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 --> 00:00:14,749
from hundreds of
MIT courses, visit

7
00:00:14,749 --> 00:00:16,165
MIT at
OpenCourseWare@oce.mit.edu.

8
00:00:21,740 --> 00:00:26,790
PROFESSOR: OK, in that
case, let's get started.

9
00:00:26,790 --> 00:00:28,650
As usual, I like
to begin by giving

10
00:00:28,650 --> 00:00:31,040
a review of what we
talked about last time.

11
00:00:31,040 --> 00:00:34,722
This time on slides instead
of on the blackboard.

12
00:00:34,722 --> 00:00:36,680
We're talking mainly
about relativistic energy,

13
00:00:36,680 --> 00:00:39,500
or relativistic energy and
momentum, and pressure,

14
00:00:39,500 --> 00:00:40,940
sometimes.

15
00:00:40,940 --> 00:00:44,440
The key equation is probably
the most famous equation

16
00:00:44,440 --> 00:00:48,220
in physics, Einstein's
e equals m c squared.

17
00:00:48,220 --> 00:00:49,967
And I gave some
numerical examples.

18
00:00:49,967 --> 00:00:52,050
I actually looked up some
more numbers since then,

19
00:00:52,050 --> 00:00:54,037
when I was revising
the lecture notes.

20
00:00:54,037 --> 00:00:55,620
So these are slightly
more up to date.

21
00:00:55,620 --> 00:00:59,340
But it's still true that
if you could burn matter

22
00:00:59,340 --> 00:01:01,840
at the rate of one
kilogram per hour,

23
00:01:01,840 --> 00:01:04,290
you would have about
one and a half times

24
00:01:04,290 --> 00:01:06,640
the total power
output of the world.

25
00:01:06,640 --> 00:01:09,710
And that's apparently
still valid in 2011.

26
00:01:09,710 --> 00:01:12,620
I only had 2008 figures,
actually, at the lecture

27
00:01:12,620 --> 00:01:14,950
last time.

28
00:01:14,950 --> 00:01:18,552
And if you imagine the
15-gallon tank of gasoline,

29
00:01:18,552 --> 00:01:21,010
and you could figure out how
much that-- what it's mass is,

30
00:01:21,010 --> 00:01:23,430
and convert that to
energy, it turns out

31
00:01:23,430 --> 00:01:25,080
that a 15-gallon
tank of gasoline

32
00:01:25,080 --> 00:01:28,570
could power the world for about
two and half days, if you could

33
00:01:28,570 --> 00:01:31,130
convert all of it into energy.

34
00:01:31,130 --> 00:01:34,330
The catch, of course, is
that we can't convert matter

35
00:01:34,330 --> 00:01:35,710
into energy.

36
00:01:35,710 --> 00:01:38,930
We can't get around the problem
that, at least at the energies

37
00:01:38,930 --> 00:01:41,600
that we deal with,
baryon number.

38
00:01:41,600 --> 00:01:44,280
And that number of protons
and neutrons is conserved,

39
00:01:44,280 --> 00:01:47,600
so we can't make protons
and neutrons disappear.

40
00:01:47,600 --> 00:01:50,760
And that means that we're
limited in what we can do.

41
00:01:50,760 --> 00:01:53,530
In particular, one of the most
efficient things we can do

42
00:01:53,530 --> 00:01:57,380
is fission uranium 235.

43
00:01:57,380 --> 00:02:01,170
But when uranium 235 undergoes
fission, less than 1/10

44
00:02:01,170 --> 00:02:04,100
of 1% of the mass is
actually converted

45
00:02:04,100 --> 00:02:08,580
into energy, which is why we
can't actually avail ourselves

46
00:02:08,580 --> 00:02:11,340
of these fantastic
numbers that would apply,

47
00:02:11,340 --> 00:02:13,690
if we could literally just
convert matter into energy.

48
00:02:16,410 --> 00:02:20,037
We went on to talk about
the relativistic definitions

49
00:02:20,037 --> 00:02:22,120
of energy and momentum,
and how they come together

50
00:02:22,120 --> 00:02:29,190
to form a Lorentz four vector,
and the underlying theme here

51
00:02:29,190 --> 00:02:32,580
is that we consider ourselves
users of special relativity.

52
00:02:32,580 --> 00:02:35,720
Most of you I know have taken
special relativity courses,

53
00:02:35,720 --> 00:02:39,370
and for those of you who
have, this is a review.

54
00:02:39,370 --> 00:02:41,820
For those you who have not,
and there are some of those

55
00:02:41,820 --> 00:02:45,070
also, no need to panic.

56
00:02:45,070 --> 00:02:47,700
I intend to tell you
every fact that you

57
00:02:47,700 --> 00:02:49,740
need to know about
special relativity.

58
00:02:49,740 --> 00:02:52,110
I won't tell you how
to derive them all,

59
00:02:52,110 --> 00:02:55,890
but I'll tell you all you'll
need to know for this class.

60
00:02:55,890 --> 00:02:57,830
So in particular, it's
useful for this class

61
00:02:57,830 --> 00:03:00,780
to recognize that energy and
momentum can be put together

62
00:03:00,780 --> 00:03:04,850
into a four vector, where
the zeroth component

63
00:03:04,850 --> 00:03:07,450
is the energy divided
by the speed of light.

64
00:03:07,450 --> 00:03:11,590
And the three spatial components
are just the three components

65
00:03:11,590 --> 00:03:13,490
of the spatial
momentum, although they

66
00:03:13,490 --> 00:03:16,400
have to be defined
relativistically.

67
00:03:16,400 --> 00:03:18,360
The relativistic
definition of momentum,

68
00:03:18,360 --> 00:03:20,430
at least how it
relates to velocity,

69
00:03:20,430 --> 00:03:22,440
is that it's equal to
gamma times the rest

70
00:03:22,440 --> 00:03:26,459
mass times the velocity, where
gamma is the famous factor

71
00:03:26,459 --> 00:03:28,000
that we've been
seeing all along when

72
00:03:28,000 --> 00:03:30,080
we've talked about relativity.

73
00:03:30,080 --> 00:03:32,360
The Lorentz
contraction factor, one

74
00:03:32,360 --> 00:03:36,180
over the square root of 1
minus v squared over c squared.

75
00:03:36,180 --> 00:03:38,080
The energy of a particle,
relativistically,

76
00:03:38,080 --> 00:03:41,750
is the same gamma, times m
0 times c squared, and it

77
00:03:41,750 --> 00:03:44,370
can also be written as
the square root of m 0

78
00:03:44,370 --> 00:03:46,970
c squared squared, plus
the momentum squared,

79
00:03:46,970 --> 00:03:50,090
times c squared.

80
00:03:50,090 --> 00:03:52,360
Since the momentum
forms a four vector,

81
00:03:52,360 --> 00:03:57,780
its Lorentz and variant square
should be Lorentz and variant,

82
00:03:57,780 --> 00:03:59,970
and that means that
the momentum squared

83
00:03:59,970 --> 00:04:04,300
minus p 0 squared should be the
same in all inertial reference

84
00:04:04,300 --> 00:04:05,650
frames.

85
00:04:05,650 --> 00:04:07,230
And that's just
equal when you put

86
00:04:07,230 --> 00:04:09,600
in what p 0 means,
the momentum squared

87
00:04:09,600 --> 00:04:12,530
minus the energy squared,
divided by c 0 squared.

88
00:04:12,530 --> 00:04:15,030
And to know what value it's
equal to in all frames,

89
00:04:15,030 --> 00:04:18,564
it's efficient to know what
it's equal to in one frame.

90
00:04:18,564 --> 00:04:19,980
And the one frame
where we do know

91
00:04:19,980 --> 00:04:22,870
what it's equal to is the
rest frame of a particle.

92
00:04:22,870 --> 00:04:25,160
So in the rest frame,
the momentum vanishes,

93
00:04:25,160 --> 00:04:28,039
and the energy is
just m 0 c squared.

94
00:04:28,039 --> 00:04:29,830
So in the rest frame,
we can evaluate this,

95
00:04:29,830 --> 00:04:34,310
and we get minus m
0 c squared squared.

96
00:04:34,310 --> 00:04:36,880
And that means that has to
be the value in every frame.

97
00:04:36,880 --> 00:04:39,280
And this in fact is
the easy way to derive

98
00:04:39,280 --> 00:04:41,450
the relationship between
energy and momentum.

99
00:04:41,450 --> 00:04:44,310
If we go back, the
equation we had

100
00:04:44,310 --> 00:04:49,210
relating energy and momentum is
really exactly that equation,

101
00:04:49,210 --> 00:04:49,710
rearranged.

102
00:04:52,850 --> 00:04:55,490
Just to give an example of how
this works, when we actually

103
00:04:55,490 --> 00:04:58,040
have energy exchanges,
I pointed out

104
00:04:58,040 --> 00:05:00,900
that we could talk about the
energy of a hydrogen atom.

105
00:05:00,900 --> 00:05:04,250
And because energy and
mass are equivalent,

106
00:05:04,250 --> 00:05:07,550
the hydrogen atom clearly
has a little bit less energy

107
00:05:07,550 --> 00:05:11,020
than an isolated proton,
plus an isolated electron.

108
00:05:11,020 --> 00:05:12,520
Because when you
bring them together

109
00:05:12,520 --> 00:05:14,880
there's a binding energy,
and that binding energy

110
00:05:14,880 --> 00:05:19,350
is called delta e, and has
a value of 13.6 electron

111
00:05:19,350 --> 00:05:22,290
volts for the ground
state of hydrogen.

112
00:05:22,290 --> 00:05:24,290
So that tells us the
mass of an hydrogen atom

113
00:05:24,290 --> 00:05:26,130
is not the sum of
the two masses,

114
00:05:26,130 --> 00:05:28,150
but rather has this
correction factor,

115
00:05:28,150 --> 00:05:32,564
because we've taken out a little
bit of energy for the binding.

116
00:05:32,564 --> 00:05:34,730
And that means we've taken
out a little bit of mass.

117
00:05:39,010 --> 00:05:42,125
OK, then we talked about the
mass density of radiation

118
00:05:42,125 --> 00:05:46,040
and how-- building up to how
that will affect the universe.

119
00:05:46,040 --> 00:05:49,280
And we said that the
mass density of radiation

120
00:05:49,280 --> 00:05:52,710
is just the energy density
divided by c squared.

121
00:05:52,710 --> 00:05:55,240
And that can be taken, really,
as a definition of what

122
00:05:55,240 --> 00:05:58,040
we call relativistic
mass, and hence

123
00:05:58,040 --> 00:06:01,070
relativistic mass density.

124
00:06:01,070 --> 00:06:03,750
But the important point is
that this mass density actually

125
00:06:03,750 --> 00:06:06,590
does gravitate the
same as any other mass

126
00:06:06,590 --> 00:06:08,380
density of the same value.

127
00:06:08,380 --> 00:06:12,490
It really does create
gravity in the same way.

128
00:06:12,490 --> 00:06:15,390
Now I mentioned that things are
much more complicated if you

129
00:06:15,390 --> 00:06:18,580
want to talk about the
gravitational field produced

130
00:06:18,580 --> 00:06:20,520
by a single moving particle.

131
00:06:20,520 --> 00:06:22,990
That's asymmetric, the
velocity of the particle

132
00:06:22,990 --> 00:06:27,720
shows up in the equations that
describe the metric surrounding

133
00:06:27,720 --> 00:06:29,570
a single moving particle.

134
00:06:29,570 --> 00:06:32,280
But if you have a gas
of particles moving

135
00:06:32,280 --> 00:06:35,270
at high velocities, where
the velocities nonetheless

136
00:06:35,270 --> 00:06:38,850
average to zero, which
tends to happen, in a gas

137
00:06:38,850 --> 00:06:42,290
at least in the rest
frame of the gas.

138
00:06:42,290 --> 00:06:46,810
Then that gas will produce
gravitational fields,

139
00:06:46,810 --> 00:06:49,650
just like a static mass density.

140
00:06:49,650 --> 00:06:53,430
Where the mass density is this
relativistic energy divided

141
00:06:53,430 --> 00:06:59,010
by c squared, the relativistic
definition of the mass density.

142
00:06:59,010 --> 00:07:03,260
It's also useful to know
that the photon-- if we want

143
00:07:03,260 --> 00:07:07,280
to describe it as a particle,
is a particle of zero rest mass.

144
00:07:07,280 --> 00:07:09,430
Which means that it
can never be at rest,

145
00:07:09,430 --> 00:07:11,350
it always moves at
the speed of light.

146
00:07:11,350 --> 00:07:15,030
And it also means that its
energy can be arbitrarily

147
00:07:15,030 --> 00:07:19,970
small, because the energy
is proportional of momentum,

148
00:07:19,970 --> 00:07:24,049
and the momentum of a photon
can be as small as you like.

149
00:07:24,049 --> 00:07:26,340
For giving frequency, of
course, the energy of a photon

150
00:07:26,340 --> 00:07:27,570
is fixed.

151
00:07:27,570 --> 00:07:30,820
It's h times nu but
if you're allowed

152
00:07:30,820 --> 00:07:34,270
to vary the frequency,
which you can do if you just

153
00:07:34,270 --> 00:07:36,450
look at it in
different frames, you

154
00:07:36,450 --> 00:07:39,710
can make the energy
as small as you like.

155
00:07:39,710 --> 00:07:42,099
And the famous equation
then, p squared minus e

156
00:07:42,099 --> 00:07:44,140
squared c squared, which
would have on the right,

157
00:07:44,140 --> 00:07:48,370
minus m 0 squared
c to the 4th m 0

158
00:07:48,370 --> 00:07:51,280
squared c squared, excuse me--
has zero on the right hand

159
00:07:51,280 --> 00:07:54,070
side, because m 0
is 0 And that means

160
00:07:54,070 --> 00:07:58,230
that for photons, the energy is
just the speed of light times

161
00:07:58,230 --> 00:07:59,220
the momentum.

162
00:07:59,220 --> 00:08:01,790
And that's a famous
relationship that photons obey.

163
00:08:06,200 --> 00:08:09,310
Now, thinking about
how this gas of photons

164
00:08:09,310 --> 00:08:15,400
will behave in the universe,
we realized immediately

165
00:08:15,400 --> 00:08:17,360
that it does not
behave the same way

166
00:08:17,360 --> 00:08:21,760
as a mass density of ordinary
non-relativistic particles.

167
00:08:21,760 --> 00:08:24,202
Which is what we have
been dealing with to date.

168
00:08:24,202 --> 00:08:27,180
The important difference
is that in both cases,

169
00:08:27,180 --> 00:08:30,370
the number density falls
off like 1 over a cubed,

170
00:08:30,370 --> 00:08:33,190
as the universe expands,
these particles are not

171
00:08:33,190 --> 00:08:35,159
created and destroyed
in significant numbers,

172
00:08:35,159 --> 00:08:37,700
they just persevere.

173
00:08:37,700 --> 00:08:40,690
So the number density of either
non-relativistic particles,

174
00:08:40,690 --> 00:08:44,470
or photons, just falls off
like one over the volume,

175
00:08:44,470 --> 00:08:48,950
as the volume increases and
the number density dilutes.

176
00:08:48,950 --> 00:08:52,380
But what makes photons
different from non-relativistic

177
00:08:52,380 --> 00:08:55,350
particles, is that a
non-relativistic particle

178
00:08:55,350 --> 00:08:57,650
will maintain the
energy of that particle

179
00:08:57,650 --> 00:09:00,200
as the universe
expands, but photons

180
00:09:00,200 --> 00:09:02,350
will redshift as the
universe expands.

181
00:09:02,350 --> 00:09:05,200
So each photon will
itself lose energy.

182
00:09:05,200 --> 00:09:09,070
And it loses energy proportional
to one over the scale factor.

183
00:09:09,070 --> 00:09:12,820
And that's just because the
frequency shift proportionally

184
00:09:12,820 --> 00:09:15,220
to the scale factor.

185
00:09:15,220 --> 00:09:17,390
And that means that
the energy per photon

186
00:09:17,390 --> 00:09:19,720
shifts, because
quantum mechanically

187
00:09:19,720 --> 00:09:21,710
we know that the
energy of a photon

188
00:09:21,710 --> 00:09:23,920
is proportional
to its frequency.

189
00:09:23,920 --> 00:09:27,000
So if the frequency
redshifts so must the energy.

190
00:09:27,000 --> 00:09:30,250
In exactly the same
way, 1 over a f t.

191
00:09:30,250 --> 00:09:31,416
Yes, question?

192
00:09:31,416 --> 00:09:34,890
AUDIENCE: You said
previously that neutrinos

193
00:09:34,890 --> 00:09:36,730
behave like radiation
in the sense

194
00:09:36,730 --> 00:09:39,810
that theta energy
falls is 1 over a.

195
00:09:39,810 --> 00:09:43,472
What is it about them
that makes this happen?

196
00:09:43,472 --> 00:09:47,452
Because there are also particles
with standard kinetic energy,

197
00:09:47,452 --> 00:09:48,077
right?

198
00:09:48,077 --> 00:09:50,410
PROFESSOR: OK, the question,
in case you didn't hear it,

199
00:09:50,410 --> 00:09:54,100
is why-- how did
neutrinos fit in here?

200
00:09:54,100 --> 00:09:56,520
I've made the claim that
neutrinos act like radiation

201
00:09:56,520 --> 00:09:59,820
in the early universe, but
neutrinos have a non-zero mass,

202
00:09:59,820 --> 00:10:02,630
so they should obey the
standard formulas for particles

203
00:10:02,630 --> 00:10:04,440
with nonzero masses.

204
00:10:04,440 --> 00:10:06,590
The answer to that
is-- there is,

205
00:10:06,590 --> 00:10:08,730
I think, a simple answer,
which is that as long

206
00:10:08,730 --> 00:10:12,760
as the energy is large
compared to the mass,

207
00:10:12,760 --> 00:10:16,480
particles with masses will still
act like massless particles.

208
00:10:16,480 --> 00:10:19,430
It doesn't really matter
if the mass is zero or not,

209
00:10:19,430 --> 00:10:23,979
the key thing, really,
is this equation.

210
00:10:23,979 --> 00:10:25,520
So if the term on
the right hand side

211
00:10:25,520 --> 00:10:28,270
is small compared to either
of the two on the left,

212
00:10:28,270 --> 00:10:30,360
it's not much different
from being zero.

213
00:10:30,360 --> 00:10:32,120
And that's what
happens for neutrinos

214
00:10:32,120 --> 00:10:33,740
in the early universe.

215
00:10:33,740 --> 00:10:36,470
And we'll see soon that if
you go to early enough times,

216
00:10:36,470 --> 00:10:39,300
it's true even for
electron-positron pairs.

217
00:10:39,300 --> 00:10:41,960
They will also act
like radiation.

218
00:10:41,960 --> 00:10:43,930
Any particle will
act like radiation

219
00:10:43,930 --> 00:10:47,390
as long as the energy is large
compared to the rest energy.

220
00:10:51,130 --> 00:10:53,530
So getting back
to the discussion

221
00:10:53,530 --> 00:10:57,340
of the early universe, if
the energy of each photon

222
00:10:57,340 --> 00:10:59,396
falls off like one
over the scale factor,

223
00:10:59,396 --> 00:11:02,020
and the number density falls off
like one over the scale factor

224
00:11:02,020 --> 00:11:04,315
cubed, it means that
the energy density,

225
00:11:04,315 --> 00:11:07,580
and hence, the mass
density of radiation

226
00:11:07,580 --> 00:11:10,540
will fall off like one
over a to the fourth,

227
00:11:10,540 --> 00:11:14,270
in contrast to the one over
a cubed, that we found when

228
00:11:14,270 --> 00:11:17,015
we were talking about
non-relativistic matter.

229
00:11:17,015 --> 00:11:19,140
And that, of course, is
going to make a difference.

230
00:11:19,140 --> 00:11:23,119
Because those issues play a
key role in our discussions

231
00:11:23,119 --> 00:11:24,410
about how the universe evolved.

232
00:11:28,080 --> 00:11:30,400
An important feature,
which we see immediately,

233
00:11:30,400 --> 00:11:35,760
is that if we extrapolate
backwards in time,

234
00:11:35,760 --> 00:11:39,050
since the radiation mass density
is falling off like one over a

235
00:11:39,050 --> 00:11:41,980
to the fourth, and the
matter density is falling off

236
00:11:41,980 --> 00:11:44,960
like one over a cubed, it means
that as you go back in time,

237
00:11:44,960 --> 00:11:46,990
the radiation becomes
more and more important

238
00:11:46,990 --> 00:11:52,020
relative to the matter, by
factor of the scale factor.

239
00:11:52,020 --> 00:11:54,190
So if we go back far
enough, we will even

240
00:11:54,190 --> 00:11:57,910
find a time when the
mass density in radiation

241
00:11:57,910 --> 00:12:00,700
equaled the mass density
in non-relativistic matter.

242
00:12:00,700 --> 00:12:03,490
And we calculated about
when that would be.

243
00:12:03,490 --> 00:12:08,030
We said that the energy
density in radiation today

244
00:12:08,030 --> 00:12:11,650
is given by this number, 7
times 10 to the minus 14,

245
00:12:11,650 --> 00:12:12,737
joules per meter cubed.

246
00:12:12,737 --> 00:12:15,070
And I just gave you this
number, I didn't derive it yet.

247
00:12:15,070 --> 00:12:17,800
We will derive it,
probably later today.

248
00:12:17,800 --> 00:12:20,010
But for now, we're
just accepting it.

249
00:12:20,010 --> 00:12:22,310
And that implies
we can calculate

250
00:12:22,310 --> 00:12:24,060
from that the ratio
of the mass densities

251
00:12:24,060 --> 00:12:26,800
in radiation and
ordinary matter.

252
00:12:26,800 --> 00:12:29,520
Here, we use the fact
that ordinary matter

253
00:12:29,520 --> 00:12:31,750
can be described
by having an omega

254
00:12:31,750 --> 00:12:34,697
ratio to the critical
density of about 0.3.

255
00:12:34,697 --> 00:12:36,780
And we know how to calculate
the critical density,

256
00:12:36,780 --> 00:12:40,420
and that allowed us to calculate
the density of ordinary matter.

257
00:12:40,420 --> 00:12:44,160
And then this ratio turned
out to be 3.1 times 10

258
00:12:44,160 --> 00:12:45,980
to the minus four.

259
00:12:45,980 --> 00:12:48,740
So radiation in
today's universe is

260
00:12:48,740 --> 00:12:51,000
almost negligible
in its contribution

261
00:12:51,000 --> 00:12:53,220
to the overall energy
balance, compared

262
00:12:53,220 --> 00:12:55,910
to non-relativistic matter.

263
00:12:55,910 --> 00:12:58,730
But if you
extrapolate backwards,

264
00:12:58,730 --> 00:13:01,800
we know that this ratio will
vary as one over the scale

265
00:13:01,800 --> 00:13:02,837
factor.

266
00:13:02,837 --> 00:13:04,420
And we could figure
out what constants

267
00:13:04,420 --> 00:13:07,890
to put this equation by
putting in the right constant,

268
00:13:07,890 --> 00:13:10,630
so that this equation gives
us the right value today.

269
00:13:10,630 --> 00:13:14,586
Where the right value today is
3.1 times 10 to the minus four.

270
00:13:14,586 --> 00:13:15,960
And notice that
this works, if we

271
00:13:15,960 --> 00:13:18,860
let t be equal to t sub
zero, this factors one

272
00:13:18,860 --> 00:13:22,020
and we get 3.1 times
10 to the minus four.

273
00:13:22,020 --> 00:13:24,200
So these two factors
together they

274
00:13:24,200 --> 00:13:27,860
have t zero and the 3.1
times 10 to the minus four,

275
00:13:27,860 --> 00:13:29,890
are just the right
factors to put in

276
00:13:29,890 --> 00:13:32,110
to give us the right
constant of proportionality

277
00:13:32,110 --> 00:13:34,516
in that equation.

278
00:13:34,516 --> 00:13:35,890
Having this
equation, we can then

279
00:13:35,890 --> 00:13:40,180
ask, how far back do we
have to go, how much we have

280
00:13:40,180 --> 00:13:44,610
to change t, for
the ratio to be one?

281
00:13:44,610 --> 00:13:47,110
And that's a
straightforward calculation.

282
00:13:47,110 --> 00:13:50,490
And the ratio of the a is
then just one over 3.1 times

283
00:13:50,490 --> 00:13:54,154
10 to the minus four, or 3,200.

284
00:13:54,154 --> 00:13:56,320
So if we talk about it in
terms of a redshift, which

285
00:13:56,320 --> 00:14:00,240
is how astronomers always talk
about distances, or times,

286
00:14:00,240 --> 00:14:05,710
we're talking about going
back to a redshift of 3,200.

287
00:14:05,710 --> 00:14:08,420
We can figure out what time,
then, corresponds to also,

288
00:14:08,420 --> 00:14:11,160
if we know how a f
t depends on time.

289
00:14:11,160 --> 00:14:14,100
And we do, approximately.

290
00:14:14,100 --> 00:14:16,260
For this calculation,
I assume for now, we

291
00:14:16,260 --> 00:14:18,316
could do better
later, and we will--

292
00:14:18,316 --> 00:14:19,940
but I assume for now
that we could just

293
00:14:19,940 --> 00:14:25,285
treat the period between matter
radiation equality, so-called t

294
00:14:25,285 --> 00:14:29,000
x and now as being
entirely described

295
00:14:29,000 --> 00:14:30,989
by a matter-dominated universe.

296
00:14:30,989 --> 00:14:32,405
That's only a crude
approximation,

297
00:14:32,405 --> 00:14:34,930
but it will get us the
right order of magnitude.

298
00:14:34,930 --> 00:14:39,000
And we'll learn how to do
better later in the course.

299
00:14:39,000 --> 00:14:43,930
So if we assume that, then t x
is just this number to the 3/2

300
00:14:43,930 --> 00:14:48,550
power, cancelling the 2/3, times
the age of universe, t naught.

301
00:14:48,550 --> 00:14:52,110
And that turned out to
be about 75,000 years.

302
00:14:52,110 --> 00:14:57,280
So somewhere in the range of
100,000 years, 50,000 years,

303
00:14:57,280 --> 00:15:02,790
is the time in the history of
the universe when radiation

304
00:15:02,790 --> 00:15:05,340
ceased to be more
important than matter.

305
00:15:05,340 --> 00:15:08,490
And for earlier times than
that, the radiation dominated.

306
00:15:08,490 --> 00:15:11,910
And that's what we refer to as
the radiation dominated era.

307
00:15:18,727 --> 00:15:19,310
Any questions?

308
00:15:24,490 --> 00:15:26,775
OK, now I think we get
on to what is really

309
00:15:26,775 --> 00:15:30,440
the important subject that
we want to understand,

310
00:15:30,440 --> 00:15:33,440
and most of this you did
yourself on the homework.

311
00:15:33,440 --> 00:15:35,914
But I'll summarize
the argument here.

312
00:15:35,914 --> 00:15:37,580
We want to understand
what this tells us

313
00:15:37,580 --> 00:15:40,250
about the Friedmann Equations.

314
00:15:40,250 --> 00:15:41,820
And first, we'd
like to understand

315
00:15:41,820 --> 00:15:43,170
what it says about pressure.

316
00:15:43,170 --> 00:15:44,820
Because it turns
out that pressure

317
00:15:44,820 --> 00:15:47,660
is the crucial
issue in determining

318
00:15:47,660 --> 00:15:52,320
how fast row falls
off as a expands.

319
00:15:52,320 --> 00:15:56,690
So if row is proportional
to one over a cubed,

320
00:15:56,690 --> 00:15:58,770
we can just differentiate
that, putting

321
00:15:58,770 --> 00:16:00,800
in a constant proportionality
temporarily, just

322
00:16:00,800 --> 00:16:02,577
to keep track of
what we're doing.

323
00:16:02,577 --> 00:16:04,410
Since we know how to
differentiate qualities

324
00:16:04,410 --> 00:16:06,590
and we're less familiar
with how to differentiate

325
00:16:06,590 --> 00:16:08,400
proportionalities.

326
00:16:08,400 --> 00:16:11,430
But what we find immediately,
is that row dot is then minus 3,

327
00:16:11,430 --> 00:16:16,435
where that 3 is that 3,
times a dot over a times row.

328
00:16:16,435 --> 00:16:18,560
On the other hand, if
row of t falls like one

329
00:16:18,560 --> 00:16:21,270
over a to the fourth,
row dot is minus 4 times

330
00:16:21,270 --> 00:16:24,710
a dot over a times row,
just by differentiation.

331
00:16:24,710 --> 00:16:26,850
So we get different
expressions from row dot,

332
00:16:26,850 --> 00:16:28,815
between radiation and matter.

333
00:16:28,815 --> 00:16:31,200
And we want to explore
the consequences

334
00:16:31,200 --> 00:16:32,280
of that difference.

335
00:16:35,370 --> 00:16:40,510
It's related to the
pressure of the gas,

336
00:16:40,510 --> 00:16:44,020
because we can relate
the pressure to row dot.

337
00:16:44,020 --> 00:16:46,950
Because we know that
as a gas expands,

338
00:16:46,950 --> 00:16:51,380
it loses energy, which is
just equal in amount to pdV.

339
00:16:51,380 --> 00:16:55,450
And we illustrated this by
a piston thought experiment,

340
00:16:55,450 --> 00:16:58,680
but it's true in general.

341
00:16:58,680 --> 00:17:03,200
So we can apply this famous
formula, dU equals minus pdV,

342
00:17:03,200 --> 00:17:05,540
to a patch of the
expanding universe.

343
00:17:05,540 --> 00:17:12,710
And by a patch I mean some fixed
region and coordinate space.

344
00:17:12,710 --> 00:17:16,180
So the total energy in that
region of coordinate space

345
00:17:16,180 --> 00:17:17,660
will be the physical
volume, which

346
00:17:17,660 --> 00:17:20,380
will be a cubed times
the coordinate volume.

347
00:17:20,380 --> 00:17:24,310
Which is going to cancel out
of this equation on both sides.

348
00:17:24,310 --> 00:17:26,140
So it's a cubed times
the coordinate volume

349
00:17:26,140 --> 00:17:30,060
times the energy density,
which is row c squared.

350
00:17:30,060 --> 00:17:32,680
The rate of change
of that is Du.

351
00:17:32,680 --> 00:17:34,180
And then on the
right hand side, we

352
00:17:34,180 --> 00:17:37,000
have minus p minus
b times dV, which

353
00:17:37,000 --> 00:17:39,410
is the rate of change
of a cubed, again

354
00:17:39,410 --> 00:17:42,390
times the coordinate volume
that we're talking about.

355
00:17:42,390 --> 00:17:45,130
But that will cancel out on
the two sides of the equation.

356
00:17:45,130 --> 00:17:47,400
So this is really
just a description

357
00:17:47,400 --> 00:17:53,250
for the universe of the dU
equals minus pdV equation.

358
00:17:53,250 --> 00:17:56,820
And this can just be rearranged,
expanding the time derivatives,

359
00:17:56,820 --> 00:17:59,960
to give us row dot, and
we get minus 3 a dot

360
00:17:59,960 --> 00:18:05,120
over a, times row
plus p over c squared.

361
00:18:05,120 --> 00:18:09,550
So this tells us how to relate
row dot to the pressure.

362
00:18:09,550 --> 00:18:11,730
And it tells us that the
formula that we started

363
00:18:11,730 --> 00:18:15,840
with a long time ago, which
just said that row fell off

364
00:18:15,840 --> 00:18:18,720
like 1 over a cubed, was
synonymous with saying

365
00:18:18,720 --> 00:18:23,060
the pressure is zero, for a gas
of non-relativistic particles,

366
00:18:23,060 --> 00:18:25,500
the pressure is negligible.

367
00:18:25,500 --> 00:18:27,350
But for radiation,
clearly if we're

368
00:18:27,350 --> 00:18:29,640
going to get a four
instead of a three,

369
00:18:29,640 --> 00:18:31,545
the pressure will
be non-negligible.

370
00:18:31,545 --> 00:18:34,060
And in fact, it implies
that the pressure

371
00:18:34,060 --> 00:18:36,770
is exactly equal to
one third of the energy

372
00:18:36,770 --> 00:18:39,025
density for a gas of radiation.

373
00:18:43,020 --> 00:18:46,880
OK, knowing that, we can now
look back at the Friedmann

374
00:18:46,880 --> 00:18:51,700
equations and ask,
how do they stand up?

375
00:18:51,700 --> 00:18:55,470
Are they still consistent, or
do we have to modify something?

376
00:18:55,470 --> 00:18:59,420
And this is really
the crucial point.

377
00:18:59,420 --> 00:19:03,059
What we know are
these three equations,

378
00:19:03,059 --> 00:19:04,600
which are the two
Friedmann equations

379
00:19:04,600 --> 00:19:08,590
and the equation for row dot.

380
00:19:08,590 --> 00:19:11,150
And we could see immediately
that those equations are not

381
00:19:11,150 --> 00:19:13,190
independent of each other.

382
00:19:13,190 --> 00:19:15,910
The easiest thing
to see is that if we

383
00:19:15,910 --> 00:19:19,700
start with the top equation,
we could differentiate it.

384
00:19:19,700 --> 00:19:21,910
And since the top
equation has a dot in it,

385
00:19:21,910 --> 00:19:25,590
when we differentiate, we'll get
an equation for a double dot.

386
00:19:25,590 --> 00:19:27,440
But the equation will
also involve row dot,

387
00:19:27,440 --> 00:19:30,360
if we take the time derivative
of that top equation.

388
00:19:30,360 --> 00:19:32,025
But if we know
what row dot is, we

389
00:19:32,025 --> 00:19:33,400
could put that
in, and in the end

390
00:19:33,400 --> 00:19:37,900
we'll get an equation for
a double dot by itself.

391
00:19:37,900 --> 00:19:40,110
And it will in fact
agree with the equation

392
00:19:40,110 --> 00:19:40,960
on the middle line.

393
00:19:40,960 --> 00:19:42,670
Again, things would
be inconsistent.

394
00:19:42,670 --> 00:19:45,600
Things are consistent, we
didn't make any mistakes.

395
00:19:45,600 --> 00:19:48,110
If we derive the
equation for a double dot

396
00:19:48,110 --> 00:19:50,210
by using the first and
third of those equations,

397
00:19:50,210 --> 00:19:53,790
then we'll get the second
of those equations.

398
00:19:53,790 --> 00:19:55,550
But the catch is
that now we want

399
00:19:55,550 --> 00:19:57,200
to modify the third
of those equations,

400
00:19:57,200 --> 00:19:59,440
the equations for row dot.

401
00:19:59,440 --> 00:20:01,210
And then the Friedmann
equations as we've

402
00:20:01,210 --> 00:20:04,634
written them will not
be consistent anymore.

403
00:20:04,634 --> 00:20:06,800
Because we'll have a different
equation for row dot,

404
00:20:06,800 --> 00:20:10,010
we'll get a different
equation for a double dot.

405
00:20:10,010 --> 00:20:12,370
So we have to decide what gives.

406
00:20:12,370 --> 00:20:15,930
What can we change to make
everything consistent?

407
00:20:15,930 --> 00:20:18,290
And here, the rigorous
way of proceeding

408
00:20:18,290 --> 00:20:20,804
is to look at general
relativity and see what it says,

409
00:20:20,804 --> 00:20:22,470
and the answer we're
going to write down

410
00:20:22,470 --> 00:20:26,079
is exactly what general
relativity says.

411
00:20:26,079 --> 00:20:27,870
But we can motivate
the answer in, I think,

412
00:20:27,870 --> 00:20:30,850
a pretty sensible
way, by noticing

413
00:20:30,850 --> 00:20:33,770
that as the universe
expands, we'd

414
00:20:33,770 --> 00:20:36,820
expect the energy density
to vary continuously,

415
00:20:36,820 --> 00:20:38,990
because energies are conserved.

416
00:20:38,990 --> 00:20:43,490
And we also expect a dot
to vary continuously,

417
00:20:43,490 --> 00:20:47,140
because basically, the
mechanics of the universe

418
00:20:47,140 --> 00:20:48,480
are like ton's laws.

419
00:20:48,480 --> 00:20:50,610
And velocities don't
change discontinuously.

420
00:20:50,610 --> 00:20:53,100
You can apply a
force, and that causes

421
00:20:53,100 --> 00:20:55,010
velocities to have
a rate of change.

422
00:20:55,010 --> 00:20:57,480
But velocities don't
change instantaneously,

423
00:20:57,480 --> 00:20:59,650
unless you somehow
apply an infinite force.

424
00:20:59,650 --> 00:21:01,910
And the same thing will
be true with the universe.

425
00:21:01,910 --> 00:21:05,101
On the other hand, accelerations
can change instantaneously.

426
00:21:05,101 --> 00:21:07,100
You could change the force
acting on a particle,

427
00:21:07,100 --> 00:21:09,090
in principle, as
fast as you want,

428
00:21:09,090 --> 00:21:10,590
and the acceleration
of the particle

429
00:21:10,590 --> 00:21:13,610
will change at that same rate.

430
00:21:13,610 --> 00:21:16,640
So if we look at
these equations,

431
00:21:16,640 --> 00:21:20,620
we would expect that the first
equation and the third equation

432
00:21:20,620 --> 00:21:23,750
would not be allowed to
involve the pressure.

433
00:21:23,750 --> 00:21:26,130
Because the pressure basically
is a measure of a force.

434
00:21:26,130 --> 00:21:29,880
Pressures can change
instantaneously.

435
00:21:29,880 --> 00:21:33,400
So what you need to
do, if we're going

436
00:21:33,400 --> 00:21:35,730
to make these equations
consistent in the presence

437
00:21:35,730 --> 00:21:39,155
of pressure-- which changes
the row dot equation,

438
00:21:39,155 --> 00:21:41,672
the only equation we can
change is the second one.

439
00:21:41,672 --> 00:21:43,130
And then we can
ask ourselves, what

440
00:21:43,130 --> 00:21:47,330
do we have to change it to make
the three equations consistent?

441
00:21:47,330 --> 00:21:49,670
And this is what you
looked at on your homework.

442
00:21:49,670 --> 00:21:51,770
And the answer is that
the a double dot equation

443
00:21:51,770 --> 00:21:54,380
has to be modified
to give the equation

444
00:21:54,380 --> 00:21:55,930
at the bottom of
the screen here.

445
00:21:55,930 --> 00:21:59,530
And this is the correct form
of the a double dot equation

446
00:21:59,530 --> 00:22:01,335
in cosmology.

447
00:22:01,335 --> 00:22:03,710
And this is what we'll be
using for the rest of the term,

448
00:22:03,710 --> 00:22:06,240
this is exactly what you would
get from general relativity.

449
00:22:06,240 --> 00:22:07,990
As long as we're talking
about homogeneous

450
00:22:07,990 --> 00:22:11,040
and isotropic universes,
this formula as exact

451
00:22:11,040 --> 00:22:14,030
as far as we know.

452
00:22:14,030 --> 00:22:16,370
OK, any questions about that?

453
00:22:16,370 --> 00:22:18,250
Yes.

454
00:22:18,250 --> 00:22:21,590
AUDIENCE: Why when we derive
that equation do we use--

455
00:22:21,590 --> 00:22:22,852
PROFESSOR: This equation?

456
00:22:22,852 --> 00:22:23,814
AUDIENCE: Or the one above that.

457
00:22:23,814 --> 00:22:24,480
PROFESSOR: Yeah.

458
00:22:24,480 --> 00:22:26,700
AUDIENCE: We use dU
equals minus pdV?

459
00:22:26,700 --> 00:22:29,060
I mean, I agree with
that, but couldn't we

460
00:22:29,060 --> 00:22:30,384
use a more complete version?

461
00:22:30,384 --> 00:22:32,175
Like, the complete
version of the first law

462
00:22:32,175 --> 00:22:37,047
of thermodynamics, that
dU equals TDS minus pdV.

463
00:22:37,047 --> 00:22:37,880
PROFESSOR: OK, yeah.

464
00:22:37,880 --> 00:22:42,330
The question was when we wrote
down dU equals minus pdV,

465
00:22:42,330 --> 00:22:46,110
why did we not include
a plus TDS term here,

466
00:22:46,110 --> 00:22:48,110
which could also be relevant.

467
00:22:48,110 --> 00:22:50,422
The answer is that
for the applications

468
00:22:50,422 --> 00:22:52,130
we're interested in--
you're quite right,

469
00:22:52,130 --> 00:22:54,470
it could be important,
but for the applications

470
00:22:54,470 --> 00:22:56,470
that we're interested in,
which is the expanding

471
00:22:56,470 --> 00:22:59,250
gas in the universe, the
expanding gas in the universe

472
00:22:59,250 --> 00:23:00,720
will be making use of this fact.

473
00:23:00,720 --> 00:23:04,370
It really does expand
adiabatically, that is,

474
00:23:04,370 --> 00:23:07,460
there's nothing
putting heat in or out,

475
00:23:07,460 --> 00:23:11,330
and everything is remaining very
close to thermal equilibrium,

476
00:23:11,330 --> 00:23:14,250
which means that entropy does
not spontaneously change.

477
00:23:14,250 --> 00:23:17,760
So the TDS term we will be
assuming is very, very small,

478
00:23:17,760 --> 00:23:19,557
and that's accurate.

479
00:23:19,557 --> 00:23:21,390
And you're right, if
that were not the case,

480
00:23:21,390 --> 00:23:23,780
there would be further
complications in terms

481
00:23:23,780 --> 00:23:25,350
of figuring out what row dot is.

482
00:23:36,570 --> 00:23:39,750
Let me point out here that
this equation actually

483
00:23:39,750 --> 00:23:43,240
does contain a somewhat
startling perhaps fact

484
00:23:43,240 --> 00:23:47,190
about gravity it says
that in the context

485
00:23:47,190 --> 00:23:48,132
of general relativity.

486
00:23:48,132 --> 00:23:49,965
And that's really the
context that we're in,

487
00:23:49,965 --> 00:23:53,350
even though we haven't learned
a lot of general relativity.

488
00:23:53,350 --> 00:23:56,310
But it says that in the
context of general relativity,

489
00:23:56,310 --> 00:23:59,700
pressures, as well
as mass densities,

490
00:23:59,700 --> 00:24:02,480
contribute to the
gravitational field.

491
00:24:02,480 --> 00:24:04,160
A double dot is
basically a measure

492
00:24:04,160 --> 00:24:08,450
of how fast gravity is
slowing down the universe.

493
00:24:08,450 --> 00:24:10,620
And this says that
there's a pressure.

494
00:24:10,620 --> 00:24:14,460
It can also help to
slow down the universe.

495
00:24:14,460 --> 00:24:20,450
Meaning that pressure itself can
create a gravitational field.

496
00:24:20,450 --> 00:24:22,210
In the early universe,
where we go back

497
00:24:22,210 --> 00:24:25,270
to this radiation
dominated period,

498
00:24:25,270 --> 00:24:28,240
we know that the pressure is
one third of the energy density.

499
00:24:28,240 --> 00:24:29,740
That says that
this pressure term

500
00:24:29,740 --> 00:24:34,240
is the same size exactly
as the mass density term.

501
00:24:34,240 --> 00:24:36,520
So in the radiation
dominated phase,

502
00:24:36,520 --> 00:24:39,580
the pressure is just
as important in effect

503
00:24:39,580 --> 00:24:43,900
for slowing down the universe
as is the mass density.

504
00:24:43,900 --> 00:24:45,730
In today's universe
it's negligible.

505
00:24:45,730 --> 00:24:47,734
Well, we'll come back that.

506
00:24:47,734 --> 00:24:49,525
The dark energy has a
non-trivial pressure,

507
00:24:49,525 --> 00:24:52,120
but the pressure of ordinary
matter in today's universe

508
00:24:52,120 --> 00:24:55,020
is negligible.

509
00:24:55,020 --> 00:24:58,140
The other important
fact about this equation

510
00:24:58,140 --> 00:25:02,237
is that energy densities,
so far as we know,

511
00:25:02,237 --> 00:25:03,070
are always positive.

512
00:25:04,887 --> 00:25:07,220
We don't know for sure what
the ultimate laws of physics

513
00:25:07,220 --> 00:25:09,770
are, but for all the laws
of physics that we know,

514
00:25:09,770 --> 00:25:12,770
energy densities are positive.

515
00:25:12,770 --> 00:25:15,590
On the other hand,
pressures actually

516
00:25:15,590 --> 00:25:17,480
can be negative for
some kinds of material.

517
00:25:17,480 --> 00:25:19,230
And we'll talk a little
bit more about how

518
00:25:19,230 --> 00:25:22,610
to get a negative
pressure later.

519
00:25:22,610 --> 00:25:25,560
But this formula tells us
that positive pressures

520
00:25:25,560 --> 00:25:27,930
act the same way as
positive mass densities,

521
00:25:27,930 --> 00:25:30,030
creating an attractive
gravitational field which

522
00:25:30,030 --> 00:25:33,370
slows down the expansion
of the universe.

523
00:25:33,370 --> 00:25:36,860
But if there could be a material
with a negative pressure,

524
00:25:36,860 --> 00:25:39,180
this same equation, which
would presumably still hold,

525
00:25:39,180 --> 00:25:41,240
and believe it
does, would tell us

526
00:25:41,240 --> 00:25:43,240
that that negative
pressure would actually

527
00:25:43,240 --> 00:25:45,130
cause the universe to
accelerate, because

528
00:25:45,130 --> 00:25:46,895
of its gravitational effects.

529
00:25:46,895 --> 00:25:49,020
Now, we're not talking
about the mechanical effects

530
00:25:49,020 --> 00:25:50,590
of the pressure.

531
00:25:50,590 --> 00:25:52,680
Mechanical effects
of pressure only

532
00:25:52,680 --> 00:25:54,910
show up when there are
pressure gradients, when

533
00:25:54,910 --> 00:25:56,350
the pressure is uneven.

534
00:25:56,350 --> 00:25:58,600
So the very large air
pressure in this room,

535
00:25:58,600 --> 00:26:01,470
which really is quite
large, we don't feel all,

536
00:26:01,470 --> 00:26:04,500
because it's acting
equally in all directions.

537
00:26:04,500 --> 00:26:07,384
Uniform pressures do
not produce forces.

538
00:26:07,384 --> 00:26:10,050
So the mechanical effects of the
pressure in the early universe,

539
00:26:10,050 --> 00:26:12,500
which we're assuming is
completely homogeneous,

540
00:26:12,500 --> 00:26:13,790
is zilch.

541
00:26:13,790 --> 00:26:15,560
There is no mechanical effect.

542
00:26:15,560 --> 00:26:17,750
But what we're seeing
in this equation

543
00:26:17,750 --> 00:26:20,736
is a gravitational effect
caused by the pressure.

544
00:26:20,736 --> 00:26:22,110
And it's obviously
gravitational,

545
00:26:22,110 --> 00:26:24,520
business the effect is
proportional to Newton's

546
00:26:24,520 --> 00:26:27,850
capital G, a
constant determining

547
00:26:27,850 --> 00:26:30,030
the strength of gravity.

548
00:26:30,030 --> 00:26:32,640
So the equation is telling
us that a positive pressure

549
00:26:32,640 --> 00:26:34,930
creates a gravitational
attraction, which

550
00:26:34,930 --> 00:26:38,310
would cause the universe to
slow down in its expansion.

551
00:26:38,310 --> 00:26:40,490
But a negative
pressure would produce

552
00:26:40,490 --> 00:26:43,650
a gravitational repulsion,
which would cause universe

553
00:26:43,650 --> 00:26:45,750
to speed up.

554
00:26:45,750 --> 00:26:48,070
And we now know that,
today-- in fact,

555
00:26:48,070 --> 00:26:51,850
for the last five billion
years or so-- our universe

556
00:26:51,850 --> 00:26:54,770
itself is actually accelerating
under the influence

557
00:26:54,770 --> 00:26:55,940
of something.

558
00:26:55,940 --> 00:26:58,600
And the only
explanation we have is

559
00:26:58,600 --> 00:27:00,530
that the something that's
causing the universe

560
00:27:00,530 --> 00:27:03,640
to accelerate is the
repulsive gravity caused

561
00:27:03,640 --> 00:27:06,640
by some kind of a negative
pressure material.

562
00:27:06,640 --> 00:27:08,330
And that negative
pressure material

563
00:27:08,330 --> 00:27:10,654
is what we call the dark energy.

564
00:27:10,654 --> 00:27:12,820
And we'll talk a little
more later about what it is.

565
00:27:12,820 --> 00:27:14,850
It's very likely
just vacuum energy.

566
00:27:14,850 --> 00:27:17,210
But we'll come back to
that later in the course.

567
00:27:17,210 --> 00:27:18,656
Yes?

568
00:27:18,656 --> 00:27:21,031
AUDIENCE: When we looked at
the toy example of the piston

569
00:27:21,031 --> 00:27:24,867
in the cavity, the
pressure of the gas

570
00:27:24,867 --> 00:27:28,080
was pushing outwards against
the wall of the container.

571
00:27:28,080 --> 00:27:32,552
But we can't have that view
of the universe, really,

572
00:27:32,552 --> 00:27:35,182
because there's no
exteriors in the universe.

573
00:27:35,182 --> 00:27:36,640
PROFESSOR: There's
no walls, right.

574
00:27:36,640 --> 00:27:40,970
AUDIENCE: So how should we view
pressure in the sense that--

575
00:27:40,970 --> 00:27:42,110
PROFESSOR: OK.

576
00:27:42,110 --> 00:27:42,610
Yeah.

577
00:27:42,610 --> 00:27:48,800
OK, the question is, in our toy
problem involving the piston,

578
00:27:48,800 --> 00:27:51,840
we had walls for the
pressure to push against.

579
00:27:51,840 --> 00:27:53,420
And that was where
the energy went.

580
00:27:53,420 --> 00:27:55,750
It went into pushing the walls.

581
00:27:55,750 --> 00:27:58,630
When we're talking about the
universe, there are no walls.

582
00:27:58,630 --> 00:28:00,020
How does that analogy work?

583
00:28:00,020 --> 00:28:02,530
What plays the
role of the walls?

584
00:28:02,530 --> 00:28:06,460
And the answer, I think, is
that the role of the walls, when

585
00:28:06,460 --> 00:28:08,690
we're talking about the
universe, first of all, you

586
00:28:08,690 --> 00:28:10,310
can ignore it if you just
took in a small region.

587
00:28:10,310 --> 00:28:12,060
You could still just
say, the small region

588
00:28:12,060 --> 00:28:15,480
is pushing out on the
regions around it.

589
00:28:15,480 --> 00:28:18,192
And I think that's enough
to make the logic clear.

590
00:28:18,192 --> 00:28:20,400
But it still leaves open
the question of, ultimately,

591
00:28:20,400 --> 00:28:22,170
where does this energy go?

592
00:28:22,170 --> 00:28:23,800
So saying it goes
from here to there

593
00:28:23,800 --> 00:28:25,383
doesn't help you
unless you know where

594
00:28:25,383 --> 00:28:27,670
it goes after it goes
from and there and there.

595
00:28:27,670 --> 00:28:29,878
So you might want to ask
the question more generally,

596
00:28:29,878 --> 00:28:32,330
where does the energy
ultimately end up?

597
00:28:32,330 --> 00:28:34,350
And then I think the
answer is that it ends up

598
00:28:34,350 --> 00:28:36,372
in gravitational
potential energy.

599
00:28:36,372 --> 00:28:38,580
You could certainly build
a toy model, where you just

600
00:28:38,580 --> 00:28:44,879
have a gas in a finite region,
self-contained under gravity.

601
00:28:44,879 --> 00:28:47,170
And then you'd have to make
up some kind of a mechanism

602
00:28:47,170 --> 00:28:48,495
to cause it to expand.

603
00:28:48,495 --> 00:28:50,120
But when you cause
it to expand, you'll

604
00:28:50,120 --> 00:28:51,500
be pulling particles
apart, which

605
00:28:51,500 --> 00:28:53,740
are attracting each
other gravitationally.

606
00:28:53,740 --> 00:28:55,640
And that means
you'll be increasing

607
00:28:55,640 --> 00:28:57,070
the gravitational
potential energy

608
00:28:57,070 --> 00:28:59,610
as you pull the gas apart.

609
00:28:59,610 --> 00:29:02,800
So I think, ultimately, the
answer is the energy imbalance

610
00:29:02,800 --> 00:29:05,100
that we seem to be
seeing here is taken up

611
00:29:05,100 --> 00:29:07,799
by the gravitational field
so that, all in all, energy

612
00:29:07,799 --> 00:29:08,465
still conserved.

613
00:29:11,797 --> 00:29:12,630
Any other questions?

614
00:29:19,930 --> 00:29:24,630
OK, in that case, let us
continue on the blackboard.

615
00:29:39,455 --> 00:29:40,955
OK, first thing I
want to look at it

616
00:29:40,955 --> 00:29:43,625
is just the behavior of a
radiation-dominated flat

617
00:29:43,625 --> 00:29:44,125
universe.

618
00:29:59,840 --> 00:30:03,520
So a flat universe
is going to obey

619
00:30:03,520 --> 00:30:09,590
H squared equals
8 pi over 3 G rho,

620
00:30:09,590 --> 00:30:24,375
and then the potential minus
kc squared over a squared.

621
00:30:27,835 --> 00:30:29,710
Hard to write dotted
lines on the blackboard.

622
00:30:29,710 --> 00:30:34,890
But this potential term is
not there, because k equals 0.

623
00:30:34,890 --> 00:30:36,850
We're talking about
the flat case.

624
00:30:36,850 --> 00:30:41,380
So for a flat case, we could
just express H in terms of rho.

625
00:30:41,380 --> 00:30:44,000
And we know how rho
behaves for radiation.

626
00:30:44,000 --> 00:30:47,180
Rho falls off as
1/a to the fourth.

627
00:30:47,180 --> 00:30:50,870
So H squared is proportional
to 1/a to the fourth.

628
00:30:50,870 --> 00:30:54,630
That means that H itself is
proportional to 1/a squared.

629
00:30:54,630 --> 00:30:57,700
So we can do that.

630
00:30:57,700 --> 00:31:14,760
a-dot over a is equal to
some constant over a squared,

631
00:31:14,760 --> 00:31:18,930
a-dot over a being H. And now
we can multiply both sides

632
00:31:18,930 --> 00:31:21,800
by a, of course.

633
00:31:21,800 --> 00:31:28,450
And we get a-dot is equal
to a constant over a.

634
00:31:33,780 --> 00:31:35,760
And this we can integrate.

635
00:31:35,760 --> 00:31:38,090
The way to integrate is
to put all the a's on one

636
00:31:38,090 --> 00:31:42,510
side and all of the
dt's on the other side.

637
00:31:42,510 --> 00:31:49,230
So we get ada,
writing this as da/dt.

638
00:31:49,230 --> 00:31:54,890
So ada is equal to
the constant times dt.

639
00:31:57,590 --> 00:31:59,090
And then, as we've
done before, when

640
00:31:59,090 --> 00:32:01,590
we're talking about matter,
it's the same calculation there.

641
00:32:01,590 --> 00:32:03,480
I just did different
power of a appears

642
00:32:03,480 --> 00:32:05,550
so we'd know how to do it.

643
00:32:05,550 --> 00:32:17,420
Integrating, we get 1/2a squared
is equal to the constant times

644
00:32:17,420 --> 00:32:24,120
t, and then plus a new constant
of integration, constant prime.

645
00:32:28,297 --> 00:32:30,630
Now we make the same argument
as we've made in the past.

646
00:32:30,630 --> 00:32:32,530
We have not yet
said anything that

647
00:32:32,530 --> 00:32:35,930
determines how our clocks
are going to be set.

648
00:32:35,930 --> 00:32:38,709
So we can choose to set our
clocks in the standard way,

649
00:32:38,709 --> 00:32:40,500
which is to set our
clocks so that t equals

650
00:32:40,500 --> 00:32:44,050
0 corresponds to the moment
where a is equal to 0.

651
00:32:44,050 --> 00:32:47,170
And if a is going to be equal
to 0 when t is equal to 0,

652
00:32:47,170 --> 00:32:50,009
it means constant prime
is going to be equal to 0.

653
00:32:50,009 --> 00:32:51,800
So by choosing the
value of constant prime,

654
00:32:51,800 --> 00:32:54,050
we really are just
determining how

655
00:32:54,050 --> 00:32:55,730
we're going to set
our clocks, how

656
00:32:55,730 --> 00:32:57,680
we're going to
choose the 0 of time.

657
00:32:57,680 --> 00:33:03,520
And we'll do that by setting
constant prime equal to 0.

658
00:33:03,520 --> 00:33:06,020
And then we get
the famous formula

659
00:33:06,020 --> 00:33:09,020
for a radiation-dominated
universe, a of t

660
00:33:09,020 --> 00:33:10,910
is just proportional to
the square root of t,

661
00:33:10,910 --> 00:33:13,570
or t to the 1/2 power.

662
00:33:13,570 --> 00:33:16,160
And this is for a
radiation-dominated flat

663
00:33:16,160 --> 00:33:28,589
universe, replacing
the t to the 2/3

664
00:33:28,589 --> 00:33:30,755
that we have for the
matter-dominated flat universe.

665
00:33:34,060 --> 00:33:36,034
Once we know that
a is proportional

666
00:33:36,034 --> 00:33:38,200
to the square root of t--
and for the flat universe,

667
00:33:38,200 --> 00:33:40,450
the constant proportionality
mean nothing, by the way.

668
00:33:40,450 --> 00:33:42,429
It's not that we haven't
been smart enough

669
00:33:42,429 --> 00:33:43,470
to figure out what it is.

670
00:33:43,470 --> 00:33:45,217
It really has no
meaning whatever.

671
00:33:45,217 --> 00:33:47,050
You could set it equal
to whatever you want,

672
00:33:47,050 --> 00:33:49,700
and it just determines
your definition

673
00:33:49,700 --> 00:33:51,980
of the notch, your
definition of how you're

674
00:33:51,980 --> 00:33:55,220
going to measure the
comoving coordinate system.

675
00:33:55,220 --> 00:33:58,390
Once we know this, we should
know pretty much everything.

676
00:33:58,390 --> 00:34:00,820
So in particular,
we can calculate

677
00:34:00,820 --> 00:34:03,976
h, which a-dot over a.

678
00:34:03,976 --> 00:34:05,660
And the constant
proportionality drops

679
00:34:05,660 --> 00:34:07,750
when we compute a-dot over a.

680
00:34:07,750 --> 00:34:09,949
As we expect, it has no meaning.

681
00:34:09,949 --> 00:34:12,415
So it should not appear in
the equation for anything

682
00:34:12,415 --> 00:34:14,889
that does have physical meaning.

683
00:34:14,889 --> 00:34:22,880
So H is just 1/2t,
the 1/2 here coming

684
00:34:22,880 --> 00:34:24,714
from differentiating
the t the 1/2 power.

685
00:34:27,600 --> 00:34:31,940
We can also compute
the horizon distance.

686
00:34:31,940 --> 00:34:37,520
So the physical horizon
distance, l sub p horizon,

687
00:34:37,520 --> 00:34:45,100
where p stands for physical,
is equal to the scale factor

688
00:34:45,100 --> 00:34:47,710
times the coordinate
horizon distance.

689
00:34:47,710 --> 00:34:49,469
And the coordinate
horizon distance

690
00:34:49,469 --> 00:34:51,400
is just the total
coordinate distance

691
00:34:51,400 --> 00:34:54,047
that light could travel from
the beginning of the universe.

692
00:34:54,047 --> 00:34:55,880
And we know the coordinate
velocity of light

693
00:34:55,880 --> 00:34:58,750
is c divided by a.

694
00:34:58,750 --> 00:35:02,300
So we just integrate that to get
the total coordinate distance.

695
00:35:02,300 --> 00:35:08,610
So it's the integral
from 0 to t of c

696
00:35:08,610 --> 00:35:13,005
over a of t prime dt prime.

697
00:35:16,338 --> 00:35:18,870
And since a of t is
just t to the 1/2,

698
00:35:18,870 --> 00:35:21,230
this is a trivial
integral to do.

699
00:35:21,230 --> 00:35:26,690
And the answer is
2 times c times t.

700
00:35:26,690 --> 00:35:28,550
So in a radiation-dominated
universe,

701
00:35:28,550 --> 00:35:32,720
the horizon's distance
is twice c times t.

702
00:35:32,720 --> 00:35:35,100
For a static universe,
the horizon distance

703
00:35:35,100 --> 00:35:36,541
would just be c times t.

704
00:35:36,541 --> 00:35:38,915
It would just be the distance
light can travel in time t,

705
00:35:38,915 --> 00:35:41,920
but more complicated in
an expanding universe.

706
00:35:41,920 --> 00:35:44,110
For the matter-dominated
case, we

707
00:35:44,110 --> 00:35:46,580
discovered that the
horizon distance was 3ct,

708
00:35:46,580 --> 00:35:47,859
if you remember.

709
00:35:47,859 --> 00:35:49,650
For the radiation-dominated
case, it's 2ct.

710
00:35:52,950 --> 00:35:56,200
And finally, an
important equation

711
00:35:56,200 --> 00:36:03,510
is that, going back to
here, where we started,

712
00:36:03,510 --> 00:36:06,300
this equation relates H to rho.

713
00:36:06,300 --> 00:36:08,530
We found out that,
merely by knowing

714
00:36:08,530 --> 00:36:10,700
the universe is
radiation-dominated,

715
00:36:10,700 --> 00:36:12,915
without even caring about
what kind of radiation

716
00:36:12,915 --> 00:36:16,090
it is, how much neutrinos,
how much photons, whatever--

717
00:36:16,090 --> 00:36:19,590
doesn't matter-- merely
by knowing the universe

718
00:36:19,590 --> 00:36:23,820
is radiation-dominated, we were
able to tell that H is 1/2t.

719
00:36:26,362 --> 00:36:28,320
And if we know what H
is, that formula tells us

720
00:36:28,320 --> 00:36:30,620
we also know what rho is.

721
00:36:30,620 --> 00:36:33,160
So without even knowing
what kind of radiation

722
00:36:33,160 --> 00:36:36,690
is contributing, we know that,
for a radiation-dominated

723
00:36:36,690 --> 00:36:44,080
universe, rho is just
equal to 3 over 32

724
00:36:44,080 --> 00:36:50,185
pi Newton's constant G times
time, little t, squared.

725
00:36:55,459 --> 00:36:57,750
It's rather amazing that we
can write down that formula

726
00:36:57,750 --> 00:37:01,500
without even knowing what kind
of radiation is contributing.

727
00:37:01,500 --> 00:37:03,180
But as long as that
radiation falls

728
00:37:03,180 --> 00:37:08,310
off as 1 over the scale
factor to the fourth,

729
00:37:08,310 --> 00:37:11,280
and as long as we know
the universe is flat,

730
00:37:11,280 --> 00:37:14,556
then we know what that
energy density has to be.

731
00:37:14,556 --> 00:37:15,930
This is crucial
here, by the way.

732
00:37:15,930 --> 00:37:17,580
The energy could be
anything if we did not

733
00:37:17,580 --> 00:37:18,996
assume that the
universe was flat.

734
00:37:21,420 --> 00:37:25,010
OK, any questions about this?

735
00:37:25,010 --> 00:37:25,958
Yes?

736
00:37:25,958 --> 00:37:28,328
AUDIENCE: If we assumed
that it was almost flat,

737
00:37:28,328 --> 00:37:31,112
would we be able to
have any bounds on it?

738
00:37:31,112 --> 00:37:32,820
PROFESSOR: OK, question
is, if we assumed

739
00:37:32,820 --> 00:37:34,195
that it was almost
flat, would we

740
00:37:34,195 --> 00:37:35,800
be able to have
any bounds on it?

741
00:37:35,800 --> 00:37:37,674
The answer is, yeah, if
you were quantitative

742
00:37:37,674 --> 00:37:39,580
about what you meant
by "almost flat,"

743
00:37:39,580 --> 00:37:42,780
you could know how almost true
that formula would have to be.

744
00:37:50,840 --> 00:37:52,350
OK, if there are
no other questions,

745
00:37:52,350 --> 00:37:54,560
I want to switch
gears slightly now

746
00:37:54,560 --> 00:37:59,690
and go back to talk about some
of the basic underlying physics

747
00:37:59,690 --> 00:38:04,240
that we are going to need,
and in particular, the physics

748
00:38:04,240 --> 00:38:05,260
of black-body radiation.

749
00:38:16,390 --> 00:38:19,560
So this is really just a little
chapter of a stat mech course

750
00:38:19,560 --> 00:38:21,900
that we're inserting
here, because we need it.

751
00:38:21,900 --> 00:38:23,761
And because it comes
from another course,

752
00:38:23,761 --> 00:38:25,760
we're not going try to
do it in complete detail.

753
00:38:25,760 --> 00:38:28,780
But I'll try to write down
formulas that make sense.

754
00:38:28,780 --> 00:38:32,270
And that will give us what
we need to know to proceed.

755
00:38:32,270 --> 00:38:35,110
So that will be the goal.

756
00:38:35,110 --> 00:38:38,310
So what is black-body radiation?

757
00:38:38,310 --> 00:38:46,330
The physical phenomenon
is that, if one imagines

758
00:38:46,330 --> 00:38:49,012
a box with a cavity
in it-- that's

759
00:38:49,012 --> 00:38:50,720
supposed to be a box
with a cavity in it,

760
00:38:50,720 --> 00:38:55,600
in case you can't recognize the
picture-- if the box is held

761
00:38:55,600 --> 00:39:00,480
at some uniform temperature
t-- t is temperature--

762
00:39:00,480 --> 00:39:05,460
then is claimed and
verified experimentally

763
00:39:05,460 --> 00:39:10,360
that the cavity will fill up
with radiation-- in this case,

764
00:39:10,360 --> 00:39:13,580
we're really just talking about
electromagnetic radiation--

765
00:39:13,580 --> 00:39:16,550
the calving will fill up with
electromagnetic radiation

766
00:39:16,550 --> 00:39:18,720
whose characteristics
would be determined solely

767
00:39:18,720 --> 00:39:21,780
by that temperature
t and will therefore

768
00:39:21,780 --> 00:39:27,270
be totally independent of the
material that makes up the box.

769
00:39:27,270 --> 00:39:29,970
Roughly speaking,
I think the way

770
00:39:29,970 --> 00:39:33,560
to think about it is to say
that the box will fill up

771
00:39:33,560 --> 00:39:36,060
with radiation at temperature t.

772
00:39:36,060 --> 00:39:38,920
And saying that the
radiation has temperature t

773
00:39:38,920 --> 00:39:41,680
is enough to completely
describe the radiation.

774
00:39:41,680 --> 00:39:43,710
It doesn't matter
what kind of a box

775
00:39:43,710 --> 00:39:46,630
that radiation is sitting in.

776
00:39:46,630 --> 00:39:58,840
So the box will fill with
radiation at temperature t.

777
00:40:09,160 --> 00:40:11,380
And that radiation is
called black-body radiation.

778
00:40:24,580 --> 00:40:27,350
Like many things in physics,
it has a variety of names,

779
00:40:27,350 --> 00:40:29,419
just to confuse us all.

780
00:40:29,419 --> 00:40:30,960
So it's also called
cavity radiation,

781
00:40:30,960 --> 00:40:32,334
which makes a lot
of sense, given

782
00:40:32,334 --> 00:40:34,570
the description we just gave.

783
00:40:34,570 --> 00:40:36,930
And it's also sometimes called
just thermal equilibrium

784
00:40:36,930 --> 00:40:37,920
radiation.

785
00:40:37,920 --> 00:40:40,180
This is radiation
at temperature t.

786
00:40:42,921 --> 00:40:44,920
I haven't really justified
the word "black-body"

787
00:40:44,920 --> 00:40:48,000
radiation yet, so let me
try to do that quickly.

788
00:40:48,000 --> 00:40:50,980
The reason why it can be called
black-body radiation-- and this

789
00:40:50,980 --> 00:40:53,447
will be important for
some things in cosmology;

790
00:40:53,447 --> 00:40:55,530
I'm not sure if it will
be important to us or not,

791
00:40:55,530 --> 00:40:57,940
but certainly important to
know-- the reason why it's

792
00:40:57,940 --> 00:41:01,010
called black-body radiation is
because we imagine inserting

793
00:41:01,010 --> 00:41:10,990
into this cavity a black
body, in the literal sense.

794
00:41:10,990 --> 00:41:13,500
What is the literal
sense of a black body?

795
00:41:13,500 --> 00:41:15,850
It's a body which is
black in the sense

796
00:41:15,850 --> 00:41:19,980
that all radiation that
hits it is absorbed.

797
00:41:19,980 --> 00:41:22,050
Now, this black body
is still going to glow.

798
00:41:22,050 --> 00:41:23,900
If you heat a piece
of iron or something

799
00:41:23,900 --> 00:41:26,220
to very high temperatures,
you see it glow.

800
00:41:26,220 --> 00:41:27,910
That glow is not reflection.

801
00:41:27,910 --> 00:41:31,240
That glow is emission
by the hot atoms

802
00:41:31,240 --> 00:41:33,860
in the piece of
iron or whatever.

803
00:41:33,860 --> 00:41:36,630
Emission is different
from reflection.

804
00:41:36,630 --> 00:41:38,580
When we say it
absorbs everything,

805
00:41:38,580 --> 00:41:40,610
we mean it does not
reflect anything.

806
00:41:40,610 --> 00:41:45,950
But it will still admit
by thermal de-excitation.

807
00:41:45,950 --> 00:41:53,770
The crucial distinction between
reflection and thermal emission

808
00:41:53,770 --> 00:41:55,694
is that reflection
is instantaneous.

809
00:41:55,694 --> 00:41:57,610
When a light beam comes
in, if it's reflected,

810
00:41:57,610 --> 00:42:01,290
it just goes back
out instantaneously.

811
00:42:01,290 --> 00:42:05,080
Emission, thermal emission,
is a slower process.

812
00:42:05,080 --> 00:42:06,810
Atoms get excited,
and eventually, they

813
00:42:06,810 --> 00:42:09,950
de-excite and emit radiation.

814
00:42:09,950 --> 00:42:10,896
So it takes time.

815
00:42:10,896 --> 00:42:12,020
And that's the distinction.

816
00:42:12,020 --> 00:42:15,140
We're going to assume that
this body is black in the sense

817
00:42:15,140 --> 00:42:16,265
that there's no reflection.

818
00:42:29,349 --> 00:42:31,140
OK, now we're going to
make use of the fact

819
00:42:31,140 --> 00:42:33,800
that we know that thermal
equilibrium works.

820
00:42:33,800 --> 00:42:37,850
That is, if you let any
isolated system sit long enough,

821
00:42:37,850 --> 00:42:41,850
it will approach a unique
state of thermal equilibrium

822
00:42:41,850 --> 00:42:44,120
determined by its constituents,
which you've put in

823
00:42:44,120 --> 00:42:50,350
to begin with, but otherwise
independent of how exactly

824
00:42:50,350 --> 00:42:53,680
you arrange those constituents.

825
00:42:53,680 --> 00:42:57,520
So if we put in, for
example, a cold black body,

826
00:42:57,520 --> 00:42:59,440
it will start to get
harder, warming up

827
00:42:59,440 --> 00:43:01,300
to the same temperature
as everything else.

828
00:43:01,300 --> 00:43:03,510
If we put in an
extra hot black body,

829
00:43:03,510 --> 00:43:06,740
it would emit energy
and start to cool down

830
00:43:06,740 --> 00:43:08,850
to the temperature
of everything else.

831
00:43:08,850 --> 00:43:11,080
But eventually,
this black body will

832
00:43:11,080 --> 00:43:13,220
be at the same temperature
as everything else.

833
00:43:13,220 --> 00:43:16,240
And we're going to be assuming
here that the box itself

834
00:43:16,240 --> 00:43:19,610
is being held at some
fixed temperature t.

835
00:43:19,610 --> 00:43:22,342
So wherever energy
exchange occurs

836
00:43:22,342 --> 00:43:23,800
because of this
black body, it will

837
00:43:23,800 --> 00:43:25,860
be absorbed by
whatever is holding

838
00:43:25,860 --> 00:43:29,070
the outer box at the
fixed temperature.

839
00:43:29,070 --> 00:43:30,900
So in the end, if
we wait long enough,

840
00:43:30,900 --> 00:43:33,480
this black body is going to
acquire the same temperature

841
00:43:33,480 --> 00:43:37,117
as everything else and
hold that temperature.

842
00:43:37,117 --> 00:43:38,700
Now, if it's holding
that temperature,

843
00:43:38,700 --> 00:43:42,977
it means that the energy input
to the box, to the black body,

844
00:43:42,977 --> 00:43:44,560
will have to be the
same as the energy

845
00:43:44,560 --> 00:43:47,444
output of the black body.

846
00:43:47,444 --> 00:43:49,735
Now, the black body is going
to be absorbing radiation,

847
00:43:49,735 --> 00:43:51,318
because we have
radiation here, and we

848
00:43:51,318 --> 00:43:53,600
said that any radiation
that hits it is absorbed.

849
00:43:53,600 --> 00:43:56,380
That was the
definition of "black."

850
00:43:56,380 --> 00:43:59,250
So it's clearly
absorbing energy.

851
00:43:59,250 --> 00:44:00,850
If it's not going
to be heating up--

852
00:44:00,850 --> 00:44:03,610
and we know that it's
not, because it's

853
00:44:03,610 --> 00:44:06,754
in thermal equilibrium; the
temperature will remain fixed--

854
00:44:06,754 --> 00:44:08,170
in order for it
to not heat up, it

855
00:44:08,170 --> 00:44:10,870
has to radiate energy, as well.

856
00:44:10,870 --> 00:44:13,220
And the energy it radiates
has to be exactly the same

857
00:44:13,220 --> 00:44:14,749
as the energy it's
absorbing once it

858
00:44:14,749 --> 00:44:15,915
reaches thermal equilibrium.

859
00:44:21,850 --> 00:44:46,158
So in equilibrium, the black
body, BB, radiates at same rate

860
00:44:46,158 --> 00:44:47,785
that it absorbs energy.

861
00:44:59,000 --> 00:45:01,810
This radiation process
is this slow process

862
00:45:01,810 --> 00:45:03,300
of thermal emission.

863
00:45:03,300 --> 00:45:06,920
There are atoms inside this
black body that are excited.

864
00:45:06,920 --> 00:45:08,870
Those atoms will
de-excite over time,

865
00:45:08,870 --> 00:45:12,500
releasing photons
that will go off.

866
00:45:12,500 --> 00:45:16,980
And the important thing
about that slow mechanism

867
00:45:16,980 --> 00:45:20,390
is that, if we imagine
taking this black body out

868
00:45:20,390 --> 00:45:24,929
of its cavity, but not waiting
long enough for its temperature

869
00:45:24,929 --> 00:45:26,720
to change-- so we'll
assume its temperature

870
00:45:26,720 --> 00:45:27,880
is still the same, t.

871
00:45:34,970 --> 00:45:42,655
So this is a picture of the same
black body at temperature t,

872
00:45:42,655 --> 00:45:44,310
but now outside the cavity.

873
00:45:51,786 --> 00:45:54,160
Its radiation rate is not
going to change when we take it

874
00:45:54,160 --> 00:45:56,743
outside the cavity, because the
radiation was caused by things

875
00:45:56,743 --> 00:46:00,250
happening inside the black body,
which are not changed when we

876
00:46:00,250 --> 00:46:02,030
put it in or out of the cavity.

877
00:46:02,030 --> 00:46:04,620
So it will continue to radiate
at exactly the same rate

878
00:46:04,620 --> 00:46:06,740
that it was radiating
when it was in the cavity.

879
00:46:06,740 --> 00:46:08,240
And that means it's
going to radiate

880
00:46:08,240 --> 00:46:11,720
at exactly the same
rate as the energy

881
00:46:11,720 --> 00:46:16,690
that it would have
absorbed if it were bathed

882
00:46:16,690 --> 00:46:19,510
by this black-body radiation.

883
00:46:19,510 --> 00:46:22,420
So essentially, it means it
will emit black-body radiation

884
00:46:22,420 --> 00:46:25,780
with exactly the intensity that
the black-body radiation would

885
00:46:25,780 --> 00:46:29,285
have on the outside if
the black body were still

886
00:46:29,285 --> 00:46:30,035
inside the cavity.

887
00:46:44,150 --> 00:46:46,980
So it radiates with exactly the
same intensity as the energy

888
00:46:46,980 --> 00:46:51,220
that it would receive if
it were inside the cavity.

889
00:46:51,220 --> 00:46:54,940
And furthermore, you
could even elaborate a bit

890
00:46:54,940 --> 00:46:57,780
on his argument to show that
the radiation that it radiates

891
00:46:57,780 --> 00:47:01,340
has exactly the same spectrum,
exactly the same decomposition

892
00:47:01,340 --> 00:47:04,320
into wavelengths, as
the black-body radiation

893
00:47:04,320 --> 00:47:05,970
inside the cavity.

894
00:47:05,970 --> 00:47:07,840
And the way to see
that is to imagine

895
00:47:07,840 --> 00:47:15,700
surrounding this black body by
absorption filters that only

896
00:47:15,700 --> 00:47:17,369
let through certain frequencies.

897
00:47:17,369 --> 00:47:19,410
And the point is that, no
matter what frequencies

898
00:47:19,410 --> 00:47:22,747
you limit going
through this filter,

899
00:47:22,747 --> 00:47:24,080
you have to stay in equilibrium.

900
00:47:24,080 --> 00:47:25,880
It will never get
hotter or colder.

901
00:47:25,880 --> 00:47:28,640
So that means that each
frequency by itself

902
00:47:28,640 --> 00:47:33,000
has to balance, has to have
exactly the same emission

903
00:47:33,000 --> 00:47:36,180
as it would have abortion
if the black body were just

904
00:47:36,180 --> 00:47:39,340
exposed to black-body
radiation surrounding it.

905
00:47:42,590 --> 00:47:45,410
So it radiates
black-body radiation.

906
00:47:45,410 --> 00:47:59,500
And the intensity
and spectrum must

907
00:47:59,500 --> 00:48:04,015
match what we call black-body
or cavity radiation.

908
00:48:10,530 --> 00:48:13,000
So the cavity radiation
has to exactly mimic

909
00:48:13,000 --> 00:48:14,754
the radiation emitted
by this black body.

910
00:48:14,754 --> 00:48:16,420
And that's the
motivation for calling it

911
00:48:16,420 --> 00:48:17,295
black-body radiation.

912
00:48:20,260 --> 00:48:22,840
Now, if this black body
absorbed some radiation

913
00:48:22,840 --> 00:48:25,766
and reflected some, then it
would emit different radiation.

914
00:48:25,766 --> 00:48:27,140
So it is important
that this body

915
00:48:27,140 --> 00:48:30,320
be black, in the sense that
it doesn't reflect anything.

916
00:48:30,320 --> 00:48:33,306
All radiation hitting
it is absorbed.

917
00:48:33,306 --> 00:48:34,680
And only under
that assumption do

918
00:48:34,680 --> 00:48:37,970
we know exactly what
it's going to emit.

919
00:48:37,970 --> 00:48:38,470
Yes?

920
00:48:38,470 --> 00:48:40,136
AUDIENCE: So is this
only true for right

921
00:48:40,136 --> 00:48:41,470
after you take it out?

922
00:48:41,470 --> 00:48:43,150
PROFESSOR: Well, it will start
to cool after you take it out.

923
00:48:43,150 --> 00:48:45,350
And as it cools, its
temperature will change.

924
00:48:45,350 --> 00:48:47,350
But if you account for
the changing temperature,

925
00:48:47,350 --> 00:48:49,187
it will be true at
anytime, actually.

926
00:48:49,187 --> 00:48:50,520
But the temperature will change.

927
00:48:50,520 --> 00:48:52,144
AUDIENCE: Because,
in the black cavity,

928
00:48:52,144 --> 00:48:53,558
it has things exciting it.

929
00:48:53,558 --> 00:48:54,932
And when you take
it out, there's

930
00:48:54,932 --> 00:48:58,708
no photons, no constant
radiation to excite it.

931
00:48:58,708 --> 00:49:00,925
So it can radiate--

932
00:49:00,925 --> 00:49:01,550
PROFESSOR: Yes.

933
00:49:01,550 --> 00:49:03,810
Once you take it out, it's
no longer being excited.

934
00:49:03,810 --> 00:49:07,327
And I think, technically,
you're right.

935
00:49:07,327 --> 00:49:09,160
Once you take it out,
it will not only cool,

936
00:49:09,160 --> 00:49:12,960
but it will cease to be
at a uniform temperature.

937
00:49:12,960 --> 00:49:14,840
And that's basically
what we're saying

938
00:49:14,840 --> 00:49:17,070
if we're saying that the
atoms that are excited

939
00:49:17,070 --> 00:49:19,570
won't necessarily be in the
right thermal distribution

940
00:49:19,570 --> 00:49:21,460
as they would be if
it was on the inside.

941
00:49:21,460 --> 00:49:23,090
That would be a
statement that is not

942
00:49:23,090 --> 00:49:25,120
any longer in
thermal equilibrium.

943
00:49:25,120 --> 00:49:27,020
But as long as the
radiation is slow,

944
00:49:27,020 --> 00:49:29,186
you could just account for
the changing temperature.

945
00:49:29,186 --> 00:49:30,646
You would know how it radiates.

946
00:49:30,646 --> 00:49:32,520
And I think that's a
very good approximation.

947
00:49:32,520 --> 00:49:34,285
Although in principle,
it will cease

948
00:49:34,285 --> 00:49:36,660
to be in thermal equilibrium,
as soon as you take it out,

949
00:49:36,660 --> 00:49:38,785
the edges will be cool,
and the center will be hot.

950
00:49:38,785 --> 00:49:41,370
And you'd have to take into
account all of those things

951
00:49:41,370 --> 00:49:43,078
to be able to understand
how it radiates.

952
00:49:47,107 --> 00:49:47,940
Any other questions?

953
00:49:54,209 --> 00:49:55,750
OK, next, I want to
talk a little bit

954
00:49:55,750 --> 00:49:59,870
about what this
black-body radiation is.

955
00:49:59,870 --> 00:50:04,140
And one can begin by
trying to understand

956
00:50:04,140 --> 00:50:06,550
it purely classically,
which, of course,

957
00:50:06,550 --> 00:50:08,160
is what happened historically.

958
00:50:08,160 --> 00:50:12,940
In the 1800s, people tried to
understand cavity radiation

959
00:50:12,940 --> 00:50:16,090
or black-body radiation using
classical physics, Maxwell's

960
00:50:16,090 --> 00:50:19,680
equations, to describe
the radiation.

961
00:50:19,680 --> 00:50:21,980
And then, in a
nutshell-- we're just

962
00:50:21,980 --> 00:50:25,090
trying to establish
basic ideas here--

963
00:50:25,090 --> 00:50:31,630
one can try to treat a field
statistical-mechanically

964
00:50:31,630 --> 00:50:34,760
by imagining not
fields in empty space,

965
00:50:34,760 --> 00:50:37,160
but fields in some
kind of a box.

966
00:50:37,160 --> 00:50:38,660
In this case, it
doesn't necessarily

967
00:50:38,660 --> 00:50:40,618
have to be the cavity
that we're talking about.

968
00:50:40,618 --> 00:50:44,200
It could be a big box that just
enclosed the system somehow

969
00:50:44,200 --> 00:50:45,756
to make it easier to talk about.

970
00:50:45,756 --> 00:50:47,130
And in the end,
you could imagine

971
00:50:47,130 --> 00:50:51,060
taking that box to infinity,
this theoretical box

972
00:50:51,060 --> 00:50:53,280
that you use to
simplify the problem.

973
00:50:53,280 --> 00:50:55,160
But once you put
the system in a box,

974
00:50:55,160 --> 00:50:58,070
then a field, like the
electromagnetic field,

975
00:50:58,070 --> 00:51:01,000
can always be broken
up into normal modes,

976
00:51:01,000 --> 00:51:04,570
standing wave patterns that have
an integer or a half integer

977
00:51:04,570 --> 00:51:07,650
number of wavelengths
inside the box.

978
00:51:07,650 --> 00:51:10,680
And no matter how complicated
the field is inside the box,

979
00:51:10,680 --> 00:51:14,290
you could always describe
it as a superposition

980
00:51:14,290 --> 00:51:17,726
of some set of standing waves.

981
00:51:17,726 --> 00:51:19,350
In general, it takes
an infinite number

982
00:51:19,350 --> 00:51:21,690
of standing wave components
to describe an arbitrary

983
00:51:21,690 --> 00:51:26,690
field-- that is, with shorter
and shorter wavelengths.

984
00:51:26,690 --> 00:51:29,960
But you can always-- and
this is Fourier's theorem--

985
00:51:29,960 --> 00:51:32,000
you can always
describe an arbitrary

986
00:51:32,000 --> 00:51:36,089
field in terms of
the standing waves.

987
00:51:36,089 --> 00:51:37,630
And that's good for
the point of view

988
00:51:37,630 --> 00:51:39,331
of thinking about
statistical mechanics,

989
00:51:39,331 --> 00:51:41,330
because you could think
about each standing wave

990
00:51:41,330 --> 00:51:43,700
almost as if it were a particle.

991
00:51:43,700 --> 00:51:45,240
It really is a
harmonic oscillator.

992
00:51:45,240 --> 00:51:47,900
So if you think you know
the statistical mechanics

993
00:51:47,900 --> 00:51:50,670
of harmonic oscillators,
each standing wave in the box

994
00:51:50,670 --> 00:51:53,990
is just a harmonic
oscillator, so simple.

995
00:51:53,990 --> 00:51:56,360
We now try to ask what
is the thermodynamics

996
00:51:56,360 --> 00:52:01,400
of this system of
harmonic oscillators.

997
00:52:01,400 --> 00:52:05,480
And the rule for harmonic
oscillator is simple.

998
00:52:05,480 --> 00:52:06,860
Stat mech tells
you that you have

999
00:52:06,860 --> 00:52:11,505
1/2 kT per degree of freedom
in thermal equilibrium.

1000
00:52:29,054 --> 00:52:30,470
The energy of a
system should just

1001
00:52:30,470 --> 00:52:35,995
be 1/2 kT per degree of freedom.

1002
00:52:42,774 --> 00:52:44,690
Having said that, all
the complicate questions

1003
00:52:44,690 --> 00:52:46,148
come about by asking
ourselves what

1004
00:52:46,148 --> 00:52:48,150
is meant by degree of freedom.

1005
00:52:48,150 --> 00:52:51,039
But for the harmonic oscillator,
that has a simple answer.

1006
00:52:51,039 --> 00:52:52,580
A harmonic oscillator
has two degrees

1007
00:52:52,580 --> 00:52:55,597
of freedom-- the kinetic energy
and the potential energy.

1008
00:52:55,597 --> 00:52:57,180
So the energy of a
harmonic oscillator

1009
00:52:57,180 --> 00:53:00,180
should just be kT per
degree of freedom.

1010
00:53:00,180 --> 00:53:03,680
And we could apply that to our
gas in the box and we could,

1011
00:53:03,680 --> 00:53:05,965
ask how much energy
should the gas absorb

1012
00:53:05,965 --> 00:53:08,309
at a given temperature?

1013
00:53:08,309 --> 00:53:10,600
What should be the energy
density of the gas at a given

1014
00:53:10,600 --> 00:53:13,330
temperature-- this
gas of photons.

1015
00:53:13,330 --> 00:53:19,710
But it was noticed in
the 1900-- the 1800s

1016
00:53:19,710 --> 00:53:24,560
that this doesn't work because
there's no limit to how short

1017
00:53:24,560 --> 00:53:26,610
the wavelengths can be.

1018
00:53:26,610 --> 00:53:29,050
And therefore, there's
not just some finite set

1019
00:53:29,050 --> 00:53:30,130
of harmonic oscillators.

1020
00:53:30,130 --> 00:53:31,630
There's an infinite
set of harmonic

1021
00:53:31,630 --> 00:53:35,240
oscillators where you have
more and more harmonic

1022
00:53:35,240 --> 00:53:37,870
oscillators at shorter
and shorter wavelengths

1023
00:53:37,870 --> 00:53:40,150
ad infinitum, no limit.

1024
00:53:40,150 --> 00:53:42,316
And that came to be
known as Jean's Paradox.

1025
00:54:18,100 --> 00:54:21,710
So what it suggests is that
if this classical stat mech

1026
00:54:21,710 --> 00:54:23,726
worked-- which obviously
it's not working.

1027
00:54:23,726 --> 00:54:25,100
But if it did
work, it would mean

1028
00:54:25,100 --> 00:54:29,810
that as you tried to put a gas
in just an empty box in contact

1029
00:54:29,810 --> 00:54:31,630
with something at a
fixed temperature,

1030
00:54:31,630 --> 00:54:35,210
the box would absorb more and
more energy without limit.

1031
00:54:35,210 --> 00:54:37,314
And ultimately, it
would presumably

1032
00:54:37,314 --> 00:54:38,855
cause the temperature
of the whatever

1033
00:54:38,855 --> 00:54:40,396
is trying to maintain
the temperature

1034
00:54:40,396 --> 00:54:43,510
to go to 0 as energy
gets siphoned off

1035
00:54:43,510 --> 00:54:46,410
to shorter and shorter
wavelengths of excitations.

1036
00:54:50,390 --> 00:54:52,520
That obviously isn't the
way the world behaves.

1037
00:54:52,520 --> 00:54:55,260
We'd all freeze to
death if it did.

1038
00:54:55,260 --> 00:54:56,880
So something has to
happen to save it.

1039
00:54:56,880 --> 00:54:59,050
And it wasn't at all
obvious for many years

1040
00:54:59,050 --> 00:55:00,446
what it was that saves it.

1041
00:55:00,446 --> 00:55:01,820
But this Jean's
Paradox turns out

1042
00:55:01,820 --> 00:55:03,194
to be saved by
quantum mechanics.

1043
00:55:19,297 --> 00:55:21,380
And the important implication
of quantum mechanics

1044
00:55:21,380 --> 00:55:31,870
is that the energy of
a harmonic oscillator

1045
00:55:31,870 --> 00:55:35,620
is no longer allowed to
have any possible value,

1046
00:55:35,620 --> 00:55:42,570
but is now quantized
as some integer times

1047
00:55:42,570 --> 00:55:45,650
h times nu, the
frequency-- h being

1048
00:55:45,650 --> 00:55:48,160
Planck's constant,
nu being frequency, n

1049
00:55:48,160 --> 00:55:53,080
being some integer where
this integer might be called

1050
00:55:53,080 --> 00:55:55,080
the excitation level of
the harmonic oscillator.

1051
00:56:04,190 --> 00:56:06,750
Depending how you choose your
0, you might have an n plus 1/2

1052
00:56:06,750 --> 00:56:07,830
there.

1053
00:56:07,830 --> 00:56:09,970
But that's not important
for us right now.

1054
00:56:09,970 --> 00:56:12,536
It will be important
later actually.

1055
00:56:12,536 --> 00:56:14,910
But for now, we'll just allow
ourselves to readjust the 0

1056
00:56:14,910 --> 00:56:19,202
and just think of it as n
times h nu, or h bar omega.

1057
00:56:19,202 --> 00:56:20,660
Now, this makes
all the difference,

1058
00:56:20,660 --> 00:56:22,140
statistical mechanically.

1059
00:56:22,140 --> 00:56:23,670
One can apply
statistical mechanics

1060
00:56:23,670 --> 00:56:25,230
using basically
the same principles

1061
00:56:25,230 --> 00:56:27,720
to the quantum
mechanical system.

1062
00:56:27,720 --> 00:56:30,850
And the key thing now is that
for the very short wavelengths,

1063
00:56:30,850 --> 00:56:32,850
which are the ones that
were giving us trouble--

1064
00:56:32,850 --> 00:56:35,050
the infinities came
to short wavelengths.

1065
00:56:35,050 --> 00:56:39,580
For the short wavelengths
where nu is high, h nu is high.

1066
00:56:39,580 --> 00:56:42,680
And it means that
there's a minimum ante

1067
00:56:42,680 --> 00:56:45,650
that you could put in to
excite those short wavelength

1068
00:56:45,650 --> 00:56:47,100
harmonic oscillators.

1069
00:56:47,100 --> 00:56:48,544
And it's a large number.

1070
00:56:48,544 --> 00:56:50,960
You either put in a large
amount of energy or none at all.

1071
00:56:50,960 --> 00:56:54,470
Quantum mechanics doesn't let
you do anything in between.

1072
00:56:54,470 --> 00:56:56,440
Now remember, the
classical mechanics answer

1073
00:56:56,440 --> 00:57:00,670
was that you have kT in
each harmonic oscillator.

1074
00:57:00,670 --> 00:57:02,960
And kT would be small
compared to h nu,

1075
00:57:02,960 --> 00:57:05,356
if we're talking about
a very short wavelength.

1076
00:57:05,356 --> 00:57:06,730
So the classic
answer is just not

1077
00:57:06,730 --> 00:57:08,970
allowed by quantum mechanics.

1078
00:57:08,970 --> 00:57:12,380
You either have to put in
nothing or an amount much, much

1079
00:57:12,380 --> 00:57:15,225
larger than the
classical answer.

1080
00:57:15,225 --> 00:57:17,850
And when you do the statistical
mechanics quantum mechanically,

1081
00:57:17,850 --> 00:57:19,710
which is not a big
deal really, you

1082
00:57:19,710 --> 00:57:23,090
find that when you're
confronted with that choice,

1083
00:57:23,090 --> 00:57:26,120
the most likely answer is
to put in no energy at all.

1084
00:57:26,120 --> 00:57:30,750
So quantum mechanics freezes out
these short wavelength modes.

1085
00:57:30,750 --> 00:57:33,330
And then the n produces
a finite energy density

1086
00:57:33,330 --> 00:57:36,660
for a gas of photons.

1087
00:57:36,660 --> 00:57:37,337
Yes?

1088
00:57:37,337 --> 00:57:39,170
AUDIENCE: Seems like
if you were to sum over

1089
00:57:39,170 --> 00:57:43,180
like all the possible
wave numbers, that-- well,

1090
00:57:43,180 --> 00:57:46,570
so the energy is inversely
related to wavelength, right?

1091
00:57:46,570 --> 00:57:49,390
So even if you quantize it,
like for large wavelengths,

1092
00:57:49,390 --> 00:57:53,324
isn't the sum still like a sum
of one over lambda wavelength,

1093
00:57:53,324 --> 00:57:56,320
with its derivative?

1094
00:57:56,320 --> 00:57:58,070
PROFESSOR: You're
saying, isn't there also

1095
00:57:58,070 --> 00:58:00,511
a divergence at the
large wavelength n?

1096
00:58:00,511 --> 00:58:01,927
AUDIENCE: Because
that sum doesn't

1097
00:58:01,927 --> 00:58:03,770
seem like it would work.

1098
00:58:03,770 --> 00:58:04,870
PROFESSOR: Right.

1099
00:58:04,870 --> 00:58:05,815
No, that's important.

1100
00:58:09,650 --> 00:58:11,850
The reason that's
not a problem is

1101
00:58:11,850 --> 00:58:15,217
that if you're talking
about the energy in a box,

1102
00:58:15,217 --> 00:58:17,050
the wavelength can't
be bigger than the box.

1103
00:58:17,050 --> 00:58:19,850
The largest possible
wavelength is twice the box so

1104
00:58:19,850 --> 00:58:22,167
that half a wavelength
fits in the box.

1105
00:58:22,167 --> 00:58:24,500
If you're talking about the
energy in the whole infinite

1106
00:58:24,500 --> 00:58:26,870
universe, then we expect
the answer to infinite.

1107
00:58:26,870 --> 00:58:28,160
And it is.

1108
00:58:28,160 --> 00:58:30,600
There's no problem with
having infinite total energy

1109
00:58:30,600 --> 00:58:32,240
if you want to have
a finite energy

1110
00:58:32,240 --> 00:58:35,010
density throughout
an infinite universe.

1111
00:58:35,010 --> 00:58:39,510
So the size of the box cuts
off the large wavelengths.

1112
00:58:39,510 --> 00:58:41,920
And quantum mechanics cuts
off the small wavelengths.

1113
00:58:41,920 --> 00:58:44,320
So in the end, one does
get a finite answer

1114
00:58:44,320 --> 00:58:47,600
for the energy density
of black-body radiation.

1115
00:58:47,600 --> 00:58:51,182
And that's crucial
for our survival,

1116
00:58:51,182 --> 00:58:52,890
crucial for the
existence of the universe

1117
00:58:52,890 --> 00:58:55,100
as we know it, and also
crucial for the calculations

1118
00:58:55,100 --> 00:58:56,058
that we're about to do.

1119
00:59:01,970 --> 00:59:13,350
OK, so when one does these
calculations initially

1120
00:59:13,350 --> 00:59:24,260
for photons only, what we'd
find is that the energy density

1121
00:59:24,260 --> 00:59:29,280
is equal to a fudge factor,
which I'm going to call g.

1122
00:59:29,280 --> 00:59:31,660
And you'll see later why I'm
introducing a fudge factor.

1123
00:59:31,660 --> 00:59:33,750
For now, g is just 2.

1124
00:59:33,750 --> 00:59:36,660
But later, we'll generalize the
application of this formula,

1125
00:59:36,660 --> 00:59:38,900
and g will have
different values.

1126
00:59:38,900 --> 00:59:40,810
But for now, we're
dealing with photons.

1127
00:59:40,810 --> 00:59:44,060
There's a factor of 2 there,
but I'm going to write 2 as g,

1128
00:59:44,060 --> 00:59:46,410
writing g equals 2 underneath.

1129
00:59:46,410 --> 00:59:49,190
And then the pi
squared over 30--

1130
00:59:49,190 --> 00:59:52,200
you can really
calculate this-- times

1131
00:59:52,200 --> 01:00:03,000
kT to the fourth power divided
by h bar c cubed, h bar

1132
01:00:03,000 --> 01:00:07,164
being Plank's constant
divided by 2 pi and little k

1133
01:00:07,164 --> 01:00:08,205
being Boltzmann constant.

1134
01:00:12,750 --> 01:00:15,950
So this is calculated just by
thinking of the gas in a box

1135
01:00:15,950 --> 01:00:19,110
as a lot of harmonic oscillators
and applying standard stat

1136
01:00:19,110 --> 01:00:20,970
mech to each
harmonic oscillator,

1137
01:00:20,970 --> 01:00:22,730
but you apply the quantum
mechanical version of the stat

1138
01:00:22,730 --> 01:00:24,105
mech to each
harmonic oscillator.

1139
01:00:31,710 --> 01:00:37,360
And you can also find, by doing
the same kind of analysis,

1140
01:00:37,360 --> 01:00:40,490
that the pressure is
1/3 the energy density,

1141
01:00:40,490 --> 01:00:43,010
which we also derived
earlier by different means.

1142
01:00:43,010 --> 01:00:46,290
And it's all consistent so you
get the same answer every time,

1143
01:00:46,290 --> 01:00:48,370
even if you think
about it differently.

1144
01:00:48,370 --> 01:00:51,480
So here, I have mine deriving
it directly from the stat mech.

1145
01:00:55,140 --> 01:00:57,400
You can also, from
the stat mech,

1146
01:00:57,400 --> 01:01:01,120
calculate the number density of
photons in thermal equilibrium.

1147
01:01:17,230 --> 01:01:21,452
And that will be equal to --
again, there's a factor of 2.

1148
01:01:21,452 --> 01:01:22,910
But this time, I'll
call the factor

1149
01:01:22,910 --> 01:01:27,750
of 2 g star, where g star
also equals 2 for photons.

1150
01:01:31,177 --> 01:01:32,760
But when we generalize
these formulas,

1151
01:01:32,760 --> 01:01:34,470
g will not necessarily
equal g star,

1152
01:01:34,470 --> 01:01:37,340
which is why I'm
giving it two names.

1153
01:01:37,340 --> 01:01:44,360
And this g star
multiplies zeta of 3,

1154
01:01:44,360 --> 01:01:46,660
where zeta refers to the
Riemann zeta function, which

1155
01:01:46,660 --> 01:01:54,000
I'll define in a second,
divided by pi squared times

1156
01:01:54,000 --> 01:02:01,145
kT cubed divided
by h bar c cubed.

1157
01:02:08,890 --> 01:02:09,390
OK.

1158
01:02:35,180 --> 01:02:39,080
OK, so I need to
define this zeta of 3.

1159
01:02:39,080 --> 01:02:50,470
It's 1 over 1 cubed plus 1 over
2 cubed plus 1 over 3 cubed

1160
01:02:50,470 --> 01:02:51,610
plus dot dot dot.

1161
01:02:51,610 --> 01:02:53,460
It's an infinite series.

1162
01:02:53,460 --> 01:02:56,690
And if you sum up
that infinite series,

1163
01:02:56,690 --> 01:03:07,144
at least to three decimal
places, it's 1.202.

1164
01:03:07,144 --> 01:03:08,560
OK, then there's
one other formula

1165
01:03:08,560 --> 01:03:10,310
that will be of interest to us.

1166
01:03:10,310 --> 01:03:12,545
And that'll be a formula
for the entropy density.

1167
01:03:25,017 --> 01:03:26,600
Now, if you've had
a stat mech course,

1168
01:03:26,600 --> 01:03:29,030
you have some idea of what
entropy density means.

1169
01:03:29,030 --> 01:03:34,150
If you have not, suffice
it to say for this class

1170
01:03:34,150 --> 01:03:37,900
that it is some measure of
the disorder in the sense

1171
01:03:37,900 --> 01:03:40,430
of the total number of
different quantum states

1172
01:03:40,430 --> 01:03:42,965
that contribute to a given
macroscopic description.

1173
01:03:46,300 --> 01:03:48,140
The more different
microstates there

1174
01:03:48,140 --> 01:03:51,530
are that contribute to a
macroscopic description,

1175
01:03:51,530 --> 01:03:53,659
the higher the entropy.

1176
01:03:53,659 --> 01:03:55,700
And the other important
thing about entropy to us

1177
01:03:55,700 --> 01:03:58,950
besides that vague definition--
which will be enough--

1178
01:03:58,950 --> 01:04:00,450
but the important
thing for us is

1179
01:04:00,450 --> 01:04:04,100
that under most circumstances,
entropy will be conserved.

1180
01:04:04,100 --> 01:04:05,930
It's conserved as
long as things stay

1181
01:04:05,930 --> 01:04:08,197
at or near thermal equilibrium.

1182
01:04:08,197 --> 01:04:10,280
And in the early universe
as the universe expands,

1183
01:04:10,280 --> 01:04:12,120
that's the case.

1184
01:04:12,120 --> 01:04:14,320
So for us, the
entropy of our gas

1185
01:04:14,320 --> 01:04:16,240
will simply be a
conserved quantity

1186
01:04:16,240 --> 01:04:17,440
that we can make use of.

1187
01:04:17,440 --> 01:04:20,246
And we will make use of
it in some important ways.

1188
01:04:20,246 --> 01:04:21,620
And we could write
down a formula

1189
01:04:21,620 --> 01:04:24,940
for the entropy
density of photons.

1190
01:04:24,940 --> 01:04:28,220
And it's g, where this g in
fact the same g as over there.

1191
01:04:28,220 --> 01:04:29,940
It is related to the energy.

1192
01:04:29,940 --> 01:04:33,050
So it's the same g that
appears in two cases, 2

1193
01:04:33,050 --> 01:04:35,626
in both cases for
protons by themselves.

1194
01:04:35,626 --> 01:04:37,750
And then there are factors
that you can calculate--

1195
01:04:37,750 --> 01:04:50,000
2 pi squared over 45 times k to
the fourth T cubed over h bar

1196
01:04:50,000 --> 01:04:52,910
c cubed.

1197
01:04:52,910 --> 01:04:55,341
OK, this time, the number
of k's and T's do not match.

1198
01:04:55,341 --> 01:04:57,090
That's mainly due to
the conventions about

1199
01:04:57,090 --> 01:04:58,460
how entropy is defined.

1200
01:04:58,460 --> 01:05:00,250
It's not really anything deep.

1201
01:05:03,970 --> 01:05:06,205
I might mention at
this point that the 2's

1202
01:05:06,205 --> 01:05:08,330
that I've been writing for
everything-- g equals 2,

1203
01:05:08,330 --> 01:05:11,330
g star equals 2-- the
reason those 2's are written

1204
01:05:11,330 --> 01:05:14,080
explicitly rather than just
absorbing the factor of 2

1205
01:05:14,080 --> 01:05:17,610
into the other factors is
that photons are characterized

1206
01:05:17,610 --> 01:05:20,380
by the fact that there are
two polarizations of photons.

1207
01:05:20,380 --> 01:05:21,880
So if I have a beam
of photons, they

1208
01:05:21,880 --> 01:05:23,350
could be right-handed
or left-handed.

1209
01:05:23,350 --> 01:05:24,891
And anything else
could be considered

1210
01:05:24,891 --> 01:05:26,802
a superposition of those two.

1211
01:05:26,802 --> 01:05:28,990
So there are two
independent polarizations.

1212
01:05:28,990 --> 01:05:31,420
And it's useful to keep
track of these formulas

1213
01:05:31,420 --> 01:05:34,990
as the amount of energy
density per polarization.

1214
01:05:34,990 --> 01:05:36,770
Thus, different
kinds of particles

1215
01:05:36,770 --> 01:05:39,190
will have different
numbers of polarizations.

1216
01:05:39,190 --> 01:05:42,620
So if we know the
amount per polarization,

1217
01:05:42,620 --> 01:05:46,334
we'll be able to more easily
apply it to other particles.

1218
01:05:46,334 --> 01:05:46,834
Yes?

1219
01:05:46,834 --> 01:05:50,280
AUDIENCE: Sorry, just to kind
of bring up the same question,

1220
01:05:50,280 --> 01:05:53,383
if we-- I read that
in the early universe,

1221
01:05:53,383 --> 01:05:55,675
the temperature is constantly
changing or it's cooling.

1222
01:05:55,675 --> 01:05:56,382
PROFESSOR: Right.

1223
01:05:56,382 --> 01:05:58,620
AUDIENCE: So then if the
temperature is changing,

1224
01:05:58,620 --> 01:06:01,982
then how can we say that
the entropy is constant?

1225
01:06:01,982 --> 01:06:03,440
PROFESSOR: OK,
important question--

1226
01:06:03,440 --> 01:06:04,560
we'll be getting
to it very soon.

1227
01:06:04,560 --> 01:06:06,643
But since you asked the
question, I'll ask it now.

1228
01:06:06,643 --> 01:06:09,230
The question was if the
universe is expanding,

1229
01:06:09,230 --> 01:06:11,270
and the entropy
density is going down

1230
01:06:11,270 --> 01:06:13,831
because it thins,
how can that happen--

1231
01:06:13,831 --> 01:06:15,580
I guess it was asked
the other way around.

1232
01:06:15,580 --> 01:06:17,142
If the temperature
is falling, how

1233
01:06:17,142 --> 01:06:18,600
can entropy be
conserved if this is

1234
01:06:18,600 --> 01:06:21,120
the formula for entropy density?

1235
01:06:21,120 --> 01:06:25,520
And the answer-- when I tell
you, you'll see it's obvious.

1236
01:06:25,520 --> 01:06:27,310
We don't expect
the entropy density

1237
01:06:27,310 --> 01:06:29,230
to be conserved if the
entropy is conserved.

1238
01:06:29,230 --> 01:06:31,790
The entropy thins out
as the universe expands.

1239
01:06:31,790 --> 01:06:35,040
So if we just had a gas
with nothing else changing,

1240
01:06:35,040 --> 01:06:38,980
we expect the entropy density
to go down like 1 over the scale

1241
01:06:38,980 --> 01:06:44,350
factor cubed, just like the
number density of particles.

1242
01:06:44,350 --> 01:06:48,541
So if s is going to go down like
1 over the scale factor cubed,

1243
01:06:48,541 --> 01:06:50,290
that would be consistent
with this formula

1244
01:06:50,290 --> 01:06:53,880
if the temperature also fell
as 1 over the scale factor.

1245
01:06:53,880 --> 01:06:55,640
So that to cubing it
made things match.

1246
01:06:55,640 --> 01:06:56,400
And that's what we'll find.

1247
01:06:56,400 --> 01:06:58,860
The temperature falls off,
like 1 over the scale factor.

1248
01:06:58,860 --> 01:07:00,240
And that's consistent
with everything

1249
01:07:00,240 --> 01:07:02,156
that we said about energy
densities and so on.

1250
01:07:18,310 --> 01:07:19,074
OK.

1251
01:07:19,074 --> 01:07:20,490
Next thing I want
to talk about is

1252
01:07:20,490 --> 01:07:23,030
neutrinos, which
I told you earlier

1253
01:07:23,030 --> 01:07:28,366
contributes in a significant
way to the radiation energy

1254
01:07:28,366 --> 01:07:29,615
density in the universe today.

1255
01:07:37,540 --> 01:07:41,880
Neutrinos are particles
which for a long time,

1256
01:07:41,880 --> 01:07:44,670
were thought to be massless.

1257
01:07:44,670 --> 01:07:47,590
Until around 2000
or so, neutrinos

1258
01:07:47,590 --> 01:07:49,260
were thought to be massless.

1259
01:07:49,260 --> 01:07:53,550
Now we know that in fact, they
have a very small mass, which

1260
01:07:53,550 --> 01:07:55,310
complicates the
description here.

1261
01:07:55,310 --> 01:07:57,140
It turns out that
cosmologically, neutrinos

1262
01:07:57,140 --> 01:07:58,930
still act as if
they were massless

1263
01:07:58,930 --> 01:08:02,020
for almost all purposes
and for really all purposes

1264
01:08:02,020 --> 01:08:05,060
that we'll be dealing with in
this class, although if we were

1265
01:08:05,060 --> 01:08:07,740
interested in the effects
of neutrinos on structure

1266
01:08:07,740 --> 01:08:10,875
formation, we'd be
interested in whether or not

1267
01:08:10,875 --> 01:08:13,550
the neutrinos have a
small mass or whether it's

1268
01:08:13,550 --> 01:08:15,100
smaller than that.

1269
01:08:15,100 --> 01:08:16,744
We know it's non-zero.

1270
01:08:16,744 --> 01:08:18,910
We don't know what the
neutrino mass is, by the way.

1271
01:08:18,910 --> 01:08:22,279
What we actually know
from observations

1272
01:08:22,279 --> 01:08:26,546
is that there are three
types of neutrinos.

1273
01:08:26,546 --> 01:08:27,754
And those are called flavors.

1274
01:08:38,279 --> 01:08:40,329
And I'll use the letter
nu for the word neutrino.

1275
01:08:43,689 --> 01:08:45,149
And those three
types of neutrinos

1276
01:08:45,149 --> 01:08:49,020
are called nu sub e, called
the electron neutrino;

1277
01:08:49,020 --> 01:08:55,130
nu subbed mu, called the muon
neutrino, and nu sub tau,

1278
01:08:55,130 --> 01:08:58,380
called the tau neutrino.

1279
01:08:58,380 --> 01:09:04,390
And these letters e, mu, and tau
link to the names of particles.

1280
01:09:04,390 --> 01:09:06,200
This is the electron neutrino.

1281
01:09:06,200 --> 01:09:07,800
This is the muon
neutrino connected

1282
01:09:07,800 --> 01:09:09,300
to a particle called
the muon, which

1283
01:09:09,300 --> 01:09:12,430
is like the electron but
heavier and different.

1284
01:09:12,430 --> 01:09:16,710
And this is linked to a particle
called the tau, which is also

1285
01:09:16,710 --> 01:09:22,055
like the electron but much more
heavier but otherwise similar

1286
01:09:22,055 --> 01:09:23,640
in its properties.

1287
01:09:23,640 --> 01:09:25,850
And the neutrinos are
linked in the sense

1288
01:09:25,850 --> 01:09:29,240
that when a neutrino
is produced, depending

1289
01:09:29,240 --> 01:09:31,370
on how you start, it is
very typically produced

1290
01:09:31,370 --> 01:09:34,939
in conjunction with one
of these other particles.

1291
01:09:34,939 --> 01:09:37,590
So an electron
neutrino is typically

1292
01:09:37,590 --> 01:09:39,634
produced in conjunction
with an electron.

1293
01:09:44,420 --> 01:09:47,220
And similarly, a muon
neutrino is typically

1294
01:09:47,220 --> 01:09:49,260
produced in conjunction
with a muon.

1295
01:09:49,260 --> 01:09:52,170
And a tau neutrino is typically
produced in conjunction

1296
01:09:52,170 --> 01:09:52,679
with a tau.

1297
01:10:01,004 --> 01:10:03,170
Now, what does this have
to do with neutrino masses?

1298
01:10:06,790 --> 01:10:10,270
We've never actually measured
the mass of neutrino.

1299
01:10:10,270 --> 01:10:14,160
So we only know that they
have mass indirectly.

1300
01:10:14,160 --> 01:10:18,050
What we have seen is one
flavor of neutrino turn

1301
01:10:18,050 --> 01:10:20,110
into another flavor.

1302
01:10:20,110 --> 01:10:22,860
And it turns out, in the
context of quantum field theory

1303
01:10:22,860 --> 01:10:25,026
and I think this does make
a certain amount of sense

1304
01:10:25,026 --> 01:10:27,390
just by intuition, if
a particle is massless,

1305
01:10:27,390 --> 01:10:29,890
it can never change
into anything.

1306
01:10:29,890 --> 01:10:31,940
The process by which
one changes into another

1307
01:10:31,940 --> 01:10:33,860
is pretty quantum
mechanical and a little hard

1308
01:10:33,860 --> 01:10:36,560
to understand anyway.

1309
01:10:36,560 --> 01:10:39,550
But if the particles
were really massless,

1310
01:10:39,550 --> 01:10:41,430
they would move at
the speed of light.

1311
01:10:41,430 --> 01:10:43,790
And if the particles were
moving at the speed of light,

1312
01:10:43,790 --> 01:10:46,680
if the particle had any kind
of a clock on the particle,

1313
01:10:46,680 --> 01:10:48,750
that clock would literally
stop with the particle

1314
01:10:48,750 --> 01:10:51,540
moving at the speed of light.

1315
01:10:51,540 --> 01:10:54,970
So if particles are massless,
any internal workings

1316
01:10:54,970 --> 01:10:58,997
that that particle
might have to be frozen.

1317
01:10:58,997 --> 01:11:00,580
That is, if it's a
clock, it has to be

1318
01:11:00,580 --> 01:11:02,820
a clock that's
stopped completely.

1319
01:11:02,820 --> 01:11:05,600
And for reasons that
are essentially that,

1320
01:11:05,600 --> 01:11:08,650
although they can be made
more formal and more rigorous,

1321
01:11:08,650 --> 01:11:11,000
a truly massless
particle could never

1322
01:11:11,000 --> 01:11:12,970
undergo any kind
of change whatever.

1323
01:11:12,970 --> 01:11:15,020
It would have to
stay exactly like it

1324
01:11:15,020 --> 01:11:16,070
looks like to start with.

1325
01:11:16,070 --> 01:11:18,850
Because it just has no time.

1326
01:11:18,850 --> 01:11:21,870
So the fact that these
neutrinos turn into each other

1327
01:11:21,870 --> 01:11:24,650
implies that they must
have a nonzero mass.

1328
01:11:24,650 --> 01:11:28,080
It must not really be moving
at quite the speed of light.

1329
01:11:28,080 --> 01:11:30,580
And that's the way
the formalism works.

1330
01:11:30,580 --> 01:11:33,250
And we could set
limits on the masses

1331
01:11:33,250 --> 01:11:36,310
based on what we know about
the transitions between one

1332
01:11:36,310 --> 01:11:38,620
kind of neutrino and another.

1333
01:11:38,620 --> 01:11:39,421
Yes?

1334
01:11:39,421 --> 01:11:41,614
AUDIENCE: How do we
explain photon decaying

1335
01:11:41,614 --> 01:11:43,030
to an electron-positron
pair then?

1336
01:11:43,030 --> 01:11:45,410
PROFESSOR: A photon decaying
to an electron-positron pair?

1337
01:11:45,410 --> 01:11:45,900
AUDIENCE: Yeah.

1338
01:11:45,900 --> 01:11:47,580
PROFESSOR: The answer is a
free photon never does decay

1339
01:11:47,580 --> 01:11:49,244
to an electron-positron pair.

1340
01:11:49,244 --> 01:11:50,660
Photons can collide
with something

1341
01:11:50,660 --> 01:11:52,510
and produce an
electron-positron pair.

1342
01:11:52,510 --> 01:11:55,760
But that collision, that's
a more complicated process.

1343
01:11:55,760 --> 01:11:57,150
What I'm saying is-- I'm sorry.

1344
01:11:57,150 --> 01:12:00,050
This process of conversion--
maybe I should have clarified--

1345
01:12:00,050 --> 01:12:02,746
happens just as the
neutrinos travel.

1346
01:12:02,746 --> 01:12:03,870
It's not due to collisions.

1347
01:12:03,870 --> 01:12:05,369
Due to collsions,
complicated things

1348
01:12:05,369 --> 01:12:08,680
can happen whether the
particle is massless or not.

1349
01:12:12,080 --> 01:12:17,740
But a massless particle simply
in transit cannot undergo any

1350
01:12:17,740 --> 01:12:19,020
kind of transition.

1351
01:12:19,020 --> 01:12:22,300
And these neutrinos are seen
to undergo transitions simply

1352
01:12:22,300 --> 01:12:24,660
being in transit
without any collisions.

1353
01:12:31,630 --> 01:12:33,720
In terms of my clock
analogy and a stopped clock,

1354
01:12:33,720 --> 01:12:35,428
I think the reason
the photon can convert

1355
01:12:35,428 --> 01:12:38,200
into electron-positron pairs
if it collides with something

1356
01:12:38,200 --> 01:12:42,525
is that when it collides, it
essentially breaks the clock.

1357
01:12:42,525 --> 01:12:45,060
You don't have a
photon that's just

1358
01:12:45,060 --> 01:12:47,580
moving along without
time anymore.

1359
01:12:50,210 --> 01:12:53,060
OK, so these
neutrinos have masses.

1360
01:12:53,060 --> 01:12:56,852
And maybe I should,
at this point,

1361
01:12:56,852 --> 01:12:59,185
write down some bounds
on these masses.

1362
01:13:05,550 --> 01:13:13,780
m squared 21 times c to the
fourth is equal to 7.50 plus

1363
01:13:13,780 --> 01:13:25,300
or minus 0.2 times 10 to the
minus 5 electron volts squared.

1364
01:13:25,300 --> 01:13:28,630
These numbers come from the
latest particle data tables,

1365
01:13:28,630 --> 01:13:32,630
which I gave references
for in the notes.

1366
01:13:32,630 --> 01:13:35,610
So this is the difference
of the mass squared.

1367
01:13:38,370 --> 01:13:44,920
And delta m 23 squared
times c to the fourth

1368
01:13:44,920 --> 01:13:47,860
to turn it into the
square of an energy

1369
01:13:47,860 --> 01:14:01,260
is 2.32 plus 0.12
minus 0.08 times 10

1370
01:14:01,260 --> 01:14:04,850
to the minus third
electron volts squared.

1371
01:14:07,112 --> 01:14:08,570
So one thing you
notice immediately

1372
01:14:08,570 --> 01:14:11,460
is that these are
incredibly small masses.

1373
01:14:11,460 --> 01:14:17,150
Remember, the proton
weighs 938 MeV,

1374
01:14:17,150 --> 01:14:18,460
three million electron volts.

1375
01:14:18,460 --> 01:14:21,620
And these are fractions
of one electron volt.

1376
01:14:21,620 --> 01:14:23,690
So by the standards
of particle physics,

1377
01:14:23,690 --> 01:14:27,060
these are unbelievably
small energies,

1378
01:14:27,060 --> 01:14:30,150
unbelievably small
mass differences.

1379
01:14:30,150 --> 01:14:31,010
But they're there.

1380
01:14:31,010 --> 01:14:34,652
They have to be there for the
physics we know to make sense.

1381
01:14:34,652 --> 01:14:36,110
The other thing
that you may notice

1382
01:14:36,110 --> 01:14:38,318
about this notation-- which
I don't want to elaborate

1383
01:14:38,318 --> 01:14:40,680
on but I'll just mention--
this is called 21.

1384
01:14:40,680 --> 01:14:42,270
This is called 23.

1385
01:14:42,270 --> 01:14:43,600
There's no 1, 2, or 3 there.

1386
01:14:43,600 --> 01:14:45,890
There's an e and a mu and a tau.

1387
01:14:45,890 --> 01:14:49,256
The complication here is
something very quantum

1388
01:14:49,256 --> 01:14:49,755
mechanical.

1389
01:14:54,050 --> 01:14:57,380
The e, mu and tau
labels are labels

1390
01:14:57,380 --> 01:15:01,000
which basically label
the neutrino according

1391
01:15:01,000 --> 01:15:03,245
to how the neutrino is created.

1392
01:15:06,100 --> 01:15:09,720
It turns out that their mass
eigenstates-- states which

1393
01:15:09,720 --> 01:15:14,090
actually have a
definite mass-- are not

1394
01:15:14,090 --> 01:15:15,800
the e, the mu or the tau.

1395
01:15:15,800 --> 01:15:17,570
In fact, if the e
had a definite mass,

1396
01:15:17,570 --> 01:15:19,278
that would be saying
that an e would just

1397
01:15:19,278 --> 01:15:21,130
propagate as a particle
with a certain mass.

1398
01:15:21,130 --> 01:15:23,920
It would not convert
into anything else.

1399
01:15:23,920 --> 01:15:26,670
The fact that an e converts
into other particles--

1400
01:15:26,670 --> 01:15:28,800
a nu sub e converts
into other particles

1401
01:15:28,800 --> 01:15:31,160
is really the statement
that nu sub e is not

1402
01:15:31,160 --> 01:15:33,222
a state with a definite mass.

1403
01:15:33,222 --> 01:15:34,930
But there are states
with definite masses

1404
01:15:34,930 --> 01:15:37,660
which could be expressed quantum
mechanically as superpositions

1405
01:15:37,660 --> 01:15:40,070
of these flavor eigenstates.

1406
01:15:40,070 --> 01:15:43,220
So nu sub 1, nu
sub 2 and nu sub 3

1407
01:15:43,220 --> 01:15:46,410
are states of neutrinos
that have definite masses.

1408
01:15:46,410 --> 01:15:48,600
And each one of them is
a superposition of nu sub

1409
01:15:48,600 --> 01:15:50,720
e, nu sub mu and nu sub tau.

1410
01:15:53,770 --> 01:15:54,360
Yes?

1411
01:15:54,360 --> 01:15:58,029
AUDIENCE: How come we don't
have a delta n from 31?

1412
01:15:58,029 --> 01:15:59,070
PROFESSOR: Good question.

1413
01:15:59,070 --> 01:16:03,080
I think it's just the
lack of knowledge.

1414
01:16:03,080 --> 01:16:05,470
I don't think there's any
reason it's not defined.

1415
01:16:05,470 --> 01:16:07,590
I'm sure it is defined.

1416
01:16:07,590 --> 01:16:09,804
I think it's just
lack of knowledge.

1417
01:16:09,804 --> 01:16:12,470
And if I knew more details about
how these things were measured,

1418
01:16:12,470 --> 01:16:13,970
I could give you a
better story about that.

1419
01:16:13,970 --> 01:16:14,845
But I don't, frankly.

1420
01:16:21,150 --> 01:16:23,620
So in the end, it's a rather
complicated quantum mechanical

1421
01:16:23,620 --> 01:16:27,070
system which we're not going
to go into any details about.

1422
01:16:30,792 --> 01:16:32,000
What more should I say today?

1423
01:16:34,750 --> 01:16:37,060
OK, let me just
mention for today

1424
01:16:37,060 --> 01:16:42,770
and we'll continue next
time after the quiz,

1425
01:16:42,770 --> 01:16:45,090
for our purposes, we're going
to treat these neutrinos

1426
01:16:45,090 --> 01:16:46,580
as if they're massless.

1427
01:16:46,580 --> 01:16:49,640
And it turns out that that's
actually extraordinarily

1428
01:16:49,640 --> 01:16:52,014
accurate from the point
of view of cosmology,

1429
01:16:52,014 --> 01:16:53,430
at least for the
kind of cosmology

1430
01:16:53,430 --> 01:16:54,410
that we're doing
where we're just

1431
01:16:54,410 --> 01:16:56,380
interested in the effect
of these neutrinos

1432
01:16:56,380 --> 01:16:59,190
on the expansion
rate of the universe.

1433
01:16:59,190 --> 01:17:03,570
And treating them as
massless particles,

1434
01:17:03,570 --> 01:17:07,620
I will shortly give
you the formulas

1435
01:17:07,620 --> 01:17:10,010
for how they contribute to
the black-body radiation.

1436
01:17:10,010 --> 01:17:12,301
But it I think there's no
point in my writing them now.

1437
01:17:12,301 --> 01:17:14,820
I'll just write them again in
the beginning of next period.

1438
01:17:14,820 --> 01:17:16,903
But they do contribute to
the black-body radiation

1439
01:17:16,903 --> 01:17:19,930
and in a way that we actually
know how to calculate.

1440
01:17:19,930 --> 01:17:23,280
And they have a
noticeable effect

1441
01:17:23,280 --> 01:17:27,050
on the evolution
of our universe.

1442
01:17:27,050 --> 01:17:28,320
OK, that's all for today.

1443
01:17:28,320 --> 01:17:30,020
Good luck on the
quiz on Thursday.

1444
01:17:30,020 --> 01:17:32,280
I'll be here to help proctor.

1445
01:17:32,280 --> 01:17:35,050
I think Tim [INAUDIBLE]
will be here too.

1446
01:17:35,050 --> 01:17:39,350
And I'll see you more intimately
either at my office hour

1447
01:17:39,350 --> 01:17:43,180
tomorrow or at lecture
a week from now.