
And that was based
on one year of data.

555
00:33:57,590 --> 00:34:20,650
And in 2011, the same
WMAP satellite team

556
00:34:20,650 --> 00:34:23,440
was based on seven
years of data,

557
00:34:23,440 --> 00:34:28,270
came up with a number
of 70.2 plus or minus

558
00:34:28,270 --> 00:34:32,429
1.4, so to very precise.

559
00:34:32,429 --> 00:34:35,380
And the most recent number
comes from a similar satellite

560
00:34:35,380 --> 00:34:38,659
to WMAP but more recent
and more powerful

561
00:34:38,659 --> 00:34:44,000
satellite called
Planck, which just

562
00:34:44,000 --> 00:34:45,765
released its data last March.

563
00:34:49,100 --> 00:34:54,319
And it came up with a somewhat
surprisingly low number 67.3

564
00:34:54,319 --> 00:35:01,201
plus or minus 1.2.

565
00:35:01,201 --> 00:35:01,700
Yes.

566
00:35:01,700 --> 00:35:04,520
AUDIENCE: The other
measurements there

567
00:35:04,520 --> 00:35:07,971
are kind of inconsistent
with one another

568
00:35:07,971 --> 00:35:11,422
and then with one measurement
sort of 20th century

569
00:35:11,422 --> 00:35:13,065
make this big jump
down suggesting

570
00:35:13,065 --> 00:35:15,314
those early guys were making
the same kind of mistake.

571
00:35:15,314 --> 00:35:15,860
What was it?

572
00:35:15,860 --> 00:35:17,080
PROFESSOR: Good question.

573
00:35:17,080 --> 00:35:20,380
The early guys were
making a big mistake

574
00:35:20,380 --> 00:35:22,710
in estimating the
distance scale,

575
00:35:22,710 --> 00:35:25,590
and I'm not sure I understand
the details of that,

576
00:35:25,590 --> 00:35:29,130
but I think it had something to
do with misidentifying Cepheid

577
00:35:29,130 --> 00:35:32,540
variables, equating two
different types that should not

578
00:35:32,540 --> 00:35:34,265
have been compared
with each other.

579
00:35:34,265 --> 00:35:36,507
But I'm not altogether
sure of the details,

580
00:35:36,507 --> 00:35:38,840
but it was definitely the
distance scale they had wrong.

581
00:35:38,840 --> 00:35:42,100
The velocities are pretty
easy to measure accurately,

582
00:35:42,100 --> 00:35:43,234
and they were very wrong.

583
00:35:43,234 --> 00:35:43,734
Yes.

584
00:35:43,734 --> 00:35:46,700
AUDIENCE: There's two
types of Cepheids,

585
00:35:46,700 --> 00:35:49,360
one has a certain period
of velocity relation

586
00:35:49,360 --> 00:35:52,264
that would give it, and it's
like a completely different

587
00:35:52,264 --> 00:35:54,406
type of star, and
so we got mixed up,

588
00:35:54,406 --> 00:35:57,678
and we got completely different
absolute magnitudes, which

589
00:35:57,678 --> 00:36:00,642
will give you two completely
different distance estimates.

590
00:36:00,642 --> 00:36:03,359
So I don't know how
far, but measuring

591
00:36:03,359 --> 00:36:06,552
Cepheids and Andromeda was
way off the distance scale

592
00:36:06,552 --> 00:36:08,052
because we thought
they were Type 1,

593
00:36:08,052 --> 00:36:09,534
but they were actually Type 2.

594
00:36:09,534 --> 00:36:11,492
AUDIENCE: I think the
difference between Type 1

595
00:36:11,492 --> 00:36:13,849
and Type 2 are a factor of
4, so that would make sense.

596
00:36:13,849 --> 00:36:14,474
AUDIENCE: Yeah.

597
00:36:14,474 --> 00:36:16,944
It's like two completely
different linear relations.

598
00:36:20,402 --> 00:36:21,880
PROFESSOR: An
intensity goes like 1

599
00:36:21,880 --> 00:36:24,270
over the distance squared,
so I think that, I mean,

600
00:36:24,270 --> 00:36:29,115
a factor of 4 in intensity I
think would mean a factor of 16

601
00:36:29,115 --> 00:36:32,090
in distance estimates.

602
00:36:32,090 --> 00:36:32,590
Yes.

603
00:36:32,590 --> 00:36:34,760
AUDIENCE: I'm following
so much that these

604
00:36:34,760 --> 00:36:36,820
are like, they both
have error bars,

605
00:36:36,820 --> 00:36:38,937
but they're not within
error of each other.

606
00:36:38,937 --> 00:36:39,645
PROFESSOR: Right.

607
00:36:39,645 --> 00:36:42,830
AUDIENCE: Well, this
is like current data.

608
00:36:42,830 --> 00:36:44,900
PROFESSOR: So what's going on?

609
00:36:44,900 --> 00:36:45,880
Nobody knows for sure.

610
00:36:45,880 --> 00:36:46,905
One thing I should
mention though

611
00:36:46,905 --> 00:36:48,863
is that these are what
are called 1 sigma error

612
00:36:48,863 --> 00:36:51,520
bars, which means
that you don't expect

613
00:36:51,520 --> 00:36:53,100
them to necessarily agree.

614
00:36:53,100 --> 00:36:56,240
You expect the right answer to
be within one sigma error bar

615
00:36:56,240 --> 00:36:58,330
2/3 of the, time
but 1/3 of the time

616
00:36:58,330 --> 00:36:59,900
it should be outside
the error bar.

617
00:36:59,900 --> 00:37:01,731
The questions is, the
error bar is on both.

618
00:37:01,731 --> 00:37:03,230
But the comparison
of this, and this

619
00:37:03,230 --> 00:37:05,170
is usually viewed
as something like 2

620
00:37:05,170 --> 00:37:08,984
and 1/2 sigma effect,
which naively, I think,

621
00:37:08,984 --> 00:37:10,400
means the probability
of something

622
00:37:10,400 --> 00:37:12,108
on the order of 1% or
something like that

623
00:37:12,108 --> 00:37:14,650
of getting errors
that large at random.

624
00:37:14,650 --> 00:37:18,590
And it's debated whether or
not it's significant or not.

625
00:37:18,590 --> 00:37:22,840
It's the abstract of
the Planck paper use

626
00:37:22,840 --> 00:37:26,680
words something like there's
a tension between their value

627
00:37:26,680 --> 00:37:27,915
and other recent values.

628
00:37:31,355 --> 00:37:32,980
When somebody does
see things like that

629
00:37:32,980 --> 00:37:36,410
happen more frequently than
the probabilities indicate,

630
00:37:36,410 --> 00:37:38,290
which I think it
proves a theorem

631
00:37:38,290 --> 00:37:42,250
that experimenters always
underestimate their error bars.

632
00:37:42,250 --> 00:37:46,390
But there's no absolute
proof of that theorem.

633
00:37:46,390 --> 00:37:48,580
So these thing were
early debatable.

634
00:37:48,580 --> 00:37:51,022
People don't know-- there
are many things that turn up

635
00:37:51,022 --> 00:37:52,605
in experimental
physics and especially

636
00:37:52,605 --> 00:37:57,150
in cosmology that turn up
regularly where people have

637
00:37:57,150 --> 00:37:59,565
different opinions about
whether or not it's pointing

638
00:37:59,565 --> 00:38:01,420
to something very
important or something

639
00:38:01,420 --> 00:38:02,530
that's going to go away.

640
00:38:02,530 --> 00:38:05,060
So very often they go
away, that's a fact.

641
00:38:05,060 --> 00:38:06,889
But you never know
in any one case,

642
00:38:06,889 --> 00:38:08,430
whether it's something
important that

643
00:38:08,430 --> 00:38:12,340
will become more definite as
for the measurements are made

644
00:38:12,340 --> 00:38:14,980
or whether it's just
a spurious effect that

645
00:38:14,980 --> 00:38:16,951
will disappear in a few years.

646
00:38:16,951 --> 00:38:17,450
Yes.

647
00:38:17,450 --> 00:38:20,887
AUDIENCE: So I
imagine in the 1940s

648
00:38:20,887 --> 00:38:23,577
when people started
saying yes, that Hubble,

649
00:38:23,577 --> 00:38:25,993
for whom the constant is named,
was off by a factor of 10.

650
00:38:25,993 --> 00:38:27,076
That's very controversial.

651
00:38:27,076 --> 00:38:29,234
Was there any kind
of sloping trend

652
00:38:29,234 --> 00:38:31,689
where people may have changed
their data to make it seem,

653
00:38:31,689 --> 00:38:35,126
oh, we're not that far
off the Hubble standards.

654
00:38:35,126 --> 00:38:37,090
Has this happened a
couple of times before?

655
00:38:37,090 --> 00:38:40,820
PROFESSOR: Question is did
people perhaps try to fudge

656
00:38:40,820 --> 00:38:43,800
their data during a period in
the middle to make it look more

657
00:38:43,800 --> 00:38:45,020
like Hubble's.

658
00:38:45,020 --> 00:38:50,720
I think, I don't
know, and there were,

659
00:38:50,720 --> 00:38:53,737
as I said, pretty much through
the middle of the 20th century

660
00:38:53,737 --> 00:38:55,820
two groups, one of which
was getting a high value,

661
00:38:55,820 --> 00:38:57,790
and one of which was
getting a low value.

662
00:38:57,790 --> 00:39:00,820
The high value is where, in
fact, disciples of Hubble,

663
00:39:00,820 --> 00:39:02,890
rather directly-- wait a minute.

664
00:39:02,890 --> 00:39:05,600
That's not right.

665
00:39:05,600 --> 00:39:08,344
The most direct disciple of
Hubble was Allan Sandage,

666
00:39:08,344 --> 00:39:10,260
and he was, in fact,
abdicating the low value.

667
00:39:13,280 --> 00:39:15,600
So the sociological
trends are not that clear.

668
00:39:15,600 --> 00:39:19,050
What is clear is that
they were way off.

669
00:39:19,050 --> 00:39:22,620
I was going to add
concerning the way offness,

670
00:39:22,620 --> 00:39:28,060
that it really does have or did
have a very significant effect

671
00:39:28,060 --> 00:39:33,470
on the history of cosmology
because when one looks at a Big

672
00:39:33,470 --> 00:39:37,180
Bang model and tries to use that
model to estimate when it all

673
00:39:37,180 --> 00:39:40,030
started, what you're doing is
you're trying to extrapolate

674
00:39:40,030 --> 00:39:42,530
backwards, ask when
was everything on top

675
00:39:42,530 --> 00:39:46,892
of each other given that things
are moving at the speed now.

676
00:39:46,892 --> 00:39:48,725
There is more that goes
into the calculation

677
00:39:48,725 --> 00:39:52,570
then just H. It depends on your
model, the matter, and things

678
00:39:52,570 --> 00:39:53,680
like that.

679
00:39:53,680 --> 00:39:56,400
But nonetheless H is obviously
a crucial ingredient there.

680
00:39:56,400 --> 00:39:58,837
The faster things are
moving now outward

681
00:39:58,837 --> 00:40:00,920
when you extrapolate
backwards, the faster they're

682
00:40:00,920 --> 00:40:04,140
moving inward and the
younger the universe is.

683
00:40:04,140 --> 00:40:08,100
And to a very good
degree of reliability,

684
00:40:08,100 --> 00:40:15,510
any age estimate-- and we'll
make age calculations later--

685
00:40:15,510 --> 00:40:17,900
but any age estimate
is proportional to 1

686
00:40:17,900 --> 00:40:24,230
over the Hubble parameter, 1
over the Hubble expansion rate.

687
00:40:24,230 --> 00:40:27,460
So if you're off by now
we would say a factor of 7

688
00:40:27,460 --> 00:40:34,090
between Hubble's value and the
current value, 70 versus 500,

689
00:40:34,090 --> 00:40:35,560
if you're off by
a factor of 7, you

690
00:40:35,560 --> 00:40:38,450
get ages for the universe,
which are factors of 7

691
00:40:38,450 --> 00:40:42,130
smaller than what you
should be getting.

692
00:40:42,130 --> 00:40:44,890
And this was noticed early on.

693
00:40:44,890 --> 00:40:47,940
People were calculating ages
of the universe and Big Bang

694
00:40:47,940 --> 00:40:50,430
models and getting numbers
like 2 billion years

695
00:40:50,430 --> 00:40:54,800
instead of 14 billion
years, a factor of 7.

696
00:40:54,800 --> 00:40:58,990
And even back in
the '20s and '30s,

697
00:40:58,990 --> 00:41:00,810
there was significant
geological evidence

698
00:41:00,810 --> 00:41:03,019
that the Earth was much
older than 2 billion years,

699
00:41:03,019 --> 00:41:05,560
and people understood something
about the evolution of stars,

700
00:41:05,560 --> 00:41:07,893
and it would seem pretty clear
that the stars were older

701
00:41:07,893 --> 00:41:10,700
than 2 billion years,
so you couldn't tolerate

702
00:41:10,700 --> 00:41:13,100
a universe that was only
2 billion years old.

703
00:41:13,100 --> 00:41:18,060
And it led to very
significant problems

704
00:41:18,060 --> 00:41:20,990
with the development
of the Big Bang theory,

705
00:41:20,990 --> 00:41:26,010
and in particular, it
certainly gave extra credence

706
00:41:26,010 --> 00:41:28,160
to what was called the
steady state theory, which

707
00:41:28,160 --> 00:41:30,451
you may have heard of, which
held that the universe was

708
00:41:30,451 --> 00:41:32,780
infinitely old and
as it expanded,

709
00:41:32,780 --> 00:41:35,190
more matter was created
in the steady state theory

710
00:41:35,190 --> 00:41:37,670
to fill in the gaps so
the density of matter

711
00:41:37,670 --> 00:41:39,630
would be constant.

712
00:41:39,630 --> 00:41:44,130
And Lemaitre himself
in his 1927 paper,

713
00:41:44,130 --> 00:41:46,910
built a very complicated,
from my standards,

714
00:41:46,910 --> 00:41:51,680
theory in order to get
the age to be compatible.

715
00:41:51,680 --> 00:41:53,300
Instead of having
a Big Bang model,

716
00:41:53,300 --> 00:41:55,890
his 1927 model was
not a Big Bang model.

717
00:41:55,890 --> 00:42:00,640
His 1927 model started out
in a static equilibrium

718
00:42:00,640 --> 00:42:02,970
where he had a positive
cosmological constant which

719
00:42:02,970 --> 00:42:04,690
produces a repulsive
gravity, like what

720
00:42:04,690 --> 00:42:08,500
we talked about in
my opening lecture,

721
00:42:08,500 --> 00:42:11,230
balancing against the
normal attractive gravity

722
00:42:11,230 --> 00:42:13,830
of ordinary matter,
producing what was almost

723
00:42:13,830 --> 00:42:15,490
a static universe
of exactly the type

724
00:42:15,490 --> 00:42:18,440
that Einstein had
been advocating.

725
00:42:18,440 --> 00:42:23,610
But Lemaitre's universe started
out with just a slightly less

726
00:42:23,610 --> 00:42:27,210
mass density than
Einstein would have had,

727
00:42:27,210 --> 00:42:30,040
so it gradually started
to get bigger and bigger.

728
00:42:30,040 --> 00:42:32,130
The force of ordinary
gravity wasn't quite enough

729
00:42:32,130 --> 00:42:34,130
to hold it together,
but when it did

730
00:42:34,130 --> 00:42:36,360
that, it starts to get
bigger and bigger very slowly

731
00:42:36,360 --> 00:42:37,822
initially and then
picks up speed

732
00:42:37,822 --> 00:42:39,280
and allows you to
have universities

733
00:42:39,280 --> 00:42:41,140
that are much older
than what you would get

734
00:42:41,140 --> 00:42:42,640
in a straightforward
Big Bang model.

735
00:42:49,290 --> 00:42:50,370
Let's go on.

736
00:42:50,370 --> 00:42:54,270
What I want to talk about next
is what this Hubble expansion

737
00:42:54,270 --> 00:42:56,810
is telling us
about the universe,

738
00:42:56,810 --> 00:42:59,692
and I want to go through
this a little bit

739
00:42:59,692 --> 00:43:01,775
carefully because it's a
very important point even

740
00:43:01,775 --> 00:43:04,402
though it's possible you've
already figured it out

741
00:43:04,402 --> 00:43:05,110
from the reading.

742
00:43:05,110 --> 00:43:08,360
I don't know for sure.

743
00:43:08,360 --> 00:43:10,837
Naively, Hubble's
law makes it sound

744
00:43:10,837 --> 00:43:13,170
like we're saying that we are
the center of the universe

745
00:43:13,170 --> 00:43:14,730
after all.

746
00:43:14,730 --> 00:43:16,830
Copernicus was really wrong.

747
00:43:16,830 --> 00:43:21,300
Everything is moving away from
us, so we must be the center.

748
00:43:21,300 --> 00:43:23,311
But that's actually
not the case.

749
00:43:23,311 --> 00:43:25,560
It turns out that when you
look at things a little bit

750
00:43:25,560 --> 00:43:28,420
carefully, and that's what
we'll do in this diagram,

751
00:43:28,420 --> 00:43:31,280
if Hubble's law looks like
it holds to one observer, it

752
00:43:31,280 --> 00:43:34,307
in fact, also looks like it
holds to any other observer

753
00:43:34,307 --> 00:43:36,140
as long as you recognize
that there's no way

754
00:43:36,140 --> 00:43:38,540
to measure absolute velocity.

755
00:43:38,540 --> 00:43:40,480
So we think that we're
at rest, but that's

756
00:43:40,480 --> 00:43:43,340
really just our definition
of the rest frame.

757
00:43:43,340 --> 00:43:45,690
If we lived on
some other galaxy,

758
00:43:45,690 --> 00:43:49,200
we would equally well
attribute the state

759
00:43:49,200 --> 00:43:51,840
of being at rest to
that other galaxy.

760
00:43:51,840 --> 00:43:55,800
And that's what's being shown
in this picture, which I Xeroxed

761
00:43:55,800 --> 00:43:58,670
from Steve Weinberg's book so
this might seem familiar to you

762
00:43:58,670 --> 00:44:01,370
if you've read that chapter yet.

763
00:44:01,370 --> 00:44:03,960
It shows just expansion
in one direction,

764
00:44:03,960 --> 00:44:07,450
but that's enough to
illustrate the point.

765
00:44:07,450 --> 00:44:10,380
And the top diagram
we imagine that we

766
00:44:10,380 --> 00:44:14,140
are living on the
galaxy labeled A.

767
00:44:14,140 --> 00:44:15,670
The other galaxies
are moving away

768
00:44:15,670 --> 00:44:18,525
from us with velocities
proportional to the distance,

769
00:44:18,525 --> 00:44:23,350
and we've spaced these galaxies
from the diagram evenly,

770
00:44:23,350 --> 00:44:25,429
so the other galaxies
are moving away at v,

771
00:44:25,429 --> 00:44:27,220
and then the next one
is moving away at 2v.

772
00:44:27,220 --> 00:44:30,494
And if we continue, it
would be 3v, 4v, et cetera,

773
00:44:30,494 --> 00:44:31,660
all the way out to infinity.

774
00:44:34,650 --> 00:44:38,735
And what we want to do
in going from A to B

775
00:44:38,735 --> 00:44:42,690
is to ask suppose we were
living in exactly this universe

776
00:44:42,690 --> 00:44:45,770
as described on line A.
But suppose we were living

777
00:44:45,770 --> 00:44:49,010
in galaxy B and considered
galaxy B to be at rest.

778
00:44:49,010 --> 00:44:53,710
So we'd describe everything
from the rest frame of galaxy B.

779
00:44:53,710 --> 00:44:55,440
Then galaxy B would
have no velocity,

780
00:44:55,440 --> 00:44:59,120
because that would
defined the rest frame.

781
00:44:59,120 --> 00:45:00,900
When you change
frames, this was all

782
00:45:00,900 --> 00:45:06,160
done in the context of
Galilean transformations.

783
00:45:06,160 --> 00:45:09,941
We'll build more
relativistic models later.

784
00:45:09,941 --> 00:45:11,940
Then the context of the
Galilean transformation,

785
00:45:11,940 --> 00:45:14,490
if you go from one
frame to a frame moving

786
00:45:14,490 --> 00:45:16,780
at a constant velocity,
the only thing you have

787
00:45:16,780 --> 00:45:20,020
to do to transform velocity
is you add to each velocity

788
00:45:20,020 --> 00:45:22,110
a fixed velocity, that
velocity difference

789
00:45:22,110 --> 00:45:24,060
between the two frames.

790
00:45:24,060 --> 00:45:28,150
So to go from the top to
the bottom picture, what

791
00:45:28,150 --> 00:45:31,990
we do in all cases is just
add a velocity, v, to the left

792
00:45:31,990 --> 00:45:34,810
to each velocity, and that
takes the velocity of v

793
00:45:34,810 --> 00:45:36,820
here when we move it
with v to the right.

794
00:45:36,820 --> 00:45:38,825
When we add a v to
the left, we get 0.

795
00:45:38,825 --> 00:45:40,407
It does the right
thing there, which

796
00:45:40,407 --> 00:45:42,270
is what defines the
transformation we're

797
00:45:42,270 --> 00:45:43,050
trying to make.

798
00:45:43,050 --> 00:45:44,883
We're trying to make
the transformation that

799
00:45:44,883 --> 00:45:47,540
brings B to rest.

800
00:45:47,540 --> 00:45:51,000
And that means
that when we add v

801
00:45:51,000 --> 00:45:53,390
to the left to
the velocity of z,

802
00:45:53,390 --> 00:45:56,540
where we already had a v to the
left, we get 2v to the left.

803
00:45:56,540 --> 00:46:00,230
When we add v to the left to
y, which had 2v to the left,

804
00:46:00,230 --> 00:46:02,480
we get 3v to the left.

805
00:46:02,480 --> 00:46:04,310
Going the other
direction, when we add v

806
00:46:04,310 --> 00:46:08,240
to the left to c, which had
a velocity 2v to the right,

807
00:46:08,240 --> 00:46:10,910
we're left with the
velocity of 1v to the right,

808
00:46:10,910 --> 00:46:14,394
and that gives us what we
have on the second row.

809
00:46:14,394 --> 00:46:15,810
And if we look
from the second row

810
00:46:15,810 --> 00:46:19,144
from the point of
view of v, the galaxy

811
00:46:19,144 --> 00:46:21,435
is one way or each moving
away from us with a velocity,

812
00:46:21,435 --> 00:46:23,811
v. The velocity is two
way and moving away

813
00:46:23,811 --> 00:46:25,810
from us with velocities
2v, et cetera.

814
00:46:25,810 --> 00:46:28,240
That's exactly the same.

815
00:46:28,240 --> 00:46:30,810
So even though the
Hubble expansion pattern

816
00:46:30,810 --> 00:46:34,180
is phrased in a way that makes
it look like you're talking

817
00:46:34,180 --> 00:46:37,360
about yourself as the center
of the universe, in fact,

818
00:46:37,360 --> 00:46:41,180
it does describe a completely
homogeneous picture.

819
00:46:41,180 --> 00:46:42,610
And it's a picture
that, in fact,

820
00:46:42,610 --> 00:46:44,940
has a very simple description.

821
00:46:44,940 --> 00:46:47,670
It's a picture of just
uniform expansion,

822
00:46:47,670 --> 00:46:51,760
and I think I have my favorite,
at least the best picture I've

823
00:46:51,760 --> 00:46:56,320
every drawn of uniform expansion
on the next slide here.

824
00:46:56,320 --> 00:47:00,020
The idea is that if you look
at some region of the universe,

825
00:47:00,020 --> 00:47:02,170
the claim-- and
the claim is just

826
00:47:02,170 --> 00:47:04,800
called homogeneous
expansion-- is

827
00:47:04,800 --> 00:47:09,330
that each picture at successive
times would look identical,

828
00:47:09,330 --> 00:47:13,390
but it would look like
a photographic blowup.

829
00:47:13,390 --> 00:47:15,450
Each picture would
just be a bigger image

830
00:47:15,450 --> 00:47:19,210
of the same picture with
one important exception,

831
00:47:19,210 --> 00:47:21,200
and I did try to
draw this correctly,

832
00:47:21,200 --> 00:47:23,439
the positions of the
galaxies-- this little

833
00:47:23,439 --> 00:47:24,980
lob there is supposed
to be a galaxy,

834
00:47:24,980 --> 00:47:29,160
by the way, in case you
can't tell my great artistry.

835
00:47:29,160 --> 00:47:33,365
The positions of each galaxy
is just expand uniformly,

836
00:47:33,365 --> 00:47:36,680
the pattern of positions,
but each individual galaxy

837
00:47:36,680 --> 00:47:37,560
does not expand.

838
00:47:37,560 --> 00:47:40,300
The individual of the
galaxies maintain their size

839
00:47:40,300 --> 00:47:43,687
as the universe undergoes
this public expansion.

840
00:47:43,687 --> 00:47:45,770
Now if we're talking about
the very early universe

841
00:47:45,770 --> 00:47:48,311
before there were any galaxies,
you would just have basically

842
00:47:48,311 --> 00:47:51,440
a uniform distribution
of matter of gas,

843
00:47:51,440 --> 00:47:53,565
and that would just uniformly
expand every molecule

844
00:47:53,565 --> 00:47:58,530
and move away from every
other molecule on average.

845
00:47:58,530 --> 00:48:02,560
So this is the picture
of Hubble expansion.

846
00:48:11,720 --> 00:48:15,550
And now what I'd like to
do is provide a description

847
00:48:15,550 --> 00:48:17,550
of how we're going to
treat this mathematically.

848
00:48:38,670 --> 00:48:40,515
If we have this
uniformly-- I'm sorry.

849
00:48:40,515 --> 00:48:41,015
Yes.

850
00:48:41,015 --> 00:48:42,638
AUDIENCE: I'm still
getting confused

851
00:48:42,638 --> 00:48:46,041
whether like the expansion
is the galaxies expanding

852
00:48:46,041 --> 00:48:50,934
into the universe or if the
universe itself is expanding.

853
00:48:50,934 --> 00:48:51,600
PROFESSOR: Yeah.

854
00:48:51,600 --> 00:48:53,800
The question in case you
didn't hear it was there's

855
00:48:53,800 --> 00:48:56,220
some confusion here about
whether we should think

856
00:48:56,220 --> 00:48:58,600
of the galaxies as
moving through space

857
00:48:58,600 --> 00:49:01,930
or whether we should think
of space itself as expanding.

858
00:49:01,930 --> 00:49:04,280
And the answer really is
that both points of view

859
00:49:04,280 --> 00:49:06,260
should be right.

860
00:49:06,260 --> 00:49:10,490
If space were like
water, then you

861
00:49:10,490 --> 00:49:12,030
could imagine
putting little dust

862
00:49:12,030 --> 00:49:15,110
in the water, little
grains of salt or something

863
00:49:15,110 --> 00:49:20,020
you can see and see if they are
carried by the water or apart.

864
00:49:20,020 --> 00:49:22,640
But there's no
way to mark space.

865
00:49:22,640 --> 00:49:25,530
It's intrinsic to the
principle of relativity

866
00:49:25,530 --> 00:49:28,020
that you can't tell if you're
moving relative to space

867
00:49:28,020 --> 00:49:29,090
or not.

868
00:49:29,090 --> 00:49:31,435
There's no meaning to
moving relative to space.

869
00:49:31,435 --> 00:49:33,810
And there's no meaning for
you to move relative to space.

870
00:49:33,810 --> 00:49:36,440
There is also no meaning for
space to move relative to you.

871
00:49:36,440 --> 00:49:38,770
they do the same thing.

872
00:49:38,770 --> 00:49:42,390
So you can't really tell,
and both points of view

873
00:49:42,390 --> 00:49:44,250
should be correct.

874
00:49:44,250 --> 00:49:46,860
There are cases where
you can tell, however,

875
00:49:46,860 --> 00:49:49,480
which is not locally
but if, for example, you

876
00:49:49,480 --> 00:49:52,820
had a closed universe, which
we'll talk about later how that

877
00:49:52,820 --> 00:49:54,616
works exactly,
then you could ask

878
00:49:54,616 --> 00:49:57,100
does the volume of a closed
universe get bigger with time

879
00:49:57,100 --> 00:49:58,700
as this Hubble
expansion takes place.

880
00:49:58,700 --> 00:50:00,020
And the answer there is yes.

881
00:50:00,020 --> 00:50:01,513
AUDIENCE: That
would mean actually,

882
00:50:01,513 --> 00:50:02,596
the universe is expanding.

883
00:50:02,596 --> 00:50:04,640
PROFESSOR: The actual
universe is expanding.

884
00:50:04,640 --> 00:50:07,880
So we will normally think
of it, globally at least,

885
00:50:07,880 --> 00:50:10,310
as the actual
universe expanding.

886
00:50:10,310 --> 00:50:12,470
That is how we will
think about it.

887
00:50:12,470 --> 00:50:16,210
But locally, there's not really
any distinction between that

888
00:50:16,210 --> 00:50:20,830
and saying that these galaxies
are just moving through space.

889
00:50:20,830 --> 00:50:21,330
Yes.

890
00:50:21,330 --> 00:50:24,753
AUDIENCE: So given that
the galaxies are actually

891
00:50:24,753 --> 00:50:27,687
two points, why
can it be claimed

892
00:50:27,687 --> 00:50:30,621
that the galaxies themselves
are not expanding?

893
00:50:30,621 --> 00:50:31,470
PROFESSOR: OK.

894
00:50:31,470 --> 00:50:33,886
How do we understand the fact
that the galaxies themselves

895
00:50:33,886 --> 00:50:35,990
are not expanding is
what you're asking.

896
00:50:35,990 --> 00:50:41,230
And I'll give you
a nutshell answer,

897
00:50:41,230 --> 00:50:44,930
and we might be talking
about it more later.

898
00:50:44,930 --> 00:50:49,180
One should imagine that
this starts out shortly

899
00:50:49,180 --> 00:50:52,840
after the Big Bang as an almost
perfectly uniform gas, which

900
00:50:52,840 --> 00:50:55,760
is just uniformly expanding,
everything moving away

901
00:50:55,760 --> 00:50:57,860
from everything else.

902
00:50:57,860 --> 00:51:00,250
But the gas is not
completely uniform.

903
00:51:00,250 --> 00:51:05,207
It has tiny ripples in
the matter density, which

904
00:51:05,207 --> 00:51:07,540
are the same ripples that we
see in the cosmic microwave

905
00:51:07,540 --> 00:51:10,486
background radiation today or
at least the cosmic ripples

906
00:51:10,486 --> 00:51:12,860
that we see in the cosmic
background radiation are caused

907
00:51:12,860 --> 00:51:17,300
by the ripples in the mass
density of the early universe.

908
00:51:17,300 --> 00:51:20,730
These ripples
eventually form galaxies

909
00:51:20,730 --> 00:51:23,530
because they're
gravitationally unstable.

910
00:51:23,530 --> 00:51:26,290
Wherever there's a
slight excess of mass,

911
00:51:26,290 --> 00:51:29,050
that will create a slightly
stronger gravitational field

912
00:51:29,050 --> 00:51:31,360
in that region pulling
in more mass creating

913
00:51:31,360 --> 00:51:34,160
a still stronger gravitational
field pulling in more mass,

914
00:51:34,160 --> 00:51:37,300
and eventually instead of having
this nice uniform distribution

915
00:51:37,300 --> 00:51:39,910
with just ripples at
1 part in 100,000,

916
00:51:39,910 --> 00:51:45,680
you eventually have huge clumps
of matter which are galaxies.

917
00:51:45,680 --> 00:51:48,280
And as you go from this
transition of things

918
00:51:48,280 --> 00:51:53,500
being almost completely
uniform and uniformly expanding

919
00:51:53,500 --> 00:51:55,390
to these lumps
that form galaxies,

920
00:51:55,390 --> 00:51:58,010
the ones that are being formed
by extra gravity pulling

921
00:51:58,010 --> 00:51:59,130
in the matter.

922
00:51:59,130 --> 00:52:01,800
And what happens is
that extra gravity

923
00:52:01,800 --> 00:52:06,120
that forms the galaxy
overcomes the Hubble expansion.

924
00:52:06,120 --> 00:52:07,920
The matter that
makes up the galaxy

925
00:52:07,920 --> 00:52:11,410
had been expanding
in the early days,

926
00:52:11,410 --> 00:52:13,430
but the gravitational
pull of the matter that

927
00:52:13,430 --> 00:52:16,370
forms the galaxy
pulls it back in.

928
00:52:16,370 --> 00:52:19,067
So the galaxy actually reaches a
maximum size and then, in fact,

929
00:52:19,067 --> 00:52:20,650
starts to get smaller
and then reaches

930
00:52:20,650 --> 00:52:24,400
equilibrium, an
equilibrium where

931
00:52:24,400 --> 00:52:28,450
the rotational motion
keeps a finite size.

932
00:52:28,450 --> 00:52:28,950
Yes.

933
00:52:28,950 --> 00:52:31,280
AUDIENCE: So the
diagrams that you're

934
00:52:31,280 --> 00:52:34,319
applying up here that all the
distance relations have been

935
00:52:34,319 --> 00:52:35,735
galaxies that are
being preserved.

936
00:52:35,735 --> 00:52:37,924
Is that a potential or is
it exactly [INAUDIBLE]?

937
00:52:37,924 --> 00:52:38,840
PROFESSOR: Well, yeah.

938
00:52:38,840 --> 00:52:40,673
It's supposed to be
just a photographic blow

939
00:52:40,673 --> 00:52:42,760
up as far as where the
relocations of dots are.

940
00:52:42,760 --> 00:52:42,850
Yeah.

941
00:52:42,850 --> 00:52:44,340
I mean is that
what you're asking?

942
00:52:44,340 --> 00:52:46,064
AUDIENCE: Well,
like will there be

943
00:52:46,064 --> 00:52:48,564
equal distance between galaxies
as a sub [INAUDIBLE] member?

944
00:52:48,564 --> 00:52:49,660
PROFESSOR: Yeah.

945
00:52:49,660 --> 00:52:51,860
I think the picture
shows that, doesn't it?

946
00:52:51,860 --> 00:52:53,700
AUDIENCE: Well, the
notches basically

947
00:52:53,700 --> 00:52:54,160
are spaces between [INAUDIBLE].

948
00:52:54,160 --> 00:52:55,530
PROFESSOR: Oh, that's right.

949
00:52:55,530 --> 00:52:56,895
That's right.

950
00:52:56,895 --> 00:52:59,041
I haven't talked
about the notches yet.

951
00:52:59,041 --> 00:53:01,540
The diagrams are supposed to
show actual physical distances.

952
00:53:01,540 --> 00:53:04,160
So the physical distance between
this galaxy and this galaxy

953
00:53:04,160 --> 00:53:07,377
is a little bit there
and much more there.

954
00:53:07,377 --> 00:53:09,710
So that's how you're supposed
to interpret that picture.

955
00:53:09,710 --> 00:53:12,860
But what I was about to
get to and you've got there

956
00:53:12,860 --> 00:53:15,580
so I'll continue,
is that the best

957
00:53:15,580 --> 00:53:19,170
way to describe this
uniformly-expanding system

958
00:53:19,170 --> 00:53:22,630
is to introduce a coordinate
system that expands with it,

959
00:53:22,630 --> 00:53:24,470
and that's what
these notches are.

960
00:53:24,470 --> 00:53:29,320
The notches are artificial
things that we create,

961
00:53:29,320 --> 00:53:33,230
and we could think of them as
just being labels on a map.

962
00:53:33,230 --> 00:53:36,375
Once we know that the
expansion is uniform this way,

963
00:53:36,375 --> 00:53:38,000
we could take any
one of these pictures

964
00:53:38,000 --> 00:53:42,950
and think of it as a map of
our region of the universe.

965
00:53:42,950 --> 00:53:50,450
And we can then get to any
other picture on the slide

966
00:53:50,450 --> 00:53:55,300
simply by converting units on
a map to physical distances

967
00:53:55,300 --> 00:53:58,700
with a different scale factor.

968
00:53:58,700 --> 00:54:02,590
So if Massachusetts was forever
getting bigger and bigger

969
00:54:02,590 --> 00:54:04,420
and we had a map
of Massachusetts,

970
00:54:04,420 --> 00:54:06,730
we would not have to throw
away that map every day

971
00:54:06,730 --> 00:54:08,690
and buy a new map.

972
00:54:08,690 --> 00:54:11,680
We could handle the expansion
of Massachusetts keeping

973
00:54:11,680 --> 00:54:14,810
the same map just crossing
out the place in the corner

974
00:54:14,810 --> 00:54:17,500
of the map where it says
1 inch equals 7 miles,

975
00:54:17,500 --> 00:54:19,520
and the next day 1
inch equals 8 miles,

976
00:54:19,520 --> 00:54:24,210
and 1 inch equals 9 miles, and
1 inch equals 10 and 1/2 miles.

977
00:54:24,210 --> 00:54:27,540
So by changing the scale
factor on the map and the use

978
00:54:27,540 --> 00:54:31,240
of the word scale factor here is
exactly the same meaning as it

979
00:54:31,240 --> 00:54:34,530
would have for a
map, you can allow

980
00:54:34,530 --> 00:54:37,830
yourself to describe
an expanding system

981
00:54:37,830 --> 00:54:40,300
without ever throwing
away the original map.

982
00:54:40,300 --> 00:54:43,110
And that's the kind
of coordinate system

983
00:54:43,110 --> 00:54:44,910
that we will be
using, and these are

984
00:54:44,910 --> 00:54:46,125
called comoving coordinates.

985
00:54:58,050 --> 00:55:05,824
And the idea here
is that galaxies

986
00:55:05,824 --> 00:55:09,920
are at-- I'm
sticking in the word

987
00:55:09,920 --> 00:55:13,381
approximately here because
none of this is exact,

988
00:55:13,381 --> 00:55:18,036
but we'll be thinking in a
toy model as it was exact.

989
00:55:18,036 --> 00:55:28,215
So galaxies are at
approximately constant values

990
00:55:28,215 --> 00:55:42,986
of the coordinates, and
the scale factor, which

991
00:55:42,986 --> 00:56:06,550
means the physical distance per
coordinate distance increases

992
00:56:06,550 --> 00:56:07,060
with time.

993
00:56:17,774 --> 00:56:21,100
So that describes this
all-important comoving

994
00:56:21,100 --> 00:56:23,840
coordinate system
that we'll be using

995
00:56:23,840 --> 00:56:26,740
for the rest of the course
to describe the expanding

996
00:56:26,740 --> 00:56:28,050
universe.

997
00:56:28,050 --> 00:56:28,550
Yes.

998
00:56:28,550 --> 00:56:31,912
AUDIENCE: Do we have to do
anything funny to the Lorentz

999
00:56:31,912 --> 00:56:35,848
transform to come to the
fact that coordinates are now

1000
00:56:35,848 --> 00:56:38,800
not moving at the same velocity
relative to each other?

1001
00:56:38,800 --> 00:56:41,090
PROFESSOR: It depends on
what questions you ask.

1002
00:56:41,090 --> 00:56:43,540
There are questions where you
do have to think carefully,

1003
00:56:43,540 --> 00:56:46,660
and we'll have one of
those shortly as probably

1004
00:56:46,660 --> 00:56:47,744
an extra credit problem.

1005
00:56:47,744 --> 00:56:50,160
But for most things, it actually
makes things very simple,

1006
00:56:50,160 --> 00:56:52,400
and you can ignore most
of the complications

1007
00:56:52,400 --> 00:56:53,830
of special relativity.

1008
00:56:53,830 --> 00:56:57,310
And we'll try to think as we
go along where we need to worry

1009
00:56:57,310 --> 00:57:00,100
about special relativity and
where we don't, and usually we

1010
00:57:00,100 --> 00:57:00,600
don't.

1011
00:57:25,990 --> 00:57:40,920
So the key relationship is
that the physical distance

1012
00:57:40,920 --> 00:57:45,120
between any two points on the
map, by physical distance,

1013
00:57:45,120 --> 00:57:47,570
I mean what it is
in the real world,

1014
00:57:47,570 --> 00:57:49,450
miles if we're talking
about Massachusetts,

1015
00:57:49,450 --> 00:57:53,630
and this is miles between
the real physical points,

1016
00:57:53,630 --> 00:57:56,830
is equal to a
time-dependent scale

1017
00:57:56,830 --> 00:58:01,500
factor times the
coordinate distance.

1018
00:58:25,400 --> 00:58:29,210
Now, here I'm going
to use conventions

1019
00:58:29,210 --> 00:58:32,220
for defining things that are
slightly different from what

1020
00:58:32,220 --> 00:58:34,210
are often used.

1021
00:58:34,210 --> 00:58:38,330
A common procedure where I think
it's done in most of the books,

1022
00:58:38,330 --> 00:58:40,940
is to think of both
the coordinate distance

1023
00:58:40,940 --> 00:58:43,620
and the physical distances being
measured in normal distance

1024
00:58:43,620 --> 00:58:48,130
units, meters, and then the
scale factor is dimensionless,

1025
00:58:48,130 --> 00:58:49,680
and it just tells
you how much you

1026
00:58:49,680 --> 00:58:53,940
have to blow up the
map to be able to make

1027
00:58:53,940 --> 00:58:56,750
it the physical map.

1028
00:58:56,750 --> 00:58:59,230
I find it significantly
easier to know

1029
00:58:59,230 --> 00:59:01,280
what you're doing
as things go along

1030
00:59:01,280 --> 00:59:05,430
to label the map in units that
are not centimeters, but are

1031
00:59:05,430 --> 00:59:08,780
what shown on the
picture as notches.

1032
00:59:08,780 --> 00:59:10,640
One advantage of
that logically is it

1033
00:59:10,640 --> 00:59:12,852
means that if you have
different copies of the map

1034
00:59:12,852 --> 00:59:14,310
that you've made
on a Xerox machine

1035
00:59:14,310 --> 00:59:17,290
with different scalings, you
have a big copy of the map

1036
00:59:17,290 --> 00:59:19,040
and a little copy of
the map, that they're

1037
00:59:19,040 --> 00:59:20,580
marked off with notches.

1038
00:59:20,580 --> 00:59:23,720
The notches grow with the
physical size of the page,

1039
00:59:23,720 --> 00:59:25,650
the scale factor is
the same no matter

1040
00:59:25,650 --> 00:59:28,570
which copy of the
map you're using.

1041
00:59:28,570 --> 00:59:30,600
But most importantly, it
allows you to, I think,

1042
00:59:30,600 --> 00:59:32,820
do dimensional tests.

1043
00:59:32,820 --> 00:59:34,620
The size of the map
is clearly something

1044
00:59:34,620 --> 00:59:38,250
that's unrelated to the
actual length of a meter,

1045
00:59:38,250 --> 00:59:41,580
it's just how many units
you put on your map.

1046
00:59:41,580 --> 00:59:45,050
So there's a clear
and logical separation

1047
00:59:45,050 --> 00:59:48,180
between what is meant by
a certain number of units

1048
00:59:48,180 --> 00:59:53,540
and distance here and a
real meter by any standards.

1049
00:59:53,540 --> 00:59:56,860
So you can keep that
straight by just imagining

1050
00:59:56,860 --> 00:59:59,870
that your map is calibrated in
some new arbitrary unit which

1051
00:59:59,870 --> 01:00:03,250
is just special to the map, and
I'm going to call that a notch.

1052
01:00:13,430 --> 01:00:14,960
So notches are just
arbitrary units

1053
01:00:14,960 --> 01:00:18,145
that you use to
mark off your map.

1054
01:00:18,145 --> 01:00:20,020
And then the physical
distance is, of course,

1055
01:00:20,020 --> 01:00:26,640
measured in meters or any other
standard unit of distance.

1056
01:00:29,620 --> 01:00:33,380
And then the scale
factor is measured

1057
01:00:33,380 --> 01:00:39,400
in units of meters per notch
instead of being dimensionless.

1058
01:00:44,190 --> 01:00:49,682
And the basic advantage of this
is that when you're all done,

1059
01:00:49,682 --> 01:00:51,390
nothing had better
have any notches in it

1060
01:00:51,390 --> 01:00:53,840
as you're calculating something
physical, that is physical

1061
01:00:53,840 --> 01:00:56,630
why you should not depend
on the size of your map.

1062
01:00:56,630 --> 01:00:58,570
So you have a nice
dimensional check

1063
01:00:58,570 --> 01:01:00,220
to make sure that
the notches drop out

1064
01:01:00,220 --> 01:01:02,810
of any physical calculation
that you try to do.

1065
01:01:07,790 --> 01:01:09,920
What I want to do
next is to show you

1066
01:01:09,920 --> 01:01:14,910
that this relationship
leads me to Hubble's law

1067
01:01:14,910 --> 01:01:19,960
and furthermore will allow us
to figure out what this Hubble

1068
01:01:19,960 --> 01:01:23,396
expansion rate is in terms of
what the scale factor is doing.

1069
01:01:36,230 --> 01:01:39,610
So it's an easy
enough calculation

1070
01:01:39,610 --> 01:01:41,720
if we're looking at
some object out there

1071
01:01:41,720 --> 01:01:46,030
and it's physical distance l
sub p is given by that formula,

1072
01:01:46,030 --> 01:01:48,200
and we want to know
what its velocity is.

1073
01:01:48,200 --> 01:01:49,900
It's velocity is
by definition, just

1074
01:01:49,900 --> 01:01:52,560
a time derivative
of that quantity.

1075
01:01:52,560 --> 01:01:58,780
So the velocity of some object,
some distance out in space,

1076
01:01:58,780 --> 01:02:08,263
is just equal to d
dt of l sub p of t,

1077
01:02:08,263 --> 01:02:17,226
and that will be da
dt times l sub c,

1078
01:02:17,226 --> 01:02:19,290
because l sub c is constant.

1079
01:02:19,290 --> 01:02:21,590
On average, all our
galaxies are at rest

1080
01:02:21,590 --> 01:02:24,290
in this coordinate system,
in this expanding coordinate

1081
01:02:24,290 --> 01:02:26,640
system.

1082
01:02:26,640 --> 01:02:29,550
Now this could be written
in a way that ends up

1083
01:02:29,550 --> 01:02:34,276
being more useful by dividing
and multiplying by a.

1084
01:02:34,276 --> 01:02:37,060
So I could write it
as 1 over a times

1085
01:02:37,060 --> 01:02:41,800
da dt times a of t times lc.

1086
01:02:44,660 --> 01:02:48,840
And the advantage of multiplying
and dividing by a this way

1087
01:02:48,840 --> 01:02:52,510
is that this quantity
is again just l

1088
01:02:52,510 --> 01:02:57,450
sub p, the physical distance.

1089
01:02:57,450 --> 01:03:01,710
So now we say that the
velocity of any distant object

1090
01:03:01,710 --> 01:03:06,070
is equal to 1 over a da dt times
the distance to that object.

1091
01:03:06,070 --> 01:03:09,040
And that is Hubble's
law, and it tells us

1092
01:03:09,040 --> 01:03:12,320
that the Hubble expansion
rate, which is itself

1093
01:03:12,320 --> 01:03:14,670
going to be a
function of time, is

1094
01:03:14,670 --> 01:03:19,010
equal to 1 over a times da dt.

1095
01:03:26,810 --> 01:03:29,060
And this allows us to
illustrate the unit check

1096
01:03:29,060 --> 01:03:32,750
that I talked about
for the first time.

1097
01:03:32,750 --> 01:03:37,005
Notice that a is measured
in meters per notch,

1098
01:03:37,005 --> 01:03:39,380
so the meters per notch here
cancels the meters per notch

1099
01:03:39,380 --> 01:03:41,005
here, and you just
get inverse time,

1100
01:03:41,005 --> 01:03:42,380
and the really
important thing is

1101
01:03:42,380 --> 01:03:44,560
that the notches have gone away.

1102
01:03:44,560 --> 01:03:46,805
And again, notches
have to go away

1103
01:03:46,805 --> 01:03:48,684
in any calculation
of any physical,

1104
01:03:48,684 --> 01:03:49,850
and that makes a nice check.

1105
01:04:03,140 --> 01:04:05,084
And once you know how
a of t is behaving,

1106
01:04:05,084 --> 01:04:07,650
you know exactly how the
Hubble expansion rate behaves.

1107
01:04:07,650 --> 01:04:12,040
It's determined by a of t.

1108
01:04:12,040 --> 01:04:15,580
You might mention
one notational item.

1109
01:04:15,580 --> 01:04:19,640
Nowadays almost everybody calls
this scale factor little a.

1110
01:04:19,640 --> 01:04:22,380
In the early days,
it was very first

1111
01:04:22,380 --> 01:04:24,885
introduced by Alexander
Friedmann who first invented

1112
01:04:24,885 --> 01:04:26,785
the equation describing
expanding universe

1113
01:04:26,785 --> 01:04:28,430
in the early 1920s.

1114
01:04:28,430 --> 01:04:31,440
He used the letter R,
capital R. Lemaitre also

1115
01:04:31,440 --> 01:04:34,570
used the letter capital R,
and I guess Einstein probably

1116
01:04:34,570 --> 01:04:37,940
used the capital
R. I'm not sure.

1117
01:04:37,940 --> 01:04:40,050
And going into
more modern times,

1118
01:04:40,050 --> 01:04:43,060
Steve Weinberg wrote a book
on gravitation and cosmology

1119
01:04:43,060 --> 01:04:44,780
which still used the
letter capital R,

1120
01:04:44,780 --> 01:04:48,040
but that was sort of
the last major work that

1121
01:04:48,040 --> 01:04:50,760
used the capital R
for the scale factor.

1122
01:04:50,760 --> 01:04:52,890
The disadvantage of
it is at the same time

1123
01:04:52,890 --> 01:04:55,930
this capital R means something
else in general relativity.

1124
01:04:55,930 --> 01:04:57,530
It's the standard
symbol for what's

1125
01:04:57,530 --> 01:05:00,270
called the scalar curvature
in general relativity.

1126
01:05:00,270 --> 01:05:04,320
So to avoid confusion between
those two quantities, nowadays

1127
01:05:04,320 --> 01:05:06,720
almost everybody calls
the scale factor little a.

1128
01:05:09,400 --> 01:05:11,450
If you look at old
A286 notes, I used

1129
01:05:11,450 --> 01:05:14,350
to follow Steve Weinberg's
textbook on gravitation

1130
01:05:14,350 --> 01:05:16,390
and cosmology and
call it capital R,

1131
01:05:16,390 --> 01:05:18,760
but now it's hopefully
all switched to little a.

1132
01:05:21,590 --> 01:05:22,090
OK.

1133
01:05:22,090 --> 01:05:24,370
Next item.

1134
01:05:24,370 --> 01:05:26,070
If we're going to
understand what

1135
01:05:26,070 --> 01:05:28,240
it would look like to live
in a universe like this,

1136
01:05:28,240 --> 01:05:29,906
we're going to need
to know how to trace

1137
01:05:29,906 --> 01:05:33,010
light rays through our
expanding universe.

1138
01:05:33,010 --> 01:05:34,720
And that turned out to be easy.

1139
01:06:06,290 --> 01:06:14,530
If I let x be equal
to a coordinate, that

1140
01:06:14,530 --> 01:06:24,600
means it's measured in
notches, and if I imagine

1141
01:06:24,600 --> 01:06:28,100
I have a light ray moving
in the x direction,

1142
01:06:28,100 --> 01:06:31,880
I can describe how that
light ray is going to move.

1143
01:06:31,880 --> 01:06:34,025
If I can write down a
formula for the dx dt.

1144
01:06:34,025 --> 01:06:35,775
Tells me how fast in
the coordinate system

1145
01:06:35,775 --> 01:06:36,705
the light ray travels.

1146
01:06:47,541 --> 01:06:49,540
Well the basic principle
that we're going to use

1147
01:06:49,540 --> 01:06:51,530
here is that light,
in fact, always

1148
01:06:51,530 --> 01:06:55,670
travels at the speed of
light at some fixed value c,

1149
01:06:55,670 --> 01:06:58,760
but c is the physical velocity
of light, the velocity measured

1150
01:06:58,760 --> 01:07:00,890
in meters per second.

1151
01:07:00,890 --> 01:07:05,080
But dx dt is the velocity
measured in notches per second

1152
01:07:05,080 --> 01:07:07,180
because our coordinate
system is marked off

1153
01:07:07,180 --> 01:07:08,660
not in meters, but in notches.

1154
01:07:08,660 --> 01:07:10,070
And that really
is very important

1155
01:07:10,070 --> 01:07:12,486
because meters are going to
be constantly changing lengths

1156
01:07:12,486 --> 01:07:14,474
relative to our
notches, and we want

1157
01:07:14,474 --> 01:07:15,890
to keep track of
things in notches

1158
01:07:15,890 --> 01:07:19,690
so we have a nice picture within
our coordinate description

1159
01:07:19,690 --> 01:07:23,310
of the universe that
we can think about.

1160
01:07:23,310 --> 01:07:25,370
So we're going to want
to know what dx dt is,

1161
01:07:25,370 --> 01:07:27,430
but it's just a unit
conversion problem.

1162
01:07:27,430 --> 01:07:30,620
dx dt is the speed of light
in notches per second.

1163
01:07:30,620 --> 01:07:34,740
We know the speed of light
in meters per second, c.

1164
01:07:34,740 --> 01:07:38,110
So to convert is just a matter
of multiplying by the scale

1165
01:07:38,110 --> 01:07:41,145
factor to convert the
units of meters to notches.

1166
01:07:41,145 --> 01:07:45,010
And here again it helps to
have this meters versus notches

1167
01:07:45,010 --> 01:07:47,430
because it guarantees that
you can't get it wrong

1168
01:07:47,430 --> 01:07:50,260
if you just check your units.

1169
01:07:50,260 --> 01:07:52,090
So this is not really
a question mark.

1170
01:07:52,090 --> 01:07:56,880
It is just c divided by
a of t, the scale factor.

1171
01:08:00,030 --> 01:08:04,750
And we can make sure we got it
right by checking our units.

1172
01:08:04,750 --> 01:08:08,230
I'm going to use brackets
to indicate units of.

1173
01:08:12,343 --> 01:08:14,217
So we're going to work
out what the units are

1174
01:08:14,217 --> 01:08:16,976
of c over a of t, trivial
problem, of course,

1175
01:08:16,976 --> 01:08:19,010
but we'll make sure we
got the right answer.

1176
01:08:19,010 --> 01:08:24,800
The units of c are, of
course, meters per second.

1177
01:08:24,800 --> 01:08:27,325
a of t we said is
meters per notch.

1178
01:08:33,689 --> 01:08:38,310
So the meters cancel, and
we get notches per second.

1179
01:08:40,608 --> 01:08:42,649
Now, I told you that you
should never get notches

1180
01:08:42,649 --> 01:08:45,190
in the physical answer because
this is not a physical answer.

1181
01:08:45,190 --> 01:08:46,970
This is a coordinate
velocity of light,

1182
01:08:46,970 --> 01:08:48,689
so it does depend
on our coordinates

1183
01:08:48,689 --> 01:08:50,130
depending on what
coordinates we've chosen.

1184
01:08:50,130 --> 01:08:52,090
So it should certainly
be notches per second

1185
01:08:52,090 --> 01:08:54,790
because x is measured in notches
and t is measured in seconds.

1186
01:08:54,790 --> 01:08:56,582
So we put in the a of
t in the right place.

1187
01:08:56,582 --> 01:08:57,956
It does belong in
the denominator

1188
01:08:57,956 --> 01:08:59,040
and not the numerator.

1189
01:08:59,040 --> 01:08:59,540
Yes.

1190
01:08:59,540 --> 01:09:03,040
AUDIENCE: Why aren't we
worrying about the fact

1191
01:09:03,040 --> 01:09:07,540
that as the universe
expands, there's

1192
01:09:07,540 --> 01:09:11,040
also a velocity component with
a light rate from its position

1193
01:09:11,040 --> 01:09:14,040
moving according the
Hubble expansion?

1194
01:09:14,040 --> 01:09:15,229
PROFESSOR: Right.

1195
01:09:15,229 --> 01:09:17,140
The reason we don't
worry about that

1196
01:09:17,140 --> 01:09:20,850
is that special
relativity tells us

1197
01:09:20,850 --> 01:09:24,229
that all inertial
observers are equivalent

1198
01:09:24,229 --> 01:09:27,700
and that the speed of light
does not depend on the cannon

1199
01:09:27,700 --> 01:09:30,240
that the light beam
was shot out of.

1200
01:09:30,240 --> 01:09:36,600
So if I'm at rest in this
expanding coordinate system,

1201
01:09:36,600 --> 01:09:38,224
I'm not really an
inertial observer

1202
01:09:38,224 --> 01:09:40,140
because there is gravity
in this whole system,

1203
01:09:40,140 --> 01:09:43,030
but we're going to ignore that.

1204
01:09:43,030 --> 01:09:44,529
If we're really
being rigorous here,

1205
01:09:44,529 --> 01:09:46,700
we have to do the full
general relatively thing.

1206
01:09:46,700 --> 01:09:49,910
But I think the intuitive
explanation is pretty obviously

1207
01:09:49,910 --> 01:09:50,410
valid.

1208
01:09:50,410 --> 01:09:52,620
It is, in fact,
rigorously valid.

1209
01:09:52,620 --> 01:09:56,440
If I'm standing still in this
expanding coordinate system,

1210
01:09:56,440 --> 01:09:58,820
I am an inertial observer.

1211
01:09:58,820 --> 01:10:00,883
And if a light
beam comes by me, I

1212
01:10:00,883 --> 01:10:03,216
should measure it's B and C,
no matter where it started,

1213
01:10:03,216 --> 01:10:05,940
no matter what's
happened in the past.

1214
01:10:05,940 --> 01:10:08,170
So the conversion between
my units of distance

1215
01:10:08,170 --> 01:10:12,070
and physical units of distance,
my coordinate distances

1216
01:10:12,070 --> 01:10:16,750
and physical distances,
is just a of t.

1217
01:10:16,750 --> 01:10:19,410
So that's the only
factor that appears

1218
01:10:19,410 --> 01:10:21,040
and this is completely rigorous.

1219
01:10:21,040 --> 01:10:23,570
One can drive this in
a more general context

1220
01:10:23,570 --> 01:10:25,530
in general relativity.

1221
01:10:25,530 --> 01:10:28,519
Here we're starting
out with a premise

1222
01:10:28,519 --> 01:10:30,060
that the light pulse
travels at speed

1223
01:10:30,060 --> 01:10:33,720
say if one had the full
theory of general relativity

1224
01:10:33,720 --> 01:10:35,200
coupled to Maxwell's
equations we

1225
01:10:35,200 --> 01:10:37,220
could really derive the
fact of exactly how rays

1226
01:10:37,220 --> 01:10:39,895
travel and would give us this.

1227
01:10:44,950 --> 01:10:47,435
So we have a very
simple equation

1228
01:10:47,435 --> 01:10:49,180
for how light rays travel.

1229
01:10:54,580 --> 01:10:57,080
Now I want to spend a little
bit of time, and this, I guess,

1230
01:10:57,080 --> 01:11:01,830
will be the last topic I'll
talk about today discussing

1231
01:11:01,830 --> 01:11:06,010
the synchronization of
clocks in this world system,

1232
01:11:06,010 --> 01:11:10,900
in this cosmological
coordinate system.

1233
01:11:10,900 --> 01:11:14,400
In special relativity,
you know that it's

1234
01:11:14,400 --> 01:11:18,130
hard to talk about synchronizing
clocks at large distances.

1235
01:11:21,100 --> 01:11:24,727
The synchronization
of clocks depends

1236
01:11:24,727 --> 01:11:26,060
on the velocity of the observer.

1237
01:11:26,060 --> 01:11:27,820
That was one of the
principles we learned

1238
01:11:27,820 --> 01:11:29,610
about when I wrote
down those three

1239
01:11:29,610 --> 01:11:33,180
kinematic properties
of special relativity.

1240
01:11:33,180 --> 01:11:36,620
So in general, in the context
of special relativity,

1241
01:11:36,620 --> 01:11:40,156
there is no universal way
of synchronize the clocks.

1242
01:11:40,156 --> 01:11:42,530
You could synchronize clocks
with respect to one observer

1243
01:11:42,530 --> 01:11:45,310
but then they would not be
synchronized with respect

1244
01:11:45,310 --> 01:11:49,690
to another observer moving
with respect to that observer.

1245
01:11:49,690 --> 01:11:52,435
In this case, we have perhaps
even a further complication

1246
01:11:52,435 --> 01:11:54,770
although in the end
everything is simple,

1247
01:11:54,770 --> 01:11:57,840
but we have the further
complication that the different

1248
01:11:57,840 --> 01:12:00,770
clocks that we're talking
about, which are clocks carried

1249
01:12:00,770 --> 01:12:03,520
by these observers that are
stationary in our comoving

1250
01:12:03,520 --> 01:12:07,120
coordinate system-- clocks
carried essentially by galaxies

1251
01:12:07,120 --> 01:12:10,050
that are uniformly
expanding-- all these clocks

1252
01:12:10,050 --> 01:12:12,860
are moving relative
to each other.

1253
01:12:12,860 --> 01:12:15,460
because the Hubble
expansion tells us that.

1254
01:12:15,460 --> 01:12:18,900
So the notion of trying
to synchronize clocks

1255
01:12:18,900 --> 01:12:20,080
seems a bit formidable.

1256
01:12:22,940 --> 01:12:25,920
Turns out however that
we can synchronize clocks

1257
01:12:25,920 --> 01:12:29,880
and one can develop a notion of
what we call cosmic time, which

1258
01:12:29,880 --> 01:12:32,180
is the time that would be
right on all these clocks

1259
01:12:32,180 --> 01:12:34,054
where all these clocks,
I mean all the clocks

1260
01:12:34,054 --> 01:12:37,630
that are stationary with
respect to the local matter,

1261
01:12:37,630 --> 01:12:40,270
in other words stationary
with respect to this comoving

1262
01:12:40,270 --> 01:12:44,040
but expanding coordinate system.

1263
01:12:44,040 --> 01:12:47,280
So why can't we
synchronize clocks?

1264
01:12:47,280 --> 01:12:51,560
What we're using as our
core assumption, which

1265
01:12:51,560 --> 01:12:53,860
is what makes
everything simple, is

1266
01:12:53,860 --> 01:12:57,830
that the model universe that
we're building this homogeneous

1267
01:12:57,830 --> 01:13:01,940
and that means that what I
would see if I was living

1268
01:13:01,940 --> 01:13:06,400
in this universe would
not depend on where I was.

1269
01:13:06,400 --> 01:13:11,510
So if I were living
on galaxy number one

1270
01:13:11,510 --> 01:13:15,820
and took out my
stopwatch and timed

1271
01:13:15,820 --> 01:13:22,020
how long it took before say
the Hubble parameter changed

1272
01:13:22,020 --> 01:13:26,390
from Hubble's value
to the current value

1273
01:13:26,390 --> 01:13:30,650
say, as an example,
any two numbers,

1274
01:13:30,650 --> 01:13:33,904
if I were living any place
else and timed the same thing,

1275
01:13:33,904 --> 01:13:35,820
how long it took for the
Hubble expansion rate

1276
01:13:35,820 --> 01:13:39,490
to change from A to B, I
would have to get exactly

1277
01:13:39,490 --> 01:13:41,675
the same time
interval, otherwise, it

1278
01:13:41,675 --> 01:13:43,190
would not be homogeneous.

1279
01:13:43,190 --> 01:13:47,370
Homogeneous means everybody
sees exactly the same thing.

1280
01:13:47,370 --> 01:13:49,050
So we all have,
no matter where we

1281
01:13:49,050 --> 01:13:53,010
live in this universe,
a common history,

1282
01:13:53,010 --> 01:13:56,510
and that means that the
only thing we don't know

1283
01:13:56,510 --> 01:14:01,140
is how to synchronize our
clocks what time on my watch

1284
01:14:01,140 --> 01:14:03,827
might correspond to
what time on your watch.

1285
01:14:03,827 --> 01:14:05,660
But if we imagine that
we could send signals

1286
01:14:05,660 --> 01:14:07,470
or that we're some
global observer watching

1287
01:14:07,470 --> 01:14:11,790
the whole thing, then we could
just tell each other let's

1288
01:14:11,790 --> 01:14:14,935
all set our clocks to noon
when the Hubble expansion

1289
01:14:14,935 --> 01:14:18,570
rate is 500 kilometers
per second per megaparsec.

1290
01:14:18,570 --> 01:14:21,420
And then we would have a
well-defined synchronization.

1291
01:14:21,420 --> 01:14:25,510
And once we synchronized
our watch that way,

1292
01:14:25,510 --> 01:14:28,890
if we each looked at how the
Hubble expansion rate changed

1293
01:14:28,890 --> 01:14:32,350
with time, we would get exactly
the same function of time

1294
01:14:32,350 --> 01:14:34,525
by this principle
of homogeneity.

1295
01:14:34,525 --> 01:14:36,320
None of us could see
anything different

1296
01:14:36,320 --> 01:14:38,070
as long as we're
measuring time intervals,

1297
01:14:38,070 --> 01:14:40,215
and we've fixed it so now
we're measuring nothing

1298
01:14:40,215 --> 01:14:43,430
but time intervals because
we've arranged to all

1299
01:14:43,430 --> 01:14:46,790
set our clocks to the same
time at some particular value

1300
01:14:46,790 --> 01:14:49,420
to the Hubble parameter.

1301
01:14:49,420 --> 01:14:52,920
So to synchronize, we can
ask what are the options.

1302
01:14:52,920 --> 01:14:54,330
I mentioned the
Hubble parameter.

1303
01:14:54,330 --> 01:14:55,870
That's certainly
one parameter that

1304
01:14:55,870 --> 01:14:59,040
can be used in principle
to synchronize clocks

1305
01:14:59,040 --> 01:15:02,770
throughout our model universe.

1306
01:15:02,770 --> 01:15:04,810
You might wonder if we
can use the scale factor

1307
01:15:04,810 --> 01:15:08,712
itself to synchronize times.

1308
01:15:08,712 --> 01:15:10,170
And the answer
there I would say is

1309
01:15:10,170 --> 01:15:14,440
no because of this
ambiguity of the notch.

1310
01:15:17,110 --> 01:15:20,230
I have no way of comparing
my notch to your notch.

1311
01:15:20,230 --> 01:15:21,780
We can compare
physical distances

1312
01:15:21,780 --> 01:15:25,070
because they're related
to physical properties

1313
01:15:25,070 --> 01:15:27,680
as the size of a hydrogen atom
is a certain physical size,

1314
01:15:27,680 --> 01:15:29,340
no matter where it
is in this universe.

1315
01:15:29,340 --> 01:15:32,450
So we could use hydrogen
atoms to measure meters,

1316
01:15:32,450 --> 01:15:35,670
and we would all be talking
about the same meters.

1317
01:15:35,670 --> 01:15:37,792
And we could use those
meters to define seconds

1318
01:15:37,792 --> 01:15:40,000
by how long it takes light
to travel through a meter,

1319
01:15:40,000 --> 01:15:41,530
and so on.

1320
01:15:41,530 --> 01:15:43,290
So meters and
seconds, we can all

1321
01:15:43,290 --> 01:15:46,216
agree on because they're
related to physical phenomena

1322
01:15:46,216 --> 01:15:47,590
that we can all
see and that will

1323
01:15:47,590 --> 01:15:51,740
be the same everywhere in our
homogeneous model universe.

1324
01:15:51,740 --> 01:15:56,180
But notches, not so, everybody
gets to make up his own notch.

1325
01:15:56,180 --> 01:15:59,470
It's just the size of the
map he happens to draw.

1326
01:15:59,470 --> 01:16:02,691
So we cannot compare
scale factors and say,

1327
01:16:02,691 --> 01:16:04,690
we'll set our clocks to
a certain time when both

1328
01:16:04,690 --> 01:16:06,745
of our scale factors
have a certain value.

1329
01:16:06,745 --> 01:16:08,370
We would get different
synchronizations

1330
01:16:08,370 --> 01:16:10,200
depending on what
choices we've made

1331
01:16:10,200 --> 01:16:12,960
about having to find a notch.

1332
01:16:12,960 --> 01:16:15,440
So the scale factor does not
work as a synchronization

1333
01:16:15,440 --> 01:16:19,630
mechanism, Hubble
expansion rates does.

1334
01:16:19,630 --> 01:16:23,840
Also we haven't really talked
about an ideal cosmic microwave

1335
01:16:23,840 --> 01:16:26,090
background, but we
certainly talked about it,

1336
01:16:26,090 --> 01:16:28,381
the cosmic microwave background
has a temperature which

1337
01:16:28,381 --> 01:16:30,010
is falling as the
universe expands,

1338
01:16:30,010 --> 01:16:34,270
so that could be used to
synchronize clocks as well.

1339
01:16:34,270 --> 01:16:36,680
Might mention in
the last 30 seconds

1340
01:16:36,680 --> 01:16:41,012
one interesting phenomenon.

1341
01:16:41,012 --> 01:16:43,157
For our universe, the
Hubble expansion rate

1342
01:16:43,157 --> 01:16:44,740
is changing with
time, the temperature

1343
01:16:44,740 --> 01:16:45,830
is changing with time.

1344
01:16:45,830 --> 01:16:49,269
There's no problem talking
about this synchronization.

1345
01:16:49,269 --> 01:16:51,060
But if you're talking
about different kinds

1346
01:16:51,060 --> 01:16:53,360
of mathematical models
of the universes,

1347
01:16:53,360 --> 01:16:55,650
you can imagine a universe
where the Hubble expansion

1348
01:16:55,650 --> 01:16:58,480
rate is just
constant, and in fact

1349
01:16:58,480 --> 01:17:00,670
that is a space that
was studied very

1350
01:17:00,670 --> 01:17:02,610
early in the history
of general relativity.

1351
01:17:02,610 --> 01:17:03,842
It's called de Sitter space.

1352
01:17:03,842 --> 01:17:06,050
And it's approximately what
happens during inflation,

1353
01:17:06,050 --> 01:17:08,400
so we'll even be talking
about de Sitter space

1354
01:17:08,400 --> 01:17:10,330
later in the course.

1355
01:17:10,330 --> 01:17:12,765
In de Sitter space,
the Hubble constant

1356
01:17:12,765 --> 01:17:14,899
is absolutely constant,
so at least one

1357
01:17:14,899 --> 01:17:17,190
of the mechanisms I mentioned
to synchronize the clocks

1358
01:17:17,190 --> 01:17:18,615
goes away.

1359
01:17:18,615 --> 01:17:21,240
There's also, in fact, no cosmic
microwave background radiation

1360
01:17:21,240 --> 01:17:24,230
in pure de Sitter space,
so that goes away.

1361
01:17:24,230 --> 01:17:26,660
You could ask, is
there anything else,

1362
01:17:26,660 --> 01:17:29,330
it turns out there
is not, so you really

1363
01:17:29,330 --> 01:17:32,730
can construct a well-defined
model of the universe,

1364
01:17:32,730 --> 01:17:35,120
the so-called de Sitter
space, where there really

1365
01:17:35,120 --> 01:17:36,879
is no way of
synthesizing clocks.

1366
01:17:36,879 --> 01:17:39,420
And you could really show that
you could make transformations

1367
01:17:39,420 --> 01:17:41,735
so that if you synchronize
the clocks one way,

1368
01:17:41,735 --> 01:17:44,760
you could make a symmetry
transformation if we take

1369
01:17:44,760 --> 01:17:46,534
all those clocks out
of synchronization

1370
01:17:46,534 --> 01:17:48,200
and otherwise the
space would be just as

1371
01:17:48,200 --> 01:17:49,845
good as what you started with.

1372
01:17:49,845 --> 01:17:51,220
So the synchronization
is subtle,

1373
01:17:51,220 --> 01:17:53,261
and it depends on having
something which actually

1374
01:17:53,261 --> 01:17:55,050
changes with time,
but that will be

1375
01:17:55,050 --> 01:17:57,830
the case where our real
universe and for the model

1376
01:17:57,830 --> 01:17:59,750
universes that we'll
be talking about.

1377
01:17:59,750 --> 01:18:01,310
So I'll stop there.

1378
01:18:01,310 --> 01:18:03,840
See you folks on Thursday.