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PROFESSOR: OK, in
that case, let's

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00:00:23,320 --> 00:00:31,590
start-- I want to begin by
giving a quick review of where

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00:00:31,590 --> 00:00:32,720
we were last time.

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00:00:32,720 --> 00:00:35,470
And then we'll
pick up from there.

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00:00:35,470 --> 00:00:37,820
Today's lecture,
the main subject

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00:00:37,820 --> 00:00:40,175
will be the-- see if this works.

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00:00:42,814 --> 00:00:44,730
The main subject will
be the spacetime metric,

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00:00:44,730 --> 00:00:46,680
which is what we'll
begin by talking about.

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00:00:46,680 --> 00:00:52,130
And later, I hope we'll be
doing the geodesic equation.

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00:00:52,130 --> 00:00:56,710
Last time, we began by
talking about open universes.

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00:00:56,710 --> 00:01:00,560
And we got to open universes
by way of closed universes.

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00:01:00,560 --> 00:01:04,680
And we started with this
Robertson-Walker form

20
00:01:04,680 --> 00:01:07,260
of the closed
universe metric, which

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00:01:07,260 --> 00:01:13,800
holds for k greater than 0
describing a closed universe.

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00:01:13,800 --> 00:01:17,850
And then we said if we want
to describe an open universe,

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00:01:17,850 --> 00:01:22,930
we can use the same equation,
but let k be less than 0.

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00:01:22,930 --> 00:01:26,230
And this metric has a name,
the Robertson-Walker metric,

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00:01:26,230 --> 00:01:28,870
which applies for k
being positive, negative,

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00:01:28,870 --> 00:01:30,980
or 0 as a special case.

27
00:01:30,980 --> 00:01:34,540
When k is 0, this just
becomes the Euclidean metric

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00:01:34,540 --> 00:01:36,400
written in polar coordinates.

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00:01:36,400 --> 00:01:38,610
So it's a flat space
for k equals 0.

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00:01:41,744 --> 00:01:43,160
Then we addressed
the question of,

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00:01:43,160 --> 00:01:46,920
why should we believe this
as a proper description

32
00:01:46,920 --> 00:01:49,370
of an open universe?

33
00:01:49,370 --> 00:01:51,209
We know how to write it.

34
00:01:51,209 --> 00:01:53,250
And we could make it look
a little better perhaps

35
00:01:53,250 --> 00:01:56,480
by introducing a kappa,
which is minus k.

36
00:01:56,480 --> 00:01:59,515
So that kappa could be
positive when k is negative.

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00:02:02,730 --> 00:02:05,470
To answer that question,
we had to define

38
00:02:05,470 --> 00:02:10,130
what criteria we had in mind for
the metric we were looking for.

39
00:02:10,130 --> 00:02:12,240
And what we're trying
to do is write down

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00:02:12,240 --> 00:02:14,820
a metric that will describe
a homogeneous and isotropic

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00:02:14,820 --> 00:02:15,389
universe.

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00:02:15,389 --> 00:02:16,930
Because from the
beginning of course,

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00:02:16,930 --> 00:02:19,460
we said that those are
the key properties that

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00:02:19,460 --> 00:02:22,180
describe, to a
good approximation,

45
00:02:22,180 --> 00:02:24,280
the universe that we live in.

46
00:02:24,280 --> 00:02:26,370
So what we want to know
is that this metric

47
00:02:26,370 --> 00:02:30,440
is homogeneous and isotropic
when kappa is positive

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00:02:30,440 --> 00:02:33,650
and k is negative, the new case.

49
00:02:33,650 --> 00:02:36,240
For the closed universe case,
we already knew those things

50
00:02:36,240 --> 00:02:38,000
because in the
closed universe case,

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00:02:38,000 --> 00:02:41,300
it's obvious because we know
we got it from a sphere.

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00:02:41,300 --> 00:02:44,350
And a sphere is clearly
homogeneous and isotropic

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00:02:44,350 --> 00:02:46,760
from the start.

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00:02:46,760 --> 00:02:51,640
So looking at this metric
for kappa positive,

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00:02:51,640 --> 00:02:54,590
we see immediately that
it's obviously isotropic,

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00:02:54,590 --> 00:02:57,130
at least about the origin.

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00:02:57,130 --> 00:02:59,740
Because if we just
sit at the origin

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00:02:59,740 --> 00:03:03,480
and looked at different
angles-- theta and phi--

59
00:03:03,480 --> 00:03:07,240
the theta phi dependence
of this metric

60
00:03:07,240 --> 00:03:10,240
is simply given by
that expression.

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00:03:10,240 --> 00:03:12,840
And this is exactly the
metric of the surface

62
00:03:12,840 --> 00:03:17,675
of a sphere whose radius happens
to be a of t times little r.

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00:03:17,675 --> 00:03:21,870
And we know that that
sphere is isotropic.

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It doesn't look
manifestly isotropic

65
00:03:23,780 --> 00:03:26,320
because when you put
theta phi coordinates

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on the surface of a sphere, you
choose a north pole and measure

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everything from that.

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00:03:30,720 --> 00:03:33,990
So your coordinates
break the isotropy.

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00:03:33,990 --> 00:03:37,620
But you know perfectly
well that the sphere itself

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is completely isotropic.

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00:03:40,290 --> 00:03:42,580
So isotropy is settled
about the origin.

72
00:03:42,580 --> 00:03:45,367
And if we're soon going
to prove homogeneity,

73
00:03:45,367 --> 00:03:47,700
it's enough to know that it's
isotropic about the origin

74
00:03:47,700 --> 00:03:50,340
because homogeneity
will demonstrate

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00:03:50,340 --> 00:03:51,950
that all points are equivalent.

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00:03:51,950 --> 00:03:54,010
So if it's isotropic
about the origin,

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00:03:54,010 --> 00:03:58,640
we will ultimately know that's
isotropic about all points.

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00:03:58,640 --> 00:04:01,340
So homogeneity is
the hard thing.

79
00:04:01,340 --> 00:04:03,860
How do we convince
ourselves that this metric

80
00:04:03,860 --> 00:04:04,960
is homogeneous?

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00:04:04,960 --> 00:04:08,340
Now, if we look at it, it
doesn't look homogeneous.

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00:04:08,340 --> 00:04:11,230
It certainly looks like
the origin is special.

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00:04:11,230 --> 00:04:13,170
That was the case for
the closed universe

84
00:04:13,170 --> 00:04:15,610
Robertson-Walker metric as well.

85
00:04:15,610 --> 00:04:17,860
So it's certainly not decisive.

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00:04:17,860 --> 00:04:21,410
But it also doesn't prove that
it's homogeneous by itself.

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00:04:21,410 --> 00:04:23,340
So we have to figure
out how we would

88
00:04:23,340 --> 00:04:25,510
prove that it's homogeneous.

89
00:04:25,510 --> 00:04:27,170
And here, we only
sketched an argument.

90
00:04:27,170 --> 00:04:29,220
We didn't really go
through it in detail

91
00:04:29,220 --> 00:04:31,319
because it gets messy.

92
00:04:31,319 --> 00:04:33,110
But I think logic of
the argument is clear.

93
00:04:33,110 --> 00:04:36,214
And it's, I think,
rather persuasive

94
00:04:36,214 --> 00:04:38,130
that this argument would
work if you wrote out

95
00:04:38,130 --> 00:04:39,740
all the equations.

96
00:04:39,740 --> 00:04:42,190
So let me go through
the argument again.

97
00:04:42,190 --> 00:04:45,000
We start by thinking about
how we would demonstrate

98
00:04:45,000 --> 00:04:48,250
that the closed
universe was homogeneous

99
00:04:48,250 --> 00:04:55,701
in a mathematical way using
algebra rather than words.

100
00:04:55,701 --> 00:04:57,950
And if we wanted to prove
that the closed universe was

101
00:04:57,950 --> 00:05:01,460
homogeneous using
algebra, we would set out

102
00:05:01,460 --> 00:05:05,030
with the goal of trying to show
that any point-- and we'll let

103
00:05:05,030 --> 00:05:10,124
r sub 0, theta sub 0, pi sub
0 denote an arbitrary point.

104
00:05:10,124 --> 00:05:11,790
What we'd like to
show is that any point

105
00:05:11,790 --> 00:05:13,830
is equivalent to the origin.

106
00:05:13,830 --> 00:05:16,190
And by equivalent to the
origin, what we really mean

107
00:05:16,190 --> 00:05:18,690
is that we could define a
new coordinate system where

108
00:05:18,690 --> 00:05:21,280
this arbitrary point
would become the origin,

109
00:05:21,280 --> 00:05:24,410
and the metric would look just
like it looked to start with.

110
00:05:24,410 --> 00:05:26,310
And then this new
arbitrary point

111
00:05:26,310 --> 00:05:28,540
will be playing
exactly the same role

112
00:05:28,540 --> 00:05:31,380
that the origin played
in the first place.

113
00:05:31,380 --> 00:05:33,270
So we're looking for
a transformation which

114
00:05:33,270 --> 00:05:37,720
will map r0, theta 0, and phi 0
to the origin while maintaining

115
00:05:37,720 --> 00:05:39,650
the form of the metric.

116
00:05:39,650 --> 00:05:41,280
And we really know
how to do that,

117
00:05:41,280 --> 00:05:44,490
because we know
from the beginning

118
00:05:44,490 --> 00:05:47,700
that this universe is
homogeneous because

119
00:05:47,700 --> 00:05:50,940
of the way we constructed
it as the coordinization

120
00:05:50,940 --> 00:05:52,280
of the surface of a sphere.

121
00:05:52,280 --> 00:05:55,550
And the sphere is
manifestly homogeneous.

122
00:05:55,550 --> 00:05:58,580
You can rotate any point on
the sphere into any other point

123
00:05:58,580 --> 00:06:01,030
just by performing a rotation,
which certainly does not

124
00:06:01,030 --> 00:06:04,720
change anything about
the metric on the sphere.

125
00:06:04,720 --> 00:06:07,280
So we want to basically
take advantage of that fact.

126
00:06:07,280 --> 00:06:09,270
And we could imagine--
and we can even

127
00:06:09,270 --> 00:06:13,010
carry it out if we have to,
but I only want to imagine it.

128
00:06:13,010 --> 00:06:17,550
I want to imagine constructing
a map from r, theta,

129
00:06:17,550 --> 00:06:20,130
and phi to some new r prime,
theta prime, phi prime.

130
00:06:20,130 --> 00:06:21,902
We want to map the entire space.

131
00:06:21,902 --> 00:06:23,360
But we want it to
have the property

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00:06:23,360 --> 00:06:27,490
that the special point--
r0, theta 0, and phi 0--

133
00:06:27,490 --> 00:06:29,840
gets mapped to the origin.

134
00:06:29,840 --> 00:06:32,830
So we want to construct
this general mapping which

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00:06:32,830 --> 00:06:34,710
has the property that
our special point is

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00:06:34,710 --> 00:06:36,710
mapped to the origin.

137
00:06:36,710 --> 00:06:40,460
And we can do that
in three steps.

138
00:06:40,460 --> 00:06:42,600
And they're shown
schematically here.

139
00:06:42,600 --> 00:06:45,470
We first simply transform
from our r, theta,

140
00:06:45,470 --> 00:06:48,930
and phi coordinates to the
four coordinates x, y, z,

141
00:06:48,930 --> 00:06:52,360
w that we, in fact, started
with, the four coordinates

142
00:06:52,360 --> 00:06:55,990
that describe the euclidean
four dimensional space in which

143
00:06:55,990 --> 00:07:00,280
this three dimensional
sphere is embedded.

144
00:07:00,280 --> 00:07:03,530
Once we have the four
dimensional space,

145
00:07:03,530 --> 00:07:06,010
we can perform
euclidean rotations

146
00:07:06,010 --> 00:07:08,270
in that four dimensional space.

147
00:07:08,270 --> 00:07:11,240
And we can perform
any rotation we want.

148
00:07:11,240 --> 00:07:14,740
And we, in principle, know how
to write those out in detail.

149
00:07:14,740 --> 00:07:19,550
And we can choose the rotation,
which maps r0, theta 0, and phi

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00:07:19,550 --> 00:07:23,520
0, keeping track of where it
went, to the values of x, y.

151
00:07:23,520 --> 00:07:25,290
Z, w that will
ultimately correspond

152
00:07:25,290 --> 00:07:28,397
to the origin of our
coordinate system

153
00:07:28,397 --> 00:07:30,730
when we get back to r prime,
theta prime, and phi prime.

154
00:07:30,730 --> 00:07:32,330
So we can arrange
for that to happen

155
00:07:32,330 --> 00:07:34,572
by choosing the
right rotation here.

156
00:07:34,572 --> 00:07:36,030
And once we've
rotated, we can then

157
00:07:36,030 --> 00:07:38,040
define r prime, theta
prime, and phi prime

158
00:07:38,040 --> 00:07:40,764
in terms of x prime, y
prime, z prime, and w prime

159
00:07:40,764 --> 00:07:42,930
in exactly the same way as
we did in the first place

160
00:07:42,930 --> 00:07:44,230
when we didn't have primes.

161
00:07:44,230 --> 00:07:46,010
We just used the
same formulas again.

162
00:07:46,010 --> 00:07:48,052
And that will ensure that
the metric and r prime,

163
00:07:48,052 --> 00:07:50,593
theta prime, and phi prime will
be just the metric that we've

164
00:07:50,593 --> 00:07:52,970
had, because it's determined
from the euclidean

165
00:07:52,970 --> 00:07:57,360
metric in the four dimensional
space in exactly the same way.

166
00:07:57,360 --> 00:07:58,700
So this does it.

167
00:07:58,700 --> 00:08:01,910
And we could, in principle,
do it all in detail.

168
00:08:01,910 --> 00:08:06,570
And we would get a
concrete expression

169
00:08:06,570 --> 00:08:08,670
for r prime, theta
prime, and phi prime

170
00:08:08,670 --> 00:08:10,780
in terms of r,
theta, and phi that

171
00:08:10,780 --> 00:08:13,690
would have the property that we
want of mapping the arbitrary

172
00:08:13,690 --> 00:08:17,050
point that we chose to become
the origin of the new system.

173
00:08:19,860 --> 00:08:21,740
So the point is that
once you've written

174
00:08:21,740 --> 00:08:24,340
that all those
equations, you know

175
00:08:24,340 --> 00:08:26,019
that they work for k positive.

176
00:08:26,019 --> 00:08:27,560
But in the end,
you'd just have a set

177
00:08:27,560 --> 00:08:29,790
of equations that define
r prime, theta prime,

178
00:08:29,790 --> 00:08:33,030
and phi prime in terms
of r, theta, and phi.

179
00:08:33,030 --> 00:08:36,159
And those equations are
just as valid for k negative

180
00:08:36,159 --> 00:08:38,950
as they are for k positive.

181
00:08:38,950 --> 00:08:41,730
And the fact that the
metric will be unchanged

182
00:08:41,730 --> 00:08:44,480
is also just as valid for
k positive and k negative,

183
00:08:44,480 --> 00:08:46,350
because the metric is
really just determined

184
00:08:46,350 --> 00:08:48,230
by derivatives of the new
coordinates with respect

185
00:08:48,230 --> 00:08:49,340
to the old coordinates.

186
00:08:49,340 --> 00:08:52,220
And those are all just
algebraic expressions.

187
00:08:52,220 --> 00:08:54,300
And if an algebraic
expression is

188
00:08:54,300 --> 00:08:56,660
an equality for
one sine of k, it

189
00:08:56,660 --> 00:09:00,330
will be an equality for
the other sine of k.

190
00:09:00,330 --> 00:09:03,190
So I think we have good
faith-- although it'd

191
00:09:03,190 --> 00:09:05,190
be more convincing perhaps
to actually write out

192
00:09:05,190 --> 00:09:08,090
all equations, but I
think we have good faith

193
00:09:08,090 --> 00:09:11,560
that the same map will
work for k less than 0.

194
00:09:11,560 --> 00:09:14,790
And by "work," it means that it
will show that any point could

195
00:09:14,790 --> 00:09:18,180
be mapped to the origin
by a metric-preserving

196
00:09:18,180 --> 00:09:21,300
transformation, which is the
key thing, which is what we need

197
00:09:21,300 --> 00:09:24,072
to show that the space
is actually homogeneous

198
00:09:24,072 --> 00:09:26,030
even though when we write
in these coordinates,

199
00:09:26,030 --> 00:09:28,370
it doesn't look homogeneous.

200
00:09:28,370 --> 00:09:30,922
So does that make
sense to everybody?

201
00:09:30,922 --> 00:09:32,630
Are there any questions
at this point?

202
00:09:38,957 --> 00:09:40,540
The next thing we
did-- ah, I'm sorry.

203
00:09:40,540 --> 00:09:42,900
I guess we-- point
out that we're not

204
00:09:42,900 --> 00:09:45,292
going to actually
show this explicitly

205
00:09:45,292 --> 00:09:47,000
because the algebra
involved in the steps

206
00:09:47,000 --> 00:09:49,800
does get very complicated.

207
00:09:49,800 --> 00:09:51,800
I wanted to mention
a few other facts

208
00:09:51,800 --> 00:09:54,150
about this
Robertson-Walker metric.

209
00:09:54,150 --> 00:09:58,800
One important fact, which we
will not show-- to show it

210
00:09:58,800 --> 00:10:01,070
would take approximately
another lecture.

211
00:10:01,070 --> 00:10:03,880
It's not an incredibly
deep mathematical fact,

212
00:10:03,880 --> 00:10:07,480
but it requires
establishing some formalism

213
00:10:07,480 --> 00:10:11,390
to handle descriptions
of curved spaces

214
00:10:11,390 --> 00:10:14,365
without yet knowing what
the metric is going to be.

215
00:10:14,365 --> 00:10:16,620
But in any case,
it can be shown--

216
00:10:16,620 --> 00:10:20,380
and we're going to store
this in back of our heads--

217
00:10:20,380 --> 00:10:24,330
that any three dimensional,
homogeneous isotropic space can

218
00:10:24,330 --> 00:10:27,880
be described by this
Robertson-Walker metric.

219
00:10:27,880 --> 00:10:29,970
Now, it's important to
realize that that does not

220
00:10:29,970 --> 00:10:32,410
mean that the Roberson-Walker
metric is the only way

221
00:10:32,410 --> 00:10:34,870
to write down a metric for
a homogeneous and isotropic

222
00:10:34,870 --> 00:10:36,906
space.

223
00:10:36,906 --> 00:10:38,780
You could choose different
coordinate systems

224
00:10:38,780 --> 00:10:41,210
that would make
things look different.

225
00:10:41,210 --> 00:10:44,390
But the point is that for any
homogeneous and isotropic three

226
00:10:44,390 --> 00:10:48,300
dimensional space, it is always
possible to assign coordinates

227
00:10:48,300 --> 00:10:50,515
who make the metric the
Roberson-Walker metric, which

228
00:10:50,515 --> 00:10:52,890
means that if you understand
the Robertson-Walker metric,

229
00:10:52,890 --> 00:10:55,220
you need to understand
anything else.

230
00:10:55,220 --> 00:10:57,690
Any homogeneous
and isotropic space

231
00:10:57,690 --> 00:11:01,190
can be described this way.

232
00:11:01,190 --> 00:11:05,556
Next, we pointed out last time--
and did a short calculation

233
00:11:05,556 --> 00:11:07,930
to demonstrate for ourselves--
that for k greater than 0,

234
00:11:07,930 --> 00:11:10,001
the universe is finite.

235
00:11:10,001 --> 00:11:11,500
And that's clear
from the beginning,

236
00:11:11,500 --> 00:11:14,000
because it was described
as a surface of a sphere

237
00:11:14,000 --> 00:11:16,630
and the surface of
a sphere is finite.

238
00:11:16,630 --> 00:11:19,870
But for k less
than or equal to 0,

239
00:11:19,870 --> 00:11:22,510
the variable little r of
the Robertson-Walker metric

240
00:11:22,510 --> 00:11:24,840
can become arbitrarily large.

241
00:11:24,840 --> 00:11:27,145
That by itself does not
imply that the space

242
00:11:27,145 --> 00:11:29,310
is necessarily infinite.

243
00:11:29,310 --> 00:11:31,110
But you could also
calculate the distance

244
00:11:31,110 --> 00:11:33,350
from the origin as a
function of little r.

245
00:11:33,350 --> 00:11:35,480
And that, you can show,
becomes arbitrarily large.

246
00:11:35,480 --> 00:11:37,420
And that does mean that
the space is literally

247
00:11:37,420 --> 00:11:39,320
infinite in size.

248
00:11:39,320 --> 00:11:42,070
So for the flat case
or the open case,

249
00:11:42,070 --> 00:11:46,540
the Robertson-Walker metric
describes an infinite universe.

250
00:11:46,540 --> 00:11:49,380
And next, so we mentioned--
and this is a homework problem,

251
00:11:49,380 --> 00:11:51,970
or an optional homework
problem on the current set--

252
00:11:51,970 --> 00:11:53,710
that the
Gauss-Bolyai-Lobachevsky

253
00:11:53,710 --> 00:11:59,160
geometry is actually simply
the open universe in the two

254
00:11:59,160 --> 00:12:03,190
dimensional case rather
than three dimensional case.

255
00:12:03,190 --> 00:12:05,920
So in the language of the
Robertson-Walker metric,

256
00:12:05,920 --> 00:12:07,420
I think it's much
easier to describe

257
00:12:07,420 --> 00:12:10,740
than in the coordinate system
that Felix Klein invented.

258
00:12:10,740 --> 00:12:13,735
But it's the same space.

259
00:12:13,735 --> 00:12:16,040
And on the homework
set, you can work out

260
00:12:16,040 --> 00:12:18,740
the mapping between
the Klein coordinates

261
00:12:18,740 --> 00:12:21,330
and the Robertson-Walker
coordinates

262
00:12:21,330 --> 00:12:22,875
to see that they're
the same space.

263
00:12:26,182 --> 00:12:26,765
Any questions?

264
00:12:30,070 --> 00:12:31,280
OK.

265
00:12:31,280 --> 00:12:35,492
Next, we changed subjects and
started talking about the topic

266
00:12:35,492 --> 00:12:37,450
that we'll be continuing
now because we did not

267
00:12:37,450 --> 00:12:40,310
finish this discussion--
the discussion

268
00:12:40,310 --> 00:12:43,190
of how to go from
this space metric

269
00:12:43,190 --> 00:12:47,280
that we now understand to
the spacetime metric, which

270
00:12:47,280 --> 00:12:50,780
is the fundamental quantity
of general relativity

271
00:12:50,780 --> 00:12:55,660
and which we'll be using to
describe our model universes.

272
00:12:55,660 --> 00:12:58,120
Time ends up not playing
an important role

273
00:12:58,120 --> 00:12:59,800
in what we'll be talking about.

274
00:12:59,800 --> 00:13:01,560
But nonetheless, it
is an important part

275
00:13:01,560 --> 00:13:04,020
of the basic formalism
of general relativity.

276
00:13:04,020 --> 00:13:05,770
And for some
questions, it's crucial

277
00:13:05,770 --> 00:13:07,240
how time enters the metric.

278
00:13:07,240 --> 00:13:10,880
So we will discuss how
time enters the metric.

279
00:13:10,880 --> 00:13:13,280
So we want to generalize the
metric from a spatial metric

280
00:13:13,280 --> 00:13:14,880
to a spacetime metric.

281
00:13:14,880 --> 00:13:16,390
And the first
thing that means is

282
00:13:16,390 --> 00:13:21,800
that we want to understand
the relativistically invariant

283
00:13:21,800 --> 00:13:24,730
interval between two
points in spacetime.

284
00:13:24,730 --> 00:13:28,760
A point in spacetime is
also called an event.

285
00:13:28,760 --> 00:13:30,940
And we begin with
special relativity

286
00:13:30,940 --> 00:13:35,480
because that's what all
this is a generalization of.

287
00:13:35,480 --> 00:13:38,920
In special relativity, one can
define the Lorentz invariant

288
00:13:38,920 --> 00:13:41,410
distance between two events.

289
00:13:41,410 --> 00:13:44,120
Here, the events are A
and B. And the coordinates

290
00:13:44,120 --> 00:13:48,710
of those events are xA,
yA, zA, and tA for event A.

291
00:13:48,710 --> 00:13:51,090
And as you would
obviously guess,

292
00:13:51,090 --> 00:13:57,150
xB, yB, zB, and tB for event
B. And the Lorentz invariant

293
00:13:57,150 --> 00:14:01,270
interval between those
events is s squared sub AB.

294
00:14:01,270 --> 00:14:04,430
And the first and most important
thing about this interval

295
00:14:04,430 --> 00:14:07,020
is that it is Lorentz invariant.

296
00:14:07,020 --> 00:14:10,320
That is, you can compute
it in any inertial frame,

297
00:14:10,320 --> 00:14:13,657
in any Lorentz frame, and
it will have the same value

298
00:14:13,657 --> 00:14:15,240
even though the
different pieces of it

299
00:14:15,240 --> 00:14:16,364
will have different values.

300
00:14:16,364 --> 00:14:18,580
The differences will cancel out.

301
00:14:18,580 --> 00:14:21,310
And in the end, when you
calculate the sum of these four

302
00:14:21,310 --> 00:14:24,125
terms to make s
squared sub ab, you

303
00:14:24,125 --> 00:14:26,847
will find they'll have the same
value in any Lorentz frame.

304
00:14:26,847 --> 00:14:28,430
And we're not going
to show that fact.

305
00:14:28,430 --> 00:14:31,730
But we're going to use that
fact, a very well known fact

306
00:14:31,730 --> 00:14:37,592
to anybody who studied special
relativity in a reasonably way.

307
00:14:37,592 --> 00:14:39,550
It's important to think
a little bit about what

308
00:14:39,550 --> 00:14:41,091
the meaning of this
peculiar quantity

309
00:14:41,091 --> 00:14:43,400
is that mixes space and time.

310
00:14:43,400 --> 00:14:47,790
And I think the easiest way to
think about the meaning of it

311
00:14:47,790 --> 00:14:50,190
is to think about special cases.

312
00:14:50,190 --> 00:14:55,570
And it's a real number which
could be positive, negative,

313
00:14:55,570 --> 00:14:56,070
or 0.

314
00:14:56,070 --> 00:14:57,444
And those are the
special cases I

315
00:14:57,444 --> 00:15:01,260
want to think about--
positive, negative, or 0.

316
00:15:01,260 --> 00:15:05,790
So if s squared is
positive, it means

317
00:15:05,790 --> 00:15:08,426
that the separation
between the two events

318
00:15:08,426 --> 00:15:09,550
is what's called spacelike.

319
00:15:09,550 --> 00:15:12,242
It's dominated by
the spatial term.

320
00:15:12,242 --> 00:15:13,700
And if that's the
case, it's always

321
00:15:13,700 --> 00:15:15,810
possible to find a
frame where the two

322
00:15:15,810 --> 00:15:17,660
events are simultaneous.

323
00:15:17,660 --> 00:15:20,760
And in that frame,
s squared is just

324
00:15:20,760 --> 00:15:23,830
the square of the distance
between the two events,

325
00:15:23,830 --> 00:15:25,280
so has a clear interpretation.

326
00:15:25,280 --> 00:15:27,840
It's just the distance
in the Lorentz frame

327
00:15:27,840 --> 00:15:30,830
in which they occur
simultaneously.

328
00:15:30,830 --> 00:15:34,920
Similarly, if s
squared is negative,

329
00:15:34,920 --> 00:15:39,590
it means it's dominated
by the negative time term.

330
00:15:39,590 --> 00:15:42,750
And in that case, the
separation is called timelike,

331
00:15:42,750 --> 00:15:44,530
as you'd guess.

332
00:15:44,530 --> 00:15:46,850
And it has the property
that it's always

333
00:15:46,850 --> 00:15:50,770
possible to find a Lorentz frame
in which the two events occur

334
00:15:50,770 --> 00:15:53,710
at exactly the same
point in space.

335
00:15:53,710 --> 00:15:58,640
And in that frame,
s squared is equal,

336
00:15:58,640 --> 00:16:00,950
up to a factor of
minus c squared,

337
00:16:00,950 --> 00:16:04,620
to the time separation
between the two events.

338
00:16:04,620 --> 00:16:07,280
So s squared measures the time
separation between the two

339
00:16:07,280 --> 00:16:10,202
events in the frame at which
they happen in the same place.

340
00:16:10,202 --> 00:16:11,910
And s squared is
actually minus c squared

341
00:16:11,910 --> 00:16:15,520
times the time
separation squared.

342
00:16:15,520 --> 00:16:17,930
And finally, if s
squared is 0, that's

343
00:16:17,930 --> 00:16:19,860
called a lightlike separation.

344
00:16:19,860 --> 00:16:21,940
And it means that the
two events are separated

345
00:16:21,940 --> 00:16:25,835
by just the right distance so
that if a light pulse left one,

346
00:16:25,835 --> 00:16:27,585
it would just arrive
at the other location

347
00:16:27,585 --> 00:16:29,970
at exactly the
time of that event.

348
00:16:29,970 --> 00:16:31,880
So the two events
could be joined

349
00:16:31,880 --> 00:16:34,500
by a light pulse that travels
at the speed of light.

350
00:16:34,500 --> 00:16:39,205
And that's the significance
of s squared being 0.

351
00:16:39,205 --> 00:16:39,955
OK, any questions?

352
00:16:42,910 --> 00:16:46,870
OK-- actually, I don't
want to that slide.

353
00:16:46,870 --> 00:16:48,990
We'll get back to that.

354
00:16:48,990 --> 00:16:52,320
OK, that finishes the
review of last lecture.

355
00:16:52,320 --> 00:16:54,160
Now, what we want
to do is continue

356
00:16:54,160 --> 00:16:56,000
talking about
spacetime intervals

357
00:16:56,000 --> 00:17:00,520
and how we fit that into the
metric, which will ultimately

358
00:17:00,520 --> 00:17:03,810
describe distances in
both space and time.

359
00:17:03,810 --> 00:17:08,055
So I'm going to work on the
blackboard for awhile now.

360
00:17:27,700 --> 00:17:35,494
So the formula that we're
starting with-- and we'll put

361
00:17:35,494 --> 00:17:42,210
dot dot dot here and minus c
squared tA minus tB squared.

362
00:17:42,210 --> 00:17:44,950
The dot dot dot means the
y term and the z term which

363
00:17:44,950 --> 00:17:48,870
you can probably imagine are
there without my writing them.

364
00:17:48,870 --> 00:17:53,970
From this expression, knowing
that what we want to do

365
00:17:53,970 --> 00:17:56,810
is to take advantage of these
geometrical ideas introduced

366
00:17:56,810 --> 00:18:02,440
by people like Gauss
which described distances

367
00:18:02,440 --> 00:18:04,080
in terms of
infinitesimal distances

368
00:18:04,080 --> 00:18:07,190
between infinitesimally
close points,

369
00:18:07,190 --> 00:18:09,300
we can write the
analogous equation

370
00:18:09,300 --> 00:18:12,010
for an infinitesimal distance.

371
00:18:12,010 --> 00:18:15,970
And that becomes ds
squared is equal to dx

372
00:18:15,970 --> 00:18:23,420
squared plus dy squared plus
dz squared minus c squared

373
00:18:23,420 --> 00:18:24,830
dt squared.

374
00:18:24,830 --> 00:18:27,280
So this would be the
Lorentz invariant separation

375
00:18:27,280 --> 00:18:29,432
between infinitesimally
separated events.

376
00:18:29,432 --> 00:18:30,890
And this is what
we're going to try

377
00:18:30,890 --> 00:18:33,995
to generalize to our
curved space situation.

378
00:18:37,850 --> 00:18:41,246
So this would be the metric
for special relativity.

379
00:18:41,246 --> 00:18:42,620
It's called the
Minkowsky metric.

380
00:18:57,590 --> 00:19:02,950
OK, now I want to move
into the general relativity

381
00:19:02,950 --> 00:19:11,000
generalization of this idea.

382
00:19:13,650 --> 00:19:16,940
And general relativity
makes use of the idea

383
00:19:16,940 --> 00:19:20,580
that Gauss originally suggested
that distances should always

384
00:19:20,580 --> 00:19:25,510
be quadratic functions of
the coordinate differentials.

385
00:19:25,510 --> 00:19:28,040
So we're going to keep that.

386
00:19:28,040 --> 00:19:29,260
Einstein kept that.

387
00:20:05,450 --> 00:20:08,290
Now, in talking about
coordinate differentials,

388
00:20:08,290 --> 00:20:12,240
we should emphasize here
that in general relativity,

389
00:20:12,240 --> 00:20:14,610
unlike special
relativity, coordinates

390
00:20:14,610 --> 00:20:19,479
are just arbitrary labels
for points in spacetime.

391
00:20:19,479 --> 00:20:21,270
In special relativity,
coordinates actually

392
00:20:21,270 --> 00:20:23,470
measure distances
and times directly,

393
00:20:23,470 --> 00:20:25,090
which is why the
metric is so simple.

394
00:20:25,090 --> 00:20:27,980
You don't really need the
metric in special relativity.

395
00:20:27,980 --> 00:20:29,820
The coordinates
themselves will tell you

396
00:20:29,820 --> 00:20:31,970
the distances and the times.

397
00:20:31,970 --> 00:20:35,190
But in general relativity,
that will not be the case.

398
00:20:35,190 --> 00:20:37,700
There's no way to do
that for a curved space

399
00:20:37,700 --> 00:20:40,120
or a curved spacetime.

400
00:20:40,120 --> 00:20:52,050
So in general relativity,
the coordinates

401
00:20:52,050 --> 00:21:00,190
are just arbitrary labels
of points in spacetime.

402
00:21:04,817 --> 00:21:06,650
And to know anything
about actual distances,

403
00:21:06,650 --> 00:21:08,696
you have to look at the metric.

404
00:21:08,696 --> 00:21:13,920
The coordinates themselves don't
tell you the actual distances.

405
00:21:13,920 --> 00:21:15,450
This immediately
implies something

406
00:21:15,450 --> 00:21:17,840
about the kinds of
coordinate transformations

407
00:21:17,840 --> 00:21:20,580
that you might want
to think about.

408
00:21:20,580 --> 00:21:24,210
In special relativity,
we have a privileged set

409
00:21:24,210 --> 00:21:28,550
of coordinates, namely the
coordinates of Lorentz frames,

410
00:21:28,550 --> 00:21:30,950
of inertial frames.

411
00:21:30,950 --> 00:21:34,030
And the physics is simple when
described in terms of those

412
00:21:34,030 --> 00:21:34,852
coordinates.

413
00:21:34,852 --> 00:21:36,560
In principle, you can
use any coordinates

414
00:21:36,560 --> 00:21:38,860
you want, even in
special relativity.

415
00:21:38,860 --> 00:21:42,560
But you never do, because the
physics is so much simpler

416
00:21:42,560 --> 00:21:44,730
in the inertial coordinates
that there's never

417
00:21:44,730 --> 00:21:49,230
any motivation for using any
other coordinate systems.

418
00:21:49,230 --> 00:21:50,810
But in general
relativity, there is

419
00:21:50,810 --> 00:21:52,890
no privileged coordinate system.

420
00:21:52,890 --> 00:21:56,570
And it's very common to make
all kinds of transformations

421
00:21:56,570 --> 00:22:00,390
of coordinates in the context
of general relativity.

422
00:22:00,390 --> 00:22:02,080
And the formalism
is set up so you

423
00:22:02,080 --> 00:22:04,980
could make any coordinate
transformation you want,

424
00:22:04,980 --> 00:22:07,460
and it's just thought
of as a relabeling

425
00:22:07,460 --> 00:22:09,930
of the points in space and time.

426
00:22:09,930 --> 00:22:12,330
And the formalism of
general relativity

427
00:22:12,330 --> 00:22:17,100
works for an arbitrary labeling
of points in spacetime.

428
00:22:17,100 --> 00:22:38,240
So in general relativity,
any coordinate transformation

429
00:22:38,240 --> 00:22:38,740
is allowed.

430
00:22:50,970 --> 00:22:55,274
But there's an
important feature of

431
00:22:55,274 --> 00:22:56,690
these coordinate
transformations--

432
00:22:56,690 --> 00:22:59,630
is that when we make a
coordinate transformation,

433
00:22:59,630 --> 00:23:05,180
we're always going to readjust
our metric so that ds squared

434
00:23:05,180 --> 00:23:08,720
between any two nearby
spacetime points

435
00:23:08,720 --> 00:23:10,400
has the same value
in the new coordinate

436
00:23:10,400 --> 00:23:12,250
system that it had in the old.

437
00:23:12,250 --> 00:23:15,265
We will always change
our metric to reflect

438
00:23:15,265 --> 00:23:16,390
our changes of coordinates.

439
00:23:28,940 --> 00:23:37,240
So ds squared must have the same
value in any coordinate system.

440
00:23:53,560 --> 00:23:55,180
So the statement
is that ds squared

441
00:23:55,180 --> 00:23:56,180
is coordinate-invariant.

442
00:24:33,310 --> 00:24:37,070
OK, to define what we mean by
ds squared, which if you notice,

443
00:24:37,070 --> 00:24:38,462
I haven't quite done yet.

444
00:24:38,462 --> 00:24:40,670
It's going to be, of course,
the analog of what we've

445
00:24:40,670 --> 00:24:43,580
been talking about in
special relativity.

446
00:24:43,580 --> 00:24:46,170
In special relativity, we
did have this special class

447
00:24:46,170 --> 00:24:49,920
of observers, inertial
observers, observers

448
00:24:49,920 --> 00:24:51,950
whose measurements
of length and time

449
00:24:51,950 --> 00:24:55,870
really corresponded
to the inertial frames

450
00:24:55,870 --> 00:24:58,880
and whose observations
are related to each other

451
00:24:58,880 --> 00:25:02,100
by Lorentz transformations.

452
00:25:02,100 --> 00:25:04,040
It's important to
start out by asking

453
00:25:04,040 --> 00:25:08,510
is there any class of
observers in general relativity

454
00:25:08,510 --> 00:25:11,100
which might play the same
roles-- the observers that

455
00:25:11,100 --> 00:25:15,150
sort of define the measurements
that you want to talk about.

456
00:25:15,150 --> 00:25:17,830
And it's clearly a little
bit more complicated

457
00:25:17,830 --> 00:25:20,700
in general relativity.

458
00:25:20,700 --> 00:25:23,390
The inertial observers
of special relativity

459
00:25:23,390 --> 00:25:25,280
are characterized
by the statement

460
00:25:25,280 --> 00:25:28,060
that there are no
forces acting on them.

461
00:25:28,060 --> 00:25:31,820
So they just travel at
a constant velocity.

462
00:25:31,820 --> 00:25:34,800
And you can always go to a
frame where that velocity is 0

463
00:25:34,800 --> 00:25:39,650
and you an talk about the rest
frame of any inertial observer.

464
00:25:39,650 --> 00:25:43,960
In general relativity,
we need to distinguish

465
00:25:43,960 --> 00:25:46,950
to some extent between
non-gravitational forces

466
00:25:46,950 --> 00:25:49,570
and gravitational forces.

467
00:25:49,570 --> 00:25:52,610
Non-gravitational forces,
like say, electrical forces,

468
00:25:52,610 --> 00:25:54,760
are treated in
general relativity

469
00:25:54,760 --> 00:25:57,980
in a way that's fundamentally
similar to the way

470
00:25:57,980 --> 00:26:01,450
that such forces are treated
in special relativity.

471
00:26:01,450 --> 00:26:03,730
But gravity is treated
totally differently.

472
00:26:03,730 --> 00:26:05,650
Gravity is really just
going to be described

473
00:26:05,650 --> 00:26:10,640
by the metric of spacetime--
by the distortion of spacetime.

474
00:26:10,640 --> 00:26:14,240
And we already know, by
way of simple examples,

475
00:26:14,240 --> 00:26:15,870
that if general
relativity actually

476
00:26:15,870 --> 00:26:18,510
works to describe the universe
that we've been talking-- which

477
00:26:18,510 --> 00:26:22,390
it'd better or we'd
be in trouble--

478
00:26:22,390 --> 00:26:23,810
we have a system
where if we just

479
00:26:23,810 --> 00:26:29,020
look at the co-moving observers,
each co-moving observer has

480
00:26:29,020 --> 00:26:32,180
no non-gravitational
forces acting on him.

481
00:26:32,180 --> 00:26:35,560
He's just sitting still
as far as he's concerned.

482
00:26:35,560 --> 00:26:37,890
But nonetheless, these
co-moving observers

483
00:26:37,890 --> 00:26:39,750
are accelerating
relative to each other

484
00:26:39,750 --> 00:26:42,950
as the universe expands and
as that expansion changes

485
00:26:42,950 --> 00:26:46,840
its expansion rate, which
we've already calculated.

486
00:26:46,840 --> 00:26:51,749
So if there's going to
be any observers that

487
00:26:51,749 --> 00:26:53,790
are going to play the role
of inertial observers,

488
00:26:53,790 --> 00:26:55,660
it's presumably going
to be a class that

489
00:26:55,660 --> 00:26:59,300
includes these
co-moving observers.

490
00:26:59,300 --> 00:27:01,590
And the question
of whether or not

491
00:27:01,590 --> 00:27:03,750
there are gravitational
forces acting

492
00:27:03,750 --> 00:27:07,500
on the co-moving
observers ends up

493
00:27:07,500 --> 00:27:09,640
depending on your point of view.

494
00:27:09,640 --> 00:27:11,500
Each co-moving
observer would think

495
00:27:11,500 --> 00:27:14,360
that there's no gravitational
forces acting on him.

496
00:27:14,360 --> 00:27:16,290
He would just be standing still.

497
00:27:16,290 --> 00:27:18,670
But he would see all these
other co-moving observers

498
00:27:18,670 --> 00:27:20,510
accelerating relative to him.

499
00:27:20,510 --> 00:27:23,510
So he would say that there are
gravitational forces acting

500
00:27:23,510 --> 00:27:27,790
on these other observers.

501
00:27:27,790 --> 00:27:31,540
So gravitational forces
in general activity

502
00:27:31,540 --> 00:27:36,540
becomes
coordinate-dependent ideas.

503
00:27:36,540 --> 00:27:39,060
And the Hubble
expansion is one example

504
00:27:39,060 --> 00:27:41,730
of that where every co-moving
observer would consider himself

505
00:27:41,730 --> 00:27:43,580
to be unaccelerating
but would see

506
00:27:43,580 --> 00:27:47,270
all the other co-moving
observers accelerating.

507
00:27:47,270 --> 00:27:50,280
The other famous
example, which is

508
00:27:50,280 --> 00:27:53,930
part of the original motivation
of general relativity,

509
00:27:53,930 --> 00:27:58,020
is the famous Einstein
elevator, which

510
00:27:58,020 --> 00:28:01,830
is also discussed
in Ryden's textbook.

511
00:28:01,830 --> 00:28:11,099
If we have an
elevator box, we could

512
00:28:11,099 --> 00:28:12,515
imagine letting
the elevator fall.

513
00:28:15,710 --> 00:28:19,410
There was a rope there,
but somebody cut it.

514
00:28:19,410 --> 00:28:21,150
And the elevator's now falling.

515
00:28:21,150 --> 00:28:22,290
And we have a person in it.

516
00:28:25,620 --> 00:28:30,210
And the person in my--
the version of the story

517
00:28:30,210 --> 00:28:32,050
that I have in
the lecture notes,

518
00:28:32,050 --> 00:28:35,550
the person's holding
a bag of groceries.

519
00:28:35,550 --> 00:28:39,410
And if the elevator
is falling freely

520
00:28:39,410 --> 00:28:43,100
and we ignore any air resistance
or any other kind of friction

521
00:28:43,100 --> 00:28:45,790
so the elevator's falling
at exactly the freefall

522
00:28:45,790 --> 00:28:48,970
rate, inside the
elevator, everything

523
00:28:48,970 --> 00:28:52,230
will be falling with
the same acceleration.

524
00:28:52,230 --> 00:28:55,680
The person could lift his
feet up off the floor,

525
00:28:55,680 --> 00:28:57,760
and he would just hover there.

526
00:28:57,760 --> 00:29:00,670
He would feel no gravity
pushing him towards the floor.

527
00:29:00,670 --> 00:29:03,650
And similarly, he could let
go of the bag of groceries,

528
00:29:03,650 --> 00:29:06,520
and they will just appear
to float in front of him

529
00:29:06,520 --> 00:29:09,330
as long as he's
undergoing this freefall.

530
00:29:09,330 --> 00:29:13,870
So the effects of gravity
have been completely removed.

531
00:29:13,870 --> 00:29:15,910
On the other hand, from
outside the elevator,

532
00:29:15,910 --> 00:29:17,964
if we use the frame of
reference of the Earth,

533
00:29:17,964 --> 00:29:19,380
we said that there
very definitely

534
00:29:19,380 --> 00:29:21,510
is a force of
gravity acting here.

535
00:29:21,510 --> 00:29:25,770
It's just acting the
same on all the objects.

536
00:29:25,770 --> 00:29:31,374
And this gets elevated into
the equivalence principle

537
00:29:31,374 --> 00:29:32,290
of general relativity.

538
00:29:40,770 --> 00:29:42,320
And maybe before
I annunciate that,

539
00:29:42,320 --> 00:29:45,930
I should consider
the other case here.

540
00:29:45,930 --> 00:29:49,580
This is one example
of how things work.

541
00:29:49,580 --> 00:29:57,680
A similar situation can
involve the same elevator,

542
00:29:57,680 --> 00:30:01,150
but this time,
let's have it just

543
00:30:01,150 --> 00:30:05,785
be sitting on the floor of the
building that it's located in.

544
00:30:05,785 --> 00:30:11,470
In that case, the
person inside would

545
00:30:11,470 --> 00:30:14,840
feel himself pushed against
the floor by gravity.

546
00:30:14,840 --> 00:30:17,190
If he was holding
his bag of groceries,

547
00:30:17,190 --> 00:30:20,370
he would notice he has to apply
force to the bag of groceries

548
00:30:20,370 --> 00:30:23,740
to stop the groceries
from falling to the floor.

549
00:30:23,740 --> 00:30:25,290
He would say that
he's being acted

550
00:30:25,290 --> 00:30:26,885
on by the force of gravity.

551
00:30:26,885 --> 00:30:29,520
That would be the
natural description.

552
00:30:29,520 --> 00:30:35,000
But we can consider
an analogous case

553
00:30:35,000 --> 00:30:40,150
where we have the same elevator
in empty space with a rocket

554
00:30:40,150 --> 00:30:41,830
ship up here that I
didn't allow myself

555
00:30:41,830 --> 00:30:46,890
room to draw tied by
cables to the elevator.

556
00:30:51,410 --> 00:30:54,690
And if the rocket ship
accelerated with acceleration

557
00:30:54,690 --> 00:30:58,040
little g, the person
inside the elevator

558
00:30:58,040 --> 00:31:01,010
would feel himself pressed
against the floor in exactly

559
00:31:01,010 --> 00:31:03,410
the same way as you would here.

560
00:31:03,410 --> 00:31:06,060
So again, we have
a situation where

561
00:31:06,060 --> 00:31:09,250
there's gravity in one case and
no gravity in the other case,

562
00:31:09,250 --> 00:31:13,370
but no difference in what
the person inside would feel.

563
00:31:13,370 --> 00:31:36,500
And that is what becomes this
principle of equivalence, which

564
00:31:36,500 --> 00:31:46,800
says that the physics
of the accelerating

565
00:31:46,800 --> 00:32:07,100
frame of the elevator in
that analogy-- so this

566
00:32:07,100 --> 00:32:22,520
is accelerating frame
but with no gravity--

567
00:32:22,520 --> 00:32:43,950
is equivalent to feeling
the gravitational field

568
00:32:43,950 --> 00:32:44,620
of the Earth.

569
00:32:51,920 --> 00:32:54,520
In short, if you were
living inside the elevator,

570
00:32:54,520 --> 00:32:58,080
you cannot tell which of those
two pictures describe the world

571
00:32:58,080 --> 00:33:00,350
that you're actually part of.

572
00:33:00,350 --> 00:33:01,780
And this is a very
deep principle.

573
00:33:01,780 --> 00:33:03,610
It has very strong implications.

574
00:33:03,610 --> 00:33:05,490
It really does mean
that everything

575
00:33:05,490 --> 00:33:08,310
you'd ever want to know
about how gravity affects

576
00:33:08,310 --> 00:33:12,540
physical systems can be
described by understanding how

577
00:33:12,540 --> 00:33:15,660
accelerations affect
the physical systems.

578
00:33:15,660 --> 00:33:17,510
So it reduces the
questions of what gravity

579
00:33:17,510 --> 00:33:20,120
does to just understanding
what happens when you're

580
00:33:20,120 --> 00:33:21,700
in an accelerating
coordinate system.

581
00:33:27,050 --> 00:33:30,070
OK, this also opens the
door for the question

582
00:33:30,070 --> 00:33:33,340
I began with-- is there a
special class of observers

583
00:33:33,340 --> 00:33:35,000
here?

584
00:33:35,000 --> 00:33:38,870
And we can identify a
special class of observers.

585
00:33:38,870 --> 00:33:41,150
But the special class
is not observers

586
00:33:41,150 --> 00:33:42,900
which have no forces
acting on them, which

587
00:33:42,900 --> 00:33:45,580
is what we would have said
in special relativity.

588
00:33:45,580 --> 00:33:47,850
But rather, the
special observers

589
00:33:47,850 --> 00:33:51,820
are the observers who have no
non-gravitational forces acting

590
00:33:51,820 --> 00:33:54,930
on them, like the
co-moving observers

591
00:33:54,930 --> 00:33:57,490
in our model of the universe.

592
00:33:57,490 --> 00:33:59,220
But gravitational
forces, you could never

593
00:33:59,220 --> 00:34:02,170
say if they're there
or not because they're

594
00:34:02,170 --> 00:34:06,610
always there in some frames
and not there in other frames.

595
00:34:06,610 --> 00:34:09,659
So we have no control
or no way of making

596
00:34:09,659 --> 00:34:13,909
any frame-invariant statements
about the force of gravity.

597
00:34:13,909 --> 00:34:18,460
So what we'll be interested
in as our primary observers

598
00:34:18,460 --> 00:34:22,350
and which we're going
to use to define things

599
00:34:22,350 --> 00:34:29,461
in this class of observers with
no non-gravitational forces--

600
00:34:29,461 --> 00:34:31,460
and those will be called
free-falling observers.

601
00:35:09,230 --> 00:35:11,790
And they're called
free-falling because you

602
00:35:11,790 --> 00:35:14,350
have no way of knowing
whether they're just observers

603
00:35:14,350 --> 00:35:16,170
for which there is
no gravity, which

604
00:35:16,170 --> 00:35:18,540
would be an example of
a free-falling observer

605
00:35:18,540 --> 00:35:20,030
by our definition.

606
00:35:20,030 --> 00:35:23,640
But the situation
is indistinguishable

607
00:35:23,640 --> 00:35:25,630
from this one,
where free-falling

608
00:35:25,630 --> 00:35:28,250
has its obvious meaning--
that the guy there

609
00:35:28,250 --> 00:35:30,761
is falling relative
to the Earth's frame.

610
00:35:30,761 --> 00:35:31,760
But he's freely falling.

611
00:35:31,760 --> 00:35:35,210
And therefore, he does not feel,
relative to his environment,

612
00:35:35,210 --> 00:35:38,009
any forces whatever.

613
00:35:38,009 --> 00:35:40,300
Now, I should emphasize that
this equivalence principle

614
00:35:40,300 --> 00:35:41,900
holds only in small regions.

615
00:35:41,900 --> 00:35:45,010
In principle, it only holds
in infinitesimal regions

616
00:35:45,010 --> 00:35:48,560
because there are, in
gravitational systems, what

617
00:35:48,560 --> 00:35:51,419
we call tidal effects,
where a tidal effect simply

618
00:35:51,419 --> 00:35:53,210
means that the
gravitational field is never

619
00:35:53,210 --> 00:35:54,464
completely uniform.

620
00:35:54,464 --> 00:35:56,380
And if the gravitational
field is not uniform,

621
00:35:56,380 --> 00:35:59,320
you do not completely cancel it
by going into the accelerating

622
00:35:59,320 --> 00:36:00,650
frame of the elevator.

623
00:36:00,650 --> 00:36:02,290
But in any infinitesimal
region, you

624
00:36:02,290 --> 00:36:05,000
can always cancel the
effects of gravity

625
00:36:05,000 --> 00:36:07,590
by going into a properly
accelerating frame.

626
00:36:07,590 --> 00:36:10,396
And that's what the
equivalence principle says.

627
00:36:10,396 --> 00:36:12,020
OK, are there any
questions about that?

628
00:36:16,580 --> 00:36:21,120
OK, this being said-- it was
a long prelude-- we can now

629
00:36:21,120 --> 00:36:23,975
define what ds squared
is supposed to represent.

630
00:36:27,950 --> 00:36:41,960
And the answer is simply that ds
squared has the same meaning as

631
00:36:41,960 --> 00:37:06,170
in special relativity except
that inertial observers are

632
00:37:06,170 --> 00:37:08,450
replaced by
free-falling observers.

633
00:37:32,970 --> 00:37:35,320
OK, so let's review
what exactly that means.

634
00:37:35,320 --> 00:37:38,804
It means that if ds
squared is positive,

635
00:37:38,804 --> 00:37:40,220
it means that there
will always be

636
00:37:40,220 --> 00:37:45,610
a class of free-falling
observers for whom those two

637
00:37:45,610 --> 00:37:50,740
events will occur
at the same time.

638
00:37:50,740 --> 00:37:53,210
And ds squared will be the
distance between those two

639
00:37:53,210 --> 00:37:56,270
events as measured by those
inertial observers-- bah,

640
00:37:56,270 --> 00:37:58,370
I said "inertial"--
free-falling observers.

641
00:38:01,330 --> 00:38:04,602
And similarly, if ds
squared is negative,

642
00:38:04,602 --> 00:38:06,935
it means there will be a class
of free-falling observers

643
00:38:06,935 --> 00:38:10,040
for whom those two events will
occur at the same location.

644
00:38:10,040 --> 00:38:14,100
And ds squared will measure, up
to a factor of minus c squared,

645
00:38:14,100 --> 00:38:18,150
the time separation squared
between those two events.

646
00:38:18,150 --> 00:38:21,114
And it will again be the
case that if ds squared is 0,

647
00:38:21,114 --> 00:38:23,030
it will mean that the
two events are separated

648
00:38:23,030 --> 00:38:25,312
by just the right distance
so that a light pulse can

649
00:38:25,312 --> 00:38:26,520
travel from one to the other.

650
00:38:31,150 --> 00:38:33,990
OK, any questions about that?

651
00:38:33,990 --> 00:38:35,740
It's was kind of a
long-winded discussion.

652
00:38:35,740 --> 00:38:36,950
But I think it does
pay to actually

653
00:38:36,950 --> 00:38:39,570
understand what df ds means
rather than just to write down

654
00:38:39,570 --> 00:38:41,940
a formula for it and say
that's like special relativity.

655
00:38:45,050 --> 00:38:48,460
OK, having said all
this, our next goal

656
00:38:48,460 --> 00:38:55,370
is to figure out how time enters
the Robertson-Walker metric

657
00:38:55,370 --> 00:38:57,260
to give us a spacetime
metric instead

658
00:38:57,260 --> 00:39:01,720
of just a spatial metric, which
we already have written down.

659
00:39:01,720 --> 00:39:03,420
And I'm going to
write down the answer

660
00:39:03,420 --> 00:39:06,110
and then describe why that
has to be the right answer.

661
00:39:06,110 --> 00:39:08,980
I think it's the easiest
way to handle it here.

662
00:39:08,980 --> 00:39:12,950
So the right answer is that when
we incorporate time and think

663
00:39:12,950 --> 00:39:14,790
of this as a metric
for spacetime,

664
00:39:14,790 --> 00:39:20,420
ds squared is going to be
minus c squared dt squared

665
00:39:20,420 --> 00:39:28,800
plus a squared of t
times dr squared over 1

666
00:39:28,800 --> 00:39:33,130
minus k r squared-- right now,
it's just the same spatial part

667
00:39:33,130 --> 00:39:37,970
that we had before-- plus
r squared d theta squared

668
00:39:37,970 --> 00:39:43,900
plus sine squared
theta d phi squared.

669
00:39:43,900 --> 00:39:45,410
End parentheses.

670
00:39:45,410 --> 00:39:48,160
End curly brackets.

671
00:39:48,160 --> 00:39:50,100
So all I've done is
I've added a minus

672
00:39:50,100 --> 00:39:56,670
c squared dt squared
term to the metric.

673
00:39:56,670 --> 00:40:00,525
Now, why is this
the right metric?

674
00:40:00,525 --> 00:40:03,810
I'm going to first consider
two special cases, which

675
00:40:03,810 --> 00:40:06,970
will verify some
of the terms there.

676
00:40:06,970 --> 00:40:08,980
And then I want to
also discuss why

677
00:40:08,980 --> 00:40:11,410
there aren't any other
terms besides the ones

678
00:40:11,410 --> 00:40:13,890
that we know have to be there.

679
00:40:13,890 --> 00:40:22,240
So first, let's just consider
the case-- case one will just

680
00:40:22,240 --> 00:40:26,080
be dt equals 0.

681
00:40:26,080 --> 00:40:29,010
If there's no time separation
between the two events,

682
00:40:29,010 --> 00:40:31,600
then we're only interested
in spatial separations.

683
00:40:31,600 --> 00:40:33,125
And we've already
talked about how

684
00:40:33,125 --> 00:40:35,260
to describe spatial
separations in a way which

685
00:40:35,260 --> 00:40:39,040
makes the description
homogeneous and isotropic.

686
00:40:39,040 --> 00:40:41,630
And we said that this is the
most general way of describing

687
00:40:41,630 --> 00:40:44,630
spatial separations that
are in a space which

688
00:40:44,630 --> 00:40:47,180
is homogeneous and isotropic.

689
00:40:47,180 --> 00:40:52,680
So from what we said previously,
this has to be the answer.

690
00:40:52,680 --> 00:40:55,810
That's how we describe
homogeneous isotropic spaces.

691
00:40:55,810 --> 00:40:57,590
So when dt vanishes,
it just reduces

692
00:40:57,590 --> 00:40:59,470
to the case we've
already discussed.

693
00:40:59,470 --> 00:41:01,770
Simple enough.

694
00:41:01,770 --> 00:41:06,300
Case two, which involves a
little bit of new thinking--

695
00:41:06,300 --> 00:41:10,730
suppose dr equals d
theta equals d phi

696
00:41:10,730 --> 00:41:15,568
equals 0 so that
only time changes.

697
00:41:15,568 --> 00:41:19,470
OK, this describes the situation
about our co-moving observers.

698
00:41:19,470 --> 00:41:22,450
They're sitting at fixed
spatial coordinates

699
00:41:22,450 --> 00:41:24,920
and evolving in time.

700
00:41:24,920 --> 00:41:30,220
And we've already said that the
thing that we call cosmic time

701
00:41:30,220 --> 00:41:32,940
is simply time as
measured by the wrist

702
00:41:32,940 --> 00:41:36,440
watches of our
co-moving observers.

703
00:41:36,440 --> 00:41:39,220
So t, if you want t to
be cosmic time, which

704
00:41:39,220 --> 00:41:42,980
we do-- we're trying to describe
a metric for our spacetime

705
00:41:42,980 --> 00:41:44,430
as we've already described it.

706
00:41:44,430 --> 00:41:46,460
We're now just trying to
write a metric for it.

707
00:41:46,460 --> 00:41:48,550
So t should be cosmic time.

708
00:41:48,550 --> 00:41:50,870
So t should be the
time as measured

709
00:41:50,870 --> 00:41:54,700
on the wrist watches of
the co-moving observers.

710
00:41:54,700 --> 00:41:57,650
And that's exactly
what this metric says.

711
00:41:57,650 --> 00:42:01,550
It says that if
ds squared defines

712
00:42:01,550 --> 00:42:04,930
the measurements of our
free-falling observers, which

713
00:42:04,930 --> 00:42:09,570
is what we said is the
definition of ds squared,

714
00:42:09,570 --> 00:42:11,820
that it is just equal
to minus c squared

715
00:42:11,820 --> 00:42:15,040
times the change in
the coordinate time.

716
00:42:15,040 --> 00:42:16,670
And coordinate time
means cosmic time

717
00:42:16,670 --> 00:42:19,270
because that's the coordinate
system we're using.

718
00:42:19,270 --> 00:42:21,380
So putting in the
minus c squared

719
00:42:21,380 --> 00:42:25,190
dt squared term is the
only way that it can be so

720
00:42:25,190 --> 00:42:29,400
that the wrist watches of
our co-moving, free-falling

721
00:42:29,400 --> 00:42:34,610
observers measure the same thing
that the coordinate measures,

722
00:42:34,610 --> 00:42:38,130
which is what we defined cosmic
time to be in the first place.

723
00:42:38,130 --> 00:42:40,090
So I think that
justifies this term.

724
00:42:40,090 --> 00:42:42,370
And notice, if you had
any coefficient here other

725
00:42:42,370 --> 00:42:45,240
than c squared, there'd
be a multiplicity

726
00:42:45,240 --> 00:42:48,020
offset between what the wrist
watches of your observers

727
00:42:48,020 --> 00:42:51,590
are measuring and what
cosmic time is ticking off.

728
00:42:51,590 --> 00:42:55,110
And we're not allowing that,
because we define cosmic time

729
00:42:55,110 --> 00:42:57,460
to be the time measured
by the wrist watches.

730
00:43:00,140 --> 00:43:03,220
OK, so that takes care
of these two cases.

731
00:43:03,220 --> 00:43:05,170
And I think it implies
that these terms have

732
00:43:05,170 --> 00:43:08,730
to be here in exactly the
form that we've written.

733
00:43:08,730 --> 00:43:10,170
And we could stop
now and pretend

734
00:43:10,170 --> 00:43:12,300
that we've solved
the whole problem.

735
00:43:12,300 --> 00:43:17,590
But I always like to be what
I consider to be thorough.

736
00:43:17,590 --> 00:43:20,150
So I like to sort of
imagine the questions that

737
00:43:20,150 --> 00:43:23,800
could pop up if people
were inquisitive.

738
00:43:23,800 --> 00:43:28,030
So you might imagine that you
have some difficult roommate

739
00:43:28,030 --> 00:43:31,070
who says, why can't I
put some other term,

740
00:43:31,070 --> 00:43:34,055
and what else could there be?

741
00:43:34,055 --> 00:43:35,680
The only thing that
we've left out here

742
00:43:35,680 --> 00:43:39,300
are terms that
involve products of dt

743
00:43:39,300 --> 00:43:41,475
with either dr, d
theta, or d phi.

744
00:43:50,150 --> 00:43:53,790
So what about terms like
the product of dr times

745
00:43:53,790 --> 00:43:59,955
dt or d theta times
dt or d phi times dt?

746
00:44:06,830 --> 00:44:07,850
Question mark.

747
00:44:11,357 --> 00:44:13,190
So one possible answer
is I looked in a book

748
00:44:13,190 --> 00:44:15,211
and it wasn't there.

749
00:44:15,211 --> 00:44:17,210
But that's not the best
of all possible answers.

750
00:44:17,210 --> 00:44:18,626
It's good to
understand why things

751
00:44:18,626 --> 00:44:21,180
are in books and why
other things or not.

752
00:44:21,180 --> 00:44:23,870
So you might want to
construct an argument of why

753
00:44:23,870 --> 00:44:25,730
these terms have to be absent.

754
00:44:25,730 --> 00:44:29,850
And the reason why those
terms have to be absent

755
00:44:29,850 --> 00:44:33,629
is because if they were there,
they would violate isotropy.

756
00:44:33,629 --> 00:44:35,420
Roughly speaking, the
notion is that if you

757
00:44:35,420 --> 00:44:38,630
have a dt times some
d spatial coordinate,

758
00:44:38,630 --> 00:44:41,020
that singles out a certain
direction in spatial

759
00:44:41,020 --> 00:44:48,310
coordinates space because dr
is not the same as minus dr. Dr

760
00:44:48,310 --> 00:44:50,790
points in a certain direction.

761
00:44:50,790 --> 00:44:55,410
To be more explicit
about that, in the notes,

762
00:44:55,410 --> 00:45:00,000
I discuss a thought
experiment which basically

763
00:45:00,000 --> 00:45:06,479
gives a concrete
realization of the asymmetry

764
00:45:06,479 --> 00:45:07,395
that I just discussed.

765
00:45:24,290 --> 00:45:28,990
So to see how those terms
explicitly violate isotropy,

766
00:45:28,990 --> 00:45:31,600
we can imagine a
thought experiment

767
00:45:31,600 --> 00:45:36,170
where we start by thinking about
some particular point in space.

768
00:45:36,170 --> 00:45:38,830
And we'll give it coordinates
r, theta, and phi.

769
00:45:38,830 --> 00:45:42,420
And I'll assume r is non-zero.

770
00:45:42,420 --> 00:45:46,790
We could then imagine that two
people sitting at this point--

771
00:45:46,790 --> 00:45:48,890
and in the Lewis Carroll
spirit, I call them

772
00:45:48,890 --> 00:45:52,730
Tweedledee and
Tweedledum-- can decide

773
00:45:52,730 --> 00:45:56,649
to do an experiment by first
synchronizing their clocks.

774
00:45:56,649 --> 00:45:58,690
And they might as well
synchronize t cosmic time,

775
00:45:58,690 --> 00:46:00,310
let's say.

776
00:46:00,310 --> 00:46:05,060
And then, one of them can go off
in the direction of positive r,

777
00:46:05,060 --> 00:46:08,800
and the other can go off in
the direction of negative r

778
00:46:08,800 --> 00:46:12,290
at the same coordinate
velocity, which I'll call v.

779
00:46:12,290 --> 00:46:14,200
And by coordinate
velocity, I mean dr dt,

780
00:46:14,200 --> 00:46:16,610
because that's the simplest
thing to talk about here.

781
00:46:16,610 --> 00:46:18,651
It many not be the same
as the physical velocity,

782
00:46:18,651 --> 00:46:19,510
but we don't care.

783
00:46:19,510 --> 00:46:22,000
It'll be the same
for both of them.

784
00:46:22,000 --> 00:46:25,970
And the experiment will be
that they will each travel

785
00:46:25,970 --> 00:46:29,900
until there's some-- until
cosmic time-- they're

786
00:46:29,900 --> 00:46:33,440
passing a lot of cosmic
time clocks as they travel.

787
00:46:33,440 --> 00:46:36,280
And they agree to travel
until cosmic time ticks off

788
00:46:36,280 --> 00:46:39,490
until some chosen final time.

789
00:46:39,490 --> 00:46:41,950
And when they each
finish the experiment

790
00:46:41,950 --> 00:46:45,882
by noticing that the cosmic
time clocks now read t sub f,

791
00:46:45,882 --> 00:46:47,340
they will look at
their own watches

792
00:46:47,340 --> 00:46:49,450
and see how much time elapsed.

793
00:46:49,450 --> 00:46:52,100
So they're basically
measuring time dilation--

794
00:46:52,100 --> 00:46:54,920
how do their wrist watch
times, when they move,

795
00:46:54,920 --> 00:46:58,900
differ from cosmic time.

796
00:46:58,900 --> 00:47:04,530
And the point is that if we
have a dr dt term in the metric,

797
00:47:04,530 --> 00:47:07,240
these people get
different values

798
00:47:07,240 --> 00:47:09,550
because the time
that they measure

799
00:47:09,550 --> 00:47:11,520
will be what they
call ds squared,

800
00:47:11,520 --> 00:47:13,720
up to a factor of
minus c squared.

801
00:47:13,720 --> 00:47:29,550
And that will include
this dt dr term.

802
00:47:32,500 --> 00:47:39,930
And dr for one of them will
be the coordinate velocity

803
00:47:39,930 --> 00:47:48,420
they chose times d cosmic
time, the amount of cosmic time

804
00:47:48,420 --> 00:47:49,740
interval they travel for.

805
00:47:49,740 --> 00:47:52,870
They've agreed on that
before they take off.

806
00:47:52,870 --> 00:47:56,710
And for the other, dr
will be negative vc times

807
00:47:56,710 --> 00:47:59,520
dt, so cosmic.

808
00:48:03,200 --> 00:48:05,540
So this term will give
a different contribution

809
00:48:05,540 --> 00:48:09,550
to the ds squared that each of
these two entities, Tweedledee

810
00:48:09,550 --> 00:48:11,400
and Tweedledum, will measure.

811
00:48:11,400 --> 00:48:13,400
And therefore, they'll
be measuring different ds

812
00:48:13,400 --> 00:48:15,240
squareds, and that means they'll
be measuring different things

813
00:48:15,240 --> 00:48:16,630
on their wrist watches.

814
00:48:16,630 --> 00:48:19,940
And that means they have
an asymmetry in directions.

815
00:48:19,940 --> 00:48:21,640
By going one direction
or the other,

816
00:48:21,640 --> 00:48:23,940
they could determine
whether their time dilation

817
00:48:23,940 --> 00:48:26,770
will be increased or decreased.

818
00:48:26,770 --> 00:48:29,520
And that, if our
universe is isotopic,

819
00:48:29,520 --> 00:48:31,100
should not be possible.

820
00:48:31,100 --> 00:48:34,070
And therefore, if we want to
write a metric which describes

821
00:48:34,070 --> 00:48:40,840
an isotropic universe, we
have to omit the dr dt term.

822
00:48:40,840 --> 00:48:42,830
And a completely
identical argument

823
00:48:42,830 --> 00:48:45,930
implies that we have to also
omit d theta dt and d phi dt.

824
00:48:54,840 --> 00:49:05,685
So isotropy implies dt
dr term is not allowed.

825
00:49:08,900 --> 00:49:13,940
Because otherwise, we would have
a Tweedledee Tweedledum time

826
00:49:13,940 --> 00:49:15,860
dilation asymmetry,
which we're not

827
00:49:15,860 --> 00:49:17,490
allowed to have an
isotropic universe.

828
00:49:21,232 --> 00:49:22,440
OK, any questions about that?

829
00:49:28,944 --> 00:49:30,610
OK, if you have no
questions about that,

830
00:49:30,610 --> 00:49:33,740
now we're ready to go
onto our next topic.

831
00:49:33,740 --> 00:49:43,370
Now that we've described
the metric of our universe,

832
00:49:43,370 --> 00:49:45,880
there it is-- the full
Robertson-Walker spacetime

833
00:49:45,880 --> 00:49:51,180
metric for a homogeneous
and isotropic universe.

834
00:49:51,180 --> 00:49:55,160
The next thing I'd like to about
is how do we calculate motion

835
00:49:55,160 --> 00:49:58,290
in a metric in the context
of general relativity.

836
00:49:58,290 --> 00:50:00,900
And our treatment here
will be completely general.

837
00:50:00,900 --> 00:50:04,300
We'll learn how to calculate
motion in an arbitrary metric.

838
00:50:04,300 --> 00:50:06,350
And we'll in fact use
the Schwarzschild metric,

839
00:50:06,350 --> 00:50:09,510
which describes spherically
symmetric objects like stars

840
00:50:09,510 --> 00:50:12,030
or even black holes
as an example.

841
00:50:15,350 --> 00:50:17,889
But our real purpose
is to understand things

842
00:50:17,889 --> 00:50:19,055
like motion in the universe.

843
00:50:22,830 --> 00:50:31,240
And I guess at this point, I
am going to-- load up screen.

844
00:50:31,240 --> 00:50:32,232
OK, good.

845
00:51:11,990 --> 00:51:17,390
OK, I'm going to just use the
equations from the lecture

846
00:51:17,390 --> 00:51:17,890
notes.

847
00:51:20,950 --> 00:51:22,500
This particular
calculation involves

848
00:51:22,500 --> 00:51:25,630
a lot of long
equations, so I think

849
00:51:25,630 --> 00:51:29,020
doing it on the blackboard would
probably be a bit too tedious.

850
00:51:29,020 --> 00:51:31,230
So I'm instead going to
just lift the equations

851
00:51:31,230 --> 00:51:33,660
and talk about them directly
from the lecture notes

852
00:51:33,660 --> 00:51:35,250
themselves.

853
00:51:35,250 --> 00:51:37,070
So what we're interested
in is thinking

854
00:51:37,070 --> 00:51:41,320
about a geodesic in
some arbitrary metric.

855
00:51:41,320 --> 00:51:44,150
And we're going to start with
the simplest possible example

856
00:51:44,150 --> 00:51:48,420
of the two dimensional spatial
metric of the same kind

857
00:51:48,420 --> 00:51:51,300
of spaces that Gauss and
Bolyai and Lobachevsky

858
00:51:51,300 --> 00:51:55,930
were talking about using
this notation of differential

859
00:51:55,930 --> 00:52:00,400
geometry of thinking of a metric
in terms of coordinates, which

860
00:52:00,400 --> 00:52:04,150
in this case, we'll
initially call x and y.

861
00:52:04,150 --> 00:52:08,190
It will generalize perfectly
straightforwardly to spacetimes

862
00:52:08,190 --> 00:52:10,700
because all the
ideas are the same.

863
00:52:10,700 --> 00:52:14,450
But it's easiest to start out
by thinking about you're simply

864
00:52:14,450 --> 00:52:16,480
talking about
measuring distances

865
00:52:16,480 --> 00:52:19,710
in a two dimensional space.

866
00:52:19,710 --> 00:52:26,240
A geodesic is defined as a line
between two points in space

867
00:52:26,240 --> 00:52:28,760
which has the property that
the length of that line

868
00:52:28,760 --> 00:52:31,520
is stationary with
respect to any variations.

869
00:52:31,520 --> 00:52:34,350
Stationary means the
first derivative vanishes.

870
00:52:34,350 --> 00:52:36,490
Now, in this two
dimensional space example,

871
00:52:36,490 --> 00:52:40,350
our stationary lines will
also always be minima.

872
00:52:40,350 --> 00:52:42,820
That is, you can minimize the
distance between two points

873
00:52:42,820 --> 00:52:44,830
by finding the
shortest possible line.

874
00:52:44,830 --> 00:52:47,125
There is no longest
possible line.

875
00:52:47,125 --> 00:52:49,500
And there aren't any saddle
points either, I don't think.

876
00:52:49,500 --> 00:52:51,850
So I think in this
case the minima--

877
00:52:51,850 --> 00:52:54,632
the stationary points will
always be minimum I believe.

878
00:52:54,632 --> 00:52:56,590
But in general, when we
have spacetime metrics,

879
00:52:56,590 --> 00:53:00,090
especially when things are
not even positive definite,

880
00:53:00,090 --> 00:53:02,790
these geodesics, the
stationary lines,

881
00:53:02,790 --> 00:53:06,140
can be either maxima or
minima or saddle points.

882
00:53:06,140 --> 00:53:11,610
So you should imagine that all
those possibilities are there.

883
00:53:11,610 --> 00:53:13,165
However, the
equations we'll derive

884
00:53:13,165 --> 00:53:14,540
will really just
be the equations

885
00:53:14,540 --> 00:53:17,419
that say that the first
order difference vanishes.

886
00:53:17,419 --> 00:53:19,460
If you vary the path a
little bit to first order,

887
00:53:19,460 --> 00:53:21,080
the length does not change.

888
00:53:21,080 --> 00:53:24,350
And that will be true for
maxima, minima, or saddle

889
00:53:24,350 --> 00:53:24,850
points.

890
00:53:24,850 --> 00:53:29,020
We won't have to care in
deriving the equations.

891
00:53:29,020 --> 00:53:31,120
So we start by imagining
a metric like that.

892
00:53:34,170 --> 00:53:35,545
And the first
thing we want to do

893
00:53:35,545 --> 00:53:39,360
is just adopt a better
notation for the metric.

894
00:53:39,360 --> 00:53:42,070
And there are two improvements.

895
00:53:42,070 --> 00:53:44,380
The first is to
number the coordinates

896
00:53:44,380 --> 00:53:46,770
instead of thinking of
them as different letters.

897
00:53:46,770 --> 00:53:49,120
So instead of talking
about x and y,

898
00:53:49,120 --> 00:53:53,860
we're going to talk about
x super one and x super 2.

899
00:53:53,860 --> 00:53:55,670
Now, these 1's and
2's have the danger

900
00:53:55,670 --> 00:53:57,860
of possibly being
confused with a power.

901
00:53:57,860 --> 00:53:59,890
We probably never write
x to the first power,

902
00:53:59,890 --> 00:54:03,410
but you might write x super 2
and think of if as x squared.

903
00:54:03,410 --> 00:54:05,530
Many times we, of
course, do that.

904
00:54:05,530 --> 00:54:08,640
So one always has to hope that
the context will make it clear

905
00:54:08,640 --> 00:54:12,290
what that index refers to.

906
00:54:12,290 --> 00:54:16,660
But here, these upper index
objects-- those superscripts

907
00:54:16,660 --> 00:54:17,850
are just indices.

908
00:54:17,850 --> 00:54:20,450
They're not powers.

909
00:54:20,450 --> 00:54:23,372
You might wonder why we tolerate
such a crazy notation when

910
00:54:23,372 --> 00:54:25,080
we could have written
them as subscripts.

911
00:54:25,080 --> 00:54:27,286
And then there would
not be this confusion.

912
00:54:27,286 --> 00:54:29,160
But the answer is that
in general relativity,

913
00:54:29,160 --> 00:54:31,368
one does make use of both
subscripts and superscripts

914
00:54:31,368 --> 00:54:33,070
in a slightly different way.

915
00:54:33,070 --> 00:54:36,280
And to some extent, you'll see
that in what we'll be doing.

916
00:54:36,280 --> 00:54:37,860
So it's useful in
general relativity

917
00:54:37,860 --> 00:54:41,650
to have two kinds of scripts.

918
00:54:41,650 --> 00:54:44,290
And the only places that seem
to exist are up and down.

919
00:54:44,290 --> 00:54:46,930
So they're superscripts
and subscripts.

920
00:54:46,930 --> 00:54:48,760
And one simply
hopes that there's

921
00:54:48,760 --> 00:54:51,390
no confusion with powers.

922
00:54:51,390 --> 00:54:54,640
OK, so step one is
number your indices

923
00:54:54,640 --> 00:54:57,140
and to number your coordinates.

924
00:54:57,140 --> 00:54:59,872
And then instead of writing
that the sum of three terms

925
00:54:59,872 --> 00:55:02,080
that we had-- and of course,
it gets to be much more.

926
00:55:02,080 --> 00:55:06,550
If you have four coordinates, it
would be 16 terms or 10 terms,

927
00:55:06,550 --> 00:55:09,245
depending on how
you collected them.

928
00:55:09,245 --> 00:55:10,620
But instead of
writing that mess,

929
00:55:10,620 --> 00:55:12,890
you can write it using
the summation notation.

930
00:55:12,890 --> 00:55:16,163
Sum from i equals 1 to
2, sum j equals 1 to 2

931
00:55:16,163 --> 00:55:23,490
of g sub ij of the
x-coordinates times dx i dx j.

932
00:55:23,490 --> 00:55:26,300
And when you sum over i and j,
you're summing over 1 and 2,

933
00:55:26,300 --> 00:55:28,210
which means you're
summing over x and y.

934
00:55:28,210 --> 00:55:29,810
And the sum includes
the x, x term,

935
00:55:29,810 --> 00:55:31,550
which is now called
the 1, 1 term.

936
00:55:31,550 --> 00:55:33,300
And the y, y term
which is now called

937
00:55:33,300 --> 00:55:36,380
the 2, 2 term,
and the x, y term,

938
00:55:36,380 --> 00:55:38,627
which is now called the
1, 2 or the 2, 1 term.

939
00:55:38,627 --> 00:55:40,210
And those are identical
to each other.

940
00:55:40,210 --> 00:55:43,220
And they just get added.

941
00:55:43,220 --> 00:55:45,780
So that shortens the
notation considerably.

942
00:55:45,780 --> 00:55:48,140
But then there's one
further simplification

943
00:55:48,140 --> 00:55:50,360
that was actually introduced
by Einstein himself.

944
00:55:50,360 --> 00:55:54,640
And it's always called the
Einstein summation convention.

945
00:55:54,640 --> 00:55:58,010
Notice that in this
equation, the letter i

946
00:55:58,010 --> 00:56:01,180
appears twice as an
index-- as an upper index

947
00:56:01,180 --> 00:56:05,390
there and as a lower index
on the metric, g sub ij.

948
00:56:05,390 --> 00:56:09,360
And the Einstein
convention is that whenever

949
00:56:09,360 --> 00:56:13,690
you have a repeated index where
one is upper and one is lower,

950
00:56:13,690 --> 00:56:16,490
you automatically sum over them
without writing the summation

951
00:56:16,490 --> 00:56:17,340
sign.

952
00:56:17,340 --> 00:56:20,200
The summation sign is implied.

953
00:56:20,200 --> 00:56:22,862
So then this equation get
simplified to that equation,

954
00:56:22,862 --> 00:56:24,820
which is the form that
we'll actually be using.

955
00:56:24,820 --> 00:56:28,450
And that's as about
as simple as it gets.

956
00:56:28,450 --> 00:56:30,055
OK, so far, that's
just notation.

957
00:56:34,509 --> 00:56:36,050
OK, now what we want
to do is to talk

958
00:56:36,050 --> 00:56:38,140
about a path between two points.

959
00:56:38,140 --> 00:56:40,879
And we want to discuss how
we're going to describe the path

960
00:56:40,879 --> 00:56:43,420
and how we're going to derive
the equations that will tell us

961
00:56:43,420 --> 00:56:45,810
that this path has the
minimum possible length, which

962
00:56:45,810 --> 00:56:47,220
is what we're trying to do.

963
00:56:47,220 --> 00:56:49,250
We're trying to find the
equations that tell us

964
00:56:49,250 --> 00:56:54,340
when a path has an extreme
value of the length.

965
00:56:54,340 --> 00:56:57,030
So to describe the
path itself, we're

966
00:56:57,030 --> 00:56:59,450
going to imagine
parameterizing it, which

967
00:56:59,450 --> 00:57:02,940
means we're going to think
of a function xi of lambda,

968
00:57:02,940 --> 00:57:05,200
where xi-- remember,
xi means 1 and 2.

969
00:57:05,200 --> 00:57:08,840
It means specifying one both x
and y as a function of lambda.

970
00:57:08,840 --> 00:57:12,070
So this is really two
functions of lambda.

971
00:57:12,070 --> 00:57:15,270
And this function
of lambda is going

972
00:57:15,270 --> 00:57:19,700
to show us how the point
that traces out this path

973
00:57:19,700 --> 00:57:23,640
varies from point A
to point B. And we'll

974
00:57:23,640 --> 00:57:25,850
allow it to vary as a
function of this parameter

975
00:57:25,850 --> 00:57:27,750
that we've introduced, lambda.

976
00:57:27,750 --> 00:57:29,500
And we'll adopt the
convention that lambda

977
00:57:29,500 --> 00:57:33,770
is 0 at one end and lambda
sub f at the other end.

978
00:57:33,770 --> 00:57:38,310
So xi of 0 will be required to
be the coordinates of point A

979
00:57:38,310 --> 00:57:40,690
If we want to start at
some specified point A,

980
00:57:40,690 --> 00:57:43,100
we want to end up at
some specified point B.

981
00:57:43,100 --> 00:57:46,150
So we'll insist that the
coordinates evaluated at lambda

982
00:57:46,150 --> 00:57:51,050
sub f are xi sub B. And
as long as xi of lambda

983
00:57:51,050 --> 00:57:54,810
is a continuous function,
which we will also insist on,

984
00:57:54,810 --> 00:57:58,756
then xi of lambda will describe
a path from A to B, which

985
00:57:58,756 --> 00:57:59,880
is what we're trying to do.

986
00:58:03,400 --> 00:58:07,460
OK, to apply the
metric and write down

987
00:58:07,460 --> 00:58:09,210
an expression for the
length of this path,

988
00:58:09,210 --> 00:58:10,501
that's what we want to do next.

989
00:58:10,501 --> 00:58:13,160
And then we want to
figure how to extremize

990
00:58:13,160 --> 00:58:15,120
that expression for the length.

991
00:58:15,120 --> 00:58:17,985
First step, we need to get
an expression for the length.

992
00:58:17,985 --> 00:58:19,360
So the metric is
written in terms

993
00:58:19,360 --> 00:58:21,470
of infinitesimal separations.

994
00:58:21,470 --> 00:58:23,950
So we want to imagine
dividing this path up

995
00:58:23,950 --> 00:58:28,887
into little segments, each
corresponding to some d lambda.

996
00:58:28,887 --> 00:58:31,220
Each little segment goes from
some lambda to some lambda

997
00:58:31,220 --> 00:58:35,060
plus d lambda, where d
lambda is infinitesimal.

998
00:58:35,060 --> 00:58:38,720
And the change in the
coordinates over that interval

999
00:58:38,720 --> 00:58:40,850
are then just the
derivative of xi

1000
00:58:40,850 --> 00:58:42,360
with respect to
lambda-- remember,

1001
00:58:42,360 --> 00:58:46,664
we have this function, xi
of lambda-- times d lambda.

1002
00:58:46,664 --> 00:58:48,080
This will give us
the differential

1003
00:58:48,080 --> 00:58:50,570
coordinates between
any two neighboring

1004
00:58:50,570 --> 00:58:53,890
points along the line.

1005
00:58:53,890 --> 00:58:57,020
Then ds squared is
defined in terms of d xi.

1006
00:58:57,020 --> 00:59:00,270
And we just plug this
formula into the expression

1007
00:59:00,270 --> 00:59:03,990
for ds squared in terms of
the infinitesimal separations.

1008
00:59:03,990 --> 00:59:05,620
So we have the metric.

1009
00:59:05,620 --> 00:59:08,490
And then where we had
previously just d xi,

1010
00:59:08,490 --> 00:59:12,170
now we have d xi d
lambda times d lambda.

1011
00:59:12,170 --> 00:59:15,940
And similarly, where we
previously had just d xj,

1012
00:59:15,940 --> 00:59:19,430
now we have the derivative
of xj with respect to lambda,

1013
00:59:19,430 --> 00:59:20,800
again, times d lambda.

1014
00:59:20,800 --> 00:59:22,800
So we have two
powers of d lambda

1015
00:59:22,800 --> 00:59:25,380
appearing in this expression.

1016
00:59:25,380 --> 00:59:29,274
ds itself will be the
square root of ds squared.

1017
00:59:29,274 --> 00:59:30,690
In this case, we
are talking about

1018
00:59:30,690 --> 00:59:32,150
positive, definite distances.

1019
00:59:32,150 --> 00:59:34,819
So we can take the square root.

1020
00:59:34,819 --> 00:59:36,360
So we put a square
root sign over it.

1021
00:59:36,360 --> 00:59:40,410
And now, we have only
one power of d lambda.

1022
00:59:40,410 --> 00:59:43,900
And this describes the
length of the segment that

1023
00:59:43,900 --> 00:59:48,650
goes from lambda to lambda plus
d lambda, the length as defined

1024
00:59:48,650 --> 00:59:51,530
by the metric.

1025
00:59:51,530 --> 00:59:53,520
The full length of the
line is obtained just

1026
00:59:53,520 --> 00:59:58,210
by integrating that from 0
to the final value of lambda.

1027
00:59:58,210 --> 01:00:03,660
So equation 540
here is what we're

1028
01:00:03,660 --> 01:00:07,930
looking for-- the expression for
the length of the line in terms

1029
01:00:07,930 --> 01:00:11,584
of the parametrization
that we've chosen.

1030
01:00:11,584 --> 01:00:15,334
OK, any questions
about that formula?

1031
01:00:15,334 --> 01:00:18,464
OK, next step-- and here's where
things get kind of complicated

1032
01:00:18,464 --> 01:00:19,880
with the algebra,
although I think

1033
01:00:19,880 --> 01:00:21,710
the ideas are still
pretty simple.

1034
01:00:21,710 --> 01:00:23,840
The next step is
to figure out how

1035
01:00:23,840 --> 01:00:27,280
we determine when
that expression is

1036
01:00:27,280 --> 01:00:29,030
at its minimum value.

1037
01:00:29,030 --> 01:00:32,220
How do we determine when the
path has the right properties

1038
01:00:32,220 --> 01:00:35,450
that we found the
minimum length?

1039
01:00:35,450 --> 01:00:48,015
So to do that, we want to
imagine varying the path.

1040
01:00:51,740 --> 01:00:54,750
We want to consider comparing
the length of the path

1041
01:00:54,750 --> 01:00:57,140
that we're thinking about to
the length of an arbitrary

1042
01:00:57,140 --> 01:00:59,140
nearby path.

1043
01:00:59,140 --> 01:01:03,570
And to do that, we can introduce
a little bit of extra notation

1044
01:01:03,570 --> 01:01:04,290
here.

1045
01:01:04,290 --> 01:01:07,490
Here's the point xA
Here's the point xB.

1046
01:01:07,490 --> 01:01:11,840
x of lambda is the path
that we're thinking about.

1047
01:01:11,840 --> 01:01:14,240
And we're asking the
question, is this the path

1048
01:01:14,240 --> 01:01:16,220
of minimum possible length?

1049
01:01:16,220 --> 01:01:18,150
And to do that, we're
going to compare it

1050
01:01:18,150 --> 01:01:21,930
with an arbitrary nearby path.

1051
01:01:21,930 --> 01:01:23,850
So the arbitrary
nearby path is what's

1052
01:01:23,850 --> 01:01:27,240
called x tilde in this diagram.

1053
01:01:27,240 --> 01:01:30,630
It starts at the same point xA
and ends at the same point xB.

1054
01:01:30,630 --> 01:01:34,200
But along the way, it deviates
by an infinitesimal amount

1055
01:01:34,200 --> 01:01:36,220
from the original path.

1056
01:01:36,220 --> 01:01:41,630
And we're going to parametrize
that by equation 541a here.

1057
01:01:41,630 --> 01:01:45,000
The tilde path will be
equal to the original path

1058
01:01:45,000 --> 01:01:49,150
plus a parameter that I'm going
to introduce called alpha times

1059
01:01:49,150 --> 01:01:52,180
a function wi of lambda,
where wi of lambda

1060
01:01:52,180 --> 01:01:54,300
is really an arbitrary function.

1061
01:01:54,300 --> 01:01:56,680
And I've introduced this
extra parameter alpha

1062
01:01:56,680 --> 01:01:58,480
just so I could say
in a simple way what

1063
01:01:58,480 --> 01:02:01,140
it means for these paths to
be infinitesimally close,

1064
01:02:01,140 --> 01:02:03,670
which just means that if alpha
has an infinitesimal value,

1065
01:02:03,670 --> 01:02:05,940
the two paths are
infinitesimally close.

1066
01:02:05,940 --> 01:02:07,920
And the function wi,
we'll think of being

1067
01:02:07,920 --> 01:02:13,140
a perfectly finite function
with values like 2 and 5,

1068
01:02:13,140 --> 01:02:17,540
not values that are
infinitesimally small.

1069
01:02:17,540 --> 01:02:24,200
OK, with this
parametrization of our paths,

1070
01:02:24,200 --> 01:02:26,695
we want to impose one
important criteria, which

1071
01:02:26,695 --> 01:02:28,820
is that the two paths are
supposed to start and end

1072
01:02:28,820 --> 01:02:34,360
at the same points, A and B.
And that means that this w super

1073
01:02:34,360 --> 01:02:37,420
i that describes the
derivation between two paths

1074
01:02:37,420 --> 01:02:39,400
have an advantage at
those two end points.

1075
01:02:39,400 --> 01:02:42,010
Or else, the paths aren't going
from the same starting point

1076
01:02:42,010 --> 01:02:43,300
to the same ending point.

1077
01:02:43,300 --> 01:02:45,008
And, certainly if you
move the endpoints,

1078
01:02:45,008 --> 01:02:46,510
you can always find
a shorter path.

1079
01:02:46,510 --> 01:02:49,350
There's no geodesic if you allow
yourself to move the endpoints.

1080
01:02:49,350 --> 01:02:52,000
So we insist therefore
that wi of 0,

1081
01:02:52,000 --> 01:02:55,690
which is wi at the
first endpoint, is 0.

1082
01:02:55,690 --> 01:02:58,310
And similarly, wi of lambda
sub f at the other endpoint

1083
01:02:58,310 --> 01:02:59,820
is also 0.

1084
01:02:59,820 --> 01:03:02,330
That's what we insist on.

1085
01:03:02,330 --> 01:03:05,270
OK, now having set
up this formalism,

1086
01:03:05,270 --> 01:03:07,190
we can now write down
a very simple equation

1087
01:03:07,190 --> 01:03:11,670
that says this path
is an extremum.

1088
01:03:11,670 --> 01:03:13,770
The path is an
extremum if ds d alpha

1089
01:03:13,770 --> 01:03:18,650
is equal to 0 for all
choices of wi of lambda.

1090
01:03:18,650 --> 01:03:21,540
OK, if it's an extremum, it
means any small variation,

1091
01:03:21,540 --> 01:03:24,180
any small variations are
proportional to alpha.

1092
01:03:24,180 --> 01:03:28,080
Any small variation
produces 0 derivative.

1093
01:03:28,080 --> 01:03:31,590
So ds d alpha should equal 0.

1094
01:03:31,590 --> 01:03:34,020
And that should be the case
for any possible deviation

1095
01:03:34,020 --> 01:03:38,480
if we really have found the
minimum possible length.

1096
01:03:38,480 --> 01:03:39,211
OK?

1097
01:03:39,211 --> 01:03:39,960
OK with everybody?

1098
01:03:42,540 --> 01:03:47,370
OK, now, it's mostly just a
lot of gore to get the answer.

1099
01:03:47,370 --> 01:03:50,370
The key step will be a
crucial integration by parts

1100
01:03:50,370 --> 01:03:52,080
that you'll see in a minute.

1101
01:03:52,080 --> 01:03:54,420
But let's just go through
the algebra together.

1102
01:03:54,420 --> 01:03:59,160
I'm going to define an auxiliary
quantity a of lambda alpha,

1103
01:03:59,160 --> 01:04:01,952
which is just the metric
times the derivatives

1104
01:04:01,952 --> 01:04:02,660
of the functions.

1105
01:04:05,930 --> 01:04:09,350
The path length of the
deviated path-- these

1106
01:04:09,350 --> 01:04:11,010
are tilde functions here.

1107
01:04:11,010 --> 01:04:14,550
So a is the integrand
for the length

1108
01:04:14,550 --> 01:04:18,270
of the perturbed
path, the tilde path.

1109
01:04:18,270 --> 01:04:21,190
So s of x tilde is
just the integral

1110
01:04:21,190 --> 01:04:22,900
of the square root
of a, d lambda.

1111
01:04:32,320 --> 01:04:34,880
OK, now we need some pieces
to carry out our derivative.

1112
01:04:34,880 --> 01:04:38,820
So I've introduced a few
auxiliary calculations here

1113
01:04:38,820 --> 01:04:41,230
that we can then put
into the big calculation.

1114
01:04:41,230 --> 01:04:43,230
We're going to need the
derivative of the metric

1115
01:04:43,230 --> 01:04:45,810
with respect to alpha.

1116
01:04:45,810 --> 01:04:48,590
Now, the metric does not
depend directly on alpha.

1117
01:04:48,590 --> 01:04:51,230
But the metric does
depend on x tilde.

1118
01:04:51,230 --> 01:04:53,150
It's evaluated at
the point x tilde

1119
01:04:53,150 --> 01:04:55,640
for any given value of lambda.

1120
01:04:55,640 --> 01:04:58,860
And x tilde depends on alpha,
because remember, x tilde

1121
01:04:58,860 --> 01:05:01,260
was equal to the original
path plus alpha times

1122
01:05:01,260 --> 01:05:04,730
this wi, the derivation.

1123
01:05:04,730 --> 01:05:07,060
So it's a chain rule
problem to figure out

1124
01:05:07,060 --> 01:05:11,200
what the derivative of gij
is with respect to alpha.

1125
01:05:11,200 --> 01:05:21,680
So it's the derivative
of gij with respect

1126
01:05:21,680 --> 01:05:24,750
to xk times the
derivative of xk tilde

1127
01:05:24,750 --> 01:05:29,250
with respect to alpha-- just
straightforward chain rule.

1128
01:05:29,250 --> 01:05:31,720
And the derivative of x
tilde with respect to alpha

1129
01:05:31,720 --> 01:05:36,810
is just this function, w super
i, or in this case, w super k.

1130
01:05:36,810 --> 01:05:39,700
That's what defines
the deviations.

1131
01:05:39,700 --> 01:05:41,790
So this is our result,
then, for the derivative

1132
01:05:41,790 --> 01:05:44,980
of gij with respect to alpha.

1133
01:05:44,980 --> 01:05:49,690
Then, we apply that to
differentiating s itself,

1134
01:05:49,690 --> 01:05:52,250
finding all the alpha's
inside that square root.

1135
01:05:52,250 --> 01:06:01,590
And scroll up a little bit so
we can see the definition of a.

1136
01:06:01,590 --> 01:06:05,050
a consists of gij times
the dx's themselves.

1137
01:06:05,050 --> 01:06:07,050
And the dx's themselves
depend on the alphas.

1138
01:06:07,050 --> 01:06:09,670
So we're going to get terms
coming from differentiating

1139
01:06:09,670 --> 01:06:11,260
those with respect to alpha.

1140
01:06:11,260 --> 01:06:13,500
And we get a term coming
from differentiating

1141
01:06:13,500 --> 01:06:16,580
the gij with respect to alpha.

1142
01:06:16,580 --> 01:06:18,860
So the whole quantity
in the integrand

1143
01:06:18,860 --> 01:06:22,580
here has a square
root operating on it.

1144
01:06:22,580 --> 01:06:24,670
So the derivative of the
square root of a quantity

1145
01:06:24,670 --> 01:06:27,330
is 1 over the square
root of the same quantity

1146
01:06:27,330 --> 01:06:29,760
times the derivative
of the quantity,

1147
01:06:29,760 --> 01:06:32,800
just differentiating
the 1/2 power of a.

1148
01:06:32,800 --> 01:06:36,010
So that gives us a 1/2 and
1 over the square root of a.

1149
01:06:36,010 --> 01:06:38,010
And then inside here,
we have the derivative

1150
01:06:38,010 --> 01:06:40,910
of a with respect to alpha.

1151
01:06:40,910 --> 01:06:44,000
And one of those terms,
we've already calculated.

1152
01:06:44,000 --> 01:06:49,140
It's this multiplied
by dx i d lambda dx j

1153
01:06:49,140 --> 01:06:51,950
d lambda, which come
along for the ride.

1154
01:06:51,950 --> 01:06:53,880
And they lose their
tilde because we're

1155
01:06:53,880 --> 01:06:57,070
trying to calculate the
derivative at alpha equals 0.

1156
01:06:57,070 --> 01:06:59,960
So once we differentiate one
factor with respect to alpha,

1157
01:06:59,960 --> 01:07:03,737
we evaluate the other
factors at alpha equals 0.

1158
01:07:03,737 --> 01:07:04,820
So that's what we've done.

1159
01:07:04,820 --> 01:07:09,780
We've evaluated the other
factors at alpha equals 0.

1160
01:07:09,780 --> 01:07:14,250
And then, when we
differentiate dx i tilde

1161
01:07:14,250 --> 01:07:18,140
with respect to d lambda, we
just get DWI d wi with respect

1162
01:07:18,140 --> 01:07:19,140
to lambda.

1163
01:07:19,140 --> 01:07:24,050
And dx j d lambda comes
along, now evaluated at alpha

1164
01:07:24,050 --> 01:07:25,200
equals 0.

1165
01:07:25,200 --> 01:07:26,920
And similarly,
the second term is

1166
01:07:26,920 --> 01:07:30,090
where we differentiated the
second factor here with respect

1167
01:07:30,090 --> 01:07:30,819
to alpha.

1168
01:07:30,819 --> 01:07:32,610
And we differentiate
with respect to alpha,

1169
01:07:32,610 --> 01:07:34,571
we bring down the w.

1170
01:07:34,571 --> 01:07:40,750
So this becomes dx i d
lambda dx j d lambda.

1171
01:07:40,750 --> 01:07:42,349
So this is the expression.

1172
01:07:42,349 --> 01:07:44,140
And now we want to
simplify it a little bit

1173
01:07:44,140 --> 01:07:46,810
and figure out how to write
down an equation which

1174
01:07:46,810 --> 01:07:48,320
tells us when it's actually 0.

1175
01:07:53,004 --> 01:07:54,920
So first, we want to
simplify it a little bit.

1176
01:07:54,920 --> 01:07:58,920
And I guess I want
to-- do they fit?

1177
01:07:58,920 --> 01:08:00,300
Almost.

1178
01:08:00,300 --> 01:08:05,960
What I want to argue is
that these last two terms

1179
01:08:05,960 --> 01:08:08,670
are really equal to each
other up to just rearranging

1180
01:08:08,670 --> 01:08:10,040
the indices.

1181
01:08:10,040 --> 01:08:12,670
Remember, i and j are
just being summed over.

1182
01:08:12,670 --> 01:08:15,410
So we could have called them
any letter we wanted, and they

1183
01:08:15,410 --> 01:08:17,670
would still just
be summed 1 and 2.

1184
01:08:17,670 --> 01:08:19,240
And in particular,
we can interchange

1185
01:08:19,240 --> 01:08:21,390
what we call i with
what we call j.

1186
01:08:21,390 --> 01:08:23,432
And then, these two terms
would become identical.

1187
01:08:23,432 --> 01:08:25,556
And we're allowed to do
that because these are just

1188
01:08:25,556 --> 01:08:26,830
what are called dummy indices.

1189
01:08:26,830 --> 01:08:29,350
They're just names of indices
that are being summed over.

1190
01:08:29,350 --> 01:08:30,766
And you get the
same sum no matter

1191
01:08:30,766 --> 01:08:33,649
what you call the index
you're summing over.

1192
01:08:33,649 --> 01:08:38,710
So those terms can be combined,
giving us just 2 times

1193
01:08:38,710 --> 01:08:41,120
either one of those
two terms we can keep.

1194
01:08:41,120 --> 01:08:43,060
And now, we only two
terms in our expression,

1195
01:08:43,060 --> 01:08:45,140
which is not bad.

1196
01:08:45,140 --> 01:08:49,204
But things are still
a little complicated.

1197
01:08:49,204 --> 01:08:51,370
And what makes them complicated
at this point, which

1198
01:08:51,370 --> 01:08:54,460
is what we have to get
rid of, is the fact

1199
01:08:54,460 --> 01:09:00,600
that w occurs as a multiplicity
factor in the first term.

1200
01:09:00,600 --> 01:09:02,390
But w is differentiated
with respect

1201
01:09:02,390 --> 01:09:04,350
to lambda in the second term.

1202
01:09:04,350 --> 01:09:07,840
And when it's written
that way, there's

1203
01:09:07,840 --> 01:09:11,930
no direct way you could see
what properties w has to have

1204
01:09:11,930 --> 01:09:14,310
or the other terms
have to have so

1205
01:09:14,310 --> 01:09:17,340
that the expression vanishes.

1206
01:09:17,340 --> 01:09:21,470
But the crucial
trick for handling

1207
01:09:21,470 --> 01:09:23,824
that particular issue-- and
it's the only real issue

1208
01:09:23,824 --> 01:09:24,490
in this problem.

1209
01:09:24,490 --> 01:09:27,620
The rest is just straightforward
or sometimes tedious

1210
01:09:27,620 --> 01:09:29,310
manipulations.

1211
01:09:29,310 --> 01:09:32,380
The key step is to
integrate by parts

1212
01:09:32,380 --> 01:09:35,050
to turn the derivative
of w expression

1213
01:09:35,050 --> 01:09:38,020
into an expression that is
just multiplicative in w.

1214
01:09:38,020 --> 01:09:41,859
And the miraculous thing is
that for the situation we have,

1215
01:09:41,859 --> 01:09:44,951
there are no boundary terms
that arise from that integration

1216
01:09:44,951 --> 01:09:45,450
by parts.

1217
01:09:49,410 --> 01:09:53,029
So here's integration by parts
spelled out in gory detail.

1218
01:09:53,029 --> 01:09:55,170
We're going to use the
famous formula that

1219
01:09:55,170 --> 01:09:57,620
says that the integral
of udv is equal to minus

1220
01:09:57,620 --> 01:10:01,930
the interval at vdu plus
the product U times V

1221
01:10:01,930 --> 01:10:05,360
evaluated at the two
endpoints and subtracted.

1222
01:10:05,360 --> 01:10:06,860
So this is just the
standard formula

1223
01:10:06,860 --> 01:10:09,700
that defines
integration by parts.

1224
01:10:09,700 --> 01:10:15,990
The U is the term that starts
out not having derivatives

1225
01:10:15,990 --> 01:10:17,390
and later acquires derivatives.

1226
01:10:17,390 --> 01:10:21,460
So that's the 1 over
root a gij dxj d lambda

1227
01:10:21,460 --> 01:10:24,710
of this-- this is the quantity
we're trying to calculate.

1228
01:10:24,710 --> 01:10:32,560
And the du will just be--
I'm sorry-- dv will just

1229
01:10:32,560 --> 01:10:34,263
be the factor which
is a differential

1230
01:10:34,263 --> 01:10:37,950
in the original expression,
d wi d lambda times d lambda

1231
01:10:37,950 --> 01:10:41,190
we're going to let
be equal to dv.

1232
01:10:41,190 --> 01:10:44,400
So this u and this dv give
us the original integral.

1233
01:10:44,400 --> 01:10:46,875
This integral is the
same as the integral

1234
01:10:46,875 --> 01:10:49,600
we're trying to evaluate.

1235
01:10:49,600 --> 01:10:53,920
Now when we integrate by parts,
we apply a derivative to U

1236
01:10:53,920 --> 01:10:57,470
to write down dU, which means
it's just the derivative

1237
01:10:57,470 --> 01:11:00,660
with respect to lambda of the
quantity in brackets times

1238
01:11:00,660 --> 01:11:02,660
d lambda.

1239
01:11:02,660 --> 01:11:04,040
That's the du.

1240
01:11:04,040 --> 01:11:08,650
And dv is just
easily integrable.

1241
01:11:08,650 --> 01:11:09,845
D is equal to wi.

1242
01:11:12,646 --> 01:11:15,020
Now, the important thing is
to look at the boundary term.

1243
01:11:15,020 --> 01:11:16,770
Because if the boundary
term went on 0,

1244
01:11:16,770 --> 01:11:19,420
we might not have
accomplished anything.

1245
01:11:19,420 --> 01:11:22,510
But the boundary term is 0
because the boundary term

1246
01:11:22,510 --> 01:11:27,270
is the product of U
times V, and V is wi.

1247
01:11:27,270 --> 01:11:30,930
And wi vanishes at both
boundaries that remember,

1248
01:11:30,930 --> 01:11:35,235
was just the condition that
the path goes between A and B.

1249
01:11:35,235 --> 01:11:37,360
When you vary the path,
you don't vary the points A

1250
01:11:37,360 --> 01:11:40,280
and B. You only vary
the path in between.

1251
01:11:40,280 --> 01:11:41,802
So wi vanishes at the endpoint.

1252
01:11:41,802 --> 01:11:43,760
So that means that v
vanishes at the endpoints.

1253
01:11:43,760 --> 01:11:45,990
So that means if the
product of U times V

1254
01:11:45,990 --> 01:11:47,800
vanishes at the end points.

1255
01:11:47,800 --> 01:11:50,310
And that means our boundary
term, our service term,

1256
01:11:50,310 --> 01:11:52,910
does not contribute.

1257
01:11:52,910 --> 01:11:56,290
So we turned the original
integral into another integral

1258
01:11:56,290 --> 01:11:59,200
where now wi appears as
the multiplicity factor

1259
01:11:59,200 --> 01:12:01,640
and it's no longer
differentiated.

1260
01:12:01,640 --> 01:12:04,410
Lots of other things get
differentiated in the process.

1261
01:12:04,410 --> 01:12:06,590
But wi gets to sit by itself.

1262
01:12:06,590 --> 01:12:09,090
And that now makes it easy
to combine these two terms

1263
01:12:09,090 --> 01:12:12,295
and see under what
circumstances the sum vanishes.

1264
01:12:16,290 --> 01:12:19,100
So the integral, after
we make this integration

1265
01:12:19,100 --> 01:12:22,420
by parts on one
of the two terms,

1266
01:12:22,420 --> 01:12:27,120
becomes this expression
where now, the w's

1267
01:12:27,120 --> 01:12:29,130
are always multiplicative.

1268
01:12:29,130 --> 01:12:31,890
And by rearranging the names
of these dummy indices--

1269
01:12:31,890 --> 01:12:35,950
as we initially have it, w has a
superscript k in the first term

1270
01:12:35,950 --> 01:12:38,370
and the subscript i
and the second term.

1271
01:12:38,370 --> 01:12:40,870
But one could rearrange
these dummy indices--

1272
01:12:40,870 --> 01:12:44,360
we could name them anything you
want-- so that in both cases,

1273
01:12:44,360 --> 01:12:47,750
w has the same index.

1274
01:12:47,750 --> 01:12:49,450
And then you can factor it out.

1275
01:12:49,450 --> 01:12:51,170
And then you get this
marvelous equation,

1276
01:12:51,170 --> 01:12:53,650
which is now very close
to being an equation

1277
01:12:53,650 --> 01:12:55,460
that we're prepared
to deal with.

1278
01:13:07,742 --> 01:13:09,200
The question we
want to address now

1279
01:13:09,200 --> 01:13:13,999
is under what circumstances
does this vanish for every wi?

1280
01:13:13,999 --> 01:13:15,540
Now, if we only know
that it vanished

1281
01:13:15,540 --> 01:13:18,300
for some particular
wi, then we would not

1282
01:13:18,300 --> 01:13:19,520
be able to say very much.

1283
01:13:19,520 --> 01:13:21,360
Because it's very
easy for an integral

1284
01:13:21,360 --> 01:13:25,960
to be nonzero all over the
place, literally everywhere

1285
01:13:25,960 --> 01:13:28,730
except maybe at some
isolated points and still

1286
01:13:28,730 --> 01:13:29,770
integrate to 0.

1287
01:13:29,770 --> 01:13:31,270
It could be positive
in some places,

1288
01:13:31,270 --> 01:13:36,050
negative in other places and
zero only at crossing points

1289
01:13:36,050 --> 01:13:38,530
and still integrate to 0.

1290
01:13:38,530 --> 01:13:43,620
But if this is going
to vanish for any wi,

1291
01:13:43,620 --> 01:13:46,150
then the claim is that the
quantity in curly brackets

1292
01:13:46,150 --> 01:13:49,460
has to vanish identically.

1293
01:13:49,460 --> 01:13:51,490
And I think the
best way to prove

1294
01:13:51,490 --> 01:13:54,550
that is to say that if
the quantity in brackets

1295
01:13:54,550 --> 01:13:57,450
did not vanish
identically, then you

1296
01:13:57,450 --> 01:14:01,270
could let wi be equal to
the quantity in brackets.

1297
01:14:01,270 --> 01:14:05,409
Remember, if this is
non-zero for any wi,

1298
01:14:05,409 --> 01:14:06,950
we have a contradiction
because we're

1299
01:14:06,950 --> 01:14:11,230
going to require this
to vanish for any wi.

1300
01:14:11,230 --> 01:14:13,770
So if the quantity in
brackets were nonzero,

1301
01:14:13,770 --> 01:14:15,599
we could let wi be
the same quantity.

1302
01:14:15,599 --> 01:14:18,140
And then we would just have the
integral of a perfect square.

1303
01:14:18,140 --> 01:14:21,290
And then clearly, the
integral would not vanish.

1304
01:14:21,290 --> 01:14:23,417
So that shows that if
the quantity in brackets

1305
01:14:23,417 --> 01:14:25,000
does not vanish, the
integral does not

1306
01:14:25,000 --> 01:14:27,809
vanish, at least for some wi.

1307
01:14:27,809 --> 01:14:29,350
And that means that
if the integral's

1308
01:14:29,350 --> 01:14:30,808
going to vanish
for every wi, which

1309
01:14:30,808 --> 01:14:34,090
is what we're trying to impose,
the quantity in curly brackets

1310
01:14:34,090 --> 01:14:35,860
has to vanish identically.

1311
01:14:35,860 --> 01:14:37,950
And that's our conclusion.

1312
01:14:37,950 --> 01:14:42,280
So that implies--
and I guess here

1313
01:14:42,280 --> 01:14:43,780
is where we're going to stop.

1314
01:14:43,780 --> 01:14:46,046
But we get to the
famous boxed equation.

1315
01:14:46,046 --> 01:14:48,170
And this really is-- we'll
simplify it a little bit

1316
01:14:48,170 --> 01:14:49,820
afterwards next time.

1317
01:14:49,820 --> 01:14:51,960
But this really is the result.

1318
01:14:51,960 --> 01:14:54,464
The geodesic equation
is that equation,

1319
01:14:54,464 --> 01:14:56,630
which is just the equation
that the quantity that we

1320
01:14:56,630 --> 01:14:59,890
had in curly brackets vanishes.

1321
01:14:59,890 --> 01:15:03,130
So if the path that
we've chosen has

1322
01:15:03,130 --> 01:15:05,490
the property that
these derivatives are

1323
01:15:05,490 --> 01:15:08,280
equal to each other-- and
notice it depends on the metric

1324
01:15:08,280 --> 01:15:09,850
and it depends on the path.

1325
01:15:09,850 --> 01:15:12,790
Because you have dx d
lambda appearing everywhere.

1326
01:15:12,790 --> 01:15:16,230
And x of lambda is the path.

1327
01:15:16,230 --> 01:15:19,820
But if this equation
holds, then that path

1328
01:15:19,820 --> 01:15:21,250
is a stationary point.

1329
01:15:21,250 --> 01:15:23,520
It's an if and only
if statement as long

1330
01:15:23,520 --> 01:15:25,610
as paths are continuous.

1331
01:15:25,610 --> 01:15:27,025
That is the geodesic equation.

1332
01:15:27,025 --> 01:15:30,242
It tells us whether or not
our path is the minimum.

1333
01:15:30,242 --> 01:15:31,950
And next time, we will
simplify it a bit.

1334
01:15:31,950 --> 01:15:34,190
And we'll look at
examples and understand

1335
01:15:34,190 --> 01:15:36,290
how the formula works.

1336
01:15:36,290 --> 01:15:39,810
So I'll see you folks
again next Tuesday.