1
00:00:00,500 --> 00:00:01,932
[SQUEAKING]

2
00:00:01,932 --> 00:00:04,347
[RUSTLING]

3
00:00:04,347 --> 00:00:06,762
[CLICKING]

4
00:00:10,920 --> 00:00:11,920
SCOTT HUGHES: All right.

5
00:00:11,920 --> 00:00:13,070
So welcome back.

6
00:00:13,070 --> 00:00:14,510
I had a little bit of a break.

7
00:00:14,510 --> 00:00:16,010
Before I get into--

8
00:00:16,010 --> 00:00:18,357
I go over my quick
recap, you hopefully all

9
00:00:18,357 --> 00:00:19,940
have seen the
announcements that we're

10
00:00:19,940 --> 00:00:23,090
going to delay the due date
of the next problem set

11
00:00:23,090 --> 00:00:24,710
until Tuesday.

12
00:00:24,710 --> 00:00:27,812
That's in part because some of
the material that appears on it

13
00:00:27,812 --> 00:00:30,020
is what we're going to talk
about in today's lecture.

14
00:00:30,020 --> 00:00:32,710
So I want to make sure you've
at least seen a lecture on all

15
00:00:32,710 --> 00:00:35,470
the topics before you try to
do some of the problems on it.

16
00:00:38,180 --> 00:00:39,648
In truth, you could probably--

17
00:00:39,648 --> 00:00:41,940
based on things I talked
about in the previous lecture,

18
00:00:41,940 --> 00:00:44,232
you could probably deduce a
lot of what you need to do,

19
00:00:44,232 --> 00:00:47,180
but still, it's good to go
over it a little bit more

20
00:00:47,180 --> 00:00:50,420
methodically.

21
00:00:50,420 --> 00:00:51,628
And I'm trying to think.

22
00:00:51,628 --> 00:00:53,170
Is there anything
else I want to say?

23
00:00:53,170 --> 00:00:53,670
Yeah.

24
00:00:53,670 --> 00:00:57,040
So then the next problem set
will be posted a week from--

25
00:00:57,040 --> 00:00:58,010
is that right?

26
00:00:58,010 --> 00:01:00,542
Yeah, I think we post
a week from today.

27
00:01:00,542 --> 00:01:02,000
So let me just get
back to what you

28
00:01:02,000 --> 00:01:03,208
were talking about last time.

29
00:01:03,208 --> 00:01:06,980
So I began introducing
some geometric concepts

30
00:01:06,980 --> 00:01:09,310
and some quantities that
we use to describe matter.

31
00:01:09,310 --> 00:01:10,870
We've done a lot
of things so far

32
00:01:10,870 --> 00:01:12,245
that's appropriate
for describing

33
00:01:12,245 --> 00:01:14,320
kind of the kinematics
of particles,

34
00:01:14,320 --> 00:01:15,790
but that's rather restrictive.

35
00:01:15,790 --> 00:01:18,550
We want to talk about a broader
class of things than that.

36
00:01:18,550 --> 00:01:20,800
And I began by introducing
something which is, again,

37
00:01:20,800 --> 00:01:24,190
fairly simple, but
it's a useful tool

38
00:01:24,190 --> 00:01:27,520
for beginning to think
about how we are going

39
00:01:27,520 --> 00:01:30,910
to mathematically categorize
certain important types

40
00:01:30,910 --> 00:01:31,740
of matter.

41
00:01:31,740 --> 00:01:33,573
Pardon me while I put
some of my notes away.

42
00:01:35,613 --> 00:01:36,280
Also, pardon me.

43
00:01:36,280 --> 00:01:37,238
I'm recovering from a--

44
00:01:37,238 --> 00:01:39,400
I caught a terrible cold
over the long weekend,

45
00:01:39,400 --> 00:01:40,810
and I'm coughing incessantly.

46
00:01:40,810 --> 00:01:45,432
So I will be occasionally
sucking on a cough drop.

47
00:01:45,432 --> 00:01:47,890
So I introduced the quantity
called the number four-vector.

48
00:01:47,890 --> 00:01:50,980
And that is given by-- imagine
you have sort of a-- it's best

49
00:01:50,980 --> 00:01:53,397
to think about this in the
context of something like dust.

50
00:01:53,397 --> 00:01:56,170
So you have some kind of
non-interacting little

51
00:01:56,170 --> 00:01:59,620
agglomeration of tiny particles.

52
00:01:59,620 --> 00:02:02,890
And in the rest frame of
an element of this dust--

53
00:02:02,890 --> 00:02:04,440
so imagine you go
into-- well, I'm

54
00:02:04,440 --> 00:02:05,710
going to define the rest
frame of that element

55
00:02:05,710 --> 00:02:07,877
by saying imagine you've
got a cubic nanometer of it

56
00:02:07,877 --> 00:02:08,860
or something like that.

57
00:02:08,860 --> 00:02:10,610
And you go into the
frame where everything

58
00:02:10,610 --> 00:02:13,690
in that cubic nanometer
is on average at rest.

59
00:02:13,690 --> 00:02:16,660
We'll call n sub 0
the rest density.

60
00:02:16,660 --> 00:02:18,070
So I forgot to write that down.

61
00:02:27,770 --> 00:02:29,240
So that's the rest density.

62
00:02:29,240 --> 00:02:30,560
The rest-- pardon me--

63
00:02:30,560 --> 00:02:31,880
number density.

64
00:02:38,310 --> 00:02:39,690
So I go into that rest frame.

65
00:02:39,690 --> 00:02:42,360
That tells me how many
little dust particles there

66
00:02:42,360 --> 00:02:45,510
are per cubic volume,
per element of volume.

67
00:02:45,510 --> 00:02:47,480
Multiply that by the
four-vector that--

68
00:02:47,480 --> 00:02:49,260
excuse me-- the four-velocity
describing that element.

69
00:02:49,260 --> 00:02:51,302
And we're going to call
that capital N. So that's

70
00:02:51,302 --> 00:02:52,740
a vector that describes--

71
00:02:52,740 --> 00:02:55,260
its components describe
the number density

72
00:02:55,260 --> 00:02:56,640
in some other
frame of reference,

73
00:02:56,640 --> 00:02:58,440
in any frame of reference.

74
00:02:58,440 --> 00:03:00,960
And the flux, talking about
how those dust particles

75
00:03:00,960 --> 00:03:04,153
flow from one
element to another.

76
00:03:04,153 --> 00:03:05,820
So we spent a little
bit of time talking

77
00:03:05,820 --> 00:03:07,830
about how to define
volume elements

78
00:03:07,830 --> 00:03:08,970
in a covariant fashion.

79
00:03:08,970 --> 00:03:10,345
I'm not going to
go through that,

80
00:03:10,345 --> 00:03:12,240
but part of the
punchline of doing that

81
00:03:12,240 --> 00:03:16,200
is that we're going to
define a sort of what

82
00:03:16,200 --> 00:03:19,360
I call a covariant formulation
of conservation of number.

83
00:03:19,360 --> 00:03:21,290
It's going to be the
spacetime divergence

84
00:03:21,290 --> 00:03:23,460
of that vector is equal to 0.

85
00:03:23,460 --> 00:03:28,040
That holds-- this equation holds
in all frames of reference.

86
00:03:28,040 --> 00:03:30,540
If I choose a particular frame
of reference, in other words,

87
00:03:30,540 --> 00:03:33,330
I define a particular time,
I define a particular set

88
00:03:33,330 --> 00:03:35,100
of spatial
coordinates, I can then

89
00:03:35,100 --> 00:03:37,260
break this up and
say that this is

90
00:03:37,260 --> 00:03:40,470
equivalent to saying that
the time derivative of the t

91
00:03:40,470 --> 00:03:44,460
component of that plus
the spatial divergence,

92
00:03:44,460 --> 00:03:47,640
the spatial confluence
is that they sum to 0.

93
00:03:47,640 --> 00:03:50,360
I can also take this thing
and integrate over a--

94
00:03:50,360 --> 00:03:54,600
oops-- it would help to have
one element defined here.

95
00:03:54,600 --> 00:03:58,800
I integrate this over a
four-dimensional volume.

96
00:04:03,160 --> 00:04:07,870
And by my conservation law,
I must get 0 when I do that.

97
00:04:07,870 --> 00:04:11,710
Again, having chosen a
particular frame, defined what

98
00:04:11,710 --> 00:04:14,290
time means, defined
what space means,

99
00:04:14,290 --> 00:04:17,740
I can then split this
out into a statement

100
00:04:17,740 --> 00:04:25,090
that the time derivative of
the volume integral of the time

101
00:04:25,090 --> 00:04:27,520
component of this
thing is balanced

102
00:04:27,520 --> 00:04:31,270
by the flux of the number
across the boundaries

103
00:04:31,270 --> 00:04:33,280
of a particular three-volume.

104
00:04:33,280 --> 00:04:35,740
I remind you this notation,
this sort of delta V3

105
00:04:35,740 --> 00:04:37,930
means the boundaries
of the V3 that's

106
00:04:37,930 --> 00:04:40,960
used under this integral.

107
00:04:40,960 --> 00:04:43,437
So we talked a little bit
about a few other four-vectors

108
00:04:43,437 --> 00:04:44,020
that are used.

109
00:04:44,020 --> 00:04:46,030
In particular I
introduced some stuff

110
00:04:46,030 --> 00:04:47,920
that we use in
electricity and magnetism.

111
00:04:47,920 --> 00:04:49,870
But I want to switch gears.

112
00:04:49,870 --> 00:04:52,750
In particular, I want
to introduce and discuss

113
00:04:52,750 --> 00:04:57,040
in some detail one of the
most important tensors

114
00:04:57,040 --> 00:05:00,040
we are going to use all term.

115
00:05:00,040 --> 00:05:04,282
So I motivated this by saying
imagine I have a cloud of dust.

116
00:05:04,282 --> 00:05:05,740
And I won't recreate
one, because I

117
00:05:05,740 --> 00:05:07,190
am getting over a cold.

118
00:05:07,190 --> 00:05:09,850
And I don't want to
breathe chalk dust.

119
00:05:09,850 --> 00:05:12,440
So I imagine I make a little
cloud of these things here.

120
00:05:12,440 --> 00:05:14,020
So far we've
characterized this thing

121
00:05:14,020 --> 00:05:15,730
by just counting the
number of dust particles

122
00:05:15,730 --> 00:05:16,360
that are in there.

123
00:05:16,360 --> 00:05:18,068
But that's not all
there is to it, right?

124
00:05:18,068 --> 00:05:19,993
That dust has other properties.

125
00:05:19,993 --> 00:05:21,910
And so the next thing
which I would like to do

126
00:05:21,910 --> 00:05:28,000
is let's consider the energy and
the momentum of every particle

127
00:05:28,000 --> 00:05:29,120
of dust in that cloud.

128
00:05:37,100 --> 00:05:38,630
So let's just
imagine, again, we're

129
00:05:38,630 --> 00:05:39,870
going to start with
something simple,

130
00:05:39,870 --> 00:05:41,578
and then we'll kind
of walk up from there

131
00:05:41,578 --> 00:05:43,850
to a more generic situation.

132
00:05:43,850 --> 00:05:45,980
Let's imagine that
my cloud consists

133
00:05:45,980 --> 00:05:49,010
of particles that
are all identical,

134
00:05:49,010 --> 00:05:53,270
and each dust particle
has the same rest mass.

135
00:06:06,910 --> 00:06:12,680
So suppose this guy
has a rest mass m.

136
00:06:12,680 --> 00:06:15,520
So one of the things
that I might want to do

137
00:06:15,520 --> 00:06:17,958
is in addition to characterizing
this dust by saying

138
00:06:17,958 --> 00:06:19,000
that the number density--

139
00:06:19,000 --> 00:06:20,488
[COUGHING] excuse me--

140
00:06:20,488 --> 00:06:22,030
in addition to saying
that the number

141
00:06:22,030 --> 00:06:25,890
density in a particular
frame is n or n0,

142
00:06:25,890 --> 00:06:28,990
let's go into the rest
frame of this thing.

143
00:06:28,990 --> 00:06:32,020
I might be interested in knowing
about the rest energy density.

144
00:06:54,470 --> 00:06:58,340
So I'm going to denote the rest
energy density by rho sub 0.

145
00:07:01,520 --> 00:07:05,165
Each particle has
a rest energy of m.

146
00:07:05,165 --> 00:07:06,990
Remember, c is equal to 1.

147
00:07:06,990 --> 00:07:13,623
So mc squared, if you want
to hold to the formula

148
00:07:13,623 --> 00:07:15,790
that we've all known and
loved since we were babies.

149
00:07:15,790 --> 00:07:17,500
So that's the rest energy.

150
00:07:17,500 --> 00:07:20,410
And then how many-- if I want
to get the rest energy density,

151
00:07:20,410 --> 00:07:23,200
I count up the number
in each volume,

152
00:07:23,200 --> 00:07:24,620
or the number per unit volume.

153
00:07:24,620 --> 00:07:26,920
So that's m times n0.

154
00:07:26,920 --> 00:07:28,990
So that's the rest energy
density of this thing.

155
00:07:28,990 --> 00:07:30,400
Great.

156
00:07:30,400 --> 00:07:33,100
Let's now ask, OK, so that's
what it looks like in the rest

157
00:07:33,100 --> 00:07:36,190
frame of this dust element.

158
00:07:36,190 --> 00:07:38,410
Let's imagine that I now
bop into a frame that

159
00:07:38,410 --> 00:07:41,516
is moving with some speed
relative to this rest frame.

160
00:08:02,850 --> 00:08:05,202
In this frame, the
energy density--

161
00:08:05,202 --> 00:08:07,410
I'm not going to say rest
energy density, because I'm

162
00:08:07,410 --> 00:08:08,940
no longer in the rest frame.

163
00:08:08,940 --> 00:08:18,800
But the energy
density in this frame,

164
00:08:18,800 --> 00:08:22,640
which I will denote by rho
without the subscript 0

165
00:08:22,640 --> 00:08:26,260
is going to be the
energy of each particle.

166
00:08:26,260 --> 00:08:31,880
So that's gamma times m.

167
00:08:31,880 --> 00:08:33,539
And the volume is
going to be length--

168
00:08:33,539 --> 00:08:34,690
it's going to be a
little bit larger.

169
00:08:34,690 --> 00:08:35,190
Sorry.

170
00:08:35,190 --> 00:08:39,480
The volume is smaller because
of a length contraction.

171
00:08:39,480 --> 00:08:46,060
And so this guy actually gets
boosted up to gamma and 0.

172
00:08:46,060 --> 00:08:53,020
So in this frame,
the energy density

173
00:08:53,020 --> 00:08:55,355
is larger than the rest energy--

174
00:08:55,355 --> 00:08:59,170
excuse me-- the rest energy
density with a gamma squared

175
00:08:59,170 --> 00:09:01,380
factor.

176
00:09:01,380 --> 00:09:02,380
Straightforward algebra.

177
00:09:02,380 --> 00:09:04,963
But I want you to stop and think
about what that's telling us.

178
00:09:08,100 --> 00:09:11,700
If this rho were a
component of a four-vector,

179
00:09:11,700 --> 00:09:14,790
is there any way I could get a
gamma squared out of this thing

180
00:09:14,790 --> 00:09:16,900
by bopping between
reference frames?

181
00:09:16,900 --> 00:09:17,710
No, right?

182
00:09:17,710 --> 00:09:20,280
When I do that, it's linear
in the Lorentz transformation

183
00:09:20,280 --> 00:09:21,150
matrix.

184
00:09:21,150 --> 00:09:23,670
There's no way with a
linear transformation

185
00:09:23,670 --> 00:09:27,420
that I would pick up
two powers of gamma.

186
00:09:27,420 --> 00:09:30,630
This is not how a-- you
guys are probably all

187
00:09:30,630 --> 00:09:33,240
used to from Newtonian physics
of thinking of energy density

188
00:09:33,240 --> 00:09:34,690
as a scalar.

189
00:09:34,690 --> 00:09:36,430
This is not a Lorentz scalar.

190
00:09:36,430 --> 00:09:39,002
A scalar is the same in
all frames of reference.

191
00:09:39,002 --> 00:09:40,335
So this is a different quantity.

192
00:09:55,380 --> 00:10:00,880
So it's neither a four-vector
component nor a scalar.

193
00:10:06,342 --> 00:10:08,050
It's got two of those
things, so it's not

194
00:10:08,050 --> 00:10:09,490
going to surprise you
that what's actually

195
00:10:09,490 --> 00:10:11,907
going on here is we are picking
out-- what we've done here

196
00:10:11,907 --> 00:10:15,450
is we've picked out a particular
component of a tensor.

197
00:10:15,450 --> 00:10:17,200
Let's think a little
bit more methodically

198
00:10:17,200 --> 00:10:19,300
about what tensor that must be.

199
00:10:19,300 --> 00:10:22,630
So when I originally wrote down
my rho 0 here, I sort of just

200
00:10:22,630 --> 00:10:25,210
argued on physical
grounds that it

201
00:10:25,210 --> 00:10:29,470
is the energy of every particle
times the number of particles

202
00:10:29,470 --> 00:10:31,870
per unit volume of that thing.

203
00:10:31,870 --> 00:10:36,160
Well, the energy that I
use in this thing doing it

204
00:10:36,160 --> 00:10:39,520
in both the rest frame and
in the other frame, those

205
00:10:39,520 --> 00:10:42,640
are the time-like components
of a particular four-vector.

206
00:10:50,740 --> 00:11:02,150
So we assembled rho
by combining energy,

207
00:11:02,150 --> 00:11:20,590
which is the time-like
component of the four-momentum

208
00:11:20,590 --> 00:11:33,270
with number density, which
is the time-like component

209
00:11:33,270 --> 00:11:35,820
of the number vector that I
just recapped a few minutes ago.

210
00:11:44,340 --> 00:11:46,570
So those are two time-like
components of four-vectors

211
00:11:46,570 --> 00:11:47,070
here.

212
00:11:47,070 --> 00:11:53,250
So that rho is something like--

213
00:11:53,250 --> 00:11:54,610
well, let's put it this way.

214
00:11:54,610 --> 00:12:00,910
I can write this as
pt nt, which tells me

215
00:12:00,910 --> 00:12:03,850
that it belongs to a tensor.

216
00:12:03,850 --> 00:12:09,020
So let's define this
as some tt of a tensor

217
00:12:09,020 --> 00:12:11,290
that I've not carefully
introduced yet.

218
00:12:11,290 --> 00:12:13,000
There is some underlying
tensor that we

219
00:12:13,000 --> 00:12:16,630
have built by looking at the--

220
00:12:16,630 --> 00:12:18,085
there are various
names for this.

221
00:12:18,085 --> 00:12:21,744
We'll call it the tensor product
of two different four-vectors.

222
00:12:52,390 --> 00:12:55,140
So if I write this in the kind
of abstract tensor notation

223
00:12:55,140 --> 00:12:58,020
that I've used occasionally--
in some of your textbooks this

224
00:12:58,020 --> 00:13:00,720
would be written with
a bold faced capital T,

225
00:13:00,720 --> 00:13:03,460
I will use a double
over line for this.

226
00:13:03,460 --> 00:13:10,110
We might say that this t is the
tensor product of the number

227
00:13:10,110 --> 00:13:12,900
vector with the four-momentum.

228
00:13:12,900 --> 00:13:20,070
Now, the number vector
is itself just n0 times

229
00:13:20,070 --> 00:13:21,750
the four-velocity.

230
00:13:21,750 --> 00:13:25,050
And my momentum, since all
the particles, I'm assuming,

231
00:13:25,050 --> 00:13:31,200
are the same is just the rest
mass times the four-momentum.

232
00:13:31,200 --> 00:13:34,440
This thing actually looks kind
of like the four-velocity times

233
00:13:34,440 --> 00:13:37,140
four-velocity with a
prefactor And that prefactor

234
00:13:37,140 --> 00:13:41,480
is nothing more than
the rest density.

235
00:13:46,390 --> 00:13:48,980
So another way to
say this is that if I

236
00:13:48,980 --> 00:13:50,690
make my cloud of
dust, and I go in

237
00:13:50,690 --> 00:13:53,330
and I look at every
little element in it,

238
00:13:53,330 --> 00:13:59,090
I take the rest energy density
of every element in that dust.

239
00:13:59,090 --> 00:14:01,255
I construct the
tensor that comes

240
00:14:01,255 --> 00:14:03,380
from making the tensor
product of the four-velocity

241
00:14:03,380 --> 00:14:05,550
of that dust with itself.

242
00:14:05,550 --> 00:14:08,570
That is what this geometric
object is equal to.

243
00:14:08,570 --> 00:14:10,940
If you want to write this
in index notation, which

244
00:14:10,940 --> 00:14:19,360
is how we will write
it 99.8% of the time,

245
00:14:19,360 --> 00:14:27,850
we would say that component
alpha beta of this quantity t

246
00:14:27,850 --> 00:14:31,720
is that rest density
times u alpha u beta.

247
00:14:35,900 --> 00:14:39,970
So this is-- in terms
of the physics we're

248
00:14:39,970 --> 00:14:41,692
going to do this
term, this is perhaps

249
00:14:41,692 --> 00:14:44,150
the most important quantity
that we're going to talk about.

250
00:14:44,150 --> 00:14:46,358
So it's worth understanding
what this thing is really

251
00:14:46,358 --> 00:14:47,300
telling us.

252
00:14:47,300 --> 00:14:52,570
So if I wanted to get these
components, the alpha beta,

253
00:14:52,570 --> 00:14:55,300
out of this thing from this
sort of abstract notation

254
00:14:55,300 --> 00:14:58,510
of the tensor, I'll remind
you that we can do this

255
00:14:58,510 --> 00:15:03,220
by taking the tensor and
plugging into its slots

256
00:15:03,220 --> 00:15:08,320
the basis 1 forms, which
tell me something about--

257
00:15:08,320 --> 00:15:10,120
basis 1 forms are
really useful for sort

258
00:15:10,120 --> 00:15:13,490
of measuring fluxes across
in a particular direction.

259
00:15:13,490 --> 00:15:17,920
And in fact, this quantity has
the geometric interpretation--

260
00:15:17,920 --> 00:15:32,760
think of this as the flux
of momentum component alpha

261
00:15:32,760 --> 00:15:36,768
in the beta direction.

262
00:15:40,240 --> 00:15:42,413
So if we look at this
component by component,

263
00:15:42,413 --> 00:15:43,330
it's worth doing that.

264
00:15:43,330 --> 00:15:46,066
So let's do the tt
or 00 component.

265
00:15:50,807 --> 00:15:52,390
So this is the one
we've already done.

266
00:15:52,390 --> 00:15:58,200
This is just rho 0 ut ut.

267
00:15:58,200 --> 00:16:00,760
It's just rho in that thing.

268
00:16:00,760 --> 00:16:02,680
According to the words
I've written down here,

269
00:16:02,680 --> 00:16:14,020
this is the flux of
pt in the t direction.

270
00:16:14,020 --> 00:16:17,080
So pt tells me about energy.

271
00:16:17,080 --> 00:16:20,330
The flux of energy in the
energy direction refers to--

272
00:16:20,330 --> 00:16:24,528
so here's my water flowing
in the time-like direction,

273
00:16:24,528 --> 00:16:26,695
just sitting there apparently
doing nothing, but no,

274
00:16:26,695 --> 00:16:28,280
it's moving through time.

275
00:16:28,280 --> 00:16:31,120
This is energy
flowing through time.

276
00:16:31,120 --> 00:16:45,370
It's energy density, all sort of
locked up in stable equilibrium

277
00:16:45,370 --> 00:16:46,330
here.

278
00:16:46,330 --> 00:16:49,385
But if somebody comes in someday
with a nice cup of anti-water

279
00:16:49,385 --> 00:16:51,760
and combines it, we get to
have all that energy released.

280
00:16:51,760 --> 00:16:54,460
And we can enjoy that
for a few femtoseconds

281
00:16:54,460 --> 00:16:57,470
before we evaporate.

282
00:16:57,470 --> 00:16:59,037
Let's look at the
other components.

283
00:17:10,740 --> 00:17:15,910
So T0i, you can
take the definition

284
00:17:15,910 --> 00:17:17,660
and plug it in in terms
of the components.

285
00:17:17,660 --> 00:17:20,450
It's obviously ut
times ui, but still use

286
00:17:20,450 --> 00:17:23,240
the definition by words.

287
00:17:23,240 --> 00:17:37,810
So this is the flux of p
sub t in the xi direction.

288
00:17:37,810 --> 00:17:40,280
This is talking
about energy moving

289
00:17:40,280 --> 00:17:41,921
in a particular direction.

290
00:17:47,110 --> 00:17:50,423
This is nothing more than
energy flux, energy sort of flux

291
00:17:50,423 --> 00:17:52,840
that we are used to think
about, not flowing through time,

292
00:17:52,840 --> 00:17:53,923
but flowing through space.

293
00:17:57,760 --> 00:18:05,380
Now, T0i, this is the
flux of momentum component

294
00:18:05,380 --> 00:18:13,520
i in the t direction.

295
00:18:13,520 --> 00:18:17,480
So this tells me about
momentum density.

296
00:18:17,480 --> 00:18:19,900
So if I had this
stuff flowing, there's

297
00:18:19,900 --> 00:18:22,030
some momentum associated
with that flow.

298
00:18:22,030 --> 00:18:24,850
You can count up the amount of
mass and every little volume.

299
00:18:24,850 --> 00:18:26,130
Divide by that volume.

300
00:18:26,130 --> 00:18:26,980
Take some ratios.

301
00:18:26,980 --> 00:18:28,615
That is the density of momentum.

302
00:18:43,810 --> 00:18:56,530
And finally, this last one,
flux of p i in the xj direction,

303
00:18:56,530 --> 00:18:58,270
there's really no
great wisdom for that.

304
00:18:58,270 --> 00:19:00,040
This is nothing more
than momentum flux.

305
00:19:00,040 --> 00:19:04,780
If I have a bucket of water
and things are sort of sloshing

306
00:19:04,780 --> 00:19:06,370
around, there's some
momentum moving,

307
00:19:06,370 --> 00:19:08,830
and the whole assembly is
moving in some direction,

308
00:19:08,830 --> 00:19:11,777
we can get a flow of
momentum going in kind

309
00:19:11,777 --> 00:19:12,860
of a non-normal direction.

310
00:19:12,860 --> 00:19:14,568
As long as it goes in
a normal direction,

311
00:19:14,568 --> 00:19:16,240
it might be moving
along its flow.

312
00:19:16,240 --> 00:19:18,115
I'm going to talk about
that in a few moments

313
00:19:18,115 --> 00:19:20,830
when I talk about a few
different kinds of stress

314
00:19:20,830 --> 00:19:22,735
energy tensors.

315
00:19:22,735 --> 00:19:24,610
The key thing which I
want you to be aware of

316
00:19:24,610 --> 00:19:26,235
is that under the
hood of this thing,

317
00:19:26,235 --> 00:19:28,360
we're going to talk about
different kinds of stress

318
00:19:28,360 --> 00:19:30,880
energy tensors for a
little later in this class.

319
00:19:30,880 --> 00:19:32,843
But this basic
interpretation of the way

320
00:19:32,843 --> 00:19:34,510
to think about the
different components,

321
00:19:34,510 --> 00:19:35,913
it holds for all of them.

322
00:19:35,913 --> 00:19:37,330
So this is good
intuition to have.

323
00:19:46,120 --> 00:19:57,403
One thing which
is worth noting is

324
00:19:57,403 --> 00:19:58,820
if you look at--
if you actually--

325
00:19:58,820 --> 00:20:00,778
let's take the form of
the stress energy tensor

326
00:20:00,778 --> 00:20:02,873
that corresponds to the dust.

327
00:20:02,873 --> 00:20:05,040
So I'm going to write out
all four of my components.

328
00:20:05,040 --> 00:20:06,165
I've already given you T00.

329
00:20:09,350 --> 00:20:18,810
T0i, I can write that as
gamma squared rho 0 vi T0i

330
00:20:18,810 --> 00:20:22,340
equals gamma squared rho 0 vi.

331
00:20:29,000 --> 00:20:32,147
This guy is gamma
squared rho 0 vi vj.

332
00:20:32,147 --> 00:20:33,730
There's two elements
of symmetry here,

333
00:20:33,730 --> 00:20:35,330
which I want to emphasize.

334
00:20:35,330 --> 00:20:41,210
Notice energy density
and momentum flux.

335
00:20:41,210 --> 00:20:42,560
Sorry.

336
00:20:42,560 --> 00:20:46,010
Energy flux and momentum
density, those words

337
00:20:46,010 --> 00:20:46,730
are important.

338
00:20:46,730 --> 00:20:48,540
Energy flux and
momentum density,

339
00:20:48,540 --> 00:20:51,290
they're exactly the same.

340
00:20:51,290 --> 00:20:54,620
That is actually
true in all physics.

341
00:20:54,620 --> 00:20:57,658
It's clouded by the fact,
though, that in most units

342
00:20:57,658 --> 00:21:00,200
that we measure things, we don't
typically use speed of light

343
00:21:00,200 --> 00:21:01,370
equal to 1.

344
00:21:01,370 --> 00:21:04,970
So when you look at these kind
of quantities in the physics

345
00:21:04,970 --> 00:21:10,875
that you are more used to,
you would find your energy,

346
00:21:10,875 --> 00:21:12,500
your energy flux,
your momentum density

347
00:21:12,500 --> 00:21:14,210
will have different
units, because you'll

348
00:21:14,210 --> 00:21:16,002
have factors of c that
come into there that

349
00:21:16,002 --> 00:21:19,510
convert from 1 momentum,
and energy, and time flow,

350
00:21:19,510 --> 00:21:21,200
and space flow.

351
00:21:21,200 --> 00:21:27,200
Also, you don't usually include
rest mass and rest energy

352
00:21:27,200 --> 00:21:30,290
in the energy flows
and momentum densities

353
00:21:30,290 --> 00:21:33,330
that you've probably been
familiar with in the past.

354
00:21:33,330 --> 00:21:35,905
So it's when we do
relativity, and we

355
00:21:35,905 --> 00:21:37,280
set the speed of
light equal to 1

356
00:21:37,280 --> 00:21:40,820
that this symmetry between
energy flux and momentum

357
00:21:40,820 --> 00:21:42,980
density becomes apparent.

358
00:21:42,980 --> 00:21:43,480
Question.

359
00:21:43,480 --> 00:21:46,294
AUDIENCE: The second one, do
you mean rho 1 will be there?

360
00:21:48,967 --> 00:21:49,800
SCOTT HUGHES: Sorry.

361
00:21:49,800 --> 00:21:50,330
This one?

362
00:21:50,330 --> 00:21:52,008
AUDIENCE: Yeah, on
the right-hand side.

363
00:21:52,008 --> 00:21:53,550
SCOTT HUGHES: On
the right-hand side.

364
00:21:53,550 --> 00:21:54,360
This is--

365
00:21:54,360 --> 00:21:55,680
AUDIENCE: Rho i.

366
00:21:55,680 --> 00:21:57,368
SCOTT HUGHES: No,
this is definitely

367
00:21:57,368 --> 00:21:58,660
what I mean to write down here.

368
00:21:58,660 --> 00:22:02,700
So this is-- what I've
done, I've taken advantage

369
00:22:02,700 --> 00:22:06,230
of the fact that I can write--

370
00:22:06,230 --> 00:22:06,730
let's see.

371
00:22:06,730 --> 00:22:07,897
Do I have it down somewhere?

372
00:22:07,897 --> 00:22:08,470
Yeah.

373
00:22:08,470 --> 00:22:08,970
Yeah.

374
00:22:08,970 --> 00:22:11,610
So what I've done
over here is I have

375
00:22:11,610 --> 00:22:18,150
written the fact-- so u sub t,
ut is just the gamma factor.

376
00:22:18,150 --> 00:22:21,280
And I didn't explicitly
write out what t--

377
00:22:21,280 --> 00:22:23,730
the i components, but you get
a similar factor that way.

378
00:22:23,730 --> 00:22:25,775
So no, that is correct that way.

379
00:22:25,775 --> 00:22:28,150
AUDIENCE: So that's why you
mean the first and the second

380
00:22:28,150 --> 00:22:28,570
are the same.

381
00:22:28,570 --> 00:22:29,050
SCOTT HUGHES: Yeah.

382
00:22:29,050 --> 00:22:29,550
Yeah.

383
00:22:29,550 --> 00:22:30,850
And I just want to emphasize--

384
00:22:30,850 --> 00:22:32,340
I mean, the key thing
which I want to emphasize

385
00:22:32,340 --> 00:22:34,950
is when you do it in the physics
that you guys have generally

386
00:22:34,950 --> 00:22:37,270
all seen up till now,
it won't look that way.

387
00:22:37,270 --> 00:22:38,580
And there's two causes of that.

388
00:22:38,580 --> 00:22:40,830
One is you're not used to
necessarily working in units

389
00:22:40,830 --> 00:22:42,110
where the speed of light is 1.

390
00:22:42,110 --> 00:22:44,850
And two, rest energy is
built into these definitions.

391
00:22:44,850 --> 00:22:46,442
That shifts things a little bit.

392
00:22:46,442 --> 00:22:48,150
The other thing which
I want to emphasize

393
00:22:48,150 --> 00:22:52,290
is notice this is symmetric
under exchange of indices.

394
00:22:52,290 --> 00:22:53,610
Now, it's obvious for dust.

395
00:22:53,610 --> 00:22:58,080
It comes right back to
the fact that these guys--

396
00:22:58,080 --> 00:22:59,610
that those indices
enter like so.

397
00:22:59,610 --> 00:23:03,780
It's an exterior product of
the four-velocity with itself.

398
00:23:03,780 --> 00:23:05,830
I'm not going to go
through in detail,

399
00:23:05,830 --> 00:23:07,920
but there is a
physical motivation

400
00:23:07,920 --> 00:23:09,870
that I will sketch
in a few moments that

401
00:23:09,870 --> 00:23:12,210
argues that it must
be symmetric like this

402
00:23:12,210 --> 00:23:14,570
in all physical cases.

403
00:23:14,570 --> 00:23:19,290
I have a detailed
sketch of this.

404
00:23:19,290 --> 00:23:21,960
I keep using the word sketch,
because my mind wants to say

405
00:23:21,960 --> 00:23:23,700
proof, but it ain't a proof.

406
00:23:23,700 --> 00:23:27,220
It's sort of a motivation.

407
00:23:27,220 --> 00:23:29,490
And so I will post it for
everyone to go through,

408
00:23:29,490 --> 00:23:32,220
but I will sort of illuminate
it in a couple of moments.

409
00:23:32,220 --> 00:23:34,470
Basically, if this is
not true, if it's not

410
00:23:34,470 --> 00:23:36,810
the case this is symmetric,
a physical absurdity

411
00:23:36,810 --> 00:23:37,740
can be set up.

412
00:23:37,740 --> 00:23:39,240
I'll describe that
in a few moments.

413
00:23:43,290 --> 00:23:49,330
So most of the
world is not dust.

414
00:23:49,330 --> 00:23:52,690
So this is a decent example for
helping to understand things.

415
00:23:52,690 --> 00:23:54,763
But we are going
to want to do more

416
00:23:54,763 --> 00:23:56,305
complicated and
interesting examples.

417
00:24:14,404 --> 00:24:14,983
Pardon me.

418
00:24:14,983 --> 00:24:15,650
Just one second.

419
00:24:15,650 --> 00:24:18,990
Let me just check
here with this page.

420
00:24:18,990 --> 00:24:19,910
Yeah.

421
00:24:19,910 --> 00:24:24,990
So for us, the way that
we are going to do this,

422
00:24:24,990 --> 00:24:27,990
there is a fairly
general recipe that one

423
00:24:27,990 --> 00:24:29,620
can imagine applying to this.

424
00:24:29,620 --> 00:24:31,828
I'm going to save it for a
little later in the class.

425
00:24:31,828 --> 00:24:34,320
It sort of borrows some
techniques from field theory.

426
00:24:34,320 --> 00:24:36,840
Basically, if you can write
down a Lagrangian density that

427
00:24:36,840 --> 00:24:39,990
describes the fifth system
that you're under study,

428
00:24:39,990 --> 00:24:42,390
there's a particular
variation you can do,

429
00:24:42,390 --> 00:24:44,850
that stress energy
tensor emerges from.

430
00:24:44,850 --> 00:24:47,190
But for now, the key
thing which I want to say

431
00:24:47,190 --> 00:24:53,780
is that we basically are going
to deduce what the stress

432
00:24:53,780 --> 00:25:03,360
energy tensor looks
like by essentially

433
00:25:03,360 --> 00:25:06,128
going into a particular frame--
we'll call it the rest frame.

434
00:25:06,128 --> 00:25:07,920
It's usually the rest
frame of some element

435
00:25:07,920 --> 00:25:10,050
of the material being studied--

436
00:25:10,050 --> 00:25:13,200
and thinking carefully about
this physical definition

437
00:25:13,200 --> 00:25:14,850
of what the different
components mean.

438
00:25:14,850 --> 00:25:16,743
This is not the most
rigorous way to do it,

439
00:25:16,743 --> 00:25:18,660
but it's a good way to
get started and develop

440
00:25:18,660 --> 00:25:19,813
some intuition.

441
00:25:47,250 --> 00:25:53,370
So let me give one example
that for many of us

442
00:25:53,370 --> 00:25:56,880
in astrophysics, this is
probably the one stress energy

443
00:25:56,880 --> 00:25:59,095
tensor that we write
down and use over,

444
00:25:59,095 --> 00:26:00,720
and over, and over
again in our career.

445
00:26:00,720 --> 00:26:02,760
And it's rare we
do anything more

446
00:26:02,760 --> 00:26:04,365
than this in many, many cases.

447
00:26:08,810 --> 00:26:10,730
This is called a perfect fluid.

448
00:26:14,140 --> 00:26:17,230
So what is perfect
about a perfect fluid?

449
00:26:17,230 --> 00:26:18,810
It begs the question here.

450
00:26:18,810 --> 00:26:22,420
What is perfect
referring to here?

451
00:26:22,420 --> 00:26:24,900
So a perfect fluid
is a fluid in which

452
00:26:24,900 --> 00:26:32,140
there is no energy flow in what
I will call the "rest" frame.

453
00:26:36,475 --> 00:26:37,850
I put rest in
quotes, because you

454
00:26:37,850 --> 00:26:40,340
have to sort of define it
from the context of what

455
00:26:40,340 --> 00:26:41,240
your fluid is here.

456
00:26:41,240 --> 00:26:42,615
Basically, what
it means is I can

457
00:26:42,615 --> 00:26:45,440
find a frame in which
each fluid element there

458
00:26:45,440 --> 00:26:46,760
is no energy flow there.

459
00:26:46,760 --> 00:26:48,430
If a frame exists
where that happens,

460
00:26:48,430 --> 00:26:51,730
there's a candidate
to be a perfect fluid.

461
00:26:51,730 --> 00:26:56,980
And I also require there
to be no lateral stresses.

462
00:27:00,460 --> 00:27:06,070
Lateral stresses
refer to this Tij

463
00:27:06,070 --> 00:27:08,960
when i and j are not
equal to one another.

464
00:27:08,960 --> 00:27:11,290
So this sort of
refers to-- imagine

465
00:27:11,290 --> 00:27:15,040
that I have some-- let's
say going into the board

466
00:27:15,040 --> 00:27:16,700
is the y direction.

467
00:27:16,700 --> 00:27:19,420
So if there's like some
y stress that is somehow

468
00:27:19,420 --> 00:27:21,340
being transported
in the x direction,

469
00:27:21,340 --> 00:27:24,123
that would be a lateral stress.

470
00:27:24,123 --> 00:27:26,290
Physically, this is actually--
that kind of a stress

471
00:27:26,290 --> 00:27:30,340
is hugely important when
one is studying fluids.

472
00:27:30,340 --> 00:27:32,110
And one characterizes
it by a quantity

473
00:27:32,110 --> 00:27:33,960
known as the viscosity.

474
00:27:33,960 --> 00:27:37,990
Viscosity tells me about how
stress gets transported--

475
00:27:37,990 --> 00:27:40,000
how momentum, rather,
gets transported

476
00:27:40,000 --> 00:27:43,585
in a non-normal direction,
against the direction

477
00:27:43,585 --> 00:27:47,040
in which the fluid is moving.

478
00:27:47,040 --> 00:27:51,600
So my perfect fluid
has no viscosity.

479
00:27:51,600 --> 00:27:52,560
And I'll just conclude.

480
00:27:52,560 --> 00:27:55,290
I want to make-- there's a
very important point here.

481
00:27:55,290 --> 00:27:59,040
A fluid that has no
viscosity is a fluid

482
00:27:59,040 --> 00:28:02,830
that doesn't get anything wet.

483
00:28:02,830 --> 00:28:05,170
So this refers to when you
pour water on yourself,

484
00:28:05,170 --> 00:28:07,045
the reason your hand
gets wet is that there's

485
00:28:07,045 --> 00:28:09,940
some viscosity that actually
prevents it-- causes there

486
00:28:09,940 --> 00:28:12,700
to be sort of a shearing
force, which causes the water

487
00:28:12,700 --> 00:28:15,030
to stick to your skin.

488
00:28:15,030 --> 00:28:16,980
So a perfect fluid,
this has been described

489
00:28:16,980 --> 00:28:20,520
as the physics of dry water.

490
00:28:20,520 --> 00:28:23,687
So it is-- anyone who's
in applied math will know.

491
00:28:23,687 --> 00:28:25,770
They'll sort of roll their
eyes and say, OK, fine,

492
00:28:25,770 --> 00:28:30,330
we'll do this sort of infant
version of fluid first.

493
00:28:30,330 --> 00:28:32,730
And then a lot of the
action and a lot of the fun

494
00:28:32,730 --> 00:28:36,990
comes from putting in viscosity
and doing the real fluids.

495
00:28:36,990 --> 00:28:39,600
For the purposes of our class,
what this boils down to is

496
00:28:39,600 --> 00:28:43,230
this tells me that the
physics of this quantity

497
00:28:43,230 --> 00:28:51,790
is totally dominated
by the fluid's energy

498
00:28:51,790 --> 00:28:55,240
density and its pressure.

499
00:28:59,380 --> 00:29:02,230
The pressure is an
isotropic spatial stress.

500
00:29:14,850 --> 00:29:20,630
So in this particular frame,
where I have no energy being

501
00:29:20,630 --> 00:29:42,110
transported, then the
stress energy tensor can be

502
00:29:42,110 --> 00:29:46,910
represented as energy density
as usual up in the-- whoops,

503
00:29:46,910 --> 00:29:47,720
over there now--

504
00:29:47,720 --> 00:29:49,100
up in the tt component.

505
00:29:52,390 --> 00:29:55,540
There is no energy flux.

506
00:29:55,540 --> 00:29:57,400
By symmetry, if
there's no energy flux,

507
00:29:57,400 --> 00:30:01,540
there is no momentum of density.

508
00:30:01,540 --> 00:30:04,840
And my spatial stresses
are totally isotropic,

509
00:30:04,840 --> 00:30:06,115
and none of them are lateral.

510
00:30:13,180 --> 00:30:22,275
So it just looks like this,
the diagonal of rho p, p, p, p.

511
00:30:26,468 --> 00:30:28,010
As I said, this is
actually something

512
00:30:28,010 --> 00:30:30,350
that we're going to use over,
and over, and over again.

513
00:30:30,350 --> 00:30:32,090
This is actually a--
so you can actually

514
00:30:32,090 --> 00:30:35,210
consider my dust
stress energy tensor

515
00:30:35,210 --> 00:30:37,580
to be a perfect fluid
with no pressure.

516
00:30:37,580 --> 00:30:41,000
So this actually
subsumes this other one.

517
00:30:41,000 --> 00:30:42,570
I will-- just
illustrative purposes,

518
00:30:42,570 --> 00:30:45,290
I'll show an example of a
case that cannot be thought

519
00:30:45,290 --> 00:30:48,450
of as a perfect fluid.

520
00:30:48,450 --> 00:30:51,530
But we will tend to use
this a lot in our class.

521
00:30:51,530 --> 00:30:53,900
And as I will demonstrate
in just a moment,

522
00:30:53,900 --> 00:30:55,455
we're going to find
that this ends up

523
00:30:55,455 --> 00:30:57,080
playing an important
role in generating

524
00:30:57,080 --> 00:30:58,130
gravitational fields.

525
00:30:58,130 --> 00:30:59,420
What's interesting
about this is you

526
00:30:59,420 --> 00:31:00,837
guys are probably
used to the idea

527
00:31:00,837 --> 00:31:03,022
that mass generates
gravity, and then

528
00:31:03,022 --> 00:31:04,730
throw in a c square
that tells you energy

529
00:31:04,730 --> 00:31:06,240
to generating gravity.

530
00:31:06,240 --> 00:31:09,740
But it's also-- we're going to
see pressure generates gravity.

531
00:31:09,740 --> 00:31:11,870
And it's connected to the
mathematical structure

532
00:31:11,870 --> 00:31:14,520
of this guy here.

533
00:31:14,520 --> 00:31:17,608
So notice I've written
an equal dot here.

534
00:31:17,608 --> 00:31:19,400
So this is just the
way this is represented

535
00:31:19,400 --> 00:31:21,800
in this particular frame.

536
00:31:21,800 --> 00:31:27,680
I would like to write this in a
more covariant form, something

537
00:31:27,680 --> 00:31:31,070
that does not rely on me
going into a particular frame

538
00:31:31,070 --> 00:31:32,218
of reference.

539
00:31:39,400 --> 00:31:46,370
So the trick which I'm
going to use for this

540
00:31:46,370 --> 00:31:48,530
is when I think about
that form there--

541
00:31:48,530 --> 00:31:53,570
so the rho piece of it, clearly,
what I'm doing there is I'm

542
00:31:53,570 --> 00:31:54,250
picking out--

543
00:31:54,250 --> 00:31:58,100
that is-- can be thought of--
as the energy density multiplied

544
00:31:58,100 --> 00:32:02,090
by the tensor product of
the four-velocity that

545
00:32:02,090 --> 00:32:04,370
describes a particular
element of this fluid.

546
00:32:06,920 --> 00:32:11,390
So I can write, again, using
this sort of abstract notation.

547
00:32:11,390 --> 00:32:16,630
This piece of it
looks like this.

548
00:32:16,630 --> 00:32:19,190
And so if I go into
the frame, if I

549
00:32:19,190 --> 00:32:22,280
go into the rest frame of
the fluid, that's just my u

550
00:32:22,280 --> 00:32:23,240
is 1 and 0.

551
00:32:23,240 --> 00:32:25,220
And the spatial
components, that builds

552
00:32:25,220 --> 00:32:27,530
my upper left-hand
corner of this tensor.

553
00:32:27,530 --> 00:32:30,170
How do I get the rest of this?

554
00:32:30,170 --> 00:32:32,270
Well, to get the rest of
it, these are all sort of

555
00:32:32,270 --> 00:32:34,220
picked out of components
of the tensor that

556
00:32:34,220 --> 00:32:37,230
are orthogonal to u.

557
00:32:37,230 --> 00:32:39,990
And I put an Easter
egg in p set 1.

558
00:32:39,990 --> 00:32:42,870
You guys developed
a tensor that allows

559
00:32:42,870 --> 00:32:44,970
me to build-- it's a
geometric object that

560
00:32:44,970 --> 00:32:48,150
allows me to describe things
that are orthogonal to a given

561
00:32:48,150 --> 00:32:50,050
four-vector.

562
00:32:50,050 --> 00:32:56,520
So if I take the
projection tensor

563
00:32:56,520 --> 00:33:05,470
that you guys built
one piece at 1,

564
00:33:05,470 --> 00:33:08,210
which looks like the
metric plus the extern--

565
00:33:08,210 --> 00:33:10,800
the tensor product of a
four-vector with itself,

566
00:33:10,800 --> 00:33:14,060
that gives me the p's that
go into that component.

567
00:33:14,060 --> 00:33:19,750
Or if I write this
in index notation,

568
00:33:19,750 --> 00:33:20,750
I can do it in two ways.

569
00:33:42,790 --> 00:33:44,290
I will also emphasize
that there are

570
00:33:44,290 --> 00:33:47,500
few moments in this class
where I sort of urge

571
00:33:47,500 --> 00:33:50,380
you to take the sort of
long-term memory synapses

572
00:33:50,380 --> 00:33:52,120
and switch them on.

573
00:33:52,120 --> 00:33:54,220
This is one of those moments.

574
00:33:54,220 --> 00:33:55,790
In a couple of
lectures, we're going

575
00:33:55,790 --> 00:33:57,790
to introduce the principle
of equivalence, which

576
00:33:57,790 --> 00:33:59,457
is the physical
principle by which we're

577
00:33:59,457 --> 00:34:01,785
going to argue how we go
from formulas that work

578
00:34:01,785 --> 00:34:03,160
in special relativity
to formulas

579
00:34:03,160 --> 00:34:05,530
that work in general relativity.

580
00:34:05,530 --> 00:34:08,020
And by invoking the
principle of equivalence,

581
00:34:08,020 --> 00:34:09,520
we're going to see
that when we want

582
00:34:09,520 --> 00:34:12,040
to describe perfect fluids
in a general spacetime,

583
00:34:12,040 --> 00:34:15,190
not just in special relativity
and general relativity,

584
00:34:15,190 --> 00:34:16,929
it's exactly this formula.

585
00:34:16,929 --> 00:34:20,280
I just need to modify
what the metric means,

586
00:34:20,280 --> 00:34:22,030
but that will allow
me to carry that over.

587
00:34:26,719 --> 00:34:28,340
Let me see.

588
00:34:28,340 --> 00:34:30,330
So my notes are a little
bit disorganized here.

589
00:34:30,330 --> 00:34:31,909
There's a piece that I--

590
00:34:31,909 --> 00:34:33,710
every year I want to
clean this bit up,

591
00:34:33,710 --> 00:34:35,210
and every year at
this time of year,

592
00:34:35,210 --> 00:34:37,520
I have 70 gajillion
administrative tasks.

593
00:34:37,520 --> 00:34:40,409
And I end up getting
behind schedule.

594
00:34:40,409 --> 00:34:42,422
So I will clean
this up on the fly.

595
00:34:42,422 --> 00:34:43,880
So there's a couple
of points which

596
00:34:43,880 --> 00:34:45,530
I want to make about this.

597
00:34:45,530 --> 00:34:47,540
Let me do this point first.

598
00:35:03,920 --> 00:35:07,370
So by virtue of taking a
graduate physics class,

599
00:35:07,370 --> 00:35:10,760
I'm confident you guys know
about Newtonian gravity.

600
00:35:10,760 --> 00:35:24,940
One can write the field
equation for Newtonian gravity

601
00:35:24,940 --> 00:35:27,130
as essentially a
differential equation

602
00:35:27,130 --> 00:35:30,040
for the potential that governs
Newtonian gravitational

603
00:35:30,040 --> 00:35:31,890
interaction.

604
00:35:31,890 --> 00:35:34,240
So let's call phi sub g
the Newtonian gravitational

605
00:35:34,240 --> 00:35:37,380
interaction.

606
00:35:37,380 --> 00:35:38,320
Whoops.

607
00:35:38,320 --> 00:35:40,240
And I can write a
field equation that's

608
00:35:40,240 --> 00:35:47,770
governing it as essentially
its Poisson's equation.

609
00:35:47,770 --> 00:35:50,830
So the Laplace operator
acting on that potential

610
00:35:50,830 --> 00:35:55,290
is up to a constant
equal to the--

611
00:35:55,290 --> 00:35:57,697
we usually learn it in
terms of mass density.

612
00:35:57,697 --> 00:36:00,280
We're working in units where the
speed of light is equal to 1.

613
00:36:00,280 --> 00:36:03,670
So it could just as well
be the energy density.

614
00:36:03,670 --> 00:36:05,927
So you've all kind of seen that.

615
00:36:05,927 --> 00:36:07,510
Now, if we think
about how we're going

616
00:36:07,510 --> 00:36:09,550
to carry this forward
and make gravity

617
00:36:09,550 --> 00:36:12,640
a relativistic
interaction, this equation

618
00:36:12,640 --> 00:36:14,440
should right away
make us suspicious,

619
00:36:14,440 --> 00:36:17,470
because we spent several
minutes earlier today

620
00:36:17,470 --> 00:36:23,240
talking about the fact
that this is not a scalar.

621
00:36:23,240 --> 00:36:24,925
This is a component of a tensor.

622
00:36:35,470 --> 00:36:37,490
A physical theory
which tries to pick out

623
00:36:37,490 --> 00:36:40,550
just one component
of a geometric tensor

624
00:36:40,550 --> 00:36:43,700
is not a healthy theory.

625
00:36:43,700 --> 00:36:46,190
It would be like if
you had sort of learned

626
00:36:46,190 --> 00:36:49,473
in E&M that there was
a preferred direction

627
00:36:49,473 --> 00:36:50,390
of the electric field.

628
00:36:50,390 --> 00:36:52,730
If there's a particular set
of Maxwell's equations for Ex,

629
00:36:52,730 --> 00:36:54,813
and a different set of
Maxwell's equations for Ey.

630
00:36:54,813 --> 00:36:57,260
That would just be nonsense.

631
00:36:57,260 --> 00:37:00,980
Nature doesn't pick out spatial
components of any one being

632
00:37:00,980 --> 00:37:04,850
particularly having some
weight over the others,

633
00:37:04,850 --> 00:37:08,220
and the same thing holds in
relativity for spacetime.

634
00:37:08,220 --> 00:37:12,380
So when we make this into
a relativistic theory,

635
00:37:12,380 --> 00:37:15,200
we're going to say if
this component of a tensor

636
00:37:15,200 --> 00:37:17,750
plays a large role in
gravity to make this

637
00:37:17,750 --> 00:37:20,730
into a geometric object that
makes a large role in gravity,

638
00:37:20,730 --> 00:37:23,060
I'm going to have to-- so
let's call this my Newtonian

639
00:37:23,060 --> 00:37:25,280
equation.

640
00:37:25,280 --> 00:37:28,340
My Einsteinian equation--

641
00:37:28,340 --> 00:37:30,170
I'm going to put an
equal sign in quotes

642
00:37:30,170 --> 00:37:32,570
here, because there's a
lot of details to fill in.

643
00:37:32,570 --> 00:37:35,540
But what's going to go on
the right-hand side of this

644
00:37:35,540 --> 00:37:39,230
has to be something that
involves the stress energy

645
00:37:39,230 --> 00:37:40,970
tensor.

646
00:37:40,970 --> 00:37:44,450
Newton picks out one component
stress energy tensor.

647
00:37:44,450 --> 00:37:47,630
Relativity doesn't let me pick
out particular components.

648
00:37:47,630 --> 00:37:50,690
So whatever I get when
I do Einstein's gravity,

649
00:37:50,690 --> 00:37:52,250
the whole stress
energy tensor is

650
00:37:52,250 --> 00:37:57,315
going to be important in setting
the source of my gravity.

651
00:37:57,315 --> 00:37:58,940
That then sort of
says, well, then what

652
00:37:58,940 --> 00:38:02,750
the heck do you do with
this left-hand side?

653
00:38:02,750 --> 00:38:06,800
That is, in fact, going to be
starting probably on Tuesday

654
00:38:06,800 --> 00:38:09,500
the subject of the next couple
of lectures, basically going

655
00:38:09,500 --> 00:38:10,980
all the way up to spring break.

656
00:38:10,980 --> 00:38:14,630
The week before spring break
is when we complete the story

657
00:38:14,630 --> 00:38:16,800
of what goes on the
right-hand side--

658
00:38:16,800 --> 00:38:18,800
excuse me-- the left-hand
side of this equation.

659
00:38:18,800 --> 00:38:21,740
But what I will tell
you is that it is indeed

660
00:38:21,740 --> 00:38:27,500
going to involve two derivatives
of a potential-like object.

661
00:38:27,500 --> 00:38:29,260
And the potential-like
object is actually

662
00:38:29,260 --> 00:38:31,693
going to turn out to be
the metric of spacetime.

663
00:38:31,693 --> 00:38:33,610
So that's kind of where
we're going with this.

664
00:38:46,120 --> 00:38:49,010
Let's do a little bit more
physics with the stress energy

665
00:38:49,010 --> 00:38:49,510
tensor.

666
00:39:17,350 --> 00:39:21,300
So I have somewhat
more detailed notes,

667
00:39:21,300 --> 00:39:24,030
which I will post online, which
I am not going to go through

668
00:39:24,030 --> 00:39:25,050
in great detail here.

669
00:39:25,050 --> 00:39:26,880
But I'm going to
kind of sketch this.

670
00:39:26,880 --> 00:39:30,522
So one is that I
would like to prove--

671
00:39:30,522 --> 00:39:32,730
again, that word is a little
bit of an overstatement,

672
00:39:32,730 --> 00:39:36,360
but at least motivate the
symmetry of this tensor.

673
00:39:45,480 --> 00:39:47,730
I'm just going to focus on
the spatial bits of this.

674
00:39:47,730 --> 00:39:49,230
That'll be enough.

675
00:39:49,230 --> 00:39:52,272
Like I said, you can kind
of see that T0i and ti0 are

676
00:39:52,272 --> 00:39:54,480
the same thing by thinking
about the physical meaning

677
00:39:54,480 --> 00:39:57,270
of four-momentum and what
a flux of four-momentum is.

678
00:39:57,270 --> 00:40:00,160
This one, there's kind of a
cute calculation you can do.

679
00:40:04,450 --> 00:40:10,665
So imagine you have some
little cube of stuff.

680
00:40:10,665 --> 00:40:15,270
It could be immersed in some
field or fluid, something

681
00:40:15,270 --> 00:40:17,520
that is described by a
field of stress energy.

682
00:40:39,370 --> 00:40:47,210
And so to start out with,
let's look at how the flux--

683
00:40:47,210 --> 00:40:49,520
remember what t alpha
beta, or really Tij

684
00:40:49,520 --> 00:40:52,337
tells me about is
the flux of momentum

685
00:40:52,337 --> 00:40:53,420
in a particular direction.

686
00:40:53,420 --> 00:40:54,860
It's telling me about
the amount of momentum

687
00:40:54,860 --> 00:40:57,360
going into this box on one side
and coming out on the other.

688
00:41:13,922 --> 00:41:15,630
So I'm going to look
at the momentum flux

689
00:41:15,630 --> 00:41:18,140
into and out of this box.

690
00:41:18,140 --> 00:41:22,770
And so what I'm going to do
is let's look at the momentum

691
00:41:22,770 --> 00:41:25,785
going into the sides that are--

692
00:41:28,530 --> 00:41:30,830
so let's call this the
top and the bottom here.

693
00:41:35,280 --> 00:41:37,350
What I'm going to do
then is number the four

694
00:41:37,350 --> 00:41:39,540
sides, the other four
sides of the box, not

695
00:41:39,540 --> 00:41:41,200
the top and the bottom--

696
00:41:41,200 --> 00:41:43,440
so let's call this side,
which is sort of facing away

697
00:41:43,440 --> 00:41:46,230
from us here, that side one.

698
00:41:46,230 --> 00:41:48,570
The one that is facing
us that sort of points

699
00:41:48,570 --> 00:41:52,380
out in the x direction,
I'll call that two.

700
00:41:52,380 --> 00:41:55,260
This side here, I
will call three.

701
00:41:55,260 --> 00:41:58,560
And the one that is on
the back I will call four.

702
00:41:58,560 --> 00:42:00,780
Apologies for the little
bit of a busy picture here.

703
00:42:00,780 --> 00:42:01,730
But just what you
want to do is sort

704
00:42:01,730 --> 00:42:03,540
of imagine there's a
cube in front of you,

705
00:42:03,540 --> 00:42:05,630
and you go around and
label the four sides.

706
00:42:22,880 --> 00:42:24,680
What we'd like to
do is calculate

707
00:42:24,680 --> 00:42:26,810
what is the force
that is flowing

708
00:42:26,810 --> 00:42:29,345
through each of these four
sides, 1, 2, 3, and 4.

709
00:42:29,345 --> 00:42:31,930
Just ignore the top and the
bottom for just a second.

710
00:42:31,930 --> 00:42:34,860
I'll describe why that
is in just a moment.

711
00:42:34,860 --> 00:42:45,175
So if I look at the force
on face one, well, face one,

712
00:42:45,175 --> 00:42:45,675
it is--

713
00:42:50,190 --> 00:42:53,800
pardon me-- I
mislabeled my sides.

714
00:42:53,800 --> 00:42:56,160
I want to be synced
up with my notes,

715
00:42:56,160 --> 00:42:57,590
and so I realize it's annoying.

716
00:42:57,590 --> 00:42:58,840
And I'm very sorry about that.

717
00:42:58,840 --> 00:43:01,030
But if I mess this
up, I will get out

718
00:43:01,030 --> 00:43:02,860
of sync with what I
have written down.

719
00:43:02,860 --> 00:43:06,530
So the one that is
facing us is side one.

720
00:43:06,530 --> 00:43:08,900
Two is on the right-hand side.

721
00:43:08,900 --> 00:43:11,320
Three is the back.

722
00:43:11,320 --> 00:43:12,860
Four is on the left-hand side.

723
00:43:31,290 --> 00:43:32,230
My apologies for that.

724
00:43:32,230 --> 00:43:33,870
But I think it's important
we get that right so I don't

725
00:43:33,870 --> 00:43:35,520
get out of sync with myself.

726
00:43:35,520 --> 00:43:37,740
So the total force
on face one is

727
00:43:37,740 --> 00:43:41,940
what I get by basically adding
up all the flux of momentum

728
00:43:41,940 --> 00:43:43,590
flowing through face one.

729
00:43:46,620 --> 00:43:51,990
So force on face one, that is--

730
00:43:51,990 --> 00:43:55,110
the i-th component
of that is what

731
00:43:55,110 --> 00:43:59,880
I get when I integrate
that over face one, which

732
00:43:59,880 --> 00:44:01,110
is normal to the x-axis.

733
00:44:04,980 --> 00:44:07,990
Each side of the
cube is little l.

734
00:44:07,990 --> 00:44:09,880
So it's Tix l squared.

735
00:44:18,250 --> 00:44:23,540
Force on face two,
now this is the one

736
00:44:23,540 --> 00:44:25,430
that is normal for the y-axis.

737
00:44:29,650 --> 00:44:34,440
And so this guy looks
like Tiy l squared.

738
00:44:37,700 --> 00:44:41,720
And you continue this F3--

739
00:44:41,720 --> 00:44:44,000
if you look at, and
you if you assume

740
00:44:44,000 --> 00:44:53,220
that this thing is small,
it's approximately the same

741
00:44:53,220 --> 00:44:54,360
as the force on F1.

742
00:44:54,360 --> 00:44:56,850
It'll become equal in the
limit of the q becoming

743
00:44:56,850 --> 00:44:57,870
infinitesimally small.

744
00:45:03,610 --> 00:45:08,220
And this is approximately the
same as the force on face two.

745
00:45:08,220 --> 00:45:14,820
So that tells us that
there's no unbalanced force

746
00:45:14,820 --> 00:45:17,760
on this thing, which is great.

747
00:45:21,882 --> 00:45:24,760
Basically, it means that there's
no unbalanced force causing

748
00:45:24,760 --> 00:45:27,750
this little element
to accelerate away.

749
00:45:27,750 --> 00:45:30,760
One of the reasons why I am
focusing on these four sides,

750
00:45:30,760 --> 00:45:33,320
though, is I would also
like to consider torques

751
00:45:33,320 --> 00:45:34,570
that are acting on this thing.

752
00:46:05,690 --> 00:46:08,273
And here's where I'm going to
skip over a couple of details

753
00:46:08,273 --> 00:46:09,690
and just leave a
few notes for you

754
00:46:09,690 --> 00:46:11,180
guys to look at, because
we're a little short on time

755
00:46:11,180 --> 00:46:11,930
with other things.

756
00:46:11,930 --> 00:46:14,250
This calculation is
straightforward, but it gets--

757
00:46:14,250 --> 00:46:15,780
I already screwed
up a detail here.

758
00:46:15,780 --> 00:46:18,760
I don't want to sort of risk
messing up a few other things.

759
00:46:18,760 --> 00:46:21,060
What I want to do is
consider the torques

760
00:46:21,060 --> 00:46:23,940
about an axis that sort
of runs through the middle

761
00:46:23,940 --> 00:46:27,450
of this thing that goes from
the top and the bottom of this.

762
00:46:27,450 --> 00:46:30,660
So go through and just
add up all the torques

763
00:46:30,660 --> 00:46:33,450
associated with these
little forces about an axis

764
00:46:33,450 --> 00:46:36,050
in the center of--

765
00:46:36,050 --> 00:46:37,800
that goes through the
center of this cube.

766
00:47:04,913 --> 00:47:06,330
So I'm going to--
like I said, I'm

767
00:47:06,330 --> 00:47:09,180
going to leave out the details,
but basically, you go through,

768
00:47:09,180 --> 00:47:11,487
and you do the usual
r cross f to get

769
00:47:11,487 --> 00:47:13,320
the forces, the torques
associated with each

770
00:47:13,320 --> 00:47:14,362
of these different faces.

771
00:47:26,690 --> 00:47:29,390
And what you'll find
when you do this

772
00:47:29,390 --> 00:47:34,710
is that there is a net
torque that looks like l

773
00:47:34,710 --> 00:47:43,030
cubed times Txy minus Tyx.

774
00:47:43,030 --> 00:47:45,373
Again, though, this scales
with the size of the cube.

775
00:47:45,373 --> 00:47:47,540
So you look at this and
kind of go, well, who cares?

776
00:47:47,540 --> 00:47:48,010
It's a little bit there.

777
00:47:48,010 --> 00:47:49,302
Maybe it spins up a little bit.

778
00:47:49,302 --> 00:47:51,580
But in the limit,
the thing going to 0,

779
00:47:51,580 --> 00:47:53,550
there's no net effect.

780
00:47:53,550 --> 00:47:56,250
Well, let's be a little
bit careful about this.

781
00:47:56,250 --> 00:47:58,710
What is the moment of
inertia of this cube?

782
00:48:08,630 --> 00:48:11,350
I don't know exactly,
but I know that it's

783
00:48:11,350 --> 00:48:14,200
going to be something like
the mass of this cube.

784
00:48:18,550 --> 00:48:22,450
We'll call that l cubed times
its mass density, or its energy

785
00:48:22,450 --> 00:48:24,730
density.

786
00:48:24,730 --> 00:48:27,790
And it's going to involve two
powers of the only length scale

787
00:48:27,790 --> 00:48:28,990
characterizing this thing.

788
00:48:31,580 --> 00:48:33,423
And there will be
some prefactor alpha.

789
00:48:33,423 --> 00:48:35,090
There will be some
number that's related

790
00:48:35,090 --> 00:48:37,065
to the geometry of this.

791
00:48:37,065 --> 00:48:38,690
An integral or two
will easily work out

792
00:48:38,690 --> 00:48:40,570
what the alpha is and
make this more precise.

793
00:48:40,570 --> 00:48:43,040
But the key bit which
I want to emphasize

794
00:48:43,040 --> 00:48:48,900
is that this is
something that scales

795
00:48:48,900 --> 00:48:50,430
as the size of the fifth power.

796
00:49:00,180 --> 00:49:04,010
So yeah, the torque
vanishes as l goes to 0,

797
00:49:04,010 --> 00:49:06,740
but it does so with
the cubed power.

798
00:49:06,740 --> 00:49:11,360
It doesn't vanish as quickly as
the moment of inertia vanishes.

799
00:49:11,360 --> 00:49:14,420
And I'll remind you,
the angular acceleration

800
00:49:14,420 --> 00:49:28,140
of the cube, theta double
dot, is the torque divided

801
00:49:28,140 --> 00:49:29,950
by the moment of inertia.

802
00:49:29,950 --> 00:49:34,650
So this is something that
is proportional to Txy

803
00:49:34,650 --> 00:49:38,820
minus Tyx divided by l squared.

804
00:49:42,330 --> 00:49:45,990
So yeah, the torque does
vanish as l goes to 0,

805
00:49:45,990 --> 00:49:47,490
but the moment of
inertia vanishes

806
00:49:47,490 --> 00:49:50,340
more rapidly as l goes to 0.

807
00:49:50,340 --> 00:49:56,450
That sort of suggests that
if we're in a screw universe,

808
00:49:56,450 --> 00:49:58,160
little microscopic
vortices are just

809
00:49:58,160 --> 00:50:01,370
going to start randomly
oscillating in any cup of water

810
00:50:01,370 --> 00:50:03,850
that you pour in front of you.

811
00:50:03,850 --> 00:50:08,260
Now, I don't have a proof
that nature abhors that,

812
00:50:08,260 --> 00:50:11,590
but it seems pretty screwy.

813
00:50:11,590 --> 00:50:15,460
And so the case that most
textbooks make at this point

814
00:50:15,460 --> 00:50:23,600
is to say, physics
indicates we must

815
00:50:23,600 --> 00:50:31,580
have Txy strictly equal
to Tyx in all cases,

816
00:50:31,580 --> 00:50:33,660
independent of what the
size of this thing is.

817
00:50:33,660 --> 00:50:37,250
That had better just be a
bloody 0 in the numerator

818
00:50:37,250 --> 00:50:39,290
in order to prevent
this physical absurdity

819
00:50:39,290 --> 00:50:40,130
from being set up.

820
00:50:49,530 --> 00:50:51,830
Repeat the exercise
by looking at torques

821
00:50:51,830 --> 00:50:53,540
around the other axes.

822
00:50:53,540 --> 00:51:04,720
And that drives you
to the statement

823
00:51:04,720 --> 00:51:06,610
that this thing
must, in general,

824
00:51:06,610 --> 00:51:07,798
be spatially symmetric.

825
00:51:07,798 --> 00:51:10,340
And again, physics of the way
that energy and momentum behave

826
00:51:10,340 --> 00:51:13,840
in relativity makes our sort of
timespace component symmetric

827
00:51:13,840 --> 00:51:14,950
as well.

828
00:51:14,950 --> 00:51:16,600
I emphasize this is not a proof.

829
00:51:16,600 --> 00:51:18,370
This is really just a
physical motivation.

830
00:51:18,370 --> 00:51:21,640
This is the kind of thing that
I'm a bloody astrophysicist.

831
00:51:21,640 --> 00:51:23,320
I like this kind of stuff.

832
00:51:23,320 --> 00:51:26,303
There is a different way of
developing the stress energy

833
00:51:26,303 --> 00:51:28,720
tensor, as I said, that comes
from a variational principle

834
00:51:28,720 --> 00:51:31,360
sort of based on
sort of quantum field

835
00:51:31,360 --> 00:51:32,920
theoretic type of methods.

836
00:51:32,920 --> 00:51:35,530
And the symmetry in
that case is manifest.

837
00:51:35,530 --> 00:51:37,360
It really does sort of come up.

838
00:51:37,360 --> 00:51:40,480
But this is a good way of just
motivating the fact that you're

839
00:51:40,480 --> 00:51:43,120
not going to have
any non-diagonal--

840
00:51:43,120 --> 00:51:45,940
or excuse-- non-symmetric
stress energy tensors.

841
00:51:45,940 --> 00:51:49,090
If you did, you would have
really bizarre matter.

842
00:51:53,140 --> 00:51:56,920
Now, the stress
energy tensor has

843
00:51:56,920 --> 00:52:00,340
this physical interpretation
that it tells me

844
00:52:00,340 --> 00:52:04,360
about the flow of energy and the
flow of momentum in spacetime.

845
00:52:04,360 --> 00:52:07,690
As such, it is the
tool by which we

846
00:52:07,690 --> 00:52:13,440
are going to put conservation
of energy and conservation

847
00:52:13,440 --> 00:52:15,420
of momentum into our theory.

848
00:52:32,043 --> 00:52:33,460
And the way we're
going to do this

849
00:52:33,460 --> 00:52:37,290
is with a remarkably
simple equation.

850
00:52:37,290 --> 00:52:41,562
The spacetime divergence
of t alpha beta--

851
00:52:41,562 --> 00:52:47,250
pardon me-- must equal 0.

852
00:52:50,130 --> 00:52:53,430
This is a covariant formulation
of both conservation of energy

853
00:52:53,430 --> 00:52:55,423
and conservation of momentum.

854
00:52:55,423 --> 00:52:57,090
And if you want to
say, well, which one?

855
00:52:57,090 --> 00:52:58,900
Is it energy or is it momentum?

856
00:52:58,900 --> 00:53:00,900
You can't say that in general.

857
00:53:00,900 --> 00:53:05,760
I can only say that after I have
picked a particular reference

858
00:53:05,760 --> 00:53:08,130
frame, because it's only
once I have defined time

859
00:53:08,130 --> 00:53:10,620
and I've defined space
that I've actually

860
00:53:10,620 --> 00:53:13,972
defined energy and momentum.

861
00:53:13,972 --> 00:53:15,930
Prior to choosing a
particular reference frame,

862
00:53:15,930 --> 00:53:17,630
all I have is four-momentum.

863
00:53:17,630 --> 00:53:20,100
One observer's energy is
another observer's superposition

864
00:53:20,100 --> 00:53:21,058
of energy and momentum.

865
00:53:23,760 --> 00:53:25,800
Once I have picked
a particular frame--

866
00:53:39,910 --> 00:53:47,830
so once I have picked a
particular frame, then

867
00:53:47,830 --> 00:53:55,960
if I evaluate d alpha
T alpha either 0 or t,

868
00:53:55,960 --> 00:53:57,790
this is what
conservation of energy

869
00:53:57,790 --> 00:53:59,090
looks like in that frame.

870
00:54:18,620 --> 00:54:21,400
So in that frame, here is
conservation of energy.

871
00:54:24,368 --> 00:54:24,910
I can fit it.

872
00:54:55,166 --> 00:54:56,010
Hang on a second.

873
00:54:56,010 --> 00:54:56,560
No, no, no.

874
00:54:56,560 --> 00:54:57,810
I had it right the first time.

875
00:55:04,540 --> 00:55:06,520
Yeah.

876
00:55:06,520 --> 00:55:07,020
Sorry.

877
00:55:07,020 --> 00:55:07,760
I was just trying
to make sure I got

878
00:55:07,760 --> 00:55:08,968
my indices lined up properly.

879
00:55:13,760 --> 00:55:16,340
So apologies you can't quite
read the bottom one so well.

880
00:55:16,340 --> 00:55:18,118
The top one is
conservation of energy

881
00:55:18,118 --> 00:55:19,160
in that particular frame.

882
00:55:19,160 --> 00:55:21,350
The second one is
conservation of momentum

883
00:55:21,350 --> 00:55:22,930
in that particular frame.

884
00:55:22,930 --> 00:55:25,670
And again, the key thing
which I want to emphasize

885
00:55:25,670 --> 00:55:29,840
is the covariant statement
basically puts both of them

886
00:55:29,840 --> 00:55:31,550
together into a single equation.

887
00:55:31,550 --> 00:55:33,657
I can only sensibly
state conservation

888
00:55:33,657 --> 00:55:35,240
of energy and
conservation of momentum

889
00:55:35,240 --> 00:55:38,150
according to some
particular observer.

890
00:55:38,150 --> 00:55:39,710
Now, I can repeat
the game that I

891
00:55:39,710 --> 00:55:42,140
had done before with
the number vector

892
00:55:42,140 --> 00:55:44,900
and turned these conservation
laws into integral equations

893
00:55:44,900 --> 00:55:45,510
as well.

894
00:55:45,510 --> 00:55:46,510
Let me do it for energy.

895
00:56:18,430 --> 00:56:21,843
So if I take that
integral equation

896
00:56:21,843 --> 00:56:24,010
and integrate over a
three-volume, or if you prefer,

897
00:56:24,010 --> 00:56:26,230
I can take the original
covariant formulation

898
00:56:26,230 --> 00:56:29,325
and integrate that
over a four-volume.

899
00:56:29,325 --> 00:56:30,700
With a little bit
of manipulation

900
00:56:30,700 --> 00:56:33,242
akin to the way I
manipulated the conserv--

901
00:56:33,242 --> 00:56:35,200
the integrals associated
with the number vector

902
00:56:35,200 --> 00:56:46,910
in the previous lecture,
you can write down

903
00:56:46,910 --> 00:56:48,320
a law that looks like this.

904
00:56:55,710 --> 00:56:57,630
And so this-- again,
what I've done here

905
00:56:57,630 --> 00:57:01,180
is I've chosen a particular
set of timeline coordinates.

906
00:57:01,180 --> 00:57:03,180
And the language that we
often use in relativity

907
00:57:03,180 --> 00:57:07,740
is we basically say I'm going
to take a single slice of time.

908
00:57:07,740 --> 00:57:11,670
And I would say that the
rate of change of energy--

909
00:57:11,670 --> 00:57:13,380
so integrate energy
over a volume.

910
00:57:13,380 --> 00:57:15,600
It's the total
energy in that V3.

911
00:57:15,600 --> 00:57:17,610
The rate of change
of that thing is

912
00:57:17,610 --> 00:57:19,530
balanced by the amount
of energy flowing

913
00:57:19,530 --> 00:57:23,970
into or out of through the
boundaries of that volume.

914
00:57:23,970 --> 00:57:25,540
Add an extra index here.

915
00:57:25,540 --> 00:57:29,160
So make this-- make
one of these be a j.

916
00:57:29,160 --> 00:57:30,863
Make this guy be a j.

917
00:57:30,863 --> 00:57:32,280
And you've got a
similar statement

918
00:57:32,280 --> 00:57:35,950
for the conservation
of momentum.

919
00:57:35,950 --> 00:57:37,620
This is a particular trick.

920
00:57:37,620 --> 00:57:41,455
I have a set of typed up
notes that just sort of clean

921
00:57:41,455 --> 00:57:42,330
this up a little bit.

922
00:57:42,330 --> 00:57:43,872
They're basically
exactly this point,

923
00:57:43,872 --> 00:57:47,160
but I'll put them online
after today's lecture.

924
00:57:47,160 --> 00:57:49,880
You're going to want to use this
on one of the p set problems

925
00:57:49,880 --> 00:57:50,380
this week.

926
00:57:50,380 --> 00:57:51,880
So you guys are
going to do a couple

927
00:57:51,880 --> 00:57:53,900
exercises that involve
integrating and actually

928
00:57:53,900 --> 00:57:58,340
finding essentially moments
of the left-hand side

929
00:57:58,340 --> 00:57:58,923
of this thing.

930
00:57:58,923 --> 00:58:00,548
You're going to do
a few exercise where

931
00:58:00,548 --> 00:58:02,870
you take advantage of this
formulation of conservation

932
00:58:02,870 --> 00:58:08,730
of energy and momentum to
derive a few identities involved

933
00:58:08,730 --> 00:58:11,000
in the stress energy
tensor, one of which

934
00:58:11,000 --> 00:58:13,880
is another Easter egg that
we're going to use quite a bit

935
00:58:13,880 --> 00:58:16,610
in a future lecture.

936
00:58:16,610 --> 00:58:24,170
So let me just wrap up our
discussion of the stress energy

937
00:58:24,170 --> 00:58:26,870
tensor by just doing
two more examples.

938
00:58:26,870 --> 00:58:28,430
And then I'm going
to sort of begin

939
00:58:28,430 --> 00:58:29,638
switching gears a little bit.

940
00:58:33,200 --> 00:58:41,130
So far I have only described--

941
00:58:41,130 --> 00:58:43,920
essentially, I've really only
described perfect fluids.

942
00:58:43,920 --> 00:58:47,370
Dust is a perfect
fluid with no pressure.

943
00:58:51,860 --> 00:58:55,670
There are a lot of other kinds
of materials that we want

944
00:58:55,670 --> 00:58:57,200
to work with in the universe.

945
00:58:57,200 --> 00:59:01,170
One of which-- it's unlikely
many of you are going to use

946
00:59:01,170 --> 00:59:05,220
this, but it's actually the
stress energy tensor on which I

947
00:59:05,220 --> 00:59:06,115
have built--

948
00:59:06,115 --> 00:59:08,282
well, I guess Alex is going
to use it a little bit--

949
00:59:08,282 --> 00:59:11,250
I have built a big
chunk of my career.

950
00:59:11,250 --> 00:59:16,135
Suppose you have-- we kind of
talked a little bit about how

951
00:59:16,135 --> 00:59:17,760
the four-velocity
and the four-momentum

952
00:59:17,760 --> 00:59:20,177
are really only good for talking
about like the kinematics

953
00:59:20,177 --> 00:59:20,727
of particles.

954
00:59:20,727 --> 00:59:22,560
Well, actually, particles
aren't a bad thing

955
00:59:22,560 --> 00:59:24,030
to focus on in some
of your studies.

956
00:59:24,030 --> 00:59:25,488
And much of my
research is actually

957
00:59:25,488 --> 00:59:30,630
based on the idea of thinking
about a binary system as one

958
00:59:30,630 --> 00:59:33,120
member being a black
hole, and its companion

959
00:59:33,120 --> 00:59:34,597
being a particle-like
object that

960
00:59:34,597 --> 00:59:36,930
is a particular limit of it,
a particle-like object that

961
00:59:36,930 --> 00:59:38,550
orbits it.

962
00:59:38,550 --> 00:59:42,573
So a really simple
stress energy tensor--

963
00:59:42,573 --> 00:59:43,990
and I'm just
throwing it out here,

964
00:59:43,990 --> 00:59:46,331
because I think it nicely
illustrates the principle.

965
00:59:50,750 --> 00:59:53,220
A point particle with rest
mass, we'll call it m0.

966
00:59:58,960 --> 01:00:01,810
And I'm going to say it's moving
on a particular world line

967
01:00:01,810 --> 01:00:03,051
through spacetime.

968
01:00:10,130 --> 01:00:11,570
So the way we
define a world line

969
01:00:11,570 --> 01:00:13,550
is we just say it's
some four-vector that

970
01:00:13,550 --> 01:00:16,670
describes the place of this
thing from some chosen origin.

971
01:00:16,670 --> 01:00:19,760
And it's generally most
convenient to parameterize it

972
01:00:19,760 --> 01:00:23,090
by the proper time of
whatever object or creature is

973
01:00:23,090 --> 01:00:26,300
moving on that world line.

974
01:00:26,300 --> 01:00:31,250
And so it's a lot like dust,
only there's no volume.

975
01:00:31,250 --> 01:00:34,050
It's sort of like
one particle of dust.

976
01:00:34,050 --> 01:00:36,290
And so the stress energy
tensor we use for this--

977
01:00:45,200 --> 01:00:47,840
let me back up for
just a second here.

978
01:00:47,840 --> 01:00:51,330
When you guys learned about
electricity and magnetism,

979
01:00:51,330 --> 01:00:53,930
one of the first things you
learn about are point charges.

980
01:00:53,930 --> 01:00:55,472
And then a little
bit later you learn

981
01:00:55,472 --> 01:00:57,170
about charge distributions.

982
01:00:57,170 --> 01:00:58,475
And you have charge densities.

983
01:00:58,475 --> 01:01:00,350
And then, usually, at
some point in a class--

984
01:01:00,350 --> 01:01:01,880
it's often like
junior level E&M--

985
01:01:01,880 --> 01:01:05,570
we say, how do you describe
the density of a point charge?

986
01:01:05,570 --> 01:01:07,930
That's where you learn about
the Dirac delta function.

987
01:01:07,930 --> 01:01:09,618
Well, if I have
a point particle,

988
01:01:09,618 --> 01:01:11,660
I'm going to need to
describe this thing's energy

989
01:01:11,660 --> 01:01:14,700
density as essentially
a Dirac delta function.

990
01:01:14,700 --> 01:01:22,530
And so what we do is imagining
that these things-- so this is

991
01:01:22,530 --> 01:01:23,970
the four-velocity of that body.

992
01:01:23,970 --> 01:01:26,670
They might be functions of time
as this guy is moving along

993
01:01:26,670 --> 01:01:27,990
here.

994
01:01:27,990 --> 01:01:36,240
What we do is we introduce
a Dirac delta function

995
01:01:36,240 --> 01:01:38,343
as this thing moves
along through spacetime.

996
01:01:38,343 --> 01:01:40,260
What this does-- you can
check the dimensions.

997
01:01:40,260 --> 01:01:42,952
This gives me exactly
what we need in order

998
01:01:42,952 --> 01:01:44,910
to have something that's
dimensionally correct,

999
01:01:44,910 --> 01:01:46,952
and has all the symmetries
and all the properties

1000
01:01:46,952 --> 01:01:51,000
that describe a particle moving
at a particular four-velocity

1001
01:01:51,000 --> 01:01:53,210
through spacetime.

1002
01:01:53,210 --> 01:01:55,827
Now, you might
want to just-- it's

1003
01:01:55,827 --> 01:01:58,160
the thing which I kind of
want to pause on for a second.

1004
01:01:58,160 --> 01:02:00,260
You go, what the hell do
you do with this, right?

1005
01:02:00,260 --> 01:02:02,330
That's kind of inconvenient.

1006
01:02:02,330 --> 01:02:04,790
Well, the trick we use to sort
of clean up that Dirac delta

1007
01:02:04,790 --> 01:02:11,940
function, it's
very much like what

1008
01:02:11,940 --> 01:02:14,670
you do when you encounter
multidimensional delta

1009
01:02:14,670 --> 01:02:17,665
functions in basic physics.

1010
01:02:17,665 --> 01:02:20,040
You just build it out of a
bunch of one-dimensional delta

1011
01:02:20,040 --> 01:02:21,039
functions.

1012
01:02:35,240 --> 01:02:37,100
Likewise, you'll
have a term with a y

1013
01:02:37,100 --> 01:02:39,270
component and the z component.

1014
01:02:39,270 --> 01:02:48,660
Then you use the rule
that if I integrate

1015
01:02:48,660 --> 01:02:55,550
a function of x against a delta
function whose argument is

1016
01:02:55,550 --> 01:03:02,490
itself a function
of x, I evaluate

1017
01:03:02,490 --> 01:03:04,980
that function at the 0's of g.

1018
01:03:11,680 --> 01:03:16,590
So let's say x0 is
where g has a 0.

1019
01:03:16,590 --> 01:03:21,198
Normalizing to the
first derivative of g

1020
01:03:21,198 --> 01:03:22,800
evaluated at that 0.

1021
01:03:34,990 --> 01:03:38,480
When you put all
of that together,

1022
01:03:38,480 --> 01:03:45,420
that basically means you can do
this somewhat abstract integral

1023
01:03:45,420 --> 01:03:46,440
formally.

1024
01:03:46,440 --> 01:03:47,210
I mean, exactly.

1025
01:03:47,210 --> 01:03:48,700
You can just do it analytically.

1026
01:03:52,460 --> 01:03:55,240
And what you do is you choose
one of the delta functions

1027
01:03:55,240 --> 01:03:56,440
to apply it to.

1028
01:03:56,440 --> 01:03:59,252
Traditionally, people apply
it to the timeline component.

1029
01:04:04,450 --> 01:04:07,540
And what is the derivative
of the world line

1030
01:04:07,540 --> 01:04:12,050
z component with
respect to proper time?

1031
01:04:12,050 --> 01:04:14,820
It's the 0 component
of the four-velocity.

1032
01:04:14,820 --> 01:04:18,895
So you just divide by the 0
component of the four-velocity,

1033
01:04:18,895 --> 01:04:21,020
and then you're left with
a three-dimensional delta

1034
01:04:21,020 --> 01:04:26,120
function for the spatial
trajectory of this thing

1035
01:04:26,120 --> 01:04:28,430
through all space.

1036
01:04:28,430 --> 01:04:31,158
So this is an example
of one that just,

1037
01:04:31,158 --> 01:04:33,200
again, kind of the intuition
is one of the things

1038
01:04:33,200 --> 01:04:34,033
I want to emphasize.

1039
01:04:34,033 --> 01:04:35,690
Notice we had-- this
has the symmetries

1040
01:04:35,690 --> 01:04:37,160
that we wanted to have.

1041
01:04:37,160 --> 01:04:38,760
It's not hard to show that this.

1042
01:04:38,760 --> 01:04:39,830
You can think of
this as essentially

1043
01:04:39,830 --> 01:04:41,180
being kind like a gamma factor.

1044
01:04:41,180 --> 01:04:42,680
This all ends up
giving me just what

1045
01:04:42,680 --> 01:04:44,972
we need for this thing to
have the right transformation

1046
01:04:44,972 --> 01:04:48,060
properties.

1047
01:04:48,060 --> 01:04:51,360
And it does, in fact,
play a role in some--

1048
01:04:51,360 --> 01:04:53,310
well, I'll say in some
research that's near

1049
01:04:53,310 --> 01:04:55,047
and dear to my heart.

1050
01:04:55,047 --> 01:04:56,130
Let me do another example.

1051
01:05:05,560 --> 01:05:08,770
Suppose you want to know the
stress energy tensor associated

1052
01:05:08,770 --> 01:05:11,800
with a given electric
and magnetic field.

1053
01:05:17,080 --> 01:05:21,420
Well, first, let me just quote
for you the exact answer, which

1054
01:05:21,420 --> 01:05:25,830
is most compactly written if
we use that Faraday tensor

1055
01:05:25,830 --> 01:05:28,230
F, which describes
electromagnetic fields

1056
01:05:28,230 --> 01:05:31,327
in a frame-independent fashion,
the way that I introduced it

1057
01:05:31,327 --> 01:05:32,160
in the last lecture.

1058
01:05:35,550 --> 01:05:37,250
So in units where
basically everything

1059
01:05:37,250 --> 01:05:44,030
but pi is set equal to 1, it
ends up turning into this.

1060
01:05:59,800 --> 01:06:01,520
So that's a bit of a mouthful.

1061
01:06:01,520 --> 01:06:03,730
Let's go in and look at
particular components of it,

1062
01:06:03,730 --> 01:06:04,270
though.

1063
01:06:04,270 --> 01:06:06,550
So let's say I go into
a particular frame.

1064
01:06:06,550 --> 01:06:09,760
I fill in my Faraday
tensor with the form

1065
01:06:09,760 --> 01:06:12,093
of electromagnetic field
that I introduced last time.

1066
01:06:12,093 --> 01:06:13,510
And I'll just go
through, and I'll

1067
01:06:13,510 --> 01:06:15,677
evaluate all the different
components of this thing.

1068
01:06:23,860 --> 01:06:40,770
So what you find when you fill
this in is your T00 component,

1069
01:06:40,770 --> 01:06:44,700
1 over 8 pi e squared
plus b squared.

1070
01:06:44,700 --> 01:06:45,660
That's good.

1071
01:06:45,660 --> 01:06:48,150
Hopefully, y'all
remember from basic E&M,

1072
01:06:48,150 --> 01:06:52,020
the energy density of an
E-field is e squared over 8 pi

1073
01:06:52,020 --> 01:06:53,537
and the right system of units.

1074
01:06:53,537 --> 01:06:55,620
Energy density of a b field
is b squared over 8 pi

1075
01:06:55,620 --> 01:06:57,570
if you work in God's units.

1076
01:06:57,570 --> 01:07:01,511
Let's do the
timespace component.

1077
01:07:04,950 --> 01:07:08,860
So again, hack through
that mess there.

1078
01:07:08,860 --> 01:07:11,135
This is going to be
something that is a vector.

1079
01:07:21,180 --> 01:07:23,910
In fact, it's the
Poynting vector.

1080
01:07:23,910 --> 01:07:25,500
Could it be anything else?

1081
01:07:25,500 --> 01:07:27,850
If you use the recipe
that I sort of suggested

1082
01:07:27,850 --> 01:07:29,350
as the easiest way
to approach this,

1083
01:07:29,350 --> 01:07:31,142
this is kind of what
you would have guessed

1084
01:07:31,142 --> 01:07:32,430
for something like that.

1085
01:07:32,430 --> 01:07:35,220
The bit that's
actually kind of hard

1086
01:07:35,220 --> 01:07:38,550
is then trying to get the
spatial stresses of this thing.

1087
01:07:38,550 --> 01:07:47,260
And here I don't have
any great intuition

1088
01:07:47,260 --> 01:07:49,330
for this one to convey to you.

1089
01:07:49,330 --> 01:07:52,090
It's derived in textbooks
like Griffiths and Jackson.

1090
01:07:52,090 --> 01:08:02,910
So I'll just quote
for you the result.

1091
01:08:02,910 --> 01:08:15,380
So you get one term that looks
like E squared plus b squared

1092
01:08:15,380 --> 01:08:17,960
on the diagonals.

1093
01:08:17,960 --> 01:08:30,787
Then there's a correction,
which looks like this.

1094
01:08:30,787 --> 01:08:32,579
I want to just quickly
call out one example

1095
01:08:32,579 --> 01:08:35,060
so you can see what the
significance of this example

1096
01:08:35,060 --> 01:08:37,090
is.

1097
01:08:37,090 --> 01:08:42,478
So suppose you have something
like a pair of capacitors.

1098
01:08:42,478 --> 01:08:44,770
And there's just a uniform
electric field between them.

1099
01:08:44,770 --> 01:08:46,353
You want to evaluate
the stress energy

1100
01:08:46,353 --> 01:08:50,529
tensor between those
pairs of capacitors.

1101
01:08:50,529 --> 01:08:52,810
So my spatial electric
field, let's say it just

1102
01:08:52,810 --> 01:08:55,180
points in the Ex direction.

1103
01:08:55,180 --> 01:08:55,930
And it's constant.

1104
01:09:13,245 --> 01:09:14,870
So when you actually
evaluate this guy,

1105
01:09:14,870 --> 01:09:15,997
there's no energy flow.

1106
01:09:15,997 --> 01:09:17,080
There's no magnetic field.

1107
01:09:17,080 --> 01:09:18,507
So there's no Poynting vector.

1108
01:09:18,507 --> 01:09:20,840
You, of course, have E squared
over 8 pi for your energy

1109
01:09:20,840 --> 01:09:21,340
density.

1110
01:09:31,270 --> 01:09:33,472
Very different from
a perfect fluid.

1111
01:09:33,472 --> 01:09:34,680
And this kind of makes sense.

1112
01:09:34,680 --> 01:09:35,680
That's sort of
telling you that there

1113
01:09:35,680 --> 01:09:37,380
is a stress that
if you have a pair

1114
01:09:37,380 --> 01:09:40,649
of plain parallel capacitors,
you tend to attract

1115
01:09:40,649 --> 01:09:42,720
the plates to each other.

1116
01:09:42,720 --> 01:09:44,736
But there is a
pressure associated

1117
01:09:44,736 --> 01:09:46,319
with that electric
field that actually

1118
01:09:46,319 --> 01:09:47,978
goes in the other directions.

1119
01:09:47,978 --> 01:09:49,020
This is your x direction.

1120
01:09:49,020 --> 01:09:51,420
This is y and z.

1121
01:09:51,420 --> 01:09:53,118
And as a consequence of this--

1122
01:09:53,118 --> 01:09:54,660
so there's some
stuff which we're not

1123
01:09:54,660 --> 01:09:56,327
going to do too much
with in this class,

1124
01:09:56,327 --> 01:09:58,260
but I may give you
some pointers on this.

1125
01:09:58,260 --> 01:10:01,050
Electric and
magnetic fields, they

1126
01:10:01,050 --> 01:10:03,240
generate pressures, at
least in certain directions.

1127
01:10:03,240 --> 01:10:06,210
They kind of generate like
an anisotropic pressure.

1128
01:10:06,210 --> 01:10:09,300
And when we start
coupling this to gravity,

1129
01:10:09,300 --> 01:10:11,490
you can get electric
fields and magnetic fields

1130
01:10:11,490 --> 01:10:17,020
that contribute non-negligibly
to the gravity of their object.

1131
01:10:17,020 --> 01:10:19,600
That is it for the
stress energy tensor.

1132
01:10:19,600 --> 01:10:22,660
As I said, we are going to
use this guy over, and over,

1133
01:10:22,660 --> 01:10:23,650
and over again.

1134
01:10:23,650 --> 01:10:26,290
And the reason for
this does come back

1135
01:10:26,290 --> 01:10:30,520
to that little motivation that
I gave probably about 45 minutes

1136
01:10:30,520 --> 01:10:34,810
ago where we sort of looked at
the Newtonian field equation,

1137
01:10:34,810 --> 01:10:37,780
and then said, picking
out a particular scalar

1138
01:10:37,780 --> 01:10:39,970
as the source of
gravity makes no sense

1139
01:10:39,970 --> 01:10:41,740
in a relativistic
covariant theory.

1140
01:10:41,740 --> 01:10:43,960
It's got to be the whole tensor.

1141
01:10:43,960 --> 01:10:47,200
And indeed, this is the
one-- well, not the E&M one,

1142
01:10:47,200 --> 01:10:49,540
but the general notion
of a stress energy tensor

1143
01:10:49,540 --> 01:10:52,130
is the one that we are
going to use for that.

1144
01:10:52,130 --> 01:10:57,570
So we have about
10 minutes left.

1145
01:10:57,570 --> 01:11:02,480
And so I would like to start
the process of switching gears.

1146
01:11:02,480 --> 01:11:05,010
Before I do that, are
there any questions?

1147
01:11:10,022 --> 01:11:10,980
I will clean the board.

1148
01:11:24,340 --> 01:11:26,060
So the reason we
are switching gears

1149
01:11:26,060 --> 01:11:30,470
is we now have probably the
most important physical tools

1150
01:11:30,470 --> 01:11:32,750
that we need in order
to start thinking

1151
01:11:32,750 --> 01:11:37,490
about making a relativistic
theory of gravity.

1152
01:11:37,490 --> 01:11:40,730
But we need a few more
mathematical tools.

1153
01:11:40,730 --> 01:11:46,910
In particular, I'm going to
argue in a couple of lectures

1154
01:11:46,910 --> 01:11:50,930
that flat spacetime is
not sufficient for us

1155
01:11:50,930 --> 01:11:52,220
to build a theory of gravity.

1156
01:11:52,220 --> 01:11:53,180
We're going to need
to-- first of all,

1157
01:11:53,180 --> 01:11:54,500
you're going to have to
understand what that means.

1158
01:11:54,500 --> 01:11:56,083
And we're not quite
ready to go there.

1159
01:11:56,083 --> 01:11:57,523
So for now, it's
just fancy words.

1160
01:11:57,523 --> 01:11:58,940
But I'm going to
have to introduce

1161
01:11:58,940 --> 01:12:01,310
some kind of a notion of
curvature into things.

1162
01:12:01,310 --> 01:12:03,050
What does that even mean really?

1163
01:12:03,050 --> 01:12:05,400
We need to have the
tools to do that.

1164
01:12:05,400 --> 01:12:09,170
And so as a prelude to
going in that direction--

1165
01:12:09,170 --> 01:12:11,453
I will call this my
prelude to curvature--

1166
01:12:15,800 --> 01:12:17,840
what we're going
to start doing is

1167
01:12:17,840 --> 01:12:29,535
flat spacetime in
curvilinear coordinates.

1168
01:12:33,720 --> 01:12:35,130
And the importance
of doing this,

1169
01:12:35,130 --> 01:12:37,350
why this is going
to be useful to us

1170
01:12:37,350 --> 01:12:40,770
is that it will introduce a--

1171
01:12:40,770 --> 01:12:42,030
it'll keep the physics simple.

1172
01:12:42,030 --> 01:12:44,130
It's still going to
be special relativity,

1173
01:12:44,130 --> 01:12:46,230
but it's now going
to special relativity

1174
01:12:46,230 --> 01:12:51,930
using a mathematical structure
in which the basis vectors are

1175
01:12:51,930 --> 01:12:53,490
no longer constant.

1176
01:12:53,490 --> 01:12:54,990
So that's going to
allow us to begin

1177
01:12:54,990 --> 01:12:57,450
making a couple of the
mathematical tools that

1178
01:12:57,450 --> 01:13:00,610
are necessary to build
gravity into this theory.

1179
01:13:00,610 --> 01:13:08,220
So we'll start by
replacing this with just

1180
01:13:08,220 --> 01:13:17,520
simple plain polar coordinates
mapped in the usual way.

1181
01:13:17,520 --> 01:13:19,710
So if we want to
go back and forth,

1182
01:13:19,710 --> 01:13:24,620
well, at least one
way transformation,

1183
01:13:24,620 --> 01:13:27,602
I build x and y from r
and phi in that usual way.

1184
01:13:27,602 --> 01:13:29,060
There's an inverse
mapping as well,

1185
01:13:29,060 --> 01:13:30,460
which involves trig functions.

1186
01:13:30,460 --> 01:13:32,970
So I'm not going
to write it down.

1187
01:13:32,970 --> 01:13:37,450
One point which I really
want to emphasize here

1188
01:13:37,450 --> 01:13:41,635
is we are going to continue
to use a coordinate basis.

1189
01:13:55,070 --> 01:14:11,780
And to remind you
what that means,

1190
01:14:11,780 --> 01:14:17,650
so a coordinate basis means that
the differential displacement

1191
01:14:17,650 --> 01:14:25,120
vector from an event
a to a nearby event b

1192
01:14:25,120 --> 01:14:27,120
is simply related
by differentials

1193
01:14:27,120 --> 01:14:32,488
of my coordinate contracted
onto my basis vectors.

1194
01:14:32,488 --> 01:14:34,030
But when I'm working
in a curvilinear

1195
01:14:34,030 --> 01:14:52,090
coordinate system like
this, that means one of them

1196
01:14:52,090 --> 01:14:55,120
has a somewhat different form
from what you are used to.

1197
01:14:55,120 --> 01:14:56,890
When you guys talk
about the differential

1198
01:14:56,890 --> 01:14:59,015
of a displacement in just
about every physics class

1199
01:14:59,015 --> 01:15:01,030
up to now, if you have
a differential angle,

1200
01:15:01,030 --> 01:15:02,450
you'd want to throw an r.

1201
01:15:02,450 --> 01:15:05,440
So this has the
dimensions of length.

1202
01:15:05,440 --> 01:15:07,090
We ain't going to do that.

1203
01:15:07,090 --> 01:15:13,867
And so what this means,
since this is an angle,

1204
01:15:13,867 --> 01:15:16,075
every component of this has
the dimensions of length.

1205
01:15:22,300 --> 01:15:28,498
That means that
my basis vector is

1206
01:15:28,498 --> 01:15:30,790
going to be something that
has the dimensions of length

1207
01:15:30,790 --> 01:15:33,310
associated with it.

1208
01:15:33,310 --> 01:15:37,525
This in turn means that this is
not going to be a normal basis.

1209
01:15:46,167 --> 01:15:48,000
That's unfortunately a
somewhat loaded word.

1210
01:15:48,000 --> 01:15:50,352
What I mean by that is that
it has not been normalized.

1211
01:15:50,352 --> 01:15:52,560
But given that you guys have
spent all of your career

1212
01:15:52,560 --> 01:15:56,090
thinking about the dot product
of a unit vector with itself--

1213
01:15:56,090 --> 01:15:58,890
of a basis vector with
itself being equal to 1,

1214
01:15:58,890 --> 01:16:01,227
the other meaning of normal
might be good for you, too.

1215
01:16:01,227 --> 01:16:03,060
The key thing which I
want to emphasize here

1216
01:16:03,060 --> 01:16:07,320
is e phi dot e phi
does not equal 1.

1217
01:16:14,690 --> 01:16:16,685
One other little bit
of notation which

1218
01:16:16,685 --> 01:16:17,810
I would like to introduce--

1219
01:16:22,830 --> 01:16:26,070
so we are going to want to
talk about transformations

1220
01:16:26,070 --> 01:16:28,508
between different
representations.

1221
01:16:28,508 --> 01:16:30,300
We've done this-- so
far, we have generally

1222
01:16:30,300 --> 01:16:33,930
focused on moving between
different reference frames.

1223
01:16:33,930 --> 01:16:35,662
I want to generalize
this notion.

1224
01:16:35,662 --> 01:16:37,620
And I'm going to tweak
my notation a little bit

1225
01:16:37,620 --> 01:16:39,930
to indicate the difference.

1226
01:16:39,930 --> 01:16:59,660
So I'm going to call
capital L alpha mu

1227
01:16:59,660 --> 01:17:07,060
bar what I get when I just look
at the variation of the alpha

1228
01:17:07,060 --> 01:17:09,960
coordinate system--

1229
01:17:09,960 --> 01:17:12,870
sorry-- the unbarred coordinate
system with the barred one.

1230
01:17:12,870 --> 01:17:14,820
So this is just--

1231
01:17:14,820 --> 01:17:16,020
there's no deep things here.

1232
01:17:16,020 --> 01:17:17,550
I just want you to be familiar
with the notation which

1233
01:17:17,550 --> 01:17:18,810
I'm going to use.

1234
01:17:18,810 --> 01:17:26,552
I will always reserve lambda
for the Lorentz Transformation.

1235
01:17:35,240 --> 01:17:38,020
So in the interest of time, I'll
just write down one of these.

1236
01:17:38,020 --> 01:17:39,728
So this sort of means
like, for instance,

1237
01:17:39,728 --> 01:17:41,410
suppose that I'm transforming.

1238
01:17:41,410 --> 01:17:47,810
Let's let barred indicate
my polar coordinates.

1239
01:17:47,810 --> 01:17:50,840
Unbarred be Cartesian.

1240
01:18:10,982 --> 01:18:13,664
These things are surprisingly
sticky sometimes.

1241
01:18:24,810 --> 01:18:28,280
So if I want to transform
in one direction

1242
01:18:28,280 --> 01:18:32,340
between my barred
and my unbarred ones,

1243
01:18:32,340 --> 01:18:34,620
this matrix will go through it.

1244
01:18:34,620 --> 01:18:35,640
And it looks like this.

1245
01:18:58,810 --> 01:19:03,900
So this is the thing which
I will call l alpha mu bar.

1246
01:19:03,900 --> 01:19:06,210
In my notes, I also give
the inverse transformation.

1247
01:19:06,210 --> 01:19:07,667
And I will actually write--

1248
01:19:07,667 --> 01:19:08,250
you know what?

1249
01:19:08,250 --> 01:19:08,610
I have minute.

1250
01:19:08,610 --> 01:19:10,068
I will write that
one down as well.

1251
01:19:13,972 --> 01:19:16,180
If you want to know how to
go in the other direction,

1252
01:19:16,180 --> 01:19:18,220
it's just the matrix
that inverts this.

1253
01:19:35,020 --> 01:19:39,520
The reason I decided to
take the extra 30 seconds

1254
01:19:39,520 --> 01:19:43,670
or so to write
this down is I want

1255
01:19:43,670 --> 01:19:46,130
to call out the fact that in
this kind of transformation,

1256
01:19:46,130 --> 01:19:48,650
because of the fact that this
is a coordinate basis that

1257
01:19:48,650 --> 01:19:52,490
has this somewhat
unusual property,

1258
01:19:52,490 --> 01:19:56,280
different elements of the matrix
have different dimensions.

1259
01:19:56,280 --> 01:19:58,910
It's a feature, not a bug.

1260
01:19:58,910 --> 01:20:01,420
So this is going to show up.

1261
01:20:01,420 --> 01:20:03,560
I'm going to do a
couple more calculations

1262
01:20:03,560 --> 01:20:06,190
with this a little
bit tomorrow-- sorry--

1263
01:20:06,190 --> 01:20:08,882
not tomorrow, on Tuesday.

1264
01:20:08,882 --> 01:20:11,090
This is going to show up in
the way the metric looks.

1265
01:20:11,090 --> 01:20:12,950
The metric is going to have
a very different character

1266
01:20:12,950 --> 01:20:14,400
in this coordinate system.

1267
01:20:14,400 --> 01:20:16,530
And we're going to
see the way in which--

1268
01:20:16,530 --> 01:20:18,800
it basically boils down
to it's going to pick up

1269
01:20:18,800 --> 01:20:20,490
a non-trivial functional form.

1270
01:20:20,490 --> 01:20:22,370
It's not going to
just be a constant.

1271
01:20:22,370 --> 01:20:24,800
And that fundamentally
reflects the fact

1272
01:20:24,800 --> 01:20:29,570
that the basis vectors
are not constant anymore.

1273
01:20:29,570 --> 01:20:31,570
We'll end it there.