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PROFESSOR: OK, starting again.

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00:00:24,690 --> 00:00:26,650
I want to begin
with a quick review

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00:00:26,650 --> 00:00:28,210
of what we said last time.

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00:00:28,210 --> 00:00:29,740
This time will be
quicker than usual

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00:00:29,740 --> 00:00:31,940
because we're not really
continuing from there,

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00:00:31,940 --> 00:00:34,300
we'll be starting a
completely new topic today.

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00:00:34,300 --> 00:00:36,320
But I thought I'd
remind you, nonetheless,

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00:00:36,320 --> 00:00:38,870
that we had a last lecture--
the universe was not

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00:00:38,870 --> 00:00:41,300
created between then and now.

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00:00:41,300 --> 00:00:43,640
And what we talked
about last time

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00:00:43,640 --> 00:00:47,850
was the spacetime
geodesic equation,

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00:00:47,850 --> 00:00:49,170
which is written this way.

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00:00:49,170 --> 00:00:52,090
It's a completely
identical to the equation

21
00:00:52,090 --> 00:00:56,470
we derived for geodesics
in pure spatial situations.

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00:00:56,470 --> 00:00:59,490
The only difference really,
is a difference in notation.

23
00:00:59,490 --> 00:01:01,390
Instead of using i
and j the tradition

24
00:01:01,390 --> 00:01:05,890
is to use mu, nu, et cetera
for the spacetime indices which

25
00:01:05,890 --> 00:01:11,780
are sum from 0 to 3, where
x super 0 is identical to t.

26
00:01:11,780 --> 00:01:14,970
And in addition, the parameter
which is sometimes called s

27
00:01:14,970 --> 00:01:17,840
when we're talking about
space, is called tau

28
00:01:17,840 --> 00:01:19,760
when we're talking
about time because what

29
00:01:19,760 --> 00:01:22,370
the parameter refers
to is the proper time

30
00:01:22,370 --> 00:01:26,000
along the trajectory of the
object whose geodesic we're

31
00:01:26,000 --> 00:01:28,810
calculating.

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00:01:28,810 --> 00:01:31,560
We then introduced the
Schwarzschild metric,

33
00:01:31,560 --> 00:01:35,350
which we did not
derive, but we claimed

34
00:01:35,350 --> 00:01:38,190
it describes the metric for
any spherically symmetric mass

35
00:01:38,190 --> 00:01:40,490
distribution as long
as you're talking

36
00:01:40,490 --> 00:01:44,910
about the region outside
where the masses are located.

37
00:01:44,910 --> 00:01:47,290
And here M is the total
mass of the object, which

38
00:01:47,290 --> 00:01:49,840
is the only thing the
metric really depends on,

39
00:01:49,840 --> 00:01:54,520
G is Newton's constant, and C
is of course the speed of light.

40
00:01:54,520 --> 00:01:56,350
The metric has an
interesting feature

41
00:01:56,350 --> 00:01:58,770
that the coefficients
of both the dt

42
00:01:58,770 --> 00:02:03,750
squared term and to dr squared
term becomes singular either 0

43
00:02:03,750 --> 00:02:06,770
or infinity, depending on
which one you're looking at,

44
00:02:06,770 --> 00:02:09,620
a particular value of the
radius called the Schwarzschild

45
00:02:09,620 --> 00:02:14,380
radius, given by this formula
2 GM divided by c squared.

46
00:02:14,380 --> 00:02:17,160
The bigger the mass the bigger
the Schwarzschild radius,

47
00:02:17,160 --> 00:02:19,380
they're proportional
to each other.

48
00:02:19,380 --> 00:02:21,900
The metric is singular
at those points,

49
00:02:21,900 --> 00:02:24,630
but I mentioned
but did not prove,

50
00:02:24,630 --> 00:02:27,250
that that particular
singularity is

51
00:02:27,250 --> 00:02:30,900
in fact what is referred to
as a coordinate singularity.

52
00:02:30,900 --> 00:02:33,530
It's a singularity that's
there only because of the way

53
00:02:33,530 --> 00:02:35,495
the coordinates were chosen.

54
00:02:35,495 --> 00:02:37,870
So there are other ways of
choosing the coordinates where

55
00:02:37,870 --> 00:02:40,650
that singularity disappears.

56
00:02:40,650 --> 00:02:43,209
There's also singularity
at r equals zero,

57
00:02:43,209 --> 00:02:44,750
and that singularity
is real, there's

58
00:02:44,750 --> 00:02:46,860
no way to remove
that singularity

59
00:02:46,860 --> 00:02:48,185
by a change of coordinates.

60
00:02:51,690 --> 00:02:56,470
However, although r sub s
is not a true singularity,

61
00:02:56,470 --> 00:02:58,470
it is a horizon.

62
00:02:58,470 --> 00:03:04,320
And by that we mean that if any
particle or even a light beam

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00:03:04,320 --> 00:03:07,720
gets inside the Schwarzschild
radius, it can never get out.

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There's no geodesic which will
take it out of the horizon.

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00:03:13,225 --> 00:03:14,950
And it's not even a
matter of geodesics,

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there are no
time-like paths even

67
00:03:17,660 --> 00:03:20,330
if you have a rocket which
would then not follow geodesic.

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00:03:20,330 --> 00:03:23,950
There's no way to get out
from inside a black hole.

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00:03:23,950 --> 00:03:28,140
We didn't show that, but
that fact is claimed.

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00:03:28,140 --> 00:03:32,660
Then we calculated the geodesic
for our radially falling

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00:03:32,660 --> 00:03:33,300
object.

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00:03:33,300 --> 00:03:35,260
We solve the problem
of an object that

73
00:03:35,260 --> 00:03:38,360
is released from rest at
some initial value r sub 0,

74
00:03:38,360 --> 00:03:41,590
and just let fall straight
down towards the center

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00:03:41,590 --> 00:03:42,580
of the sphere.

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00:03:42,580 --> 00:03:47,660
And the the equation
describing the geodesic

77
00:03:47,660 --> 00:03:51,020
is just a special case
of the general equation

78
00:03:51,020 --> 00:03:52,680
that we had a few slides ago.

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00:03:52,680 --> 00:03:54,990
And we need only look
at the radial component

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00:03:54,990 --> 00:03:57,630
if we want to track how the
radius changes with time.

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00:03:57,630 --> 00:04:01,270
So there was a free index
mu in the generic form

82
00:04:01,270 --> 00:04:05,700
of the equation, we're setting
mu equal to the r variable.

83
00:04:05,700 --> 00:04:08,420
And then the equation
reduces to this form

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00:04:08,420 --> 00:04:12,450
and we know what these G sub
t t's and G sub r r's are,

85
00:04:12,450 --> 00:04:15,060
they come from the equation
for the Schwarzschild metric

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00:04:15,060 --> 00:04:16,910
on the previous slide.

87
00:04:16,910 --> 00:04:18,750
And that equation
can be manipulated

88
00:04:18,750 --> 00:04:21,125
and eventually it simplifies
to something extraordinarily

89
00:04:21,125 --> 00:04:22,050
simple.

90
00:04:22,050 --> 00:04:25,530
It's just the statement that
d squared r d tau squared is

91
00:04:25,530 --> 00:04:29,920
equal to minus GM over r
squared, which looks exactly

92
00:04:29,920 --> 00:04:32,380
like the Newtonian equation
for something falling

93
00:04:32,380 --> 00:04:35,107
in a spherically symmetric
gravitational field.

94
00:04:35,107 --> 00:04:36,690
But it's not really
the same equation,

95
00:04:36,690 --> 00:04:39,356
it just looks like it's the same
equation, because the variables

96
00:04:39,356 --> 00:04:41,180
both a different
meanings. r and tau

97
00:04:41,180 --> 00:04:43,840
both have different meanings
from the r and t that

98
00:04:43,840 --> 00:04:46,950
would have appeared in
the Newtonian calculation.

99
00:04:46,950 --> 00:04:49,200
The r variable that
appears here is not really

100
00:04:49,200 --> 00:04:50,707
the distance from the origin.

101
00:04:50,707 --> 00:04:52,790
If you wanted to know the
distance from the origin

102
00:04:52,790 --> 00:04:55,160
you'd have to integrate
the metric singular even--

103
00:04:55,160 --> 00:04:57,600
if not even a well defined
distance to the origin

104
00:04:57,600 --> 00:05:00,510
because the origin singular.

105
00:05:00,510 --> 00:05:04,000
And the tau that appears
here is a time variable,

106
00:05:04,000 --> 00:05:07,064
but it's not the time that would
be read on any fixed clock,

107
00:05:07,064 --> 00:05:08,480
rather it's the
time that would be

108
00:05:08,480 --> 00:05:11,070
read on the wristwatch
of the person falling

109
00:05:11,070 --> 00:05:14,180
into the spherically symmetric
object, which we might consider

110
00:05:14,180 --> 00:05:15,000
to be a black hole.

111
00:05:18,010 --> 00:05:22,250
And we were able to
solve this equation

112
00:05:22,250 --> 00:05:24,660
by using essentially
conservation of energy

113
00:05:24,660 --> 00:05:27,261
techniques, or at least what
would be called conservation

114
00:05:27,261 --> 00:05:29,260
of energy if we were doing
the Newtonian version

115
00:05:29,260 --> 00:05:31,420
of the problem, which is
the same equation even

116
00:05:31,420 --> 00:05:34,740
though the variables have
a different interpretation.

117
00:05:34,740 --> 00:05:37,240
So we were able to calculate
not r as a function of tau,

118
00:05:37,240 --> 00:05:40,570
but at least tau
a function of r.

119
00:05:40,570 --> 00:05:43,770
And we got that equation,
which is a little complicated,

120
00:05:43,770 --> 00:05:45,320
but the interesting
thing about it

121
00:05:45,320 --> 00:05:49,960
is that it gives finite answers
for every value of r going

122
00:05:49,960 --> 00:05:53,020
all the way down
to r equals zero.

123
00:05:53,020 --> 00:05:55,810
So it means that in a
finite amount of time,

124
00:05:55,810 --> 00:05:59,770
as seen by the person
falling into the black hole,

125
00:05:59,770 --> 00:06:02,740
the person would reach r
equals zero, at which point

126
00:06:02,740 --> 00:06:04,710
he would disappear
into the singularity.

127
00:06:04,710 --> 00:06:06,780
He'd actually be ripped
apart as he approached

128
00:06:06,780 --> 00:06:08,950
the singularity
because of tidal forces

129
00:06:08,950 --> 00:06:12,480
which pull more strongly
on the front part of him

130
00:06:12,480 --> 00:06:16,360
than on the back part of him,
stretching the object out

131
00:06:16,360 --> 00:06:19,440
in the radial direction.

132
00:06:19,440 --> 00:06:23,830
However, curiously,
if one calculates

133
00:06:23,830 --> 00:06:25,820
what this trajectory
looks like as a function

134
00:06:25,820 --> 00:06:29,300
of the coordinate time,
t, we did actually

135
00:06:29,300 --> 00:06:31,240
do that calculation
but we looked

136
00:06:31,240 --> 00:06:35,230
at how it would
behave in the limit

137
00:06:35,230 --> 00:06:37,720
as you approached-- as
the particle approached

138
00:06:37,720 --> 00:06:39,010
the Schwarzschild horizon.

139
00:06:39,010 --> 00:06:41,850
And we discovered it would take
an infinite amount of time,

140
00:06:41,850 --> 00:06:44,980
as seen from the outside,
for the in falling object

141
00:06:44,980 --> 00:06:47,700
to reach the horizon, let
go through the horizon

142
00:06:47,700 --> 00:06:49,500
and get to r equals zero.

143
00:06:49,500 --> 00:06:52,280
So from the outside, it
looks like the object never

144
00:06:52,280 --> 00:06:54,210
actually falls into
the black hole,

145
00:06:54,210 --> 00:06:57,220
but just gets closer and closer
and closer as t approaches

146
00:06:57,220 --> 00:06:58,240
infinity.

147
00:06:58,240 --> 00:07:01,830
So it's an example of a very
highly distorted spacetime,

148
00:07:01,830 --> 00:07:04,400
where you can see very
different pictures depending

149
00:07:04,400 --> 00:07:08,250
on which observer you're
trying to describe

150
00:07:08,250 --> 00:07:09,350
the observations of.

151
00:07:12,130 --> 00:07:15,220
And I think that's it.

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00:07:15,220 --> 00:07:17,350
Any questions about any of that?

153
00:07:17,350 --> 00:07:20,510
I guess I'll put that back up.

154
00:07:23,680 --> 00:07:24,400
OK.

155
00:07:24,400 --> 00:07:27,030
On your homework you'll be
applying this geodesic equation

156
00:07:27,030 --> 00:07:32,560
to a model universe, to
Robertson-Walker universe,

157
00:07:32,560 --> 00:07:35,645
and this will serve only as an
example for those calculations.

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00:07:35,645 --> 00:07:38,270
I guess there's also a homework
problem about the Schwarzschild

159
00:07:38,270 --> 00:07:40,460
metric, that orbits in
the Schwarzschild metric

160
00:07:40,460 --> 00:07:42,389
that you'll be working at.

161
00:07:42,389 --> 00:07:43,930
It's all in principal
straightforward

162
00:07:43,930 --> 00:07:46,950
if you just look at
equations and follow

163
00:07:46,950 --> 00:07:49,030
what the equations tell
you, thinking carefully

164
00:07:49,030 --> 00:07:52,570
about what the variables mean.

165
00:07:52,570 --> 00:07:53,560
OK.

166
00:07:53,560 --> 00:07:58,240
In that case, let's get
started on today's work,

167
00:07:58,240 --> 00:07:59,944
which is a change of gear.

168
00:07:59,944 --> 00:08:01,860
We're now going to be
talking about black body

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00:08:01,860 --> 00:08:05,060
radiation and its
effect on the universe.

170
00:08:05,060 --> 00:08:07,890
I should say that
my original plan was

171
00:08:07,890 --> 00:08:09,990
to get into this set
of lectures notes--

172
00:08:09,990 --> 00:08:12,710
lecture notes six, which have
not been handed out yet--

173
00:08:12,710 --> 00:08:15,222
to get into those last time
and to finish them today.

174
00:08:15,222 --> 00:08:16,930
I don't think that's
going to be possible

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00:08:16,930 --> 00:08:18,638
because I didn't get
into them last time,

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00:08:18,638 --> 00:08:21,720
and I don't think I'll be
able to finish them today.

177
00:08:21,720 --> 00:08:24,710
But I would like today to be
sort of the closing for what's

178
00:08:24,710 --> 00:08:27,010
needed for the problem set
due Monday and for the quiz

179
00:08:27,010 --> 00:08:28,370
next week.

180
00:08:28,370 --> 00:08:30,710
So, shortly after
today's lecture

181
00:08:30,710 --> 00:08:32,230
I will send you an
email telling you

182
00:08:32,230 --> 00:08:34,630
where the cutoff is
as far as the reading

183
00:08:34,630 --> 00:08:35,740
and the lecture notes.

184
00:08:35,740 --> 00:08:40,059
And I also hope to post the
lecture notes by tomorrow.

185
00:08:40,059 --> 00:08:42,919
And I also will be posting
a set of review problems

186
00:08:42,919 --> 00:08:44,770
as we had for quiz one.

187
00:08:44,770 --> 00:08:47,180
And I hope to get
that done by tomorrow.

188
00:08:47,180 --> 00:08:50,130
You may have noticed that not
all of my hopes are filled,

189
00:08:50,130 --> 00:08:54,310
but I do my best, and I'll try.

190
00:08:54,310 --> 00:08:56,270
OK are there any logistic
questions or anything

191
00:08:56,270 --> 00:08:57,760
before we go on?

192
00:08:57,760 --> 00:08:59,680
Yes.

193
00:08:59,680 --> 00:09:01,520
AUDIENCE: Can you
post the solutions

194
00:09:01,520 --> 00:09:04,901
to the previous
problem set, the one

195
00:09:04,901 --> 00:09:07,310
that we turned in on Friday?

196
00:09:07,310 --> 00:09:11,310
PROFESSOR: Oh, um,
yes I can, I should.

197
00:09:11,310 --> 00:09:14,530
OK I will.

198
00:09:14,530 --> 00:09:17,010
OK, that's a third item I
should try to get done today.

199
00:09:17,010 --> 00:09:19,020
Thanks for reminding me.

200
00:09:19,020 --> 00:09:21,849
And the solutions
to the problem set

201
00:09:21,849 --> 00:09:23,390
that you'll be
handing in Monday will

202
00:09:23,390 --> 00:09:25,264
be posted very shortly
after you hand them in

203
00:09:25,264 --> 00:09:27,080
so that people can
start talking about them

204
00:09:27,080 --> 00:09:29,070
and prepare for the quiz.

205
00:09:29,070 --> 00:09:29,570
Yes.

206
00:09:29,570 --> 00:09:33,285
AUDIENCE: Do we have any day at
which the videos might be up?

207
00:09:33,285 --> 00:09:35,660
PROFESSOR: Ah, I've request
about that and all I was told

208
00:09:35,660 --> 00:09:38,250
was that they're
doing their best.

209
00:09:38,250 --> 00:09:43,510
So, I did look into it, but
I don't know the answer.

210
00:09:48,370 --> 00:09:51,867
I hope that you'll have all
the videos available to study

211
00:09:51,867 --> 00:09:54,200
for the quiz, but I don't
know if that's going to happen

212
00:09:54,200 --> 00:09:54,699
or not.

213
00:09:59,060 --> 00:10:00,920
OK.

214
00:10:00,920 --> 00:10:26,287
in that case, the new topic
is Blackbody Radiation

215
00:10:26,287 --> 00:10:27,870
and the Early History
of the Universe.

216
00:10:39,290 --> 00:10:41,750
So far we've dealt
with a universe which

217
00:10:41,750 --> 00:10:44,330
contains only
non-relativistic matter

218
00:10:44,330 --> 00:10:46,460
and that, as we said
from the beginning,

219
00:10:46,460 --> 00:10:50,160
describes our universe for
the bulk of its history.

220
00:10:50,160 --> 00:10:51,890
But in the early
period, the universe

221
00:10:51,890 --> 00:10:53,580
was in fact dominated
by radiation,

222
00:10:53,580 --> 00:10:55,390
as we will now be calculating.

223
00:10:55,390 --> 00:10:57,550
And in the more recent
period, the universe

224
00:10:57,550 --> 00:10:59,129
is dominated by
dark energy, which

225
00:10:59,129 --> 00:11:00,920
we'll be talking about
immediately after we

226
00:11:00,920 --> 00:11:04,090
finish talking about radiation.

227
00:11:04,090 --> 00:11:08,070
So the important point here is
that even though we don't think

228
00:11:08,070 --> 00:11:11,220
of light as having
mass, light certainly

229
00:11:11,220 --> 00:11:13,640
has energy and
relativistically we

230
00:11:13,640 --> 00:11:16,200
know that energy and
mass are equivalent.

231
00:11:16,200 --> 00:11:20,470
The key equation that actually
dominates today's lecture

232
00:11:20,470 --> 00:11:24,610
is perhaps the most
famous equation physics,

233
00:11:24,610 --> 00:11:30,260
E equals MC squared, energy
and mass are equivalent.

234
00:11:30,260 --> 00:11:32,689
And the numbers--
I'll just give you

235
00:11:32,689 --> 00:11:33,980
some numbers for this equation.

236
00:11:33,980 --> 00:11:35,930
You've probably already
aware that the numbers

237
00:11:35,930 --> 00:11:38,560
are kind of out of sight.

238
00:11:38,560 --> 00:11:45,190
One kilogram, a point
to that equation,

239
00:11:45,190 --> 00:11:50,390
is equivalent to-- I don't
have any figures to give you--

240
00:11:50,390 --> 00:11:56,860
8.9876, in case you really want
to know it accurately, times 10

241
00:11:56,860 --> 00:12:00,850
to the 16th, most important
to know the exponent there,

242
00:12:00,850 --> 00:12:01,350
joules.

243
00:12:04,010 --> 00:12:07,284
And it's also perhaps
interesting to translate this

244
00:12:07,284 --> 00:12:08,700
into the kind of
energy units that

245
00:12:08,700 --> 00:12:12,100
are use when we talk
about power consumption

246
00:12:12,100 --> 00:12:14,350
in practical situations.

247
00:12:14,350 --> 00:12:33,240
It corresponds to 2.497 times
10 to the 10th kilowatt hours,

248
00:12:33,240 --> 00:12:35,720
which is a lot of
kilowatt hours.

249
00:12:35,720 --> 00:12:38,390
An interesting comparison, is
the total power consumption

250
00:12:38,390 --> 00:12:41,040
of the world, which
turns out to be

251
00:12:41,040 --> 00:12:44,320
comparable to a kilograms
worth of things.

252
00:12:44,320 --> 00:12:47,410
I looked things up
in the Wikipedia,

253
00:12:47,410 --> 00:12:49,460
and it told me that
in 2008-- which

254
00:12:49,460 --> 00:12:51,300
is the most recent year
it had numbers for,

255
00:12:51,300 --> 00:12:54,940
which is a little surprising,
that's so far in the past--

256
00:12:54,940 --> 00:12:57,300
the total world
power consumption

257
00:12:57,300 --> 00:13:14,040
for the year equaled about,
I'm rounding off here,

258
00:13:14,040 --> 00:13:17,020
about 150 petawatt hours.

259
00:13:24,760 --> 00:13:26,820
Now, I always have to look
up peta when I see it,

260
00:13:26,820 --> 00:13:28,653
I haven't quite learned
what peta means yet.

261
00:13:28,653 --> 00:13:36,250
But they translated this, this
means 150 trillion kilowatt

262
00:13:36,250 --> 00:13:37,430
hours.

263
00:13:37,430 --> 00:13:45,930
So 150 times 10 to the
12th kilowatt hours.

264
00:13:45,930 --> 00:13:48,370
And if we divide this by the
number of hours in the year--

265
00:13:48,370 --> 00:13:50,475
to ask how much
is per hour, which

266
00:13:50,475 --> 00:13:52,850
is I think a natural thing to
think about if you're using

267
00:13:52,850 --> 00:13:57,610
kilowatt hours to measure the
power, especially the energy--

268
00:13:57,610 --> 00:14:05,290
the power that goes with
this is about 17 times 10

269
00:14:05,290 --> 00:14:08,900
to the ninth kilowatts.

270
00:14:08,900 --> 00:14:18,250
And therefore 17 times 10 to
the 9th kilowatt hours per hour.

271
00:14:18,250 --> 00:14:20,080
And if you compare
these two numbers,

272
00:14:20,080 --> 00:14:27,910
it means that if you could
convert one kilogram per hour

273
00:14:27,910 --> 00:14:31,110
into pure energy,
that would be about

274
00:14:31,110 --> 00:14:51,680
equal to 1.5 times the
world's power usage in 2008.

275
00:14:51,680 --> 00:14:55,370
So if you could convert
matter completely to energy,

276
00:14:55,370 --> 00:14:57,780
as they do on Star
Trek, it would

277
00:14:57,780 --> 00:14:59,590
mean that you could
fill up your tank

278
00:14:59,590 --> 00:15:03,290
of your typical
American car and power

279
00:15:03,290 --> 00:15:04,515
the world for about two days.

280
00:15:07,476 --> 00:15:08,850
But of course we
can't undo this,

281
00:15:08,850 --> 00:15:12,670
that's the important
fact concerning power.

282
00:15:12,670 --> 00:15:15,620
Only a small
fraction of the mass

283
00:15:15,620 --> 00:15:18,010
of the uranium in
a nuclear reactor

284
00:15:18,010 --> 00:15:22,050
is actually converted as
power, a fraction of a percent.

285
00:15:22,050 --> 00:15:23,730
So you don't get
nearly as much power

286
00:15:23,730 --> 00:15:25,340
as this calculation
would indicate,

287
00:15:25,340 --> 00:15:26,790
but in principle
this much energy

288
00:15:26,790 --> 00:15:31,620
is contained in the matter that
we have around us all the time.

289
00:15:49,730 --> 00:15:50,230
OK.

290
00:15:50,230 --> 00:15:51,896
I want to introduce
a few formulas which

291
00:15:51,896 --> 00:15:53,660
we'll be using sooner
or later concerning

292
00:15:53,660 --> 00:15:57,190
the relativistic treatment
of momentum and energy,

293
00:15:57,190 --> 00:15:58,940
which is what we're
getting into here.

294
00:15:58,940 --> 00:16:03,070
And we're not trying to derive
relativity in this course,

295
00:16:03,070 --> 00:16:07,110
so I'm just trying to quote the
results that we'll be needing.

296
00:16:07,110 --> 00:16:16,910
So, it is useful to introduce
an energy momentum four-vector,

297
00:16:16,910 --> 00:16:22,680
which has a 0-th component
and an i-th component,

298
00:16:22,680 --> 00:16:27,280
where i refers to the
spatial indices 1, 2, and 3.

299
00:16:27,280 --> 00:16:32,130
And sometimes I might
write this as p zero p

300
00:16:32,130 --> 00:16:35,050
with a vector sign over it,
where the vector sign indicates

301
00:16:35,050 --> 00:16:36,750
the three components 1, 2, 3.

302
00:16:39,610 --> 00:16:43,340
The momentum here
is the momentum.

303
00:16:43,340 --> 00:16:45,415
It differs in its
relationship to velocity

304
00:16:45,415 --> 00:16:47,040
from what we have
from Newton, and I'll

305
00:16:47,040 --> 00:16:48,880
write that in a minute,
but this momentum

306
00:16:48,880 --> 00:16:52,330
is the conserved physical
momentum of an object.

307
00:16:52,330 --> 00:16:55,080
And p0 is also conserved.

308
00:16:55,080 --> 00:17:05,140
p0 is just an abbreviation
for the energy divided by C.

309
00:17:05,140 --> 00:17:07,630
And this quantity forms
a four dimensional

310
00:17:07,630 --> 00:17:09,530
vector in special relativity.

311
00:17:09,530 --> 00:17:11,589
And when we say that
it's a four-vector,

312
00:17:11,589 --> 00:17:13,609
we're actually making
a definite statement

313
00:17:13,609 --> 00:17:17,519
about how it transforms from
one Lorentz frame to another.

314
00:17:17,519 --> 00:17:20,190
a four-vector is
something which transforms

315
00:17:20,190 --> 00:17:24,890
in exactly the same way as x
super mu, four spatial and time

316
00:17:24,890 --> 00:17:25,975
coordinates transform.

317
00:17:29,140 --> 00:17:31,410
And in particular,
we learned that there

318
00:17:31,410 --> 00:17:35,040
was an invariant associated with
spacetime transformations this

319
00:17:35,040 --> 00:17:38,530
s squared, which was x squared
plus y squared plus z squared

320
00:17:38,530 --> 00:17:41,150
minus C squared t squared.

321
00:17:41,150 --> 00:17:45,170
And the same thing
will happen here.

322
00:17:45,170 --> 00:17:56,270
p squared, which means
the Lorentz invariant

323
00:17:56,270 --> 00:18:08,000
square of the four-vector is the
Lorentz [INAUDIBLE] again here,

324
00:18:08,000 --> 00:18:12,880
and it's again equal to the
sum of the squares of the 3

325
00:18:12,880 --> 00:18:19,450
spatial components minus the
square of the time component.

326
00:18:24,790 --> 00:18:34,280
And that can be written out as
the square of the momentum--

327
00:18:34,280 --> 00:18:38,000
spatial momentum-- minus
E squared C squared.

328
00:18:42,460 --> 00:18:44,925
And the claim is that this
is also Lorentz invariant.

329
00:19:01,990 --> 00:19:04,670
And we could figure out what
Lorentz invariant quantity

330
00:19:04,670 --> 00:19:08,511
it's equal to by
having knowledge

331
00:19:08,511 --> 00:19:10,010
that this is the
same in all frames.

332
00:19:10,010 --> 00:19:12,550
We can evaluate it in
the simplest frame.

333
00:19:12,550 --> 00:19:15,165
And the simplest frame would be
the rest frame of the object.

334
00:19:15,165 --> 00:19:17,290
In the rest frame of the
object the p would be zero

335
00:19:17,290 --> 00:19:20,670
and this would then just be
minus E squared over C squared.

336
00:19:20,670 --> 00:19:25,800
E squared would be M
squared C to the fourth,

337
00:19:25,800 --> 00:19:36,810
so that implies that this is
equal to minus M0 C squared

338
00:19:36,810 --> 00:19:43,960
squared where M0 is often
called the rest mass.

339
00:19:49,670 --> 00:19:53,030
And when I say it's often called
the the rest mass, what I mean

340
00:19:53,030 --> 00:19:54,950
is that nobody ever
mistakes the word "rest

341
00:19:54,950 --> 00:19:57,840
mass" to mean anything else,
if anybody says "rest mass,"

342
00:19:57,840 --> 00:20:00,150
and he knows what he's
talking about, he means this.

343
00:20:00,150 --> 00:20:02,230
But this is often sometimes
just called the mass

344
00:20:02,230 --> 00:20:04,440
because sometimes people
only talk about masses

345
00:20:04,440 --> 00:20:06,714
as being rest masses.

346
00:20:06,714 --> 00:20:08,130
But I'll try to
call this the rest

347
00:20:08,130 --> 00:20:10,780
mass because I will use the
word mass in other ways.

348
00:20:19,490 --> 00:20:23,130
We could also relate this
momentum to the velocity,

349
00:20:23,130 --> 00:20:27,220
and in doing that we will again
encounter this factor of gamma

350
00:20:27,220 --> 00:20:29,900
that we found
kinematically earlier.

351
00:20:29,900 --> 00:20:31,083
AUDIENCE: Yes, question.

352
00:20:31,083 --> 00:20:34,190
Did you forget to divide
by C squared there?

353
00:20:34,190 --> 00:20:42,420
PROFESSOR: Um, yes.

354
00:20:42,420 --> 00:20:45,330
There's too many C's here.

355
00:20:45,330 --> 00:20:46,220
Absolutely.

356
00:20:46,220 --> 00:20:48,380
It should have
units of momentum.

357
00:20:48,380 --> 00:20:52,030
So it should have units of
mass times velocity squared.

358
00:20:52,030 --> 00:20:52,920
Thank you.

359
00:20:52,920 --> 00:20:53,420
Thank you.

360
00:20:59,030 --> 00:21:04,769
So, the quantity gamma,
is the same quantity

361
00:21:04,769 --> 00:21:06,560
we encounter at the
beginning of the course

362
00:21:06,560 --> 00:21:08,650
when we talked about
time dilation and Lorentz

363
00:21:08,650 --> 00:21:12,070
contraction, it just depends
on the velocity and approaches

364
00:21:12,070 --> 00:21:14,830
infinity as the velocity
approaches the speed of light.

365
00:21:14,830 --> 00:21:19,360
And the physical momentum of
a particle, relativistically

366
00:21:19,360 --> 00:21:23,750
is equal to gamma
times M sub 0 times V.

367
00:21:23,750 --> 00:21:27,421
Where V is the ordinary
velocity-- this

368
00:21:27,421 --> 00:21:29,170
is special relativity,
we're not concerned

369
00:21:29,170 --> 00:21:32,800
with coordinate velocity
versus physical velocity yet--

370
00:21:32,800 --> 00:21:34,920
and gamma is that factor.

371
00:21:34,920 --> 00:21:39,040
So the momentum is larger than
what you would get a la Newton.

372
00:21:39,040 --> 00:21:43,260
The energy can also
be written down.

373
00:21:43,260 --> 00:21:48,480
And this formula
is one expression

374
00:21:48,480 --> 00:21:51,285
we can use to find the energy
in terms of the momentum.

375
00:21:59,380 --> 00:22:04,570
The energy in terms
of the momentum

376
00:22:04,570 --> 00:22:15,420
is M0 C squared squared
plus p squared C squared.

377
00:22:15,420 --> 00:22:18,830
And it also can be written
in terms of the velocity

378
00:22:18,830 --> 00:22:22,610
as just gamma times M0 C squared
where the velocity appears

379
00:22:22,610 --> 00:22:23,220
in the gamma.

380
00:22:39,390 --> 00:22:44,430
And a special case of this is
when the particle is at rest.

381
00:22:44,430 --> 00:22:49,660
Might as well write this, the
energy of a particle at rest,

382
00:22:49,660 --> 00:22:54,180
which we might call E0, is just
M0 C squared, which gets us

383
00:22:54,180 --> 00:22:58,350
back to where we started
with E equals MC squared.

384
00:23:01,624 --> 00:23:03,040
Now, I might just
say a quick word

385
00:23:03,040 --> 00:23:05,946
about where these formulas come
from, what idea underlies them.

386
00:23:05,946 --> 00:23:07,320
I'm not going to
make any attempt

387
00:23:07,320 --> 00:23:10,080
to derive them because
we just don't have time.

388
00:23:10,080 --> 00:23:13,510
It'd be easy to drive them,
but we want to do other things.

389
00:23:13,510 --> 00:23:15,550
But logically,
where they come from

390
00:23:15,550 --> 00:23:20,650
is simply the observation,
by Einstein originally,

391
00:23:20,650 --> 00:23:24,290
that if one has the Lorentz
transformations relating

392
00:23:24,290 --> 00:23:29,260
what one inertial observer sees
to another inertial observer,

393
00:23:29,260 --> 00:23:32,425
if one used those
transformations but used

394
00:23:32,425 --> 00:23:35,820
the Newtonian definitions
of energy and momentum,

395
00:23:35,820 --> 00:23:38,430
then you would find immediately
that if energy and momentum

396
00:23:38,430 --> 00:23:41,770
were conserved in one frame you
then know how to calculate what

397
00:23:41,770 --> 00:23:44,210
happens in other frames by
using the transformations,

398
00:23:44,210 --> 00:23:47,220
you'd find it would not
be conserved other frames.

399
00:23:47,220 --> 00:23:49,710
So if the conservation
of energy and momentum

400
00:23:49,710 --> 00:23:53,150
are to be a universal principle
of physics, which Einstein

401
00:23:53,150 --> 00:23:57,080
wanted to maintain, it would
be necessary to redefine

402
00:23:57,080 --> 00:23:58,630
energy and momentum.

403
00:23:58,630 --> 00:24:01,220
Now they're defined in
ways so that they approach

404
00:24:01,220 --> 00:24:04,110
the Newtonian values
for small velocities,

405
00:24:04,110 --> 00:24:07,030
but for velocities of the
order of the speed of light

406
00:24:07,030 --> 00:24:08,000
they're different.

407
00:24:08,000 --> 00:24:10,734
And they have the property--
not completely obvious from what

408
00:24:10,734 --> 00:24:13,150
we wrote, well it is actually
completely obvious from what

409
00:24:13,150 --> 00:24:14,780
we wrote-- they
have the property

410
00:24:14,780 --> 00:24:17,770
that if it's
conserved in on frame,

411
00:24:17,770 --> 00:24:19,420
it's conserved in all frames.

412
00:24:19,420 --> 00:24:21,291
And what makes it
obvious-- maybe

413
00:24:21,291 --> 00:24:23,290
the connections between
these different formulas

414
00:24:23,290 --> 00:24:25,400
are not obvious--
but I did tell you

415
00:24:25,400 --> 00:24:29,900
that p super mu transforms
as a four-vector,

416
00:24:29,900 --> 00:24:33,100
meaning it transforms the same
way as x super mu transforms,

417
00:24:33,100 --> 00:24:35,030
which are linear
transformations.

418
00:24:35,030 --> 00:24:38,517
And that's enough to guarantee
that if p mu is conserved

419
00:24:38,517 --> 00:24:40,600
in one frame, it has to
be conserved in all frames

420
00:24:40,600 --> 00:24:43,210
because delta p mu,
the change in p mu,

421
00:24:43,210 --> 00:24:45,815
would also be a four-vector,
and if a four-vector vanishes

422
00:24:45,815 --> 00:24:47,705
in one frame, it
vanishes in all frames.

423
00:24:56,940 --> 00:24:57,440
OK.

424
00:24:57,440 --> 00:24:59,773
Well I wanted to give you
sort of a quick example of how

425
00:24:59,773 --> 00:25:00,925
this works in practice.

426
00:25:13,170 --> 00:25:15,775
So I to just talk about the
energetics of a hydrogen atom.

427
00:25:25,770 --> 00:25:30,820
A hydrogen atom
consists of a proton,

428
00:25:30,820 --> 00:25:38,582
with a mass that we'll call
M sub p, and an electron,

429
00:25:38,582 --> 00:25:40,040
with the mass that
we'll call M sub

430
00:25:40,040 --> 00:25:46,680
E. And, as, if you imagine
starting with the electron

431
00:25:46,680 --> 00:25:50,890
and proton arbitrarily far
apart and bring them together,

432
00:25:50,890 --> 00:25:53,492
what you can discover
experimentally-- of if you know

433
00:25:53,492 --> 00:25:54,950
quantum mechanics
you can calculate

434
00:25:54,950 --> 00:25:58,960
theoretically--
energy is released,

435
00:25:58,960 --> 00:26:01,160
because you're releasing
potential energy as you

436
00:26:01,160 --> 00:26:03,430
bring the electron
into the atom.

437
00:26:03,430 --> 00:26:08,490
And the amount of
energy released

438
00:26:08,490 --> 00:26:20,070
is 13.6 electron volts.

439
00:26:20,070 --> 00:26:22,220
And the important
E equals MC squared

440
00:26:22,220 --> 00:26:24,010
implication which I
want to point out here

441
00:26:24,010 --> 00:26:26,544
is that loss of energy.

442
00:26:26,544 --> 00:26:28,460
This energy would be
extracted from the system

443
00:26:28,460 --> 00:26:30,192
as you made the hydrogen atom.

444
00:26:30,192 --> 00:26:32,400
The fact that you've extracted
energy from the system

445
00:26:32,400 --> 00:26:34,530
means that now the system
should have less energy

446
00:26:34,530 --> 00:26:36,314
than it had to start with.

447
00:26:36,314 --> 00:26:38,230
Initially it had the
rest energy of the proton

448
00:26:38,230 --> 00:26:42,760
and the rest of electronic, M
sub p C squared and M sub E C

449
00:26:42,760 --> 00:26:43,620
squared.

450
00:26:43,620 --> 00:26:45,530
Now it has less
energy by delta E,

451
00:26:45,530 --> 00:26:48,250
and that means it also
has to have less mass.

452
00:26:48,250 --> 00:26:50,120
So the mass of a
hydrogen atom is not

453
00:26:50,120 --> 00:26:52,190
the sum of the mass of
the proton and electron,

454
00:26:52,190 --> 00:26:54,300
as it would be in
Newtonian mechanics,

455
00:26:54,300 --> 00:26:56,940
but is less by an
amount proportional

456
00:26:56,940 --> 00:27:01,717
to this energy given off,
delta E, 13.6 electron volts.

457
00:27:01,717 --> 00:27:03,300
And just putting in
the C squares in I

458
00:27:03,300 --> 00:27:07,120
hope the right places, the
mass of a hydrogen atom

459
00:27:07,120 --> 00:27:09,780
will be equal to
the mass of a proton

460
00:27:09,780 --> 00:27:12,040
plus the mass of an
electron, but then

461
00:27:12,040 --> 00:27:18,800
minus the binding energy
expressed in mass units-- delta

462
00:27:18,800 --> 00:27:19,760
E over C squared.

463
00:27:36,420 --> 00:27:36,920
OK.

464
00:27:36,920 --> 00:27:38,900
So I guess, just
probably one more topic

465
00:27:38,900 --> 00:27:41,940
I want to talk about in terms of
just basic special relativity,

466
00:27:41,940 --> 00:27:45,840
and this actually get's
into general relativity.

467
00:27:45,840 --> 00:27:48,190
I wanted to find the
relativistic mass

468
00:27:48,190 --> 00:27:53,570
of any system just being its
energy divided by C squared.

469
00:27:58,970 --> 00:28:01,310
And this means that the
relativistic mass of a particle

470
00:28:01,310 --> 00:28:03,960
increases with its velocity.

471
00:28:03,960 --> 00:28:06,060
The energy of a
single moving particle

472
00:28:06,060 --> 00:28:08,280
is gamma times M0 C squared.

473
00:28:08,280 --> 00:28:09,940
That would say the
by this definition

474
00:28:09,940 --> 00:28:15,390
the relativistic mass of that
particle is gamma times M0.

475
00:28:15,390 --> 00:28:18,390
And I might mention that this
concept of relativistic mass

476
00:28:18,390 --> 00:28:22,410
is disparaged in many books
on special relativity.

477
00:28:22,410 --> 00:28:24,650
It's certainly a concept
that you can do without,

478
00:28:24,650 --> 00:28:27,840
so people who emotionally
are bothered by it

479
00:28:27,840 --> 00:28:29,720
can get along without
it because it is just

480
00:28:29,720 --> 00:28:31,350
the energy divided by C squared.

481
00:28:31,350 --> 00:28:34,750
And in fact a lot of work
in special relativity

482
00:28:34,750 --> 00:28:37,150
is done in units
where C is equal to 1

483
00:28:37,150 --> 00:28:39,060
and then it is just the energy.

484
00:28:39,060 --> 00:28:41,000
Especially if you
use C equals 1 you

485
00:28:41,000 --> 00:28:44,245
could dispense with this
concept of relativistic mass.

486
00:28:44,245 --> 00:28:45,870
We're not going to
be using C equals 1,

487
00:28:45,870 --> 00:28:48,940
so the phrase
relativistic mass will

488
00:28:48,940 --> 00:28:50,650
allow us to abbreviate
E divided by E

489
00:28:50,650 --> 00:28:53,460
squared in a convenient way.

490
00:28:53,460 --> 00:28:55,650
But the important thing
is not the definitions,

491
00:28:55,650 --> 00:28:57,233
the important thing
is what properties

492
00:28:57,233 --> 00:28:59,400
does this relativistic
mass have,

493
00:28:59,400 --> 00:29:01,710
whether or not one chooses
to call it relativistic mass

494
00:29:01,710 --> 00:29:03,920
or E divided by C squared.

495
00:29:03,920 --> 00:29:07,030
And it as an important
property concerning

496
00:29:07,030 --> 00:29:10,980
the gravitational field
created by matter.

497
00:29:10,980 --> 00:29:15,420
Now the gravitational field
of a single moving object

498
00:29:15,420 --> 00:29:17,130
is complicated.

499
00:29:17,130 --> 00:29:19,880
If we were talking
about, say, a moving star

500
00:29:19,880 --> 00:29:22,440
that was moving at a
velocity large enough

501
00:29:22,440 --> 00:29:26,947
so we care about relativity, the
way we calculate that actually

502
00:29:26,947 --> 00:29:28,947
as we start with the
Schwarzschild metric, which

503
00:29:28,947 --> 00:29:32,190
will describe the metric of
the star outside of the matter

504
00:29:32,190 --> 00:29:33,310
if it were stationary.

505
00:29:33,310 --> 00:29:35,101
And then you can just
make a transformation

506
00:29:35,101 --> 00:29:35,960
to a moving frame.

507
00:29:35,960 --> 00:29:37,501
You're allowed to
use any coordinates

508
00:29:37,501 --> 00:29:39,730
you want in general
relativity, so transforming

509
00:29:39,730 --> 00:29:43,680
to coordinates that describe
the moving frame is no problem.

510
00:29:43,680 --> 00:29:47,480
But it distorts the field in a
complicated way, nonetheless,

511
00:29:47,480 --> 00:29:49,330
a way that you can deal with.

512
00:29:51,940 --> 00:29:53,690
And what you find, of
course, is that what

513
00:29:53,690 --> 00:29:56,590
you get would be asymmetric
once you transform to the moving

514
00:29:56,590 --> 00:30:00,520
frame, it would show the
signs of the velocity

515
00:30:00,520 --> 00:30:03,500
that you used to transform
from the original spherically

516
00:30:03,500 --> 00:30:06,597
symmetric Schwarzschild
metric to the new frame.

517
00:30:06,597 --> 00:30:09,180
So the bottom line is that the
gravitational field of a moving

518
00:30:09,180 --> 00:30:13,330
object is not isotropic, it's
more complicated than that,

519
00:30:13,330 --> 00:30:16,310
just as the electric
field would be.

520
00:30:16,310 --> 00:30:19,860
But if we have a
gas of particles,

521
00:30:19,860 --> 00:30:24,160
which is pretty much what we
have in the early universe.

522
00:30:24,160 --> 00:30:35,290
If we have a gas of particles
in a box moving every which way,

523
00:30:35,290 --> 00:30:38,575
then if we thought of this
box as being an object

524
00:30:38,575 --> 00:30:40,700
that we're only going to
look at from the outside--

525
00:30:40,700 --> 00:30:43,880
a black box in the
classic use of the phrase

526
00:30:43,880 --> 00:30:51,320
black box-- the mass
of the black box

527
00:30:51,320 --> 00:30:56,352
really would just be the sum
of the relativistic masses

528
00:30:56,352 --> 00:30:57,060
of the particles.

529
00:31:00,250 --> 00:31:05,300
And the isotropy of the metric
that any one particle would

530
00:31:05,300 --> 00:31:08,670
generate would be canceled
by averaging or summing

531
00:31:08,670 --> 00:31:11,230
over all the particles going
every which direction, because

532
00:31:11,230 --> 00:31:17,162
on average the velocity of
particles inside the box is 0.

533
00:31:17,162 --> 00:31:19,620
So this relativistic mass, when
you're talking about a gas,

534
00:31:19,620 --> 00:31:22,780
really is the mass per particle.

535
00:31:22,780 --> 00:31:24,970
And if you divide
that by the volume

536
00:31:24,970 --> 00:31:26,820
really you do get
the mass density,

537
00:31:26,820 --> 00:31:28,720
which is a relevant
mass density in terms

538
00:31:28,720 --> 00:31:30,960
of talking about how
this matter would

539
00:31:30,960 --> 00:31:33,300
generate gravitational fields.

540
00:31:33,300 --> 00:31:33,800
Yes.

541
00:31:33,800 --> 00:31:35,299
AUDIENCE: So why
do some people have

542
00:31:35,299 --> 00:31:37,447
emotional problems with it?

543
00:31:37,447 --> 00:31:40,030
PROFESSOR: I think some people
have emotional problems with it

544
00:31:40,030 --> 00:31:42,860
because when one thinks
about pedagogy in a course,

545
00:31:42,860 --> 00:31:45,900
for example, one worries
about people confusing it

546
00:31:45,900 --> 00:31:49,067
with the rest mass.

547
00:31:49,067 --> 00:31:50,150
I think that's the reason.

548
00:31:54,390 --> 00:31:56,950
And I guess there are other
possible sources of confusion,

549
00:31:56,950 --> 00:31:59,020
so your question is a good one.

550
00:31:59,020 --> 00:32:02,340
Another source of confusion
is that this mass does not

551
00:32:02,340 --> 00:32:04,679
fit into an F
equals ma equation.

552
00:32:04,679 --> 00:32:06,220
So in calling this
the mass you might

553
00:32:06,220 --> 00:32:09,920
suggest to students who aren't
paying attention to every word

554
00:32:09,920 --> 00:32:12,880
that you say, you might
go ahead and put F equals

555
00:32:12,880 --> 00:32:14,440
ma for this mass,
that does not work.

556
00:32:18,420 --> 00:32:20,600
So it has some of the
properties of a mass,

557
00:32:20,600 --> 00:32:22,369
but not-- by no
means all of them.

558
00:32:27,719 --> 00:32:29,260
But in particular
for us is important

559
00:32:29,260 --> 00:32:31,634
because we're going to be
interested in the gravitational

560
00:32:31,634 --> 00:32:33,710
field of a gas
and then it really

561
00:32:33,710 --> 00:32:35,730
is the mass that
determines that.

562
00:32:40,640 --> 00:32:42,950
In the more formal language
in general relativity,

563
00:32:42,950 --> 00:32:46,182
it's the mass-- it's
the energy density, that

564
00:32:46,182 --> 00:32:48,640
appears in the equations that
produce gravitational fields,

565
00:32:48,640 --> 00:32:50,395
and then this really
is the energy density

566
00:32:50,395 --> 00:32:54,230
except for a factor
of C squared.

567
00:32:54,230 --> 00:32:55,020
OK, any questions?

568
00:32:55,020 --> 00:32:57,680
Because now I'm going
to leave this formalism

569
00:32:57,680 --> 00:33:02,921
and get into what role
this radiation could

570
00:33:02,921 --> 00:33:04,045
play in the early universe.

571
00:33:32,800 --> 00:33:37,106
So now I'd like to talk about
radiation in particular,

572
00:33:37,106 --> 00:33:39,450
and for now I mean
electromagnetic radiation

573
00:33:39,450 --> 00:33:42,530
just ordinary photons.

574
00:33:42,530 --> 00:33:46,330
And we're not accustomed
to thinking of light

575
00:33:46,330 --> 00:33:47,980
as having mass,
but we know light

576
00:33:47,980 --> 00:33:51,740
has energy and energy is
related to mass by a factor of C

577
00:33:51,740 --> 00:33:52,560
squared.

578
00:33:52,560 --> 00:33:57,000
So we can write down
the formula that

579
00:33:57,000 --> 00:34:01,330
says that rho is equal to
u divided by C squared.

580
00:34:01,330 --> 00:34:02,940
Where u is the
energy density, which

581
00:34:02,940 --> 00:34:04,525
we know electromagnetic
fields have.

582
00:34:10,699 --> 00:34:13,670
And rho will be the mass
density of radiation.

583
00:34:52,710 --> 00:34:57,810
Now, photons have
zero rest mass,

584
00:34:57,810 --> 00:35:01,630
so if we apply for example
this formula for a photon

585
00:35:01,630 --> 00:35:04,469
we would set M sub 0 equal to 0.

586
00:35:04,469 --> 00:35:06,510
And we said that photons
have zero rest mass what

587
00:35:06,510 --> 00:35:09,930
we mean is that
there's no lower limit

588
00:35:09,930 --> 00:35:11,961
to the energy a photon can have.

589
00:35:11,961 --> 00:35:13,460
In general, the
rest mass determines

590
00:35:13,460 --> 00:35:15,640
the lowest possible energy
a particle could have,

591
00:35:15,640 --> 00:35:17,020
which is when it's at rest.

592
00:35:17,020 --> 00:35:19,360
Photon can never
be rest and there's

593
00:35:19,360 --> 00:35:21,990
no lower limit to what
its energy can be.

594
00:35:27,350 --> 00:35:36,390
So for photons, M0 is
equal to 0 and that

595
00:35:36,390 --> 00:35:41,860
implies that the
energy of the photon

596
00:35:41,860 --> 00:35:44,790
is just C times the
magnitude of its momentum.

597
00:35:51,372 --> 00:35:53,955
And this formula one can derive
just for electromagnetic waves

598
00:35:53,955 --> 00:35:56,130
is purely classical
EM-- you don't

599
00:35:56,130 --> 00:35:58,057
have to be talking
about photons,

600
00:35:58,057 --> 00:36:00,390
but since it's true for a
classical electromagnetic wave

601
00:36:00,390 --> 00:36:02,890
it had better also be true
for photons because we think

602
00:36:02,890 --> 00:36:05,210
of this classical
electromagnetic waves

603
00:36:05,210 --> 00:36:09,340
as really being
made out of photons.

604
00:36:09,340 --> 00:36:11,520
So the energy that
exists in the universe

605
00:36:11,520 --> 00:36:13,970
in the form of
electromagnetic radiation

606
00:36:13,970 --> 00:36:18,230
will have an energy density,
which we know how to calculate.

607
00:36:18,230 --> 00:36:21,540
And we know to calculate the
momentum of any given photon.

608
00:36:21,540 --> 00:36:24,260
Now that will average
to zero, but nonetheless

609
00:36:24,260 --> 00:36:28,010
if we imagine talking about
a box of photons, which

610
00:36:28,010 --> 00:36:29,745
are bouncing off
the walls, the fact

611
00:36:29,745 --> 00:36:31,120
that each photon
carries momentum

612
00:36:31,120 --> 00:36:34,030
means that there will be
a pressure on the walls.

613
00:36:34,030 --> 00:36:36,450
And we will be interested
in that pressure,

614
00:36:36,450 --> 00:36:40,040
we'll be calculating a
formula for it in a minute.

615
00:36:40,040 --> 00:36:42,910
So, what we now
want to talk about

616
00:36:42,910 --> 00:36:48,555
is what happens when we put a
photon gas into the universe

617
00:36:48,555 --> 00:36:49,930
and allow the
universe to expand.

618
00:37:24,326 --> 00:37:26,200
What happens to the
radiation energy density,

619
00:37:26,200 --> 00:37:29,960
or equivalently mass density,
as the universe expands?

620
00:37:35,860 --> 00:37:38,260
This turns out to be
a very easy question

621
00:37:38,260 --> 00:37:40,770
to answer if we
think of the energy

622
00:37:40,770 --> 00:37:43,960
as being made out of photons.

623
00:37:43,960 --> 00:37:47,690
We would get an equivalent
identical answer

624
00:37:47,690 --> 00:37:52,057
if we use classical
Maxwell's equations

625
00:37:52,057 --> 00:37:53,640
and talked about how
energy density is

626
00:37:53,640 --> 00:37:55,954
a [INAUDIBLE] of
electromagnetic fields behaved.

627
00:37:55,954 --> 00:37:57,870
It would be more work
actually do it that way,

628
00:37:57,870 --> 00:38:00,140
but we would get
the same answer.

629
00:38:00,140 --> 00:38:02,210
In terms of photos,
we could simply

630
00:38:02,210 --> 00:38:09,010
notice that the number
density of photons-- photons

631
00:38:09,010 --> 00:38:11,690
are not going to disappear
as the universe expands,

632
00:38:11,690 --> 00:38:13,350
the number density
will just keep

633
00:38:13,350 --> 00:38:16,160
the same number of photons,
but as the universe expands

634
00:38:16,160 --> 00:38:19,172
those photons will
occupy a larger volume.

635
00:38:19,172 --> 00:38:20,630
So it's exactly
the same as what we

636
00:38:20,630 --> 00:38:23,380
said about
non-relativistic matter,

637
00:38:23,380 --> 00:38:26,840
the number density of
photons will fall like 1

638
00:38:26,840 --> 00:38:31,380
over the cube of the
scale factor, which just

639
00:38:31,380 --> 00:38:33,035
says that photons are conserved.

640
00:38:41,200 --> 00:38:46,520
And the volume of any
region grows like a cubed.

641
00:38:46,520 --> 00:38:49,477
I should also mention
that I'm using gamma here.

642
00:38:49,477 --> 00:38:51,060
Gamma of course also
sometimes meant 1

643
00:38:51,060 --> 00:38:53,435
over the square-root of 1
minus V squared over C squared,

644
00:38:53,435 --> 00:38:55,570
but besides that
use, gamma is also

645
00:38:55,570 --> 00:38:57,340
just a label that means photons.

646
00:39:00,920 --> 00:39:02,820
It comes from the
idea of gamma rays,

647
00:39:02,820 --> 00:39:05,180
but it's actually
used in this context

648
00:39:05,180 --> 00:39:08,092
for any kind of a
photon no matter

649
00:39:08,092 --> 00:39:09,550
what its frequency
is, whether it's

650
00:39:09,550 --> 00:39:13,010
a gamma ray or an x-ray or
visible light, or infrared.

651
00:39:13,010 --> 00:39:13,510
Yes.

652
00:39:13,510 --> 00:39:15,930
AUDIENCE: Is this
assumed for time average?

653
00:39:15,930 --> 00:39:18,350
Because photons can
be absorbed, right?

654
00:39:18,350 --> 00:39:20,286
I mean, at least like
in a small [? slice ?]

655
00:39:20,286 --> 00:39:23,680
can't there be a decided
non-relativistic matter?

656
00:39:23,680 --> 00:39:25,090
PROFESSOR: Is this
true on the--?

657
00:39:25,090 --> 00:39:28,650
Well, the validity of
this formula-- you're

658
00:39:28,650 --> 00:39:32,130
right this formula's not
exact, photons can be absorbed.

659
00:39:32,130 --> 00:39:34,940
But in terms of what happens
as the universe evolves,

660
00:39:34,940 --> 00:39:37,450
that's a very, very,
very minor process,

661
00:39:37,450 --> 00:39:39,700
especially when we're talking
about the early universe

662
00:39:39,700 --> 00:39:43,180
when there isn't really
anything around to absorb them.

663
00:39:43,180 --> 00:39:46,055
So, especially for the early
universe and even pretty well

664
00:39:46,055 --> 00:39:48,180
today, if we're talking
about the cosmic background

665
00:39:48,180 --> 00:39:51,380
radiation, which is the
bulk of the photons,

666
00:39:51,380 --> 00:39:55,390
this formula's a very
good approximation.

667
00:39:55,390 --> 00:39:55,890
Yes.

668
00:39:55,890 --> 00:39:59,320
AUDIENCE: Even though the
photons in the early universe

669
00:39:59,320 --> 00:40:01,280
created a lot of
massive particles,

670
00:40:01,280 --> 00:40:05,710
[INAUDIBLE] didn't that
affect the expansion?

671
00:40:05,710 --> 00:40:09,200
PROFESSOR: Are you're
asking, will the photons

672
00:40:09,200 --> 00:40:11,540
be important if there's a
lot of mass of particles?

673
00:40:11,540 --> 00:40:12,210
Is that your question?

674
00:40:12,210 --> 00:40:13,835
AUDIENCE: Would the
photons, won't they

675
00:40:13,835 --> 00:40:17,060
decay into matter antimatter?

676
00:40:17,060 --> 00:40:20,040
PROFESSOR: Will the photon
decay into matter antimatter?

677
00:40:20,040 --> 00:40:22,860
No, not really.

678
00:40:22,860 --> 00:40:26,670
It is, in principle,
possible for two photons

679
00:40:26,670 --> 00:40:30,750
to collide and produce an
electron positron pair.

680
00:40:30,750 --> 00:40:34,700
It's actually a rather small
cross section for that.

681
00:40:34,700 --> 00:40:38,510
And all of these processes
in the early universe

682
00:40:38,510 --> 00:40:40,400
will rapidly reach
an equilibrium,

683
00:40:40,400 --> 00:40:42,880
which we'll be talking about
more a little later, where

684
00:40:42,880 --> 00:40:45,640
there'll be just as many
photons converting into E plus E

685
00:40:45,640 --> 00:40:48,700
minus pairs as there will be
E plus E minus pairs colliding

686
00:40:48,700 --> 00:40:51,220
and making photons.

687
00:40:51,220 --> 00:40:55,360
So the early universe is
assumed to reach equilibrium

688
00:40:55,360 --> 00:40:58,817
very quickly, and all the
description we'll be giving

689
00:40:58,817 --> 00:41:00,650
will be a description
of the universe that's

690
00:41:00,650 --> 00:41:03,070
in thermal equilibrium
with these processes will

691
00:41:03,070 --> 00:41:05,110
tend to iron out.

692
00:41:05,110 --> 00:41:06,610
We will learn that
they don't always

693
00:41:06,610 --> 00:41:08,693
cancel each other because
the universe is cooling,

694
00:41:08,693 --> 00:41:11,935
and that means it can't be
exactly in thermal equilibrium.

695
00:41:11,935 --> 00:41:14,310
And there are some cases where
the effect of that cooling

696
00:41:14,310 --> 00:41:16,700
is significant and we'll
be talking about those.

697
00:41:16,700 --> 00:41:18,750
But for the most part,
if the cooling is slow,

698
00:41:18,750 --> 00:41:22,769
which it is for the most part
compared to other processes,

699
00:41:22,769 --> 00:41:24,435
everything stays in
thermal equilibrium.

700
00:41:28,990 --> 00:41:29,490
OK.

701
00:41:29,490 --> 00:41:30,190
Those are good questions.

702
00:41:30,190 --> 00:41:31,820
We've gotten a little ahead of
what I wanted to talk about.

703
00:41:31,820 --> 00:41:33,882
So for now I'm just
imagining a free photon gas,

704
00:41:33,882 --> 00:41:35,340
which is an excellent
approximation

705
00:41:35,340 --> 00:41:36,964
for the early universe.

706
00:41:36,964 --> 00:41:38,630
And those photons
just continue to exist

707
00:41:38,630 --> 00:41:41,300
as the universe expands,
so their number density

708
00:41:41,300 --> 00:41:44,401
falls off as 1 over a cubed.

709
00:41:44,401 --> 00:41:45,900
But there's another
affect that goes

710
00:41:45,900 --> 00:41:47,880
on which is that the
photons are redshifting.

711
00:41:47,880 --> 00:41:50,050
And we already know
about that, but now we're

712
00:41:50,050 --> 00:41:53,720
going to take it into account
in terms of the energy balances.

713
00:41:53,720 --> 00:42:00,710
So we know that the frequency
of a photon at some time t2

714
00:42:00,710 --> 00:42:05,410
divide by its frequency
at some time t1,

715
00:42:05,410 --> 00:42:16,530
and here nu equals
frequency, is just

716
00:42:16,530 --> 00:42:19,820
diluted by the expansion
of the universe.

717
00:42:19,820 --> 00:42:22,040
This ratio is 1
over 1 plus z, where

718
00:42:22,040 --> 00:42:27,050
z is the redshift
between these two times.

719
00:42:27,050 --> 00:42:30,670
But written out in
more detail which

720
00:42:30,670 --> 00:42:32,880
is a formula we'll
actually be using,

721
00:42:32,880 --> 00:42:36,240
is just a of t1
divided by a of t2.

722
00:42:39,660 --> 00:42:42,290
When the scale factor doubles,
all the frequency is half.

723
00:42:45,180 --> 00:42:47,200
And that means that all
the photons are lowering

724
00:42:47,200 --> 00:42:52,026
in frequency, and we also know
that photons are quantized.

725
00:43:03,160 --> 00:43:05,850
The energy of a photon
can't be any old thing,

726
00:43:05,850 --> 00:43:12,620
but in fact, the energy of a
photon is equal to h times nu.

727
00:43:35,550 --> 00:43:37,705
Where little h is what's
called Planck's constant.

728
00:43:46,770 --> 00:43:51,420
And numerically there
are various units

729
00:43:51,420 --> 00:43:56,950
you could express it
in, but it's 4.136 times

730
00:43:56,950 --> 00:44:01,875
10 to the minus 15th
electron volt seconds.

731
00:44:05,360 --> 00:44:07,620
So if you measure a
frequency an inverse seconds,

732
00:44:07,620 --> 00:44:09,875
you get an energy in electron
volts from that formula.

733
00:44:17,040 --> 00:44:19,475
The important thing
for now though is

734
00:44:19,475 --> 00:44:24,440
that this says that the
energy of each photon-- being

735
00:44:24,440 --> 00:44:26,690
proportional to its frequency,
and the frequency being

736
00:44:26,690 --> 00:44:29,760
proportional to 1 over
the scale factor--

737
00:44:29,760 --> 00:44:34,950
the energy of each photons is
proportional to 1 over a of t.

738
00:44:39,820 --> 00:44:41,620
And then the total
energy density

739
00:44:41,620 --> 00:44:44,610
of photons in
photon on gas which

740
00:44:44,610 --> 00:44:48,360
I'll call u sub gamma, the
energy density of the gas,

741
00:44:48,360 --> 00:44:51,940
can be thought of as the
number density of photons

742
00:44:51,940 --> 00:44:55,440
times the energy of each photon.

743
00:44:55,440 --> 00:44:57,150
And the number
density is falling off

744
00:44:57,150 --> 00:45:01,180
like 1 over a cubed, the energy
is falling off like 1 over a.

745
00:45:01,180 --> 00:45:04,170
And therefore, this
is proportional to 1

746
00:45:04,170 --> 00:45:05,500
over a to the fourth.

747
00:45:12,780 --> 00:45:16,020
So as the universe
expands, the density

748
00:45:16,020 --> 00:45:19,230
of non-relativistic matter
falls off like 1 over a cubed--

749
00:45:19,230 --> 00:45:22,060
as we've been talking
about some time now--

750
00:45:22,060 --> 00:45:24,550
but the energy of
radiation falls off faster,

751
00:45:24,550 --> 00:45:26,110
like 1 over a to
the fourth because

752
00:45:26,110 --> 00:45:27,568
of the red shifting
of the photons.

753
00:45:55,820 --> 00:46:00,170
OK, now once we know
this, we can ask ourselves

754
00:46:00,170 --> 00:46:03,920
what happens if we look at
our universe going backwards,

755
00:46:03,920 --> 00:46:07,440
knowing where we are
now where we come from?

756
00:46:07,440 --> 00:46:10,340
And if the energy
density of photons

757
00:46:10,340 --> 00:46:15,630
is falling off faster than
the energy density of matter,

758
00:46:15,630 --> 00:46:17,750
it would mean that the
ratio is getting smaller

759
00:46:17,750 --> 00:46:19,722
as we go forward in time.

760
00:46:19,722 --> 00:46:21,430
But that of course
implies that the ratio

761
00:46:21,430 --> 00:46:24,397
gets larger as we go
backwards in time.

762
00:46:24,397 --> 00:46:26,230
So as we go backwards
in time, the radiation

763
00:46:26,230 --> 00:46:28,610
becomes more and more
important, and there actually

764
00:46:28,610 --> 00:46:32,590
is going to be a time when the
radiation will equal the matter

765
00:46:32,590 --> 00:46:36,390
and at earlier times the
radiation will dominate.

766
00:46:36,390 --> 00:46:41,300
Today I'll just give
you a number for now,

767
00:46:41,300 --> 00:46:42,990
we'll learn later
how to calculate it,

768
00:46:42,990 --> 00:46:46,440
but for today the
total radiation energy

769
00:46:46,440 --> 00:46:54,100
density in the universe
is equal to 7.01 times 10

770
00:46:54,100 --> 00:46:57,885
to the minus 14 joules
per meter cubed.

771
00:47:07,162 --> 00:47:09,245
And this actually includes
two kinds of radiation,

772
00:47:09,245 --> 00:47:14,980
it includes photons
and also neutrinos,

773
00:47:14,980 --> 00:47:17,220
which at least in
the early universe

774
00:47:17,220 --> 00:47:20,057
behaved just like radiation.

775
00:47:20,057 --> 00:47:22,640
And we'll be talking more about
neutrinos later so don't worry

776
00:47:22,640 --> 00:47:25,820
if you don't have any
idea what a neutrino is.

777
00:47:29,240 --> 00:47:32,090
But for now it's just another
contribution to the radiation,

778
00:47:32,090 --> 00:47:34,494
and we can measure
basically this is all

779
00:47:34,494 --> 00:47:35,910
based on measuring
the temperature

780
00:47:35,910 --> 00:47:38,007
of the cosmic microwave
background radiation--

781
00:47:38,007 --> 00:47:40,090
and we'll learn later how
to make that conversion.

782
00:47:40,090 --> 00:47:41,140
But once you measure
the temperature

783
00:47:41,140 --> 00:47:42,530
of the cosmic
background radiation

784
00:47:42,530 --> 00:47:45,010
and have a theory about how
many neutrinos there should be,

785
00:47:45,010 --> 00:47:46,718
that's actually all
theoretical and we'll

786
00:47:46,718 --> 00:47:48,320
talk about that later as well.

787
00:47:48,320 --> 00:47:49,730
One can determine
what the energy

788
00:47:49,730 --> 00:47:54,310
density of that radiation is.

789
00:47:54,310 --> 00:48:02,270
And it corresponds to a mass
density just dividing by C

790
00:48:02,270 --> 00:48:13,030
squared of 7.80 times 10 to the
minus 31 kilograms per meter

791
00:48:13,030 --> 00:48:14,234
cubed.

792
00:48:14,234 --> 00:48:15,650
And when I think
of mass densities

793
00:48:15,650 --> 00:48:19,020
I always like to think of in
centimeters per-- excuse me,

794
00:48:19,020 --> 00:48:21,684
grams per centimeter
cubed because I'm

795
00:48:21,684 --> 00:48:24,100
used to the density of water
being one gram per centimeter

796
00:48:24,100 --> 00:48:26,420
cubed and I like to be able
to make that comparison.

797
00:48:26,420 --> 00:48:31,220
So just making that conversion,
usually I use SI units,

798
00:48:31,220 --> 00:48:35,090
but some things just seem to
make more sense in other units.

799
00:48:35,090 --> 00:48:41,026
So it's 10 to the minus 34
grams per centimeter cubed,

800
00:48:41,026 --> 00:48:44,270
so 10 to the minus 34, or
maybe 10 to the minus 33,

801
00:48:44,270 --> 00:48:47,260
times the density of water.

802
00:48:47,260 --> 00:48:50,130
And this is incredibly
low even compared

803
00:48:50,130 --> 00:48:53,260
to the critical density
of our universe,

804
00:48:53,260 --> 00:48:54,710
and the actual
density we know is

805
00:48:54,710 --> 00:48:57,020
very near this critical density.

806
00:48:57,020 --> 00:48:59,800
Let me remind you that
the critical density

807
00:48:59,800 --> 00:49:03,180
we derived a formula for,
and when we put numbers

808
00:49:03,180 --> 00:49:05,050
into that formula
we found that was

809
00:49:05,050 --> 00:49:10,310
equal to 1.88 times
little h0 squared,

810
00:49:10,310 --> 00:49:13,670
which is Hubble's units--
Hubble's constant in units

811
00:49:13,670 --> 00:49:17,820
of 100 kilometers per
second per megaparsec--

812
00:49:17,820 --> 00:49:21,830
I'll write that in
a second-- times 10

813
00:49:21,830 --> 00:49:25,605
to the minus 29 grams
per centimeter cubed.

814
00:49:45,610 --> 00:49:48,770
So just writing down the
equation for little h sub 0

815
00:49:48,770 --> 00:49:54,390
is where capital H sub 0,
Hubble's expansion rate,

816
00:49:54,390 --> 00:50:02,320
is equal to 100
times little h sub 0

817
00:50:02,320 --> 00:50:07,150
kilometers per second
per megaparsec.

818
00:50:15,410 --> 00:50:17,298
Yep.

819
00:50:17,298 --> 00:50:20,868
AUDIENCE: What's
the, I guess, what's

820
00:50:20,868 --> 00:50:23,010
the motivation of
normalizing the Hubble

821
00:50:23,010 --> 00:50:24,930
constant in this way?

822
00:50:24,930 --> 00:50:26,900
PROFESSOR: Well, I think
the real motivation is

823
00:50:26,900 --> 00:50:30,780
that the astronomers like these
peculiar units of kilometers

824
00:50:30,780 --> 00:50:34,230
per second per megaparsec, but
if your favorite unit is called

825
00:50:34,230 --> 00:50:36,740
a kilometer per
second per megaparsec

826
00:50:36,740 --> 00:50:39,670
you don't want to have to
say those units very often.

827
00:50:39,670 --> 00:50:42,600
So H0 is dimensionless, so
it's a dimensionless way

828
00:50:42,600 --> 00:50:44,160
of talking about
the Hubble constant.

829
00:50:49,006 --> 00:50:50,630
But that's the only
importance, there's

830
00:50:50,630 --> 00:50:52,930
no real-- no deep
significance to it.

831
00:50:52,930 --> 00:50:56,750
But it's a standard notation,
so it's worth knowing.

832
00:50:56,750 --> 00:51:02,820
And then finally,
we can write down

833
00:51:02,820 --> 00:51:10,320
how much the radiation
contributes to omega-- omega

834
00:51:10,320 --> 00:51:14,510
sub r, r is going to
indicate radiation.

835
00:51:14,510 --> 00:51:17,164
The notation, by
the way, will be--

836
00:51:17,164 --> 00:51:19,390
and I realized I've already
violated that notation.

837
00:51:19,390 --> 00:51:22,551
This really should have been r.

838
00:51:22,551 --> 00:51:26,410
Well, I'm not going to change
it, it'll get too messy.

839
00:51:26,410 --> 00:51:29,080
But I'm going to start using the
notation where gamma indicates

840
00:51:29,080 --> 00:51:32,199
photons and little r
indicates radiation.

841
00:51:32,199 --> 00:51:34,740
And the difference is that there
are other kinds of radiation

842
00:51:34,740 --> 00:51:37,130
besides photons, in
particular we've already

843
00:51:37,130 --> 00:51:41,020
added in neutrinos in part of
what we're calling radiation.

844
00:51:41,020 --> 00:51:44,190
So omega sub r, which now
includes photons and neutrinos,

845
00:51:44,190 --> 00:51:47,560
is just defined to be the
mass density radiation divided

846
00:51:47,560 --> 00:51:49,650
by the critical density.

847
00:51:49,650 --> 00:51:52,830
And that turns out, when
you combine these numbers,

848
00:51:52,830 --> 00:52:00,035
to be 4.15 times 10 to the minus
5 little h0 to the minus 2.

849
00:52:03,870 --> 00:52:13,230
And then for h0 equals 0.67,
which is the Planck satellite

850
00:52:13,230 --> 00:52:17,460
value for little
h0, we finally get

851
00:52:17,460 --> 00:52:28,270
omega sub r is equal to 9.2
times 10 to the minus 5.

852
00:52:28,270 --> 00:52:30,500
So roughly 10 to the minus 4.

853
00:52:30,500 --> 00:52:33,110
The fraction of the mass
density, or energy density

854
00:52:33,110 --> 00:52:39,020
today is about 10 to the minus
4 fraction and radiation.

855
00:52:39,020 --> 00:52:41,620
And actually as I write
this, this actually

856
00:52:41,620 --> 00:52:46,000
calls to mind another reason for
defining little h sub 0, which

857
00:52:46,000 --> 00:52:48,920
is that if you write formulas
in terms of little h sub 0,

858
00:52:48,920 --> 00:52:51,640
they remain valid between
one year and next year.

859
00:52:51,640 --> 00:52:55,100
The observational value
of the Hubble parameter

860
00:52:55,100 --> 00:52:58,510
is still floating
around and differs,

861
00:52:58,510 --> 00:53:00,800
for example, every time
I teach this course.

862
00:53:00,800 --> 00:53:02,640
So the formulas in
terms of little h sub 0

863
00:53:02,640 --> 00:53:06,380
stay, and then you plug in the
current value of little h sub 0

864
00:53:06,380 --> 00:53:12,250
to get the best value that
one can currently write down.

865
00:53:12,250 --> 00:53:16,135
Now we know the Hubble constant
to within a few percent, which

866
00:53:16,135 --> 00:53:18,960
is much better than it used to
be, but it's still floating.

867
00:53:18,960 --> 00:53:22,340
The Planck value was somewhat
lower than the previous except

868
00:53:22,340 --> 00:53:28,520
a value, which was about 0.70.

869
00:53:28,520 --> 00:53:29,430
OK.

870
00:53:29,430 --> 00:53:34,400
So we know enough information
to extrapolate backwards

871
00:53:34,400 --> 00:53:37,120
and calculate when
this radiation would've

872
00:53:37,120 --> 00:53:39,960
equaled the energy
density of matter.

873
00:53:55,667 --> 00:53:57,250
Because we know how
the ratio changes.

874
00:53:57,250 --> 00:53:59,740
It changes by a factor
of a, the scale factor,

875
00:53:59,740 --> 00:54:02,130
because the energy density
of non-relativistic matter

876
00:54:02,130 --> 00:54:03,892
is falling off like
1 over a cubed.

877
00:54:03,892 --> 00:54:06,100
The energy density of
radiation is flowing off like 1

878
00:54:06,100 --> 00:54:07,240
over a to the fourth.

879
00:54:07,240 --> 00:54:09,340
So the ratio between
them just changes

880
00:54:09,340 --> 00:54:15,260
by a factor of a,
decreasing as a gets larger.

881
00:54:15,260 --> 00:54:23,390
So we can write RHO radiation
of t divided by RHO matter of t.

882
00:54:26,740 --> 00:54:28,750
The m there means
non-relativistic matter.

883
00:54:41,740 --> 00:54:46,640
This is just equal
to the current value.

884
00:54:46,640 --> 00:54:52,290
And the current value is
gotten by taking that number

885
00:54:52,290 --> 00:55:02,320
and, where I forgot to write
down is that omega matter today

886
00:55:02,320 --> 00:55:09,780
is about 0.30.

887
00:55:09,780 --> 00:55:16,960
So the ratio of the two, omega
radiation over omega matter,

888
00:55:16,960 --> 00:55:24,000
which is the same as RHO
radiation over RHO matter,

889
00:55:24,000 --> 00:55:35,380
is about 3.1 times
10 to the minus 4.

890
00:55:38,700 --> 00:55:42,160
And that number is about
to appear in this equation.

891
00:55:42,160 --> 00:55:44,280
If we want the ratio
as a function of time,

892
00:55:44,280 --> 00:55:51,500
we start with this value today,
3.1 times 10 to the minus 4,

893
00:55:51,500 --> 00:55:56,050
and then we can just multiply
that by the scale factor

894
00:55:56,050 --> 00:55:59,710
today divided by the
scale factor at the time

895
00:55:59,710 --> 00:56:02,410
that we want to know it, because
we know it falls off as 1

896
00:56:02,410 --> 00:56:03,420
over the scale factor.

897
00:56:08,080 --> 00:56:10,090
And by putting an a
of t0 here, this just

898
00:56:10,090 --> 00:56:11,840
guarantees that if
we let t equal t0,

899
00:56:11,840 --> 00:56:14,630
we get this number, which is
the right ratio for today.

900
00:56:21,880 --> 00:56:22,380
OK.

901
00:56:22,380 --> 00:56:25,250
Now it's just a
matter of arithmetic.

902
00:56:25,250 --> 00:56:37,090
We also know that for a
matter-dominated universe.

903
00:56:37,090 --> 00:56:42,910
And for now, we're going to
estimate when the radiation

904
00:56:42,910 --> 00:56:44,881
energy will equal
the matter density.

905
00:56:44,881 --> 00:56:46,130
This will only be an estimate.

906
00:56:46,130 --> 00:56:47,713
We're going to
estimate it by assuming

907
00:56:47,713 --> 00:56:50,230
that we can approximate the
universe as matter dominated

908
00:56:50,230 --> 00:56:55,500
between now, all the way
back until that time.

909
00:56:55,500 --> 00:56:58,110
There are two errors
in that calculation.

910
00:56:58,110 --> 00:57:01,600
We're not taking into account
here the era of acceleration

911
00:57:01,600 --> 00:57:03,600
where dark energy is
playing a significant role.

912
00:57:03,600 --> 00:57:05,110
We'll learn later
how to do that.

913
00:57:05,110 --> 00:57:08,110
And also, as we approach
this time when they're

914
00:57:08,110 --> 00:57:10,960
making equal contributions,
we will run into a regime

915
00:57:10,960 --> 00:57:13,050
where the contribution
of the radiation

916
00:57:13,050 --> 00:57:14,930
itself will be relevant.

917
00:57:14,930 --> 00:57:17,310
So this is only an estimate.

918
00:57:17,310 --> 00:57:20,257
But what we're
going to do is we're

919
00:57:20,257 --> 00:57:21,840
going to assume that
we can treat this

920
00:57:21,840 --> 00:57:25,530
as a matter-dominated universe
with a of t proportional

921
00:57:25,530 --> 00:57:28,524
to the 2/3.

922
00:57:28,524 --> 00:57:30,440
And then we can plug
numbers into this formula

923
00:57:30,440 --> 00:57:32,960
and ask, when was the ratio 1?

924
00:57:32,960 --> 00:57:34,720
And when the ratio
was 1, we call

925
00:57:34,720 --> 00:57:39,100
that the time of equality, using
the subscript EQ for equality

926
00:57:39,100 --> 00:57:43,110
to indicate anything having to
do with that crossing point.

927
00:57:43,110 --> 00:57:49,590
And what we find is
that the z of equality,

928
00:57:49,590 --> 00:57:57,085
and z is just the ratio
of the a's, is-- according

929
00:57:57,085 --> 00:58:01,120
to this calculation, it
would be 3.1 times 10

930
00:58:01,120 --> 00:58:05,230
to the minus 4 minus 1.

931
00:58:05,230 --> 00:58:07,604
I fibbed when I said that
z is the ratio of the a's.

932
00:58:07,604 --> 00:58:08,770
It's offset by a little bit.

933
00:58:08,770 --> 00:58:11,470
It's 1 plus z that's
the ratio of the a's.

934
00:58:11,470 --> 00:58:14,440
And that's why there's
a minus 1 there.

935
00:58:14,440 --> 00:58:19,900
And numerically, this is
about equal to 32,000.

936
00:58:19,900 --> 00:58:22,670
So if we look back in the
history of the universe,

937
00:58:22,670 --> 00:58:26,870
we can define looking back
in terms of the redshift.

938
00:58:26,870 --> 00:58:31,650
If you look back to a redshift
of 3,200, we get to the time

939
00:58:31,650 --> 00:58:35,210
when matter and radiation
had the same energy density.

940
00:58:38,140 --> 00:58:41,916
And we can know what time
that is if we assume t

941
00:58:41,916 --> 00:58:44,966
to the 2/3, which again is
only a crude approximation.

942
00:58:44,966 --> 00:58:47,340
We don't necessarily expect
to get the right answer here.

943
00:58:47,340 --> 00:58:49,423
But we expect to get the
right order of magnitude.

944
00:58:57,410 --> 00:59:02,500
So t-equality, according
to this situation,

945
00:59:02,500 --> 00:59:21,100
would be about 75,000 years
after the Big Bang-- just

946
00:59:21,100 --> 00:59:23,150
converting the scale
factor that we just

947
00:59:23,150 --> 00:59:26,823
calculated to a time using that
formula, treating this as t

948
00:59:26,823 --> 00:59:29,000
to the 2/3.

949
00:59:29,000 --> 00:59:32,080
So this says that about
75,000 years after the Big

950
00:59:32,080 --> 00:59:35,341
Bang, the energy densities
of matter and radiation

951
00:59:35,341 --> 00:59:35,840
were equal.

952
00:59:35,840 --> 00:59:37,256
In the earlier
times, the universe

953
00:59:37,256 --> 00:59:38,420
was radiation dominated.

954
00:59:38,420 --> 00:59:41,521
The radiation exceeded the
matter in its energy density.

955
00:59:41,521 --> 00:59:42,020
Yes?

956
00:59:42,020 --> 00:59:45,020
AUDIENCE: It seems like
zEQ is relatively about 31

957
00:59:45,020 --> 00:59:45,520
[INAUDIBLE].

958
00:59:52,404 --> 00:59:54,320
PROFESSOR: You might be
right if we would just

959
00:59:54,320 --> 00:59:55,278
look to these formulas.

960
00:59:55,278 --> 00:59:57,460
When I calculated
this at home, I

961
00:59:57,460 --> 00:59:59,170
kept more decimal
places all the way

962
00:59:59,170 --> 01:00:03,120
and rounded off each answer
to one significant figure.

963
01:00:03,120 --> 01:00:05,750
And that's not the same as
taking the answer to one

964
01:00:05,750 --> 01:00:07,380
significant figure
and calculating

965
01:00:07,380 --> 01:00:09,320
and then rounding off
to ones in every figure.

966
01:00:09,320 --> 01:00:13,610
So I think there's always an
ambiguity of, roughly speaking,

967
01:00:13,610 --> 01:00:15,794
1 in the last decimal
place whenever

968
01:00:15,794 --> 01:00:16,960
you're rounding numbers off.

969
01:00:16,960 --> 01:00:18,932
AUDIENCE: So we just
plug that [INAUDIBLE]

970
01:00:23,370 --> 01:00:25,572
PROFESSOR: 3,220--
starting with this

971
01:00:25,572 --> 01:00:27,780
or starting with a more
accurate number than the 3.1.

972
01:00:27,780 --> 01:00:28,780
AUDIENCE: Just 1/3.1.

973
01:00:28,780 --> 01:00:29,660
PROFESSOR: 1/3.1.

974
01:00:29,660 --> 01:00:30,160
OK.

975
01:00:30,160 --> 01:00:31,370
So, OK?

976
01:00:31,370 --> 01:00:34,256
AUDIENCE: Sorry.

977
01:00:34,256 --> 01:00:37,238
AUDIENCE: Wait no,
[INAUDIBLE] like we have 1

978
01:00:37,238 --> 01:00:38,585
over 3 times 10 negative 4.

979
01:00:38,585 --> 01:00:40,040
That's 3 times 10--

980
01:00:40,040 --> 01:00:41,156
PROFESSOR: It's 3.1.

981
01:00:41,156 --> 01:00:42,560
AUDIENCE: You have 1 over 3.1.

982
01:00:42,560 --> 01:00:43,476
PROFESSOR: 1 over 3.1.

983
01:00:43,476 --> 01:00:44,747
So you have to divide.

984
01:00:44,747 --> 01:00:46,118
AUDIENCE:I do that all the time.

985
01:00:46,118 --> 01:00:48,414
AUDIENCE: [INAUDIBLE]

986
01:00:48,414 --> 01:00:50,080
PROFESSOR: So apparently
it's even right

987
01:00:50,080 --> 01:00:52,100
if you just calculate with that.

988
01:00:52,100 --> 01:00:54,300
But there is actually
some ambiguity.

989
01:00:54,300 --> 01:00:57,630
The numbers I'm
giving you probably

990
01:00:57,630 --> 01:01:00,620
have some uncertainty
of 1 in the last digit,

991
01:01:00,620 --> 01:01:03,880
depending on how you calculate.

992
01:01:03,880 --> 01:01:05,610
But the number they
give you, I think

993
01:01:05,610 --> 01:01:08,600
they're the ones that
you get if you start

994
01:01:08,600 --> 01:01:13,524
using this and the 0.67 and
from then on do everything

995
01:01:13,524 --> 01:01:14,940
to large numbers
in decimal places

996
01:01:14,940 --> 01:01:16,280
and round off at each stage.

997
01:01:16,280 --> 01:01:18,639
You'll get the numbers
I've given you.

998
01:01:18,639 --> 01:01:20,430
In any case, all these
are really, at best,

999
01:01:20,430 --> 01:01:21,450
order of magnitude estimates.

1000
01:01:21,450 --> 01:01:24,110
So worrying about whether or
not the last figure is accurate

1001
01:01:24,110 --> 01:01:26,570
is not a big deal.

1002
01:01:26,570 --> 01:01:27,737
Yes?

1003
01:01:27,737 --> 01:01:29,195
AUDIENCE: I don't
really understand

1004
01:01:29,195 --> 01:01:32,093
how you got t [INAUDIBLE]
without telling was a of t dot

1005
01:01:32,093 --> 01:01:33,059
is [INAUDIBLE].

1006
01:01:35,626 --> 01:01:36,500
PROFESSOR: I'm sorry.

1007
01:01:36,500 --> 01:01:38,166
There is actually a
piece of information

1008
01:01:38,166 --> 01:01:39,520
I used I forgot to write here.

1009
01:01:39,520 --> 01:01:41,140
You're absolutely right.

1010
01:01:41,140 --> 01:01:46,654
I used t0 is equal to 13.8
times 10 to the nine years.

1011
01:01:46,654 --> 01:01:48,320
And then everything
can be related to t0

1012
01:01:48,320 --> 01:01:51,214
if you know how things
are proportional to t.

1013
01:01:51,214 --> 01:01:52,130
You're absolute right.

1014
01:01:52,130 --> 01:01:53,910
I did not give you
all the information

1015
01:01:53,910 --> 01:01:55,201
necessary for that calculation.

1016
01:01:57,970 --> 01:01:58,940
Now I think I have.

1017
01:01:58,940 --> 01:02:01,000
I haven't done the
arithmetic for you.

1018
01:02:01,000 --> 01:02:04,361
But otherwise it's all there.

1019
01:02:04,361 --> 01:02:04,860
OK.

1020
01:02:04,860 --> 01:02:10,960
Now I might mention
that in Ryden's book,

1021
01:02:10,960 --> 01:02:14,610
she does the calculation taking
into account everything--

1022
01:02:14,610 --> 01:02:17,660
matter, radiation,
cosmological constants.

1023
01:02:17,660 --> 01:02:28,100
And her number for
t-equality is 47,000 years,

1024
01:02:28,100 --> 01:02:30,980
which verifies that we have
the right order of magnitude.

1025
01:02:30,980 --> 01:02:32,480
And actually, the
biggest difference

1026
01:02:32,480 --> 01:02:34,840
between her number
and my number is not

1027
01:02:34,840 --> 01:02:37,090
that she's taken into account
these more sophisticated

1028
01:02:37,090 --> 01:02:39,330
things, but rather that
she used a different value

1029
01:02:39,330 --> 01:02:41,604
for the Hubble expansion
rate than I'm using.

1030
01:02:41,604 --> 01:02:44,020
She's using a value that was
current at the time she wrote

1031
01:02:44,020 --> 01:02:47,350
her book, which was
like '72, I think.

1032
01:02:47,350 --> 01:02:50,339
h0 equals 0.72 instead
of [INAUDIBLE] 0.67.

1033
01:02:50,339 --> 01:02:52,380
And that does make a
significant difference here.

1034
01:02:55,020 --> 01:02:57,700
But either of these numbers
are, I think, probably

1035
01:02:57,700 --> 01:03:00,420
within the range of uncertainty
of when it really happened.

1036
01:03:00,420 --> 01:03:05,450
But it's on that scale, on the
scale of 50,000 years, 100,000

1037
01:03:05,450 --> 01:03:08,980
years, something of that order.

1038
01:03:08,980 --> 01:03:11,260
So there was a
significantly long period

1039
01:03:11,260 --> 01:03:13,260
compared to human lifetime
when the universe was

1040
01:03:13,260 --> 01:03:14,630
radiation dominated.

1041
01:03:14,630 --> 01:03:17,340
But it's a very small fraction
of the overall history

1042
01:03:17,340 --> 01:03:18,970
of the universe,
but nonetheless does

1043
01:03:18,970 --> 01:03:23,340
have important features that
happened during that time

1044
01:03:23,340 --> 01:03:23,970
period.

1045
01:03:23,970 --> 01:03:25,860
Now, if we want to
understand those features,

1046
01:03:25,860 --> 01:03:29,290
we have to understand how a
radiation-dominated universe

1047
01:03:29,290 --> 01:03:31,830
evolves, which is what
we're going to get to next.

1048
01:03:48,641 --> 01:03:49,140
OK.

1049
01:03:49,140 --> 01:03:50,724
The next little
chapter than I'm going

1050
01:03:50,724 --> 01:03:56,600
to be talking
about-- the dynamics

1051
01:03:56,600 --> 01:03:58,295
of a radiation-dominated
universe.

1052
01:04:09,895 --> 01:04:11,520
This is a chapter
that you more or less

1053
01:04:11,520 --> 01:04:13,487
get to work out the
equations for yourself

1054
01:04:13,487 --> 01:04:14,820
on one of the homework problems.

1055
01:04:14,820 --> 01:04:16,780
That's part of this week's set.

1056
01:04:16,780 --> 01:04:21,460
So I will try here
to outline the logic.

1057
01:04:21,460 --> 01:04:24,550
But because all the calculations
are in the homework,

1058
01:04:24,550 --> 01:04:28,300
I will basically skip the
calculations themselves,

1059
01:04:28,300 --> 01:04:30,340
and let you do
them for yourselves

1060
01:04:30,340 --> 01:04:32,460
as part of the homework.

1061
01:04:32,460 --> 01:04:41,670
But where we start is we have
written down Friedman equations

1062
01:04:41,670 --> 01:04:43,730
for the matter-dominated case.

1063
01:04:43,730 --> 01:04:46,095
And I'll start by reminding
us what those were.

1064
01:05:09,690 --> 01:05:12,870
And then in addition to
these two equations, which

1065
01:05:12,870 --> 01:05:16,120
describe our expanding universe,
which we've derived sometime

1066
01:05:16,120 --> 01:05:19,540
ago for-- and to
remind us here, this

1067
01:05:19,540 --> 01:05:23,035
is for a matter-dominated
universe.

1068
01:05:27,100 --> 01:05:30,070
And matter-dominated means
non-relativistic matter

1069
01:05:30,070 --> 01:05:31,400
dominated.

1070
01:05:31,400 --> 01:05:34,230
And going along with
these equations,

1071
01:05:34,230 --> 01:05:38,970
we also know that RHO of t
for non-relativistic matter

1072
01:05:38,970 --> 01:05:42,340
falls off like one
over a cubed of t.

1073
01:05:47,190 --> 01:05:50,740
This can be converted into a
differential equation for RHO.

1074
01:05:50,740 --> 01:05:53,840
That is, we can calculate
Rho dot from this equation.

1075
01:05:53,840 --> 01:05:57,270
And the way to see that
is probably most easily

1076
01:05:57,270 --> 01:06:06,410
to start on a new blackboard
and write that equation not as

1077
01:06:06,410 --> 01:06:08,620
a proportionality, since
it's hard to differentiate

1078
01:06:08,620 --> 01:06:10,575
a proportionality,
but we can write it

1079
01:06:10,575 --> 01:06:13,430
in an arbitrary constant
of proportionality.

1080
01:06:13,430 --> 01:06:15,490
And then it becomes an equality.

1081
01:06:15,490 --> 01:06:18,080
So I'm going to write
the equation as RHO of t

1082
01:06:18,080 --> 01:06:23,680
is equal to some constant,
b, divided by a cubed of t.

1083
01:06:23,680 --> 01:06:25,890
And this we know how
to differentiate.

1084
01:06:25,890 --> 01:06:29,850
We can write Rho dot is
equal to minus b over a

1085
01:06:29,850 --> 01:06:33,050
to the fourth of t times a dot.

1086
01:06:36,810 --> 01:06:41,060
And that is equal to minus 3.

1087
01:06:41,060 --> 01:06:43,360
I'm sorry--there's a 3 here.

1088
01:06:43,360 --> 01:06:48,130
Minus 3 times a dot over
a times the original RHO.

1089
01:06:52,130 --> 01:06:53,860
So we can forget the
intermediate steps,

1090
01:06:53,860 --> 01:06:56,860
and we just arrived
at the equation

1091
01:06:56,860 --> 01:07:01,756
that RHO dot is equal to minus
3 a dot over a times RHO.

1092
01:07:01,756 --> 01:07:03,380
And we can think of
that as going along

1093
01:07:03,380 --> 01:07:05,630
with equations one and two.

1094
01:07:05,630 --> 01:07:07,850
Maybe I'll even
give that a number.

1095
01:07:07,850 --> 01:07:10,770
Equation three
will be RHO dot is

1096
01:07:10,770 --> 01:07:15,285
equal to minus 3 a
dot over a times RHO.

1097
01:07:18,850 --> 01:07:21,970
Now for radiation,
there will be a 4 here.

1098
01:07:21,970 --> 01:07:24,540
The 4 will arrive the same
way as the 3 arrives there.

1099
01:07:24,540 --> 01:07:28,740
It's just the power that
appeared in the factor of a.

1100
01:07:28,740 --> 01:07:30,540
So for radiation, this
last formula we know

1101
01:07:30,540 --> 01:07:33,480
is going to be modified,
which is the key point.

1102
01:07:33,480 --> 01:07:35,920
Note that these three
formulas are not

1103
01:07:35,920 --> 01:07:38,750
independent of each other.

1104
01:07:38,750 --> 01:07:42,050
If we know, for example,
equation one, which

1105
01:07:42,050 --> 01:07:44,006
is an equation for
a dot, we could

1106
01:07:44,006 --> 01:07:45,630
differentiate that
with respect to time

1107
01:07:45,630 --> 01:07:47,980
and get an equation
for a double dot.

1108
01:07:47,980 --> 01:07:51,120
When we do that, everything
has to be differentiated.

1109
01:07:51,120 --> 01:07:53,390
So it involves
differentiating a,

1110
01:07:53,390 --> 01:07:57,170
but that just expresses things
in terms of derivatives of a.

1111
01:07:57,170 --> 01:07:59,980
But the new quantity that
gets introduced is RHO.

1112
01:07:59,980 --> 01:08:02,570
If we wanted to differentiate
this equation with respect

1113
01:08:02,570 --> 01:08:06,300
to time, we have to
know what RHO dot is.

1114
01:08:06,300 --> 01:08:07,130
But we do.

1115
01:08:07,130 --> 01:08:09,850
That's what equation
three tells us.

1116
01:08:09,850 --> 01:08:13,620
So we can differentiate equation
one, use equation three,

1117
01:08:13,620 --> 01:08:15,661
and we can derive an
equation for a double dot.

1118
01:08:15,661 --> 01:08:17,244
And if these equations
are consistent,

1119
01:08:17,244 --> 01:08:18,500
it'd better be equation two.

1120
01:08:18,500 --> 01:08:19,279
And it will be.

1121
01:08:19,279 --> 01:08:21,500
You can check it.

1122
01:08:21,500 --> 01:08:23,790
And actually, I think any
two of these equations

1123
01:08:23,790 --> 01:08:25,760
can be used to derive the third.

1124
01:08:25,760 --> 01:08:27,414
Those equations
just are-- really

1125
01:08:27,414 --> 01:08:28,830
a set of two
independent equations

1126
01:08:28,830 --> 01:08:30,640
and one dependent equation.

1127
01:08:30,640 --> 01:08:35,210
You can shuffle it
any way you want.

1128
01:08:35,210 --> 01:08:39,340
But, now what we want
to do is to consider

1129
01:08:39,340 --> 01:08:40,779
a different kind of matter.

1130
01:08:40,779 --> 01:08:42,380
Instead of
non-relativistic matter,

1131
01:08:42,380 --> 01:08:45,569
we're considering photon matter.

1132
01:08:45,569 --> 01:08:47,210
And in particular,
we know that it's

1133
01:08:47,210 --> 01:08:48,615
going to change equation three.

1134
01:08:56,939 --> 01:09:01,220
So for radiation,
3 gets modified

1135
01:09:01,220 --> 01:09:10,620
into 3 prime, which
is the equation that

1136
01:09:10,620 --> 01:09:16,890
says that RHO dot is equal to
minus 4 a dot over a times RHO.

1137
01:09:36,630 --> 01:09:41,010
So how are we going to
fix these equations?

1138
01:09:41,010 --> 01:09:43,080
Now they're inconsistent.

1139
01:09:43,080 --> 01:09:46,521
If we change three and don't
change either one or two,

1140
01:09:46,521 --> 01:09:48,520
we know that we're
inconsistent, because any two

1141
01:09:48,520 --> 01:09:51,649
of those equations can be
used to derive the third.

1142
01:09:51,649 --> 01:09:53,210
So we're in trouble.

1143
01:09:53,210 --> 01:09:55,230
Either equations
one or two will also

1144
01:09:55,230 --> 01:09:58,260
have to be modified if we're
going to modify equation three.

1145
01:10:15,660 --> 01:10:16,160
OK.

1146
01:10:16,160 --> 01:10:20,040
Before we go on, I'd like to
say a little more about why

1147
01:10:20,040 --> 01:10:23,032
this equation is different
from that equation.

1148
01:10:23,032 --> 01:10:24,490
One might think
that it should just

1149
01:10:24,490 --> 01:10:26,412
be governed by the
conservation of energy.

1150
01:10:26,412 --> 01:10:28,120
After all, we just
write down an equation

1151
01:10:28,120 --> 01:10:32,540
for RHO dot-- how energy
density changes with time.

1152
01:10:32,540 --> 01:10:35,030
Shouldn't conservation
of energy determine that?

1153
01:10:35,030 --> 01:10:36,850
It does.

1154
01:10:36,850 --> 01:10:39,660
But there is an extra element
to conservation of energy

1155
01:10:39,660 --> 01:10:42,950
that we need to
take into account,

1156
01:10:42,950 --> 01:10:45,540
and that is the pressure
of the gas affects

1157
01:10:45,540 --> 01:10:48,510
what happens to its
energy as it expands.

1158
01:10:48,510 --> 01:10:50,470
So before we get back
to the early universe,

1159
01:10:50,470 --> 01:10:52,680
I just want to consider a
gas in a piston chamber.

1160
01:10:56,180 --> 01:11:01,990
And I'm going to let the piston
have an area a, and inside

1161
01:11:01,990 --> 01:11:05,750
we're going to have a volume
v. Just to define our notation.

1162
01:11:09,260 --> 01:11:11,340
If we have a gas
inside a piston chamber

1163
01:11:11,340 --> 01:11:15,390
and let the piston chamber
enlarge by pulling out

1164
01:11:15,390 --> 01:11:21,330
on the piston, the gas has
a pressure, in general,

1165
01:11:21,330 --> 01:11:24,570
and that pressure will
exert a force on the piston.

1166
01:11:24,570 --> 01:11:27,910
So if I allow the piston
to move to the right,

1167
01:11:27,910 --> 01:11:30,150
that gas will be
exerting a force

1168
01:11:30,150 --> 01:11:32,680
on the piston in the
direction that it's moving.

1169
01:11:32,680 --> 01:11:35,920
And that means the gas will
be doing work on the piston.

1170
01:11:35,920 --> 01:11:39,080
So, by our ordinary notions
of Newtonian conservation

1171
01:11:39,080 --> 01:11:43,310
of energy, we would know that
the gas would lose energy,

1172
01:11:43,310 --> 01:11:47,490
and we can even calculate
how much energy it loses.

1173
01:11:47,490 --> 01:11:51,870
And the formula is easy enough
to get in a Newtonian context.

1174
01:11:51,870 --> 01:11:55,280
It's just du is equal
to minus the pressure

1175
01:11:55,280 --> 01:11:58,970
of the gas times the
change in volume.

1176
01:11:58,970 --> 01:12:00,550
A famous formula.

1177
01:12:00,550 --> 01:12:02,550
And this just comes
about by saying

1178
01:12:02,550 --> 01:12:06,040
that the work that's done is
the force times the distance.

1179
01:12:06,040 --> 01:12:08,850
The force is the
pressure times the area.

1180
01:12:08,850 --> 01:12:12,710
And the volume is the
area times the distance.

1181
01:12:12,710 --> 01:12:14,460
And putting those
things together,

1182
01:12:14,460 --> 01:12:16,204
you get this
formula immediately.

1183
01:12:16,204 --> 01:12:18,120
Now this formula is
actually much more general

1184
01:12:18,120 --> 01:12:21,310
than the quasi derivation
that I just showed.

1185
01:12:21,310 --> 01:12:23,750
It works no matter what
the shape of the gas is.

1186
01:12:23,750 --> 01:12:25,740
If you put a gas in
any kind of a container

1187
01:12:25,740 --> 01:12:31,560
and let that container enlarge,
even in an irregular way,

1188
01:12:31,560 --> 01:12:33,770
the work that the gas
will do will always

1189
01:12:33,770 --> 01:12:35,620
be equal to minus the
pressure of the gas

1190
01:12:35,620 --> 01:12:39,144
times the change in the volume.

1191
01:12:39,144 --> 01:12:40,810
We can apply this to
the early universe.

1192
01:12:40,810 --> 01:12:43,780
It actually works.

1193
01:12:43,780 --> 01:12:45,455
The difference
between our two cases

1194
01:12:45,455 --> 01:12:48,696
is that our non-relativistic
matter has no pressure at all.

1195
01:12:48,696 --> 01:12:50,820
We're just talking about
particles sitting at rest.

1196
01:12:50,820 --> 01:12:52,420
They're not bouncing
off of any walls.

1197
01:12:52,420 --> 01:12:54,249
They're not creating
any pressure,

1198
01:12:54,249 --> 01:12:56,290
while the photons are
moving around all the time.

1199
01:12:56,290 --> 01:12:57,820
And if you imagine
a box of them,

1200
01:12:57,820 --> 01:12:59,820
they'd be hitting against
the walls of that box,

1201
01:12:59,820 --> 01:13:00,680
exerting a pressure.

1202
01:13:05,990 --> 01:13:08,170
And we're now in a position
to relate the pressure

1203
01:13:08,170 --> 01:13:11,460
to the difference between
the 3's and the 4's that

1204
01:13:11,460 --> 01:13:15,955
appear in those two equations
for RHO dot in the two cases.

1205
01:13:22,000 --> 01:13:27,190
To apply this naive idea
to a piece of the universe,

1206
01:13:27,190 --> 01:13:36,800
we can imagine choosing-- we're
going to choose some fixed

1207
01:13:36,800 --> 01:13:42,370
volume in our co-moving
coordinate system.

1208
01:13:53,842 --> 01:13:55,800
So our box, the volume
that we're talking about

1209
01:13:55,800 --> 01:13:57,633
will actually be expanding
with the universe

1210
01:13:57,633 --> 01:13:59,900
but be fixed in
co-moving coordinates.

1211
01:13:59,900 --> 01:14:08,410
And the physical volume
therefore of our box

1212
01:14:08,410 --> 01:14:13,590
will be a cubed of t times
the coordinate volume

1213
01:14:13,590 --> 01:14:18,560
of the box-- the volume
and not just cubed.

1214
01:14:18,560 --> 01:14:20,610
And this volume will
be independent of time.

1215
01:14:20,610 --> 01:14:21,911
There's time dependents there.

1216
01:14:21,911 --> 01:14:23,160
There's time dependents there.

1217
01:14:23,160 --> 01:14:25,225
The physical volume of
our box will be enlarging.

1218
01:14:28,870 --> 01:14:33,540
The total energy in our
box, the total gas energy,

1219
01:14:33,540 --> 01:14:35,840
which I'll call
capital U, will just

1220
01:14:35,840 --> 01:14:40,180
be the physical volume
times the energy density.

1221
01:14:40,180 --> 01:14:42,530
The energy density is
energy per physical volume.

1222
01:14:46,380 --> 01:14:50,660
And we can now
apply this formula

1223
01:14:50,660 --> 01:14:55,000
using this U and this v.

1224
01:14:55,000 --> 01:14:57,076
And here again is one
of these cases where

1225
01:14:57,076 --> 01:14:58,950
I'm going to be skipping
steps because you're

1226
01:14:58,950 --> 01:15:01,160
going to be doing it in
detail on the homework.

1227
01:15:01,160 --> 01:15:03,140
But by putting these
equations together,

1228
01:15:03,140 --> 01:15:08,680
what you'll find is that d dt
of a cubed times RHO times c

1229
01:15:08,680 --> 01:15:14,080
squared-- this is just d dt
of a cubed times the energy

1230
01:15:14,080 --> 01:15:18,170
density-- basically,
the left hand side

1231
01:15:18,170 --> 01:15:25,890
of that equation divided by dt
and divided by v coordinates--

1232
01:15:25,890 --> 01:15:32,881
is equal to minus p
times d dt of a cubed.

1233
01:15:32,881 --> 01:15:35,130
And this is just the PDV
term from the right hand side

1234
01:15:35,130 --> 01:15:38,117
of that equation rewritten
in terms of the variables.

1235
01:15:38,117 --> 01:15:39,700
And you'll be doing
this for homework.

1236
01:15:39,700 --> 01:15:42,609
I'm just getting
straight the factors

1237
01:15:42,609 --> 01:15:43,900
to make sure I have them right.

1238
01:15:58,021 --> 01:15:58,520
OK.

1239
01:15:58,520 --> 01:16:00,311
Reshuffling that
equation-- and again, this

1240
01:16:00,311 --> 01:16:02,140
is a homework problem--
you can turn that

1241
01:16:02,140 --> 01:16:04,590
into an equation for RHO dot.

1242
01:16:04,590 --> 01:16:09,120
And what you'll get
is minus 3 a dot

1243
01:16:09,120 --> 01:16:15,420
over a times RHO plus
p over c squared.

1244
01:16:24,690 --> 01:16:27,110
And now we can see how
our two cases emerge.

1245
01:16:27,110 --> 01:16:29,134
If the pressure is 0,
we get minus 3 a dot

1246
01:16:29,134 --> 01:16:30,550
over 8 times RHO,
which is what we

1247
01:16:30,550 --> 01:16:32,700
had for non-relativistic matter.

1248
01:16:32,700 --> 01:16:34,660
And the photon gas is
going to have a pressure,

1249
01:16:34,660 --> 01:16:36,285
and we could read
off from this formula

1250
01:16:36,285 --> 01:16:39,370
to know what the pressure has
to be to turn the 3 into a 4.

1251
01:16:39,370 --> 01:16:44,290
The pressure has to be
a factor of a third.

1252
01:16:44,290 --> 01:16:46,810
So you determine for this
that the pressure for light

1253
01:16:46,810 --> 01:16:53,050
is 1/3 of the energy density,
or 1/3 times RHO c squared.

1254
01:16:53,050 --> 01:16:55,869
And that's what you need
to turn the 3 into a 4.

1255
01:16:55,869 --> 01:16:57,910
So we now have indirectly
calculated the pressure

1256
01:16:57,910 --> 01:17:00,076
of light, and this agrees
with any other calculation

1257
01:17:00,076 --> 01:17:03,090
for the pressure of
light that you might do.

1258
01:17:03,090 --> 01:17:05,670
It's by no means the
only way to calculate it.

1259
01:17:05,670 --> 01:17:09,060
And now finally,
we're in a position--

1260
01:17:09,060 --> 01:17:11,200
and we'll just do
this quickly to decide

1261
01:17:11,200 --> 01:17:13,920
how to modify these equations.

1262
01:17:13,920 --> 01:17:16,690
Now, we're not in a position
to determine that rigorously.

1263
01:17:16,690 --> 01:17:20,410
It can be determined rigorously
by doing general relativity,

1264
01:17:20,410 --> 01:17:23,800
which we're not
doing at that level.

1265
01:17:23,800 --> 01:17:26,690
But we can still
motivate the answer.

1266
01:17:26,690 --> 01:17:28,360
One of these two
equations is going

1267
01:17:28,360 --> 01:17:30,850
to have to be changed
to accommodate

1268
01:17:30,850 --> 01:17:34,470
a more general
expression for RHO dot.

1269
01:17:34,470 --> 01:17:36,740
The top equation
we know is really

1270
01:17:36,740 --> 01:17:39,030
an equation for
conservation of energy.

1271
01:17:39,030 --> 01:17:41,930
That's how we got it
in the Newtonian case

1272
01:17:41,930 --> 01:17:46,110
where little k ended up
being partial to the energy

1273
01:17:46,110 --> 01:17:47,640
in the Newtonian case.

1274
01:17:47,640 --> 01:17:49,140
But this is basically
a conservation

1275
01:17:49,140 --> 01:17:50,440
of energy equation.

1276
01:17:50,440 --> 01:17:52,120
And that's what you
expect, just given

1277
01:17:52,120 --> 01:17:54,290
your general notion
of mechanics as well.

1278
01:17:54,290 --> 01:17:57,560
If you have a second
order equation,

1279
01:17:57,560 --> 01:17:59,930
a second order differential
is with respect to time.

1280
01:17:59,930 --> 01:18:01,110
That's the force equation.

1281
01:18:01,110 --> 01:18:02,860
And if you have a first
order differential

1282
01:18:02,860 --> 01:18:06,010
equation with respect to time,
1/2 mv squared plus v of r

1283
01:18:06,010 --> 01:18:07,070
equals constant.

1284
01:18:07,070 --> 01:18:09,300
That's energy conservation.

1285
01:18:09,300 --> 01:18:10,250
Same thing here.

1286
01:18:10,250 --> 01:18:13,910
And we know that energy
cannot suddenly change.

1287
01:18:13,910 --> 01:18:18,780
If we imagine-- I guess the
first experiment I want to do

1288
01:18:18,780 --> 01:18:20,730
is imagining somehow
there's an explosion

1289
01:18:20,730 --> 01:18:22,920
throughout all of space.

1290
01:18:22,920 --> 01:18:27,670
I imagine putting pieces
of TNT throughout space

1291
01:18:27,670 --> 01:18:30,560
and arranging for, at the same
cosmic time, for all of them

1292
01:18:30,560 --> 01:18:31,200
to be ignited.

1293
01:18:31,200 --> 01:18:34,014
And that would suddenly change
the pressure of the universe,

1294
01:18:34,014 --> 01:18:35,805
but it would not change
the energy density.

1295
01:18:35,805 --> 01:18:38,662
The energy density
would be conserved.

1296
01:18:38,662 --> 01:18:40,620
So the bottom line is
that pressures can change

1297
01:18:40,620 --> 01:18:43,950
discontinuously, but
energy densities cannot.

1298
01:18:43,950 --> 01:18:48,640
And since this equation is
the conservation of energy

1299
01:18:48,640 --> 01:18:51,660
equation, we'd expected that
nothing can change suddenly

1300
01:18:51,660 --> 01:18:54,050
here, that the pressure
term cannot contribute here,

1301
01:18:54,050 --> 01:18:56,574
because if it did, the pressure
term would changed suddenly.

1302
01:18:56,574 --> 01:18:58,740
Nothing else in this equation
would change suddenly.

1303
01:18:58,740 --> 01:19:01,350
There would be no way the
equation could be satisfied.

1304
01:19:01,350 --> 01:19:05,130
But if we added a pressure
term to the second equation,

1305
01:19:05,130 --> 01:19:08,900
that would allow the pressure
to change discontinuously

1306
01:19:08,900 --> 01:19:10,135
as the TNT went off.

1307
01:19:10,135 --> 01:19:12,260
And that would change a
double dot discontinuously.

1308
01:19:12,260 --> 01:19:14,530
And there's nothing wrong
with a double dot changing

1309
01:19:14,530 --> 01:19:15,350
discontinuously.

1310
01:19:15,350 --> 01:19:17,780
If you suddenly the apply
a new force to a particle,

1311
01:19:17,780 --> 01:19:21,350
you suddenly change its second
derivative of its motion.

1312
01:19:21,350 --> 01:19:23,500
You suddenly change
its acceleration.

1313
01:19:23,500 --> 01:19:24,510
So that's OK.

1314
01:19:24,510 --> 01:19:26,690
So any pressure to
this term make sense.

1315
01:19:26,690 --> 01:19:29,490
Adding pressure to this
equation does not make sense.

1316
01:19:29,490 --> 01:19:31,370
And then we can just
ask, what do you

1317
01:19:31,370 --> 01:19:33,910
have to do to this equation if
we're going to add a pressure

1318
01:19:33,910 --> 01:19:36,280
term to make all
three equations now

1319
01:19:36,280 --> 01:19:40,220
consistent with the new
equation for RHO dot?

1320
01:19:40,220 --> 01:19:42,970
It's your homework problem
to answer that question,

1321
01:19:42,970 --> 01:19:44,949
but the homework
tells you the answer,

1322
01:19:44,949 --> 01:19:46,990
and I'll write the answer
on the board right now,

1323
01:19:46,990 --> 01:19:50,830
and then we'll consider
today's lecture over.

1324
01:19:50,830 --> 01:19:57,840
The bottom line is that
equation number one

1325
01:19:57,840 --> 01:20:02,750
has to be modified
into one prime, which

1326
01:20:02,750 --> 01:20:05,930
says that equation number
two has to be modified.

1327
01:20:05,930 --> 01:20:06,960
What am I talking about?

1328
01:20:13,570 --> 01:20:17,890
And the new equation
is a double dot

1329
01:20:17,890 --> 01:20:23,600
is equal to minus 4
pi over 3 G times RHO

1330
01:20:23,600 --> 01:20:28,880
plus 3p over c squared times a.

1331
01:20:34,140 --> 01:20:37,070
And now we have a consistent
set of Friedman equations,

1332
01:20:37,070 --> 01:20:38,570
and these are the
Friedman equations

1333
01:20:38,570 --> 01:20:40,790
that we would have gotten
if we had done everything

1334
01:20:40,790 --> 01:20:43,330
using general relativity
from the beginning.

1335
01:20:43,330 --> 01:20:44,870
And we'll stop there.

1336
01:20:44,870 --> 01:20:49,380
And we will meet
again next Tuesday.

1337
01:20:49,380 --> 01:20:51,170
And I'll send you an
email about-- there

1338
01:20:51,170 --> 01:20:54,575
will be at least one homework
problem on the problems set

1339
01:20:54,575 --> 01:20:58,446
that will have to be held over
to the following problems set.

1340
01:20:58,446 --> 01:20:59,820
I'll send you an
email about that

1341
01:20:59,820 --> 01:21:01,960
and post it on the website.