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PROFESSOR: We do
have a little bit

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00:00:25,000 --> 00:00:27,041
of finishing to do because
we didn't quite finish

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00:00:27,041 --> 00:00:29,480
the dynamics of homogeneous
expansion last time.

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00:00:29,480 --> 00:00:33,040
So we'll begin by finishing that
after a brief review of where

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00:00:33,040 --> 00:00:34,300
we were last time.

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00:00:34,300 --> 00:00:38,100
And then we'll move on to
discuss non-euclidean spaces,

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00:00:38,100 --> 00:00:41,930
which I hope will be still
the bulk of today's lecture.

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00:00:41,930 --> 00:00:42,430
OK.

16
00:00:42,430 --> 00:00:44,080
In that case, let's get going.

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00:00:44,080 --> 00:00:45,610
Again, as I said,
I want to begin

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00:00:45,610 --> 00:00:47,270
by just reviewing
some of the things we

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00:00:47,270 --> 00:00:48,462
talked about last time.

20
00:00:48,462 --> 00:00:50,420
And you should consider
this a good opportunity

21
00:00:50,420 --> 00:00:52,961
to ask questions if you discover
there are things that you're

22
00:00:52,961 --> 00:00:55,780
not really sure you understood
as well as you'd like.

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00:00:55,780 --> 00:00:58,590
We were talking about the
evolution of a closed universe.

24
00:00:58,590 --> 00:01:02,050
And to summarize that
calculation we first

25
00:01:02,050 --> 00:01:04,260
re-shuffled the
first order Friedmann

26
00:01:04,260 --> 00:01:08,370
equation, the equation for a
dot over a quantity squared,

27
00:01:08,370 --> 00:01:11,310
by bringing all the d t's to
one side and all the d a's

28
00:01:11,310 --> 00:01:15,210
on the other side after doing
a little bit of rescaling.

29
00:01:15,210 --> 00:01:17,580
And we got this equation.

30
00:01:17,580 --> 00:01:20,700
Which we then said
we can integrate.

31
00:01:20,700 --> 00:01:25,230
And the integral from
time will go from time 0

32
00:01:25,230 --> 00:01:29,440
to some arbitrary final time
that we called t tilde sub f,

33
00:01:29,440 --> 00:01:32,310
where the "tilde" indicates
that we multiplied by c.

34
00:01:32,310 --> 00:01:37,520
And the sub f means it's the
final time of our calculation.

35
00:01:37,520 --> 00:01:40,200
And on the other side
we have to integrate

36
00:01:40,200 --> 00:01:42,730
with corresponding
limits of integration.

37
00:01:42,730 --> 00:01:45,880
Corresponding to t equals
0 is a equals zero.

38
00:01:45,880 --> 00:01:48,150
So the lower limit
of integration is 0.

39
00:01:48,150 --> 00:01:50,660
And the final limit
of integration

40
00:01:50,660 --> 00:01:54,460
is just the value of a tilde
at the final time, whatever

41
00:01:54,460 --> 00:01:56,680
that is.

42
00:01:56,680 --> 00:01:59,670
Then we discovered
that we could actually

43
00:01:59,670 --> 00:02:02,950
do the integral on the right
if we made a substitution.

44
00:02:02,950 --> 00:02:06,290
And in the lecture last time
we did it in two stages.

45
00:02:06,290 --> 00:02:09,160
But the combined
substitution is just

46
00:02:09,160 --> 00:02:11,790
to replace a tilde
by cosine theta,

47
00:02:11,790 --> 00:02:14,980
according to this formula, if
we combine the two substitutions

48
00:02:14,980 --> 00:02:17,120
that we made last time.

49
00:02:17,120 --> 00:02:20,940
And if we do that we could
integrate the right hand side.

50
00:02:20,940 --> 00:02:22,990
And the integral
then gives us t tilde

51
00:02:22,990 --> 00:02:25,360
f is equal to the
integral of that,

52
00:02:25,360 --> 00:02:27,630
which is just this expression.

53
00:02:27,630 --> 00:02:33,830
And the expression below that
is just the substitution itself.

54
00:02:33,830 --> 00:02:37,530
How to relate a tilde to theta,
according to substitution

55
00:02:37,530 --> 00:02:38,790
which gave us that formula.

56
00:02:38,790 --> 00:02:40,930
So these two formulas
together allow

57
00:02:40,930 --> 00:02:44,490
us to determine t tilde
sub f, and a tilde

58
00:02:44,490 --> 00:02:48,020
sub f in terms of
theta and alpha.

59
00:02:48,020 --> 00:02:50,760
And once we had that we
realized we no longer needed

60
00:02:50,760 --> 00:02:53,840
to keep those sub f's
because that was really

61
00:02:53,840 --> 00:02:56,880
just a way of keeping
track of our notation

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00:02:56,880 --> 00:02:57,970
during the calculation.

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00:02:57,970 --> 00:02:59,800
It holds for any theta sub f.

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00:02:59,800 --> 00:03:01,700
So it holds for any theta.

65
00:03:01,700 --> 00:03:04,650
So then we just rewrote those
same formulas without the sub

66
00:03:04,650 --> 00:03:05,400
f's.

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00:03:05,400 --> 00:03:08,160
And here we wrote it
removing the tildes,

68
00:03:08,160 --> 00:03:09,970
replacing them by
their definitions.

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00:03:09,970 --> 00:03:11,700
And this was the final answer.

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00:03:11,700 --> 00:03:16,660
This describes the evolution
of a closed universe expressed

71
00:03:16,660 --> 00:03:18,540
in this parametric form.

72
00:03:18,540 --> 00:03:21,180
That is, we were not
able to explicitly write

73
00:03:21,180 --> 00:03:24,730
a as a function of t, which
is what would have liked

74
00:03:24,730 --> 00:03:28,460
to determine how the expansion
varied as a function of time.

75
00:03:28,460 --> 00:03:31,300
But instead we introduced
the auxiliary variable theta,

76
00:03:31,300 --> 00:03:33,340
often called the
development angle.

77
00:03:33,340 --> 00:03:36,340
And in terms of theta we
can express both t and a,

78
00:03:36,340 --> 00:03:40,450
and thereby indirectly have
an unambiguous relationship

79
00:03:40,450 --> 00:03:43,470
between a and t.

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00:03:43,470 --> 00:03:44,760
Any questions about that?

81
00:03:47,650 --> 00:03:48,150
OK.

82
00:03:48,150 --> 00:03:51,230
Then we noticed that
those were, in fact,

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00:03:51,230 --> 00:03:53,350
the equations of a cycloid.

84
00:03:53,350 --> 00:03:55,990
And I won't go through
the argument again,

85
00:03:55,990 --> 00:03:58,830
but the key point is
that the evolution

86
00:03:58,830 --> 00:04:04,330
of our closed universe scale
factor, as a function of time,

87
00:04:04,330 --> 00:04:05,990
is described by
what would happen

88
00:04:05,990 --> 00:04:11,760
if you had a disk rolling
on the horizontal axis

89
00:04:11,760 --> 00:04:13,940
with a point marked on the
disk which was initially

90
00:04:13,940 --> 00:04:17,149
straight down, and then as
the disk rolls that point

91
00:04:17,149 --> 00:04:19,220
traces out a cycloid.

92
00:04:19,220 --> 00:04:22,440
And that is exactly the
evolution of a closed universe.

93
00:04:22,440 --> 00:04:26,140
It starts at size zero,
goes to a maximum size,

94
00:04:26,140 --> 00:04:28,920
and then contracts again in
a completely symmetric way.

95
00:04:28,920 --> 00:04:31,550
The contraction phase
is the mirror image

96
00:04:31,550 --> 00:04:32,900
of the expansion phase.

97
00:04:39,470 --> 00:04:43,060
And then we went on to calculate
the age of a closed universe.

98
00:04:43,060 --> 00:04:45,880
And that was really more
of an exercise in algebra

99
00:04:45,880 --> 00:04:47,570
than anything else.

100
00:04:47,570 --> 00:04:51,480
The age is really given by
this formula to start with.

101
00:04:51,480 --> 00:04:55,040
This expresses the age, t,
in terms of alpha and theta.

102
00:04:55,040 --> 00:04:57,570
And the only problem with
that is that nobody really

103
00:04:57,570 --> 00:04:59,464
knows what alpha and theta mean.

104
00:04:59,464 --> 00:05:02,130
It's much more useful to have an
expression for the age in terms

105
00:05:02,130 --> 00:05:05,420
of things that astronomers
directly measure.

106
00:05:05,420 --> 00:05:10,884
And one needs two things
to replace alpha and theta.

107
00:05:10,884 --> 00:05:13,050
And the kinds of things
astronomers directly measure

108
00:05:13,050 --> 00:05:14,640
are things like the
Hubble parameter

109
00:05:14,640 --> 00:05:19,010
and the mass density, where they
often express the mass density

110
00:05:19,010 --> 00:05:22,190
in terms of omega where omega
is the mass density divided

111
00:05:22,190 --> 00:05:23,440
by the critical density.

112
00:05:23,440 --> 00:05:25,850
And that's what I chose
to do in the formulas

113
00:05:25,850 --> 00:05:28,070
that we went on to derive.

114
00:05:28,070 --> 00:05:30,664
So our goal is simply to
take that formula for t

115
00:05:30,664 --> 00:05:33,080
and figure out how to express
the alpha and the theta that

116
00:05:33,080 --> 00:05:36,310
appear in it so that we
can get an expression

117
00:05:36,310 --> 00:05:39,090
in terms of h and omega.

118
00:05:39,090 --> 00:05:42,040
And to do that, we'll
go one step at a time.

119
00:05:42,040 --> 00:05:45,590
We started by just writing
down the Friedmann equation.

120
00:05:45,590 --> 00:05:48,560
And the Friedmann
equation has an a in it.

121
00:05:48,560 --> 00:05:50,299
And everything
else in it is rho,

122
00:05:50,299 --> 00:05:52,840
which can be expressed in terms
of omega without any trouble,

123
00:05:52,840 --> 00:05:54,670
and h itself.

124
00:05:54,670 --> 00:05:56,970
So everything else in
it consists of variables

125
00:05:56,970 --> 00:06:00,700
that we're accepting to be
part of our final answer.

126
00:06:00,700 --> 00:06:04,080
So we could solve the Friedmann
equations for a or a tilde

127
00:06:04,080 --> 00:06:07,440
and that will allow
us to eliminate

128
00:06:07,440 --> 00:06:11,810
any a tilde's from
our expressions.

129
00:06:11,810 --> 00:06:14,270
Next, alpha was
originally defined

130
00:06:14,270 --> 00:06:17,520
in terms of this formula which
only involves rho and a tilde.

131
00:06:17,520 --> 00:06:19,260
We now know what
to do with a tilde.

132
00:06:19,260 --> 00:06:21,314
We could substitute
that formula.

133
00:06:21,314 --> 00:06:22,980
And rho, we always
knew what to do with.

134
00:06:22,980 --> 00:06:26,030
We could express that
in terms of omega.

135
00:06:26,030 --> 00:06:28,730
So we can instantly
solve that equation

136
00:06:28,730 --> 00:06:31,090
and get an equation for alpha
in terms of the quantities

137
00:06:31,090 --> 00:06:33,840
that we want to appear
in our final answer.

138
00:06:33,840 --> 00:06:36,310
And there's still one
more thing we need.

139
00:06:36,310 --> 00:06:38,410
We need an expression for theta.

140
00:06:38,410 --> 00:06:41,230
And theta we can get by
looking at the other of those

141
00:06:41,230 --> 00:06:43,720
two parametric
equations, the one that's

142
00:06:43,720 --> 00:06:47,820
not the "t equals" equation, but
rather the "a equals" equation.

143
00:06:47,820 --> 00:06:51,730
And in this equation we know
everything except theta itself.

144
00:06:51,730 --> 00:06:55,766
So we could substitute for a
over root k from up here- that.

145
00:06:55,766 --> 00:06:57,140
And on the right
hand side we can

146
00:06:57,140 --> 00:07:00,060
place alpha by that
expression and then the 1

147
00:07:00,060 --> 00:07:01,730
minus cosine theta stays.

148
00:07:01,730 --> 00:07:04,730
And now we could solve
this equation for theta.

149
00:07:04,730 --> 00:07:07,600
I might just mention that,
in lecture last time I

150
00:07:07,600 --> 00:07:09,640
actually mis-copied
this equation.

151
00:07:09,640 --> 00:07:11,864
I forgot the factor of
omega in the numerator.

152
00:07:11,864 --> 00:07:13,280
So if any of you
were taking notes

153
00:07:13,280 --> 00:07:15,220
you can go back and
correct your notes.

154
00:07:15,220 --> 00:07:19,420
But it's correct in
the printed notes.

155
00:07:19,420 --> 00:07:21,739
And now it's
correct on a screen.

156
00:07:21,739 --> 00:07:23,780
So this you could either
solve for cosine theta--

157
00:07:23,780 --> 00:07:26,066
initially what I did was
to solve for cosine theta.

158
00:07:26,066 --> 00:07:27,440
But then it turned
out to be more

159
00:07:27,440 --> 00:07:29,110
useful to know
what sine theta is,

160
00:07:29,110 --> 00:07:31,870
because sine theta appears
in that parametric expression

161
00:07:31,870 --> 00:07:33,540
for the age.

162
00:07:33,540 --> 00:07:35,880
So if you know cosine
theta you can, of course,

163
00:07:35,880 --> 00:07:38,070
get sine theta because
sine theta is just

164
00:07:38,070 --> 00:07:41,330
the square root of 1 minus
cosine squared theta.

165
00:07:41,330 --> 00:07:42,482
And that's what we did.

166
00:07:42,482 --> 00:07:44,190
And that's how we got
a square root here.

167
00:07:44,190 --> 00:07:46,023
And since square roots
can have either sign,

168
00:07:46,023 --> 00:07:47,710
there's a plus or
minus there which

169
00:07:47,710 --> 00:07:52,660
depends on where you are in
the evolution of our universe.

170
00:07:52,660 --> 00:07:54,770
The right hand side
here is always positive

171
00:07:54,770 --> 00:07:57,100
because we define that
square root symbol

172
00:07:57,100 --> 00:07:59,600
to mean the positive
square root,

173
00:07:59,600 --> 00:08:03,790
where the plus or minus, in
the end, tells you the sine.

174
00:08:03,790 --> 00:08:07,470
And sine theta itself--
we know what theta does.

175
00:08:07,470 --> 00:08:09,120
It goes from 0 to 2 pi.

176
00:08:09,120 --> 00:08:11,070
So sine theta
starts out positive

177
00:08:11,070 --> 00:08:14,640
and after theta crosses
pi, sine theta is

178
00:08:14,640 --> 00:08:18,100
negative for the second
half of the evolution.

179
00:08:18,100 --> 00:08:20,780
Meaning, sine theta is negative
for the contracting phase

180
00:08:20,780 --> 00:08:23,005
and positive for
the expanding phase.

181
00:08:26,480 --> 00:08:29,780
We then put all that together
into the formula for CT,

182
00:08:29,780 --> 00:08:32,150
which is alpha theta
minus sine theta.

183
00:08:32,150 --> 00:08:34,950
The alpha becomes this factor.

184
00:08:34,950 --> 00:08:37,970
And the theta minus sine
theta becomes the arc sine

185
00:08:37,970 --> 00:08:41,570
of this expression, minus
or plus that expression,

186
00:08:41,570 --> 00:08:44,770
where this is just sine
theta from that formula.

187
00:08:44,770 --> 00:08:48,620
And it's minus sine theta, which
is why the plus minus becomes

188
00:08:48,620 --> 00:08:50,480
minus or plus.

189
00:08:50,480 --> 00:08:52,830
The signs get a little tricky,
but it's, in principle,

190
00:08:52,830 --> 00:08:53,910
straightforward.

191
00:08:53,910 --> 00:08:55,670
It all just follows
from this formula.

192
00:08:55,670 --> 00:08:57,045
And if you know
the sine of theta

193
00:08:57,045 --> 00:09:00,550
you have all your
problems solved.

194
00:09:00,550 --> 00:09:02,580
So we put all that
together into a table.

195
00:09:02,580 --> 00:09:04,260
This is just a copy
of the same formula

196
00:09:04,260 --> 00:09:07,600
that we had on the
previous slide.

197
00:09:07,600 --> 00:09:09,840
Theta is what appears in
this right hand column.

198
00:09:09,840 --> 00:09:12,705
It's indicated as the sine
inverse of some expression.

199
00:09:12,705 --> 00:09:16,265
It refers to that
expression-- an abbreviation.

200
00:09:16,265 --> 00:09:17,390
So we know what theta does.

201
00:09:17,390 --> 00:09:21,690
Theta just goes from 0
to 2 pi in quadrants.

202
00:09:21,690 --> 00:09:24,470
At least, it pays to divide
it up into quadrants.

203
00:09:24,470 --> 00:09:29,280
Sine theta is always
positive for the first half

204
00:09:29,280 --> 00:09:30,860
and negative for
the second half.

205
00:09:30,860 --> 00:09:34,530
And that means that we have the
plus sign, or the upper sign,

206
00:09:34,530 --> 00:09:36,900
for the first half of
the motion and the minus

207
00:09:36,900 --> 00:09:39,830
sign for the second
half of the motion.

208
00:09:39,830 --> 00:09:42,480
Omega we could just
calculate in terms of theta.

209
00:09:42,480 --> 00:09:44,846
No problem about filling
in the omega column.

210
00:09:44,846 --> 00:09:46,970
And we know that we're
expanding for the first half

211
00:09:46,970 --> 00:09:48,590
and contracting for
the second half,

212
00:09:48,590 --> 00:09:50,447
so that really
completes the table.

213
00:09:50,447 --> 00:09:52,280
And the important thing
when you're actually

214
00:09:52,280 --> 00:09:55,830
using this formula is
to decide what theta is.

215
00:09:55,830 --> 00:09:57,910
And once you have that,
the sine of that--

216
00:09:57,910 --> 00:09:59,510
the inverse sine
of that-- well, no.

217
00:09:59,510 --> 00:10:00,010
Excuse me.

218
00:10:00,010 --> 00:10:02,080
Theta itself appears there,
and the sine of theta

219
00:10:02,080 --> 00:10:03,130
appears there.

220
00:10:03,130 --> 00:10:06,550
And theta itself you
have to figure out

221
00:10:06,550 --> 00:10:08,910
which is the relevant
value in terms

222
00:10:08,910 --> 00:10:11,200
of where you are
in the evolution.

223
00:10:11,200 --> 00:10:14,830
The point is that sine
theta itself goes--

224
00:10:14,830 --> 00:10:18,930
does not uniquely
determine what theta is.

225
00:10:18,930 --> 00:10:20,600
OK.

226
00:10:20,600 --> 00:10:22,240
That, I think,
completes our discussion

227
00:10:22,240 --> 00:10:25,900
of the evolution of
closed universes.

228
00:10:25,900 --> 00:10:28,650
I think it completes everything
that we did last time.

229
00:10:28,650 --> 00:10:29,775
So are there any questions?

230
00:10:33,830 --> 00:10:34,790
OK.

231
00:10:34,790 --> 00:10:36,300
Good.

232
00:10:36,300 --> 00:10:38,820
To finish our discussion
of the evolution

233
00:10:38,820 --> 00:10:41,360
of matter-dominated
universes, we

234
00:10:41,360 --> 00:10:44,310
go on to discuss open universes.

235
00:10:44,310 --> 00:10:47,450
And open universes are
really the same algebra

236
00:10:47,450 --> 00:10:48,620
as closed universes.

237
00:10:48,620 --> 00:10:51,760
They just differ
by the sine of k.

238
00:10:51,760 --> 00:10:55,210
Because one doesn't like to
deal with negative numbers,

239
00:10:55,210 --> 00:10:58,170
I defined kappa to
be equal to minus k,

240
00:10:58,170 --> 00:11:00,220
so that for our
open universes kappa

241
00:11:00,220 --> 00:11:03,730
was positive while
k was negative.

242
00:11:03,730 --> 00:11:07,200
Then I used a different
substitution for a tilde.

243
00:11:07,200 --> 00:11:09,500
Instead of a tilde
being a divided

244
00:11:09,500 --> 00:11:11,800
by the square root of
k, which in this case

245
00:11:11,800 --> 00:11:14,410
would be the square root of a
negative number-- which, one

246
00:11:14,410 --> 00:11:16,201
doesn't like to deal
with imaginary numbers

247
00:11:16,201 --> 00:11:18,410
if one doesn't have any need to.

248
00:11:18,410 --> 00:11:21,000
So instead, for
the open universe

249
00:11:21,000 --> 00:11:23,870
I'm defining a tilde to
be a divided by the square

250
00:11:23,870 --> 00:11:28,310
root of kappa, so that
a tilde is again real.

251
00:11:28,310 --> 00:11:30,341
Then I'm going to skip
all the algebra here.

252
00:11:30,341 --> 00:11:32,840
There's a little bit more of
it shown in the printed lecture

253
00:11:32,840 --> 00:11:34,880
notes.

254
00:11:34,880 --> 00:11:38,020
But there are no
new concepts here.

255
00:11:38,020 --> 00:11:40,280
Everything is really
the same, conceptually,

256
00:11:40,280 --> 00:11:42,880
as it was for the
closed universe.

257
00:11:42,880 --> 00:11:44,460
One difference is
that this time,

258
00:11:44,460 --> 00:11:46,350
instead of getting
trigonometric functions,

259
00:11:46,350 --> 00:11:49,900
we find that we get
hypergeometric functions.

260
00:11:49,900 --> 00:11:51,360
Hyper-- yeah.

261
00:11:51,360 --> 00:11:54,770
Hyper trigonometric functions,
I think, is the right word.

262
00:11:54,770 --> 00:11:59,110
That is, sinhs and coshs
instead of sines and cosines.

263
00:11:59,110 --> 00:12:00,360
The formulas are very similar.

264
00:12:00,360 --> 00:12:04,060
These are the formulas we get
for the open universe case,

265
00:12:04,060 --> 00:12:06,780
compared to those formulas
for the closed universe case.

266
00:12:09,330 --> 00:12:12,200
We get a change in
the order of-- instead

267
00:12:12,200 --> 00:12:15,106
of theta minus sine theta, we
get sinh theta minus theta.

268
00:12:15,106 --> 00:12:16,730
But that's what you
have to get if it's

269
00:12:16,730 --> 00:12:18,460
going to turn out
to be positive.

270
00:12:18,460 --> 00:12:20,140
Sine theta is always
less than theta.

271
00:12:20,140 --> 00:12:22,080
So this is a positive quantity.

272
00:12:22,080 --> 00:12:23,900
Sinh theta is always
greater than theta,

273
00:12:23,900 --> 00:12:26,160
so this is a positive quantity.

274
00:12:26,160 --> 00:12:27,840
And same for the second lines.

275
00:12:27,840 --> 00:12:30,090
Cosh theta minus 1
is always positive,

276
00:12:30,090 --> 00:12:32,870
and one minus cosine
theta is always positive.

277
00:12:32,870 --> 00:12:35,120
So you really know which
order to write them in just

278
00:12:35,120 --> 00:12:36,720
by knowing that you
want to write down

279
00:12:36,720 --> 00:12:39,910
something that's positive.

280
00:12:39,910 --> 00:12:43,810
So these formulas describe the
evolution of the open universe

281
00:12:43,810 --> 00:12:47,810
exactly the same way as those
formulas describe the evolution

282
00:12:47,810 --> 00:12:51,390
of a closed,
matter-dominated universe.

283
00:12:51,390 --> 00:12:52,620
So any questions?

284
00:12:55,610 --> 00:12:56,110
OK.

285
00:12:56,110 --> 00:12:58,780
Next step, just repeating what
we did for the closed universe,

286
00:12:58,780 --> 00:13:02,271
we can derive a formula for
the age of an open universe.

287
00:13:02,271 --> 00:13:03,770
And again, it's
really just a matter

288
00:13:03,770 --> 00:13:06,380
of substituting into
the formula we already

289
00:13:06,380 --> 00:13:10,500
have to be able to re-express it
in terms of useful quantities.

290
00:13:10,500 --> 00:13:12,940
Which, again, I choose to
be the Hubble expansion

291
00:13:12,940 --> 00:13:15,252
rate and omega.

292
00:13:15,252 --> 00:13:19,380
And here I've put together all
three formulas for the age.

293
00:13:19,380 --> 00:13:20,870
The flat universe,
the first one we

294
00:13:20,870 --> 00:13:24,110
did, where t is just h
inverse if we bring it

295
00:13:24,110 --> 00:13:25,570
to the other side, times 2/3.

296
00:13:28,200 --> 00:13:33,300
And the open universe on the
top, and the closed universe

297
00:13:33,300 --> 00:13:36,260
on the bottom.

298
00:13:36,260 --> 00:13:39,000
Now one of the, perhaps
surprising, things

299
00:13:39,000 --> 00:13:43,910
that one finds here is that
all three of these expressions

300
00:13:43,910 --> 00:13:46,752
look fairly different
from each other.

301
00:13:46,752 --> 00:13:48,210
And you might think
that that would

302
00:13:48,210 --> 00:13:51,730
give some kind of a jagged,
discontinuous curve.

303
00:13:51,730 --> 00:13:55,310
But you can go ahead and
plot it, which I did.

304
00:13:55,310 --> 00:13:56,690
And there's the plot.

305
00:13:56,690 --> 00:13:59,850
It's one nice, smooth curve.

306
00:13:59,850 --> 00:14:04,050
And we won't go into this
in detail, but many of you

307
00:14:04,050 --> 00:14:07,570
have had courses in complex
functions, functions

308
00:14:07,570 --> 00:14:09,140
of a complex variable.

309
00:14:09,140 --> 00:14:11,810
If you know about functions
of a complex variable

310
00:14:11,810 --> 00:14:16,870
you can tell that these are,
in fact, all the same function.

311
00:14:16,870 --> 00:14:20,320
That is, if omega is,
say, bigger than 1,

312
00:14:20,320 --> 00:14:22,512
this formula can be
evaluated straightforwardly.

313
00:14:22,512 --> 00:14:24,220
It involves only things
like square roots

314
00:14:24,220 --> 00:14:26,280
of positive numbers.

315
00:14:26,280 --> 00:14:28,410
But you could also try
evaluating this formula

316
00:14:28,410 --> 00:14:30,420
for omega bigger than 1.

317
00:14:30,420 --> 00:14:33,020
And then you have square roots
of negative numbers appearing.

318
00:14:33,020 --> 00:14:34,520
But square roots
of negative numbers

319
00:14:34,520 --> 00:14:36,910
are OK if you know
about complex numbers.

320
00:14:36,910 --> 00:14:38,727
They're just purely imaginary.

321
00:14:38,727 --> 00:14:40,810
And then you get trigonometric
functions, and even

322
00:14:40,810 --> 00:14:43,710
inverse trigonometric functions,
or inverse hyperbolic trig

323
00:14:43,710 --> 00:14:46,580
functions of
imaginary arguments.

324
00:14:46,580 --> 00:14:48,090
But all those are well-defined.

325
00:14:48,090 --> 00:14:50,380
And if you work through
what the definitions are,

326
00:14:50,380 --> 00:14:52,750
the top line really is
identical to the bottom line.

327
00:14:52,750 --> 00:14:55,300
Those really are
the same function.

328
00:14:55,300 --> 00:14:58,720
And that's why one expects
that, when you plot them

329
00:14:58,720 --> 00:15:05,380
they will join together
smoothly, as they clearly do.

330
00:15:05,380 --> 00:15:08,090
The point in the middle
here is the first age

331
00:15:08,090 --> 00:15:10,190
that we derived for
the flat universe

332
00:15:10,190 --> 00:15:14,200
where omega is equal to 1
and h t is just equal to 2/3.

333
00:15:14,200 --> 00:15:16,290
That is, t is equal
to 2/3 h inverse

334
00:15:16,290 --> 00:15:19,450
for a flat universe,
which is the middle dot.

335
00:15:19,450 --> 00:15:22,770
And the closed universes
are on the right

336
00:15:22,770 --> 00:15:24,470
and open universes
are on the left.

337
00:15:26,821 --> 00:15:27,320
OK.

338
00:15:27,320 --> 00:15:31,310
Any questions about
these age calculations?

339
00:15:31,310 --> 00:15:32,650
Yes?

340
00:15:32,650 --> 00:15:38,530
AUDIENCE: I noticed that,
for the open solution,

341
00:15:38,530 --> 00:15:42,450
there's a case where you
get some imaginary numbers?

342
00:15:46,940 --> 00:15:50,610
PROFESSOR: If you
use these formulas

343
00:15:50,610 --> 00:15:52,950
you don't get any
imaginary numbers.

344
00:15:52,950 --> 00:15:55,000
But if you tried to use,
say, the bottom formula

345
00:15:55,000 --> 00:15:56,970
for a value of
omega less than 1 it

346
00:15:56,970 --> 00:15:58,930
would give you
imaginary numbers.

347
00:15:58,930 --> 00:16:00,570
And it would, in
fact, if you trace

348
00:16:00,570 --> 00:16:03,310
through what those
imaginary numbers do,

349
00:16:03,310 --> 00:16:06,200
it would give you the
formula on the top.

350
00:16:06,200 --> 00:16:08,407
So it's all consistent.

351
00:16:08,407 --> 00:16:10,240
It's most straightforward
to use the formula

352
00:16:10,240 --> 00:16:12,410
on the top for omega less
than 1 and the formula

353
00:16:12,410 --> 00:16:14,250
on the bottom for
omega greater than 1.

354
00:16:14,250 --> 00:16:16,310
And then one never
confronts imaginary numbers.

355
00:16:16,310 --> 00:16:17,300
AUDIENCE: OK.

356
00:16:23,219 --> 00:16:24,510
PROFESSOR: Any other questions?

357
00:16:27,970 --> 00:16:29,900
OK.

358
00:16:29,900 --> 00:16:31,270
Where are we going next?

359
00:16:31,270 --> 00:16:33,580
Finally, just actually
one last graph

360
00:16:33,580 --> 00:16:36,490
on the evolution of
matter-dominated universes,

361
00:16:36,490 --> 00:16:39,280
which is the final
form of a of t

362
00:16:39,280 --> 00:16:41,790
for a matter-dominated universe.

363
00:16:41,790 --> 00:16:45,320
If you re-scale things
by dividing a over root

364
00:16:45,320 --> 00:16:49,860
k by alpha, and c t over
alpha-- if you look back

365
00:16:49,860 --> 00:16:51,524
at our equations--
let me go back

366
00:16:51,524 --> 00:16:52,690
to where the equations were.

367
00:16:52,690 --> 00:16:54,810
We're really just
graphing these equations.

368
00:16:54,810 --> 00:16:56,900
If I divide this
equation by alpha,

369
00:16:56,900 --> 00:16:58,670
I just get a pure
function of theta

370
00:16:58,670 --> 00:17:00,320
with dimensionless variables.

371
00:17:00,320 --> 00:17:03,820
And similarly here, if I
divide a over root k by alpha

372
00:17:03,820 --> 00:17:07,666
I just get 1 minus cosine
theta, which is a pure number.

373
00:17:07,666 --> 00:17:09,040
So that's what
I've chosen to do.

374
00:17:09,040 --> 00:17:13,560
And the dimensionality works the
same for the open case as well.

375
00:17:13,560 --> 00:17:15,720
So it allows you to
draw a plot which

376
00:17:15,720 --> 00:17:17,300
is just independent
of parameters.

377
00:17:17,300 --> 00:17:19,410
All the parameters are
absorbed into the way

378
00:17:19,410 --> 00:17:21,550
the axes are defined,
which are both

379
00:17:21,550 --> 00:17:24,109
defined as
dimensionless numbers.

380
00:17:24,109 --> 00:17:26,980
And in that case
the closed universe

381
00:17:26,980 --> 00:17:29,310
survives for a duration of 2 pi.

382
00:17:29,310 --> 00:17:32,710
The axis here, at least, has
the same duration as theta.

383
00:17:32,710 --> 00:17:36,530
It's not actually theta, because
t is not linear in theta.

384
00:17:36,530 --> 00:17:38,640
But this point
does change by 2 pi

385
00:17:38,640 --> 00:17:41,440
as theta goes from 0 to 2 pi.

386
00:17:41,440 --> 00:17:45,100
And one can see here, the
three possible curves.

387
00:17:45,100 --> 00:17:49,220
A closed universe which
starts and then falls back,

388
00:17:49,220 --> 00:17:52,250
a flat universe which
goes off to the right

389
00:17:52,250 --> 00:17:55,390
and has actually a constant
slope as you go out here,

390
00:17:55,390 --> 00:17:59,584
and an open universe which
behaves slightly differently.

391
00:17:59,584 --> 00:18:01,000
Actually, I think
I was wrong when

392
00:18:01,000 --> 00:18:03,090
I said the flat universe
has a constant slope.

393
00:18:03,090 --> 00:18:07,590
But the open and the flat case
are similar to each other.

394
00:18:07,590 --> 00:18:09,880
They both go off to infinity,
but in different ways.

395
00:18:12,460 --> 00:18:16,250
And all three of them merge
as you go backwards in time.

396
00:18:16,250 --> 00:18:19,110
That's not something that
might have been obvious

397
00:18:19,110 --> 00:18:21,030
before we wrote
down the equations.

398
00:18:21,030 --> 00:18:23,470
But in very early
times all universes

399
00:18:23,470 --> 00:18:26,510
look like they're
flat universes,

400
00:18:26,510 --> 00:18:28,850
if you go to early enough times.

401
00:18:28,850 --> 00:18:30,590
And that actually is
an important point

402
00:18:30,590 --> 00:18:33,380
which we'll talk
about later in terms

403
00:18:33,380 --> 00:18:36,170
of what's called the flatness
problem of cosmology.

404
00:18:36,170 --> 00:18:41,550
That basically is the
consequences of that fact.

405
00:18:41,550 --> 00:18:42,130
Yes?

406
00:18:42,130 --> 00:18:43,558
AUDIENCE: Why
didn't you just say

407
00:18:43,558 --> 00:18:45,938
all of them looks like
open universes [INAUDIBLE]?

408
00:18:45,938 --> 00:18:46,890
PROFESSOR: [LAUGHS]

409
00:18:46,890 --> 00:18:49,280
AUDIENCE: I mean, what's
special about flat universes?

410
00:18:49,280 --> 00:18:50,820
PROFESSOR: Well,
actually, there is

411
00:18:50,820 --> 00:18:52,028
something special about flat.

412
00:18:52,028 --> 00:18:55,340
Which is that, if we plotted
omega as the function of time

413
00:18:55,340 --> 00:18:58,430
all of them approach omega
equals 1 as time goes to 0.

414
00:18:58,430 --> 00:19:00,320
So there is a very
definite meaning

415
00:19:00,320 --> 00:19:02,290
of saying that they're
all approaching flat,

416
00:19:02,290 --> 00:19:06,660
rather than they're all
approaching open or closed.

417
00:19:06,660 --> 00:19:08,394
Good question, though.

418
00:19:08,394 --> 00:19:09,810
Because from this
graph you really

419
00:19:09,810 --> 00:19:13,260
can't tell anything
special about the flat.

420
00:19:13,260 --> 00:19:15,880
Any other questions?

421
00:19:15,880 --> 00:19:16,618
Yes?

422
00:19:16,618 --> 00:19:21,100
AUDIENCE: So does this mean
that the open and flat solutions

423
00:19:21,100 --> 00:19:24,274
will extend indefinitely?

424
00:19:24,274 --> 00:19:24,940
PROFESSOR: Yeah.

425
00:19:24,940 --> 00:19:27,290
The open and flat solutions
extend indefinitely in time.

426
00:19:27,290 --> 00:19:29,094
That's right, they do.

427
00:19:29,094 --> 00:19:31,260
And one can see that from
the formulas or the graph.

428
00:19:34,040 --> 00:19:34,961
Yes?

429
00:19:34,961 --> 00:19:37,847
AUDIENCE: So for plotting
omega as a function of time,

430
00:19:37,847 --> 00:19:39,530
I'm a little bit
confused as to how

431
00:19:39,530 --> 00:19:44,750
it changes as-- for example,
an open universe expands,

432
00:19:44,750 --> 00:19:47,867
or a closed universe--
because it seemed

433
00:19:47,867 --> 00:19:50,600
like, from our calculations,
that when the universe was

434
00:19:50,600 --> 00:19:53,600
expanding, omega was increasing?

435
00:19:53,600 --> 00:19:57,390
Is that-- at least
for a closed universe?

436
00:19:57,390 --> 00:19:58,890
PROFESSOR: That is
true for a closed

437
00:19:58,890 --> 00:20:00,570
universe during the
expanding phase.

438
00:20:00,570 --> 00:20:02,611
For a closed universe
during the expanding phase,

439
00:20:02,611 --> 00:20:03,950
omega does increase.

440
00:20:03,950 --> 00:20:05,880
It starts out as 1
and then it rises

441
00:20:05,880 --> 00:20:08,910
to infinity when the universe
reaches its maximum size

442
00:20:08,910 --> 00:20:10,675
and is about to turn
around and go back.

443
00:20:10,675 --> 00:20:12,800
Because it doesn't mean
the mass density increases.

444
00:20:12,800 --> 00:20:14,290
That's maybe what's
confusing you.

445
00:20:14,290 --> 00:20:17,690
The actual mass density always
decreases as it expands,

446
00:20:17,690 --> 00:20:21,540
but the critical density
decreases even faster.

447
00:20:21,540 --> 00:20:26,400
So the ratio, omega, actually
rises for a closed universe

448
00:20:26,400 --> 00:20:27,015
as it expands.

449
00:20:30,420 --> 00:20:32,850
For an open universe
it's the reverse.

450
00:20:32,850 --> 00:20:35,100
For an open universe
omega starts

451
00:20:35,100 --> 00:20:37,660
out being 1 at
early times, as it

452
00:20:37,660 --> 00:20:41,040
does for any
matter-dominated universe.

453
00:20:41,040 --> 00:20:42,900
And as the universe
expands, omega

454
00:20:42,900 --> 00:20:47,710
gets smaller and smaller
for an open universe.

455
00:20:47,710 --> 00:20:51,410
And it follows-- I don't
have them on slides here,

456
00:20:51,410 --> 00:20:52,950
but we do have in
the notes formulas

457
00:20:52,950 --> 00:20:56,280
that we derived, that we did
on the blackboard, that give us

458
00:20:56,280 --> 00:20:58,920
omega as a function of theta.

459
00:20:58,920 --> 00:21:01,040
And those formulas, you
can just look at them

460
00:21:01,040 --> 00:21:03,180
and see how omega behaves
as the universe evolves.

461
00:21:03,180 --> 00:21:06,400
Because as the universe
evolves, theta just increases.

462
00:21:06,400 --> 00:21:09,210
And those formulas
do trivially show

463
00:21:09,210 --> 00:21:11,080
what I said about
how omega evolves.

464
00:21:14,970 --> 00:21:17,270
Any other questions?

465
00:21:17,270 --> 00:21:17,954
Yes?

466
00:21:17,954 --> 00:21:21,356
AUDIENCE: Why a over alpha root
k, but for a flat universe k

467
00:21:21,356 --> 00:21:22,390
is 0?

468
00:21:22,390 --> 00:21:23,390
PROFESSOR: That's right.

469
00:21:23,390 --> 00:21:25,570
I didn't really say that,
but for the flat universe

470
00:21:25,570 --> 00:21:26,986
there's an arbitrary
normalization

471
00:21:26,986 --> 00:21:30,150
that one chooses in
drawing this graph.

472
00:21:30,150 --> 00:21:32,180
And it was really an
arbitrary choice for me

473
00:21:32,180 --> 00:21:35,660
to draw the flat case
to join on smoothly

474
00:21:35,660 --> 00:21:37,300
with the open and closed cases.

475
00:21:37,300 --> 00:21:40,040
I could have put any constant
in front of t to the 2/3.

476
00:21:40,040 --> 00:21:42,270
And you're right, then
they would not necessarily

477
00:21:42,270 --> 00:21:45,480
mesh unless I chose that
constant in the right way.

478
00:21:51,882 --> 00:21:52,382
Yes?

479
00:21:52,382 --> 00:21:54,847
AUDIENCE: Based on that, is
there a particular reason

480
00:21:54,847 --> 00:21:58,291
you decided to chose [INAUDIBLE]
a flat universe looks

481
00:21:58,291 --> 00:21:58,791
like this?

482
00:21:58,791 --> 00:22:00,434
Is there a particular
thing you're

483
00:22:00,434 --> 00:22:02,270
trying to show in
choosing [INAUDIBLE]?

484
00:22:02,270 --> 00:22:02,570
PROFESSOR: OK.

485
00:22:02,570 --> 00:22:04,445
The question is, is
there a particular reason

486
00:22:04,445 --> 00:22:07,550
why I chose the
normalization that I chose?

487
00:22:07,550 --> 00:22:09,440
If I did not choose
it, it would only

488
00:22:09,440 --> 00:22:12,500
differ by an overall factor.

489
00:22:12,500 --> 00:22:14,420
So it would still
look-- the flat line

490
00:22:14,420 --> 00:22:16,610
by itself would look
indistinguishable, really.

491
00:22:16,610 --> 00:22:18,740
It would just be
higher or lower.

492
00:22:18,740 --> 00:22:21,180
So the only question is how
it meshes with the others.

493
00:22:21,180 --> 00:22:22,620
And since the flat
case really is

494
00:22:22,620 --> 00:22:24,590
the borderline between
the other two cases,

495
00:22:24,590 --> 00:22:27,380
and since this constant
that appears in front of t

496
00:22:27,380 --> 00:22:30,370
to the 2/3 has no
physical meaning whatever,

497
00:22:30,370 --> 00:22:32,300
it seems the
sensible thing to do

498
00:22:32,300 --> 00:22:36,470
is to plot it so that
it looks like the limit

499
00:22:36,470 --> 00:22:37,780
of an open or closed universe.

500
00:22:37,780 --> 00:22:40,300
Because physically it is the
same as the limit of an open

501
00:22:40,300 --> 00:22:42,175
or a closed universe
coming from either side.

502
00:22:49,930 --> 00:22:52,029
Any other questions?

503
00:22:52,029 --> 00:22:53,070
Those are good questions.

504
00:22:56,410 --> 00:22:56,910
OK.

505
00:22:56,910 --> 00:23:00,450
In that case, we are
finished with the evolution

506
00:23:00,450 --> 00:23:05,330
of matter-dominated
universes, and ready to start

507
00:23:05,330 --> 00:23:09,560
talking about
non-euclidean spaces.

508
00:23:09,560 --> 00:23:14,640
So what we'll be doing next
is kind of a mini introduction

509
00:23:14,640 --> 00:23:19,350
to general relativity, which
how non-euclidean spaces

510
00:23:19,350 --> 00:23:21,790
enter physics.

511
00:23:21,790 --> 00:23:24,530
Now, needless to say,
general relativity

512
00:23:24,530 --> 00:23:29,234
is an entire course
separate from this course.

513
00:23:29,234 --> 00:23:31,150
And of course that even
has more prerequisites

514
00:23:31,150 --> 00:23:33,510
that this course
has, so we're not

515
00:23:33,510 --> 00:23:36,070
going to duplicate
what would be taught

516
00:23:36,070 --> 00:23:38,760
in the general relativity class.

517
00:23:38,760 --> 00:23:40,440
But it turns out
that the discussion

518
00:23:40,440 --> 00:23:42,690
of general relativity does,
in fact, divide pretty

519
00:23:42,690 --> 00:23:46,769
cleanly into two major issues.

520
00:23:46,769 --> 00:23:49,060
And we will be dealing with
one of those issues but not

521
00:23:49,060 --> 00:23:50,360
the other.

522
00:23:50,360 --> 00:23:53,710
In particular, what we
will be doing in this class

523
00:23:53,710 --> 00:23:58,790
is learning how the formulas
of general relativity

524
00:23:58,790 --> 00:24:01,680
is used to describe
curved spaces.

525
00:24:01,680 --> 00:24:06,960
And we will learn how particles
move in curved spaces.

526
00:24:06,960 --> 00:24:10,630
So we'll be able to
analyze trajectories

527
00:24:10,630 --> 00:24:12,880
in any curbed space
if somebody tells you

528
00:24:12,880 --> 00:24:16,080
what the curved space
itself looks like.

529
00:24:16,080 --> 00:24:19,810
What we will not be
doing is we won't even

530
00:24:19,810 --> 00:24:26,010
attempt to describe how general
relativity predicts that matter

531
00:24:26,010 --> 00:24:28,740
should cause space to curve.

532
00:24:28,740 --> 00:24:31,200
That would be left
entirely for the GR course

533
00:24:31,200 --> 00:24:34,260
that you may or may not take.

534
00:24:34,260 --> 00:24:35,800
But it will not
be discussed here.

535
00:24:35,800 --> 00:24:37,299
There's only one
point where we will

536
00:24:37,299 --> 00:24:38,560
need a result of that sort.

537
00:24:38,560 --> 00:24:41,860
We'll want to know how the
matter in our Friedman Roberson

538
00:24:41,860 --> 00:24:48,120
Walker universe
affects the curvature.

539
00:24:48,120 --> 00:24:50,090
And there I would just
give you the result.

540
00:24:50,090 --> 00:24:51,590
I'll try to make
it sound plausible.

541
00:24:51,590 --> 00:24:53,770
But I won't make
any pretense being

542
00:24:53,770 --> 00:24:55,260
able to drive that result.

543
00:24:55,260 --> 00:24:59,260
That is, we will not be able
to drive how much space curves

544
00:24:59,260 --> 00:25:02,445
as a consequence of the
matter that's in it.

545
00:25:02,445 --> 00:25:03,820
But we will write
down the answer

546
00:25:03,820 --> 00:25:04,840
so you at least know
what the answer is

547
00:25:04,840 --> 00:25:06,555
for a homogeneous
isotropic universe.

548
00:25:09,600 --> 00:25:14,100
So here's a picturesque
slide about curve space.

549
00:25:14,100 --> 00:25:17,050
Four years ago, I
think it was, a postdoc

550
00:25:17,050 --> 00:25:19,900
here named Mustafa Amin
gave this lecture for me

551
00:25:19,900 --> 00:25:21,110
because I was out of town.

552
00:25:21,110 --> 00:25:24,249
And he had much more colorful
transparencies than I ever do.

553
00:25:24,249 --> 00:25:26,290
So I'll be using some of
his transparencies here.

554
00:25:26,290 --> 00:25:30,840
And this is one of
his opening slides.

555
00:25:30,840 --> 00:25:33,380
So this is what we want to
talk about-- curved space

556
00:25:33,380 --> 00:25:38,590
as illustrated in
that nifty picture.

557
00:25:38,590 --> 00:25:42,410
So I want to begin with a kind
of an historical introduction.

558
00:25:42,410 --> 00:25:45,740
To be honest, I'm pretty
much following the logic

559
00:25:45,740 --> 00:25:48,620
of the first chapter of Steve
Weinberg's General Relativity

560
00:25:48,620 --> 00:25:50,370
book.

561
00:25:50,370 --> 00:25:52,496
Non-Euclidean geometry
of course starts

562
00:25:52,496 --> 00:25:54,370
by thinking about
Euclidean geometry and then

563
00:25:54,370 --> 00:25:58,060
how one might be
move away from it.

564
00:25:58,060 --> 00:26:00,100
And historically, there's
kind of a clear cut

565
00:26:00,100 --> 00:26:03,130
path, which was followed.

566
00:26:03,130 --> 00:26:09,110
Euclid based his geometry, as
described in Euclid's elements,

567
00:26:09,110 --> 00:26:11,587
in terms of five postulates.

568
00:26:11,587 --> 00:26:13,670
The first of which is that
a straight line segment

569
00:26:13,670 --> 00:26:15,960
can be drawn joining
any two points.

570
00:26:15,960 --> 00:26:17,460
Sounds clear enough.

571
00:26:17,460 --> 00:26:19,460
Second, any straight
line segment

572
00:26:19,460 --> 00:26:22,610
can be extended indefinitely
in a straight line.

573
00:26:22,610 --> 00:26:26,670
Also sounds obvious, which is
what Euclid was banking on.

574
00:26:26,670 --> 00:26:28,900
Third, given a
straight line segment,

575
00:26:28,900 --> 00:26:32,980
a circle can be drawn having
the segment as a radius and one

576
00:26:32,980 --> 00:26:35,389
endpoint as the center.

577
00:26:35,389 --> 00:26:37,430
That also-- you can imagine
yourself doing that--

578
00:26:37,430 --> 00:26:39,920
seems straightforward enough.

579
00:26:39,920 --> 00:26:41,600
But then we come to
the fifth postulate,

580
00:26:41,600 --> 00:26:43,770
which still sounds
pretty obvious.

581
00:26:43,770 --> 00:26:48,430
But it's certainly much more
complicated than the others.

582
00:26:48,430 --> 00:26:52,010
The fifth postulate says that
if a straight line falling

583
00:26:52,010 --> 00:26:57,640
on two straight lines makes the
interior angle on the same side

584
00:26:57,640 --> 00:27:02,010
are less than two right angles,
the two straight lines--

585
00:27:02,010 --> 00:27:03,860
if produced
indefinitely-- meet on

586
00:27:03,860 --> 00:27:08,240
that-- I think
this is mis-typed.

587
00:27:08,240 --> 00:27:09,110
Mustafa typed it.

588
00:27:09,110 --> 00:27:11,316
I guess he's typed
this one, too.

589
00:27:11,316 --> 00:27:12,440
This one shows the picture.

590
00:27:18,240 --> 00:27:19,990
Yeah, this should be
on that side in which

591
00:27:19,990 --> 00:27:22,146
the angles are less
than two right angles.

592
00:27:22,146 --> 00:27:24,270
So let me just explain it,
independent of the text.

593
00:27:27,060 --> 00:27:29,600
The question is,
what happens if you

594
00:27:29,600 --> 00:27:32,530
have one line-- the line that's
shown more or less vertical

595
00:27:32,530 --> 00:27:35,800
here-- and two lines
that cross it such

596
00:27:35,800 --> 00:27:38,820
that the interior angles--
here shown as a and b--

597
00:27:38,820 --> 00:27:41,490
are on one side less than pi.

598
00:27:41,490 --> 00:27:44,757
Less than two right angles is
the way you could describe it.

599
00:27:44,757 --> 00:27:46,340
Then, as you can see
from the picture,

600
00:27:46,340 --> 00:27:47,940
these lines will
meet on this side

601
00:27:47,940 --> 00:27:50,500
and will not meet on that side.

602
00:27:50,500 --> 00:27:52,920
And that's what
the postulate says

603
00:27:52,920 --> 00:27:55,590
that under those circumstances
where two lines cross a given

604
00:27:55,590 --> 00:27:58,030
line such that the
sum of the two angles

605
00:27:58,030 --> 00:28:01,127
adds up to less
than pi on one side

606
00:28:01,127 --> 00:28:02,585
that the lines will
be on that side

607
00:28:02,585 --> 00:28:04,681
and will not be
on the other side.

608
00:28:04,681 --> 00:28:05,180
Yes?

609
00:28:05,180 --> 00:28:06,928
AUDIENCE: What
was his motivation

610
00:28:06,928 --> 00:28:08,428
for making this the
fifth postulate?

611
00:28:08,428 --> 00:28:10,284
It seems kind of arbitrary.

612
00:28:10,284 --> 00:28:13,000
PROFESSOR: OK, the question is
what was Euclid's motivation

613
00:28:13,000 --> 00:28:16,120
for making this the
fifth postulate?

614
00:28:16,120 --> 00:28:19,080
Well, I have to admit I
haven't had many conversations

615
00:28:19,080 --> 00:28:20,920
with Euclid so I'm
not sure I know

616
00:28:20,920 --> 00:28:22,430
the answer to that question.

617
00:28:22,430 --> 00:28:25,170
But it was what was needed to
basically complete geometry

618
00:28:25,170 --> 00:28:26,680
as we know it.

619
00:28:26,680 --> 00:28:29,140
So much of what you've
learned in geometry

620
00:28:29,140 --> 00:28:31,850
would not be there if there
was not something equivalent

621
00:28:31,850 --> 00:28:32,624
to this postulate.

622
00:28:32,624 --> 00:28:34,790
But actually what I'm going
to be talking about next

623
00:28:34,790 --> 00:28:36,960
is there has been
discovered there

624
00:28:36,960 --> 00:28:41,620
are a number of substitutes
for this fifth postulate.

625
00:28:41,620 --> 00:28:43,332
Mathematicians studied
for a long time

626
00:28:43,332 --> 00:28:44,790
whether or not this
postulate could

627
00:28:44,790 --> 00:28:47,840
be derived from the others
since it seems so much more

628
00:28:47,840 --> 00:28:49,630
complicated than the others.

629
00:28:49,630 --> 00:28:53,730
And there was a strong
desire during thousands

630
00:28:53,730 --> 00:28:56,620
of years really-- at
least over 1,000 years--

631
00:28:56,620 --> 00:29:00,830
among mathematicians to try
to prove the fifth postulate

632
00:29:00,830 --> 00:29:02,930
from the first four postulates.

633
00:29:02,930 --> 00:29:05,250
And nobody ever
succeeded in doing that.

634
00:29:05,250 --> 00:29:09,070
And we now are pretty clear that
it's not possible to do that.

635
00:29:09,070 --> 00:29:10,660
That the postulate
is independent

636
00:29:10,660 --> 00:29:13,710
of the other postulates.

637
00:29:13,710 --> 00:29:15,770
It was discovered along
the way that there

638
00:29:15,770 --> 00:29:18,750
are a number of equivalent
statements to fifth postulate.

639
00:29:18,750 --> 00:29:22,457
And you could equally well have
chosen any one of these four

640
00:29:22,457 --> 00:29:24,540
statements that are
illustrated in these pictures.

641
00:29:24,540 --> 00:29:27,264
And I'll give you
words one by one

642
00:29:27,264 --> 00:29:29,680
for what these alternative
versions of the fifth postulate

643
00:29:29,680 --> 00:29:31,490
would be.

644
00:29:31,490 --> 00:29:34,860
A, up here, illustrated there,
says that if a straight line

645
00:29:34,860 --> 00:29:38,660
intersects one of two parallels,
meaning two parallel lines--

646
00:29:38,660 --> 00:29:40,160
so this is the one
line intersecting

647
00:29:40,160 --> 00:29:41,334
two parallel lines.

648
00:29:41,334 --> 00:29:43,750
If it intersects one of them
as the heavy part of the line

649
00:29:43,750 --> 00:29:46,900
shows, then the theorem says
that if you continue that line

650
00:29:46,900 --> 00:29:49,740
it will always
intersect the other.

651
00:29:49,740 --> 00:29:52,880
And certainly obvious from
the picture, that's the way

652
00:29:52,880 --> 00:29:54,240
it works.

653
00:29:54,240 --> 00:29:56,850
But that's equivalent
to the fifth postulate

654
00:29:56,850 --> 00:30:01,280
and not provable from the
other four postulates.

655
00:30:01,280 --> 00:30:03,700
A second statement--
b is the one

656
00:30:03,700 --> 00:30:05,960
that's illustrated there--
is the one that I remember

657
00:30:05,960 --> 00:30:08,170
learning when I was in
high school, I think,

658
00:30:08,170 --> 00:30:12,330
which says that if you have one
line and another line parallel

659
00:30:12,330 --> 00:30:16,550
to it-- or rather, I'm sorry
another point off the line

660
00:30:16,550 --> 00:30:19,200
that there's one and only
one line through that point

661
00:30:19,200 --> 00:30:22,990
parallel to the original line.

662
00:30:22,990 --> 00:30:29,110
So that is yet another statement
of this famous fifth postulate.

663
00:30:29,110 --> 00:30:31,710
Number c is less obvious.

664
00:30:31,710 --> 00:30:33,590
But it turns out
that once you go away

665
00:30:33,590 --> 00:30:35,580
from Euclidean geometry,
your space always

666
00:30:35,580 --> 00:30:37,760
has a built in scale.

667
00:30:37,760 --> 00:30:41,210
So things are not scalable.

668
00:30:41,210 --> 00:30:43,785
One example I might mention at
this point of occurred space

669
00:30:43,785 --> 00:30:45,930
is, say, the
surface of a sphere.

670
00:30:45,930 --> 00:30:49,720
And the service of a
sphere has some fixed size.

671
00:30:49,720 --> 00:30:52,114
So if you have a
figure of one size,

672
00:30:52,114 --> 00:30:54,030
and you wanted, on the
surface of this sphere,

673
00:30:54,030 --> 00:30:57,060
to make a figure 5 times bigger,
it might not fit on the sphere

674
00:30:57,060 --> 00:30:57,560
anymore.

675
00:30:57,560 --> 00:31:00,722
So you can't always
make a figure bigger

676
00:31:00,722 --> 00:31:01,930
on the surface of the sphere.

677
00:31:01,930 --> 00:31:05,000
In fact, you could
never make a figure

678
00:31:05,000 --> 00:31:07,960
bigger without
bending in some way

679
00:31:07,960 --> 00:31:10,110
on the surface of the sphere.

680
00:31:10,110 --> 00:31:14,100
So that gives rise to
this third statement

681
00:31:14,100 --> 00:31:17,960
of the fifth postulate, which
is that, given any figure,

682
00:31:17,960 --> 00:31:21,440
there exists a figure
similar to it of any size.

683
00:31:21,440 --> 00:31:24,040
And by similar it means
that for polygons they're

684
00:31:24,040 --> 00:31:27,250
similar if the corresponding
angles are equal to each other

685
00:31:27,250 --> 00:31:30,130
as they're supposed to
be on those two images.

686
00:31:30,130 --> 00:31:33,310
And the corresponding sides
are proportional to each other.

687
00:31:33,310 --> 00:31:37,390
So a similar figure
is just a blow up--

688
00:31:37,390 --> 00:31:39,750
a rescaling-- of
the original figure.

689
00:31:39,750 --> 00:31:42,940
And you can only do it if
the fifth postulate holds.

690
00:31:42,940 --> 00:31:46,032
You can do it on a flat space
but not on a curved space.

691
00:31:46,032 --> 00:31:50,910
And I think that is a
less obvious version

692
00:31:50,910 --> 00:31:52,950
of the fifth postulate.

693
00:31:52,950 --> 00:31:56,540
And finally, the
fifth postulate is

694
00:31:56,540 --> 00:31:58,870
linked to the
behavior of triangles.

695
00:31:58,870 --> 00:32:00,760
We all learned in
Euclidean geometry

696
00:32:00,760 --> 00:32:05,500
that the sum of the three angles
of a triangle is 180 degrees.

697
00:32:05,500 --> 00:32:08,760
That is a crucial theorem
of Euclidean geometry that

698
00:32:08,760 --> 00:32:11,440
depends directly on the fifth
postulate and is, in fact,

699
00:32:11,440 --> 00:32:12,912
equivalent to the
fifth postulate.

700
00:32:12,912 --> 00:32:15,120
So you could assume it and
forget the fifth postulate

701
00:32:15,120 --> 00:32:17,720
and still prove everything.

702
00:32:17,720 --> 00:32:21,570
So the fact that--
actually, I'm sorry.

703
00:32:24,470 --> 00:32:26,136
It is equivalent to
the fifth postulate.

704
00:32:26,136 --> 00:32:27,719
But you don't need
to assume that it's

705
00:32:27,719 --> 00:32:30,630
true for every triangle to
prove the fifth postulate.

706
00:32:30,630 --> 00:32:33,880
It turns out that sufficient
and mathematicians always

707
00:32:33,880 --> 00:32:36,400
look for the minimum
axiom to be able to prove

708
00:32:36,400 --> 00:32:37,560
what they want to prove.

709
00:32:37,560 --> 00:32:39,930
The minimum version of
the axiom for triangles

710
00:32:39,930 --> 00:32:43,220
is to simply assume that
there's just one triangle who's

711
00:32:43,220 --> 00:32:45,690
angles add up 180 degrees.

712
00:32:45,690 --> 00:32:48,500
And if there exist just one
triangle whose angles add up

713
00:32:48,500 --> 00:32:50,210
to 180 degrees, then
the fifth postulate

714
00:32:50,210 --> 00:32:54,490
has to hold-- turns out--
which is not so obvious, again.

715
00:32:54,490 --> 00:32:56,410
So these are all
different, equivalent ways

716
00:32:56,410 --> 00:32:58,370
of staying in the
fifth postulate.

717
00:32:58,370 --> 00:33:02,730
But it's pretty clear
now that it's not

718
00:33:02,730 --> 00:33:08,500
possible to prove the fifth
postulate from the first four.

719
00:33:08,500 --> 00:33:09,640
Any questions about that?

720
00:33:14,200 --> 00:33:17,070
OK, so a little
bit of history now.

721
00:33:17,070 --> 00:33:19,670
The first person, apparently,
to seriously explore

722
00:33:19,670 --> 00:33:23,980
the possibility that the
fifth postulate might be wrong

723
00:33:23,980 --> 00:33:29,062
was a Jesuit priest named
Giovanni Geralamo Saccheri.

724
00:33:29,062 --> 00:33:30,520
I'm sure other
people can pronounce

725
00:33:30,520 --> 00:33:32,740
it much better than I can.

726
00:33:32,740 --> 00:33:36,230
And in 1733, which is
the same year he died,

727
00:33:36,230 --> 00:33:41,190
he published a study of
what geometry would be like

728
00:33:41,190 --> 00:33:44,660
if the fifth postulate
did not hold.

729
00:33:44,660 --> 00:33:47,469
And he titled it- this
is a lot title, which

730
00:33:47,469 --> 00:33:49,010
I don't know how to
pronounce really,

731
00:33:49,010 --> 00:33:50,790
but in English it's
apparently translated

732
00:33:50,790 --> 00:33:55,620
as Euclid Freed of Every Flaw,
treating the fifth postulate

733
00:33:55,620 --> 00:34:03,010
as kind of a flaw in Euclid's
axiomization of geometry.

734
00:34:03,010 --> 00:34:06,150
Saccheri was, in fact, convinced
that the fifth postulate

735
00:34:06,150 --> 00:34:07,330
was true.

736
00:34:07,330 --> 00:34:09,489
He didn't really want to
consider the possibility

737
00:34:09,489 --> 00:34:11,449
that it was false.

738
00:34:11,449 --> 00:34:13,600
But he was nonetheless
exploring the possibility

739
00:34:13,600 --> 00:34:15,440
that it was false
because he understood

740
00:34:15,440 --> 00:34:17,409
the concept of a proof
by contradiction.

741
00:34:17,409 --> 00:34:19,159
He was looking for
a contradiction

742
00:34:19,159 --> 00:34:23,570
to be able to prove that
mathematics would not

743
00:34:23,570 --> 00:34:26,290
be consistent if you assume
the fifth postulate was false.

744
00:34:26,290 --> 00:34:27,706
And therefore, the
fifth postulate

745
00:34:27,706 --> 00:34:29,540
would be proven to be true.

746
00:34:29,540 --> 00:34:31,110
So he was exploring
what would happen

747
00:34:31,110 --> 00:34:32,929
if the fifth
postulate was false,

748
00:34:32,929 --> 00:34:36,530
looking all the time to
find some inconsistency,

749
00:34:36,530 --> 00:34:37,886
and was not able to find any.

750
00:34:37,886 --> 00:34:39,510
So he considered all
of this a failure.

751
00:34:39,510 --> 00:34:41,440
But he published it anyway.

752
00:34:41,440 --> 00:34:45,380
And that's the
publication front page.

753
00:34:47,889 --> 00:34:50,316
The next person to enter
the stage-- or actually

754
00:34:50,316 --> 00:34:51,690
three people
together, but I only

755
00:34:51,690 --> 00:34:54,580
have a nice
transparency on Gauss.

756
00:34:54,580 --> 00:34:57,300
Gauss, Bolyai, and
Lobachevsky went on

757
00:34:57,300 --> 00:35:00,460
to seriously explore the
possibility of geometry

758
00:35:00,460 --> 00:35:03,450
without the fifth
postulate, actually assuming

759
00:35:03,450 --> 00:35:05,180
that the fifth
postulate is false,

760
00:35:05,180 --> 00:35:09,950
developing what we call GBL,
Gauss-Bolyai-Lobachevsky

761
00:35:09,950 --> 00:35:11,440
geometry.

762
00:35:11,440 --> 00:35:13,420
Gauss was a German
mathematician.

763
00:35:13,420 --> 00:35:14,920
These were the years he lived.

764
00:35:14,920 --> 00:35:17,170
He, in fact, was born the
son of a poor, working class

765
00:35:17,170 --> 00:35:19,700
parents, which I found
a little surprising.

766
00:35:19,700 --> 00:35:23,120
We kind of think of scholars
in those early years

767
00:35:23,120 --> 00:35:26,580
as being gentleman who
were part of the nobility.

768
00:35:26,580 --> 00:35:28,180
But Gauss was not
but, nonetheless,

769
00:35:28,180 --> 00:35:30,570
went on to be one of the
most important mathematicians

770
00:35:30,570 --> 00:35:31,842
of his age.

771
00:35:31,842 --> 00:35:33,550
One of the other things
that surprised me

772
00:35:33,550 --> 00:35:35,883
and to be honest I just learned
all this from Wikipedia.

773
00:35:35,883 --> 00:35:37,670
I'm no real expert
on the history.

774
00:35:37,670 --> 00:35:40,430
But they gave a list
of Gauss' students.

775
00:35:40,430 --> 00:35:43,870
And they included that Richard
Dedekind, Bernhard Riemann,

776
00:35:43,870 --> 00:35:47,930
Peter Gustav Lejeune Dirichlet,
which is the name I remember,

777
00:35:47,930 --> 00:35:50,160
Kirchhoff, and Mobius.

778
00:35:50,160 --> 00:35:52,920
So quite a list of
famous mathematicians.

779
00:35:52,920 --> 00:35:55,770
So I have to admit, when
I read that, I was just

780
00:35:55,770 --> 00:35:59,450
about to send off an email
to all of my former students

781
00:35:59,450 --> 00:36:01,470
saying, look, what's
happening here?

782
00:36:01,470 --> 00:36:05,680
You're not competing
at all. [LAUGHING]

783
00:36:05,680 --> 00:36:07,470
But I decided not to do that.

784
00:36:07,470 --> 00:36:08,530
It would be impolite.

785
00:36:08,530 --> 00:36:09,220
And who knows?

786
00:36:09,220 --> 00:36:10,720
Maybe 100 years
from now my students

787
00:36:10,720 --> 00:36:12,053
will be as famous as these guys.

788
00:36:12,053 --> 00:36:13,480
You never know.

789
00:36:13,480 --> 00:36:17,250
We can plan, I hope.

790
00:36:17,250 --> 00:36:20,170
OK, so the other guys involved
in this and they were all

791
00:36:20,170 --> 00:36:25,030
working independently were
Bolyai who was, I think,

792
00:36:25,030 --> 00:36:29,370
a Prussian military man,
actually, and Lobachevsky

793
00:36:29,370 --> 00:36:31,610
who was also a professional
mathematician working

794
00:36:31,610 --> 00:36:33,500
in the university.

795
00:36:33,500 --> 00:36:39,910
The three of them independently
developed a geometry

796
00:36:39,910 --> 00:36:42,954
in which the fifth postulate
was assumed to be false.

797
00:36:42,954 --> 00:36:44,495
There are two ways
it could be false.

798
00:36:48,530 --> 00:36:51,070
In version with the triangles,
for example, a triangle

799
00:36:51,070 --> 00:36:53,960
could have more than or
less than 180 degrees.

800
00:37:01,150 --> 00:37:04,660
Since there were assuming that
the fifth postulate was false,

801
00:37:04,660 --> 00:37:07,710
it meant they had to be assuming
that every version that we just

802
00:37:07,710 --> 00:37:09,240
talked about of
the fifth postulate

803
00:37:09,240 --> 00:37:12,035
is false because they are
all equivalent to each other

804
00:37:12,035 --> 00:37:15,130
and these people realize that.

805
00:37:15,130 --> 00:37:32,780
So in particular in the
Gauss-Bolyai-Lobachevsky

806
00:37:32,780 --> 00:37:54,420
geometry, there are
infinitely many lines parallel

807
00:37:54,420 --> 00:37:55,195
to a given line.

808
00:38:02,210 --> 00:38:11,980
And no figures of
different size are similar.

809
00:38:24,960 --> 00:38:42,205
And the sum of the
angles of a triangle

810
00:38:42,205 --> 00:38:47,890
are always less than
180 degrees or pi,

811
00:38:47,890 --> 00:38:50,488
depending on whether you're
a radian person or a degree

812
00:38:50,488 --> 00:38:50,988
person.

813
00:38:53,860 --> 00:38:58,030
Now I should mention here
that the surface of the sphere

814
00:38:58,030 --> 00:38:59,690
is, in fact, a
perfectly good example

815
00:38:59,690 --> 00:39:02,420
of the non-Euclidean geometry.

816
00:39:02,420 --> 00:39:04,520
But for some reason it
was not taken seriously

817
00:39:04,520 --> 00:39:08,970
by mathematicians until
long after these guys

818
00:39:08,970 --> 00:39:10,620
were doing their work.

819
00:39:10,620 --> 00:39:14,130
And part of the reason, I guess,
is that the surfaces three

820
00:39:14,130 --> 00:39:17,350
evaluates not just
one of Euclid's axioms

821
00:39:17,350 --> 00:39:19,925
but two if we go back
to Euclid's axioms.

822
00:39:25,516 --> 00:39:28,670
The second of Euclid's axioms
said that any straight line

823
00:39:28,670 --> 00:39:30,480
segment can be
extended indefinitely

824
00:39:30,480 --> 00:39:31,697
in a straight line.

825
00:39:31,697 --> 00:39:33,530
And the surface of the
sphere and the analog

826
00:39:33,530 --> 00:39:35,950
of the straight line
is a great circle.

827
00:39:35,950 --> 00:39:37,540
And if you extend
the great circle,

828
00:39:37,540 --> 00:39:39,670
it comes back on itself.

829
00:39:39,670 --> 00:39:43,090
So the surface of the sphere
violates the fifth postulate.

830
00:39:43,090 --> 00:39:45,660
But it also violates
the second postulate.

831
00:39:45,660 --> 00:39:47,670
But still perfectly
consistent geometry,

832
00:39:47,670 --> 00:39:50,120
and it is a
non-Euclidean geometry.

833
00:39:50,120 --> 00:39:52,820
And on the surface
of this sphere,

834
00:39:52,820 --> 00:39:56,330
these statements
all kind of reverse.

835
00:39:56,330 --> 00:39:59,050
Instead of having infinitely
many lines parallel to a given

836
00:39:59,050 --> 00:40:03,450
line, you have no lines that
are parallel to a given line.

837
00:40:03,450 --> 00:40:06,620
Remember, lines are great
circles and lines are parallel

838
00:40:06,620 --> 00:40:07,780
if they never meet.

839
00:40:07,780 --> 00:40:10,170
But any two great circles meet.

840
00:40:10,170 --> 00:40:13,460
So there are no lines
parallel to a given line

841
00:40:13,460 --> 00:40:17,730
in the geometry of the
surface of the sphere.

842
00:40:17,730 --> 00:40:19,990
It's, again, true that no
figures of different size

843
00:40:19,990 --> 00:40:20,490
are similar.

844
00:40:20,490 --> 00:40:25,210
That has to be true
for any any geometry

845
00:40:25,210 --> 00:40:27,300
where the fifth
postulate was false.

846
00:40:27,300 --> 00:40:29,370
The last one,
again, has a choice.

847
00:40:29,370 --> 00:40:31,080
And it's the opposite
choice for a sphere

848
00:40:31,080 --> 00:40:34,920
as for the
Gauss-Bolyai-Lobachevsky

849
00:40:34,920 --> 00:40:36,550
geometry.

850
00:40:36,550 --> 00:40:38,610
In the Gauss-Bolyai-Lobachevsky
Bolyai- geometry,

851
00:40:38,610 --> 00:40:41,586
the sum of the angles is
always less than 180 degrees.

852
00:40:41,586 --> 00:40:43,210
If you picture a
triangle and a sphere,

853
00:40:43,210 --> 00:40:47,010
you can imagine that the
edges look like they bulge.

854
00:40:47,010 --> 00:40:49,350
And the sum of the
angles on the surface

855
00:40:49,350 --> 00:40:50,960
of the sphere of a
triangle are always

856
00:40:50,960 --> 00:40:53,640
greater than 180 degrees.

857
00:40:53,640 --> 00:40:55,140
The easiest way to
convince yourself

858
00:40:55,140 --> 00:40:58,390
that that's true in at
least one important case

859
00:40:58,390 --> 00:40:59,870
is to imagine a triangle.

860
00:40:59,870 --> 00:41:02,130
Here's a sphere.

861
00:41:02,130 --> 00:41:04,372
Everybody see the sphere?

862
00:41:04,372 --> 00:41:05,330
There's the North Pole.

863
00:41:05,330 --> 00:41:08,440
There's an equator.

864
00:41:08,440 --> 00:41:10,240
Imagine a triangle
where one vertex

865
00:41:10,240 --> 00:41:14,330
is at the North Pole and the
liver disease or on the equator

866
00:41:14,330 --> 00:41:16,300
and the triangle
looks like this.

867
00:41:16,300 --> 00:41:18,420
And these are both 90
degree angles down here.

868
00:41:18,420 --> 00:41:20,660
So you already have 180 degrees.

869
00:41:20,660 --> 00:41:22,430
And any angle you
have on top just

870
00:41:22,430 --> 00:41:24,410
adds to the 180
degrees putting you

871
00:41:24,410 --> 00:41:29,430
above the Euclidean
value of 180 degrees.

872
00:41:29,430 --> 00:41:31,810
So for a sphere it's
always the opposite

873
00:41:31,810 --> 00:41:33,465
of this greater
than 180 degrees.

874
00:41:37,250 --> 00:41:41,390
OK, continuing with Gauss,
Bolyai and Lobachevsky,

875
00:41:41,390 --> 00:41:47,650
their work was based on
exploring the axioms of Euclid,

876
00:41:47,650 --> 00:41:51,890
assuming the reverse
of the fifth postulate

877
00:41:51,890 --> 00:41:55,000
in any one of its forms.

878
00:41:55,000 --> 00:41:58,870
And they proved
theorems and wrote,

879
00:41:58,870 --> 00:42:02,810
sort of, their own versions
of Euclid's elements.

880
00:42:02,810 --> 00:42:05,940
But that still left
open the question

881
00:42:05,940 --> 00:42:08,940
whether or not all of this
was really consistent.

882
00:42:08,940 --> 00:42:12,109
That is, from the
thinking of Saccheri

883
00:42:12,109 --> 00:42:14,400
that we already talked about,
there's always the chance

884
00:42:14,400 --> 00:42:16,441
that you might find the
contradiction even if you

885
00:42:16,441 --> 00:42:18,490
haven't found a
contradiction yet.

886
00:42:18,490 --> 00:42:20,910
And Gauss, Bolyai
and Lobachevsky

887
00:42:20,910 --> 00:42:23,340
didn't really have any
way of answering that.

888
00:42:23,340 --> 00:42:25,140
They didn't really
have any way of knowing

889
00:42:25,140 --> 00:42:26,514
whether if they
continued further

890
00:42:26,514 --> 00:42:28,970
they might find
some contradiction.

891
00:42:28,970 --> 00:42:32,690
So what they did certainly
extracted the properties

892
00:42:32,690 --> 00:42:34,570
of this non-Euclidean geometry.

893
00:42:34,570 --> 00:42:37,860
But it did not really prove
that the non-Euclidean geometry

894
00:42:37,860 --> 00:42:40,320
was consistent.

895
00:42:40,320 --> 00:42:45,880
That happened slightly
later in an argument

896
00:42:45,880 --> 00:42:49,580
by Felix Klein, who was the
same Klein as the Klein bottle,

897
00:42:49,580 --> 00:42:50,080
by the way.

898
00:43:00,710 --> 00:43:02,720
And that was his
famous paper on this

899
00:43:02,720 --> 00:43:05,850
was published in 1870 somewhat
later than the early work

900
00:43:05,850 --> 00:43:07,320
that we're talking about.

901
00:43:07,320 --> 00:43:12,640
And what he did is he gave an
actual construction of the GBL

902
00:43:12,640 --> 00:43:14,250
geometry.

903
00:43:14,250 --> 00:43:15,780
And by construction
I mean in terms

904
00:43:15,780 --> 00:43:19,530
of coordinates using coordinate
geometry ideas, which

905
00:43:19,530 --> 00:43:22,740
were originally
developed by Descartes.

906
00:43:22,740 --> 00:43:25,780
That's why we call them
Cartesian coordinate systems.

907
00:43:25,780 --> 00:43:31,580
So what he gave as a description
of the Gauss-Bolyai-Lobachevsky

908
00:43:31,580 --> 00:43:38,880
geometry was a space
that consisted of a disk

909
00:43:38,880 --> 00:43:46,340
with coordinates x and y just
as Descartes would have done.

910
00:43:46,340 --> 00:43:49,011
He restricts himself to
the inside of the disk.

911
00:43:49,011 --> 00:43:50,510
So he restricts
himself to x squared

912
00:43:50,510 --> 00:43:55,130
plus y squared less than 1.

913
00:43:55,130 --> 00:43:59,540
And what he gave was a
function of two points

914
00:43:59,540 --> 00:44:02,390
in this disk where the
function represents

915
00:44:02,390 --> 00:44:04,690
the distance between
those two points.

916
00:44:04,690 --> 00:44:06,600
He decided that
distances don't have

917
00:44:06,600 --> 00:44:08,680
to be Euclidean
distances if we're

918
00:44:08,680 --> 00:44:12,060
trying to explore
non-Euclidean geometry.

919
00:44:12,060 --> 00:44:15,230
So he invented his
own distance function.

920
00:44:15,230 --> 00:44:18,290
And it's pretty
complicated looking.

921
00:44:18,290 --> 00:44:24,160
The function he came up with was
that the cosh of the distance

922
00:44:24,160 --> 00:44:33,400
between points 1 and 2 divided
by sum number a-- and a

923
00:44:33,400 --> 00:44:43,470
could be any number-- is
equal to 1 minus x1 times x2--

924
00:44:43,470 --> 00:44:45,620
these are the two
x-coordinates for points 1

925
00:44:45,620 --> 00:44:59,680
and 2-- minus y1 y2 over
the square root of 1 minus

926
00:44:59,680 --> 00:45:03,890
x1 squared minus y1 squared.

927
00:45:03,890 --> 00:45:06,530
Sorry, it's the product of two
square roots in the denominator

928
00:45:06,530 --> 00:45:07,030
here.

929
00:45:09,547 --> 00:45:11,380
And the second square
root is the same thing

930
00:45:11,380 --> 00:45:18,245
for the two coordinates 1 minus
x2 squared minus y2 squared.

931
00:45:24,610 --> 00:45:29,500
So this formula is certainly
not obvious to anybody.

932
00:45:29,500 --> 00:45:34,030
But Klein figured out
that this actually

933
00:45:34,030 --> 00:45:38,520
does-- the geometry
described by these formulas--

934
00:45:38,520 --> 00:45:42,070
reproduce completely
the postulates

935
00:45:42,070 --> 00:45:44,440
of the Gauss-Bolyai-Lobachevsky
geometry,

936
00:45:44,440 --> 00:45:47,010
including the failure of
Euclid's fifth postulate.

937
00:45:50,050 --> 00:45:52,175
And since this boils
down to just algebra,

938
00:45:52,175 --> 00:45:54,340
if algebra is
consistent, it proves

939
00:45:54,340 --> 00:45:55,760
that the
Gauss-Bolyai-Lobachevsky

940
00:45:55,760 --> 00:45:57,914
geometry is consistent.

941
00:45:57,914 --> 00:46:00,330
Now, as I understand it, nobody
can prove that any of this

942
00:46:00,330 --> 00:46:01,288
actually is consistent.

943
00:46:01,288 --> 00:46:03,660
People prove
relative consistency.

944
00:46:03,660 --> 00:46:08,170
So in the assumption that
algebra is consistent,

945
00:46:08,170 --> 00:46:10,304
theorems about the real
numbers are consistent.

946
00:46:10,304 --> 00:46:11,720
Felix Klein was
able to prove that

947
00:46:11,720 --> 00:46:13,950
the Gauss-Bolyai-Lobachevsky
geometry is consistent.

948
00:46:16,115 --> 00:46:17,490
And this was really
the beginning

949
00:46:17,490 --> 00:46:18,740
of coordinate geometry.

950
00:46:18,740 --> 00:46:19,930
I'm sure all of you
are rather familiar

951
00:46:19,930 --> 00:46:20,971
with coordinate geometry.

952
00:46:20,971 --> 00:46:22,840
It's a standard topic
now in math courses

953
00:46:22,840 --> 00:46:24,520
and even in high school.

954
00:46:24,520 --> 00:46:26,510
And this is really
where it began.

955
00:46:26,510 --> 00:46:29,610
Euclid had no idea
that it was any value

956
00:46:29,610 --> 00:46:33,070
in trying to represent geometric
quantities as equations.

957
00:46:33,070 --> 00:46:37,250
Euclid did everything
in terms of theorems.

958
00:46:37,250 --> 00:46:39,300
But this opened up
a whole new door

959
00:46:39,300 --> 00:46:40,610
for how to discuss geometry.

960
00:46:46,800 --> 00:46:52,840
So the geometry is a slide that
just shows the same formula.

961
00:46:52,840 --> 00:46:56,562
And I guess this is supposed
to be an image of the disk.

962
00:46:56,562 --> 00:46:58,020
One thing I should
point out, which

963
00:46:58,020 --> 00:47:01,290
I forgot to point out, which is
that although this disk looks

964
00:47:01,290 --> 00:47:06,230
finite, it really is an infinite
space that's being described.

965
00:47:06,230 --> 00:47:08,360
And one can see that
by looking carefully

966
00:47:08,360 --> 00:47:09,880
at the distance functions.

967
00:47:09,880 --> 00:47:12,540
As either one of
these two points--

968
00:47:12,540 --> 00:47:17,470
point 1 or point 2--
approaches the boundary

969
00:47:17,470 --> 00:47:20,730
x square plus y square
equals 1, this square

970
00:47:20,730 --> 00:47:23,080
root denominator blows up.

971
00:47:23,080 --> 00:47:27,680
So the distance between a
point and another point that's

972
00:47:27,680 --> 00:47:30,400
approaching the boundary
goes to infinity

973
00:47:30,400 --> 00:47:32,960
as that point
approaches the boundary.

974
00:47:32,960 --> 00:47:35,160
So boundary's actually
infinitely far away

975
00:47:35,160 --> 00:47:37,900
even though in
coordinates they are still

976
00:47:37,900 --> 00:47:40,970
x squared plus y
squared equals 1.

977
00:47:40,970 --> 00:47:43,405
So this introduces
another important concept,

978
00:47:43,405 --> 00:47:45,446
which we'll be using in
general relativity, which

979
00:47:45,446 --> 00:47:47,665
is the coordinates don't
represent distances.

980
00:47:47,665 --> 00:47:49,040
Distances could
be very different

981
00:47:49,040 --> 00:47:50,737
from the way the
coordinates look.

982
00:47:50,737 --> 00:47:52,320
So that boundary,
even though it looks

983
00:47:52,320 --> 00:47:55,580
like it's right there
on the blackboard,

984
00:47:55,580 --> 00:47:57,580
is actually very far away
from the other points.

985
00:48:07,300 --> 00:48:10,850
OK, so after Klein,
the important new idea

986
00:48:10,850 --> 00:48:12,690
that Klein introduced
was, first of all,

987
00:48:12,690 --> 00:48:15,290
the explicit construction
but also the general idea

988
00:48:15,290 --> 00:48:17,960
that you can describe geometry
not by giving postulates

989
00:48:17,960 --> 00:48:20,270
but instead by actually
doing a construction where

990
00:48:20,270 --> 00:48:23,550
you've given names to the
points in terms of coordinates

991
00:48:23,550 --> 00:48:26,060
and you describe what
happens geometrically

992
00:48:26,060 --> 00:48:28,950
in terms of some distance
function which describes

993
00:48:28,950 --> 00:48:30,925
the distance between
two arbitrary points.

994
00:48:37,780 --> 00:48:44,940
And Gauss went on to make
two other very important

995
00:48:44,940 --> 00:48:46,610
observations about
geometry, which

996
00:48:46,610 --> 00:48:49,210
has become essential to
our current understanding

997
00:48:49,210 --> 00:48:51,660
of non-Euclidean geometry.

998
00:48:51,660 --> 00:48:54,550
So let me mention two other
ideas that Gauss introduced.

999
00:49:00,560 --> 00:49:03,180
The first one was
the distinction

1000
00:49:03,180 --> 00:49:07,160
between what he called
inner and outer properties

1001
00:49:07,160 --> 00:49:09,470
of a curved surface.

1002
00:49:09,470 --> 00:49:11,510
His curved spaces were
all two dimensional,

1003
00:49:11,510 --> 00:49:12,515
so they were surfaces.

1004
00:49:24,800 --> 00:49:26,610
So this is most easily
described for say

1005
00:49:26,610 --> 00:49:28,470
a surface of a sphere
where we can all

1006
00:49:28,470 --> 00:49:30,770
visualize what
we're talking about.

1007
00:49:30,770 --> 00:49:33,780
The surface of a sphere we
visualize in three Euclidean

1008
00:49:33,780 --> 00:49:36,910
dimensions and we
think of its properties

1009
00:49:36,910 --> 00:49:39,650
as being determined by that
three dimensional space.

1010
00:49:39,650 --> 00:49:42,280
And the geometry of that
three dimensional space.

1011
00:49:42,280 --> 00:49:43,988
And of course the
three dimensional space

1012
00:49:43,988 --> 00:49:47,000
is Euclidean that we're
embedding our sphere into.

1013
00:49:47,000 --> 00:49:50,780
But the non-Euclidean
aspects are all

1014
00:49:50,780 --> 00:49:54,830
seen in the geometry of the
two dimensional surface.

1015
00:49:54,830 --> 00:49:57,690
Figures drawn on the surface
as if the rest of the three

1016
00:49:57,690 --> 00:50:00,640
dimensional space did not exist.

1017
00:50:00,640 --> 00:50:05,580
And that is this key idea of
inner versus outer properties.

1018
00:50:05,580 --> 00:50:07,330
The outer properties
of the sphere

1019
00:50:07,330 --> 00:50:10,000
are properties that relate
to the three dimensional

1020
00:50:10,000 --> 00:50:12,992
space in which the
sphere is embedded.

1021
00:50:12,992 --> 00:50:14,910
But what Gauss realized
is that there's

1022
00:50:14,910 --> 00:50:18,650
a perfectly well defined
mathematics contained entirely

1023
00:50:18,650 --> 00:50:21,620
in the two dimensional space
of the surface of the sphere.

1024
00:50:21,620 --> 00:50:24,214
You could discuss it
without making any reference

1025
00:50:24,214 --> 00:50:26,630
to the three dimensional space
around it if you wanted to,

1026
00:50:26,630 --> 00:50:28,530
it's just a little
more complicated to be

1027
00:50:28,530 --> 00:50:29,770
able to do that.

1028
00:50:29,770 --> 00:50:34,860
But we will in fact be doing
it explicitly very shortly.

1029
00:50:34,860 --> 00:50:37,442
And all it amounts to
from our point of view

1030
00:50:37,442 --> 00:50:39,650
is assigning coordinates to
the points on the surface

1031
00:50:39,650 --> 00:50:43,770
of the sphere and the distance
function for those coordinates.

1032
00:50:43,770 --> 00:50:45,900
And then one has
a full description

1033
00:50:45,900 --> 00:50:48,150
of this two dimensional
space consisting

1034
00:50:48,150 --> 00:50:51,110
of the surface of the sphere
which no longer needs make

1035
00:50:51,110 --> 00:50:53,650
any reference to
the third dimension

1036
00:50:53,650 --> 00:50:57,270
that you imagined the
sphere embedded in.

1037
00:50:57,270 --> 00:51:01,260
So the study of the properties
of that two dimensional space

1038
00:51:01,260 --> 00:51:03,990
is the study of the inner
properties of the space.

1039
00:51:03,990 --> 00:51:06,300
And if you care about
how it's embedded,

1040
00:51:06,300 --> 00:51:08,300
that's called the
outer properties.

1041
00:51:08,300 --> 00:51:14,130
And Gauss made it clear in
the way he described things

1042
00:51:14,130 --> 00:51:17,240
that from his point of view, the
real thing that mathematicians

1043
00:51:17,240 --> 00:51:19,710
should be doing is studying
the inner properties.

1044
00:51:19,710 --> 00:51:23,636
The outer properties are
not that interesting.

1045
00:51:23,636 --> 00:51:25,510
So that's one key idea
that Gauss introduced.

1046
00:51:29,160 --> 00:51:33,660
And the second is the idea
of what we now call a metric.

1047
00:51:36,480 --> 00:51:40,030
And there are really
two pieces to this.

1048
00:51:40,030 --> 00:51:43,410
The first of which
is that instead

1049
00:51:43,410 --> 00:51:47,020
of giving macroscopic distances,
which is what Klein did,

1050
00:51:47,020 --> 00:51:49,420
he told you how to write down
a formula for the distance

1051
00:51:49,420 --> 00:51:52,550
between any two
points, Gauss realized

1052
00:51:52,550 --> 00:51:56,290
that things could
become more interesting

1053
00:51:56,290 --> 00:51:59,450
if instead of trying to
immediately write down

1054
00:51:59,450 --> 00:52:02,860
a function for the distance
between two arbitrary points,

1055
00:52:02,860 --> 00:52:06,070
you can find your attention
to very nearby points.

1056
00:52:06,070 --> 00:52:09,030
And consider the
distance between two

1057
00:52:09,030 --> 00:52:11,070
far away points
as just the length

1058
00:52:11,070 --> 00:52:14,577
of a line that joins them
where every little segment

1059
00:52:14,577 --> 00:52:17,160
in the line is a short distance
which is governed by the rules

1060
00:52:17,160 --> 00:52:18,969
that you've made up
for short distances.

1061
00:52:18,969 --> 00:52:20,510
So if you understand
short distances,

1062
00:52:20,510 --> 00:52:23,124
long distances are
obtained just by adding.

1063
00:52:23,124 --> 00:52:24,040
That's the basic idea.

1064
00:52:26,760 --> 00:52:28,230
So only short
distances are needed.

1065
00:52:28,230 --> 00:52:30,770
And then there's
another important idea

1066
00:52:30,770 --> 00:52:33,910
which is that the short
distances themselves

1067
00:52:33,910 --> 00:52:36,490
for an interesting
class of spaces

1068
00:52:36,490 --> 00:52:38,930
should not be some arbitrary
function of the coordinates

1069
00:52:38,930 --> 00:52:44,549
of the two points but should in
fact be a quadratic function.

1070
00:52:44,549 --> 00:52:46,340
And in two dimensions,
a quadratic function

1071
00:52:46,340 --> 00:52:49,190
mean something like
this, where gxx is just

1072
00:52:49,190 --> 00:52:52,410
any function of x and y, x
and y are the two coordinates

1073
00:52:52,410 --> 00:52:53,800
of the space.

1074
00:52:53,800 --> 00:52:57,164
gxy is also just any
function of x and u.

1075
00:52:57,164 --> 00:53:02,830
gyy is any function and gxx
multiplies the x squared,

1076
00:53:02,830 --> 00:53:09,310
xy multiplies the x times dy and
g sub yy multiplies dy squared.

1077
00:53:09,310 --> 00:53:13,260
And there's no terms portion
for dx by itself or dy by itself

1078
00:53:13,260 --> 00:53:15,075
or dz by itself.

1079
00:53:15,075 --> 00:53:16,450
And there's no
terms proportional

1080
00:53:16,450 --> 00:53:19,760
to the cubes of those
quantities, it's all quadratic.

1081
00:53:19,760 --> 00:53:21,850
That's the assumption.

1082
00:53:21,850 --> 00:53:24,020
Now what underlies
that assumption is not

1083
00:53:24,020 --> 00:53:27,160
that all spaces have
to have this property.

1084
00:53:27,160 --> 00:53:31,110
This does narrow one down to
a particular class of spaces.

1085
00:53:31,110 --> 00:53:33,810
But one way of
characterizing that class

1086
00:53:33,810 --> 00:53:38,240
is that what Gauss understood
is that that class, the class

1087
00:53:38,240 --> 00:53:40,710
of spaces described
by a metric like this,

1088
00:53:40,710 --> 00:53:43,040
are precisely the class of
spaces which are locally

1089
00:53:43,040 --> 00:53:48,630
Euclidean, meaning that even
though the surface is curved,

1090
00:53:48,630 --> 00:53:53,890
any tiny little patch of it
looks flat and can be covered

1091
00:53:53,890 --> 00:53:56,110
by Euclidean coordinates,
redefining coordinates

1092
00:53:56,110 --> 00:53:59,020
in that one little patch only
where the metric in the one

1093
00:53:59,020 --> 00:54:01,620
little patch would just be
the Euclidean metric given

1094
00:54:01,620 --> 00:54:04,750
by the Pythagorean theorem
of Euclidean geometry,

1095
00:54:04,750 --> 00:54:08,070
dx prime squared
plus dy squared,

1096
00:54:08,070 --> 00:54:10,870
which would be the distance
function for a flat space

1097
00:54:10,870 --> 00:54:13,720
and ordinary
Cartesian coordinates.

1098
00:54:13,720 --> 00:54:16,360
And it turns out
that saying that this

1099
00:54:16,360 --> 00:54:19,270
is true in every
microscopic region

1100
00:54:19,270 --> 00:54:22,760
is equivalent to saying that
the metric over the entire space

1101
00:54:22,760 --> 00:54:27,000
can be written as
a quadratic form,

1102
00:54:27,000 --> 00:54:29,710
meaning one that looks
exactly like that.

1103
00:54:29,710 --> 00:54:31,580
And spaces that
have this property

1104
00:54:31,580 --> 00:54:34,242
are now usually called
Riemannian spaces,

1105
00:54:34,242 --> 00:54:36,840
even though it was Gauss
who first identified them.

1106
00:54:39,390 --> 00:54:41,990
And all the spaces that
we deal with in physics,

1107
00:54:41,990 --> 00:54:44,840
in particular the spaces we
deal with general relativity,

1108
00:54:44,840 --> 00:54:48,860
will be Riemannian spaces.

1109
00:54:48,860 --> 00:54:51,110
Or sometimes they're called
pseudo Riemannian spaces

1110
00:54:51,110 --> 00:54:53,840
because time is treated a little
bit differently in physics.

1111
00:54:53,840 --> 00:54:55,460
And the word
"Riemannian" was really

1112
00:54:55,460 --> 00:54:58,430
built on spatial geometry.

1113
00:54:58,430 --> 00:55:01,117
But the word
"pseudo" only changes

1114
00:55:01,117 --> 00:55:03,700
the fact that this becomes the
Lorenz metric, if you know what

1115
00:55:03,700 --> 00:55:06,750
that means, instead of
the Euclidean metric.

1116
00:55:06,750 --> 00:55:08,910
But the same idea
holds, which is

1117
00:55:08,910 --> 00:55:10,900
that the spaces that
we're interested in

1118
00:55:10,900 --> 00:55:16,060
are spaces which locally
look exactly like flat space.

1119
00:55:16,060 --> 00:55:20,570
And that implies that globally,
you can always write down

1120
00:55:20,570 --> 00:55:27,680
a metric function which is
quadratic, as Gauss said

1121
00:55:27,680 --> 00:55:37,540
So the metric should be
a quadratic function that

1122
00:55:37,540 --> 00:55:39,910
specifies the distance only
between infinitesimally

1123
00:55:39,910 --> 00:55:44,030
separated points, not
finitely separated points

1124
00:55:44,030 --> 00:55:47,610
and should have the
form of ds squared

1125
00:55:47,610 --> 00:55:52,560
is equal to some
function of x and y times

1126
00:55:52,560 --> 00:56:02,200
dx squared plus a different
function of x and y times dx

1127
00:56:02,200 --> 00:56:13,520
times dy plus a different
function of x and y times dy

1128
00:56:13,520 --> 00:56:14,020
squared.

1129
00:56:14,020 --> 00:56:17,946
So I'm just writing the same
form that was on the side.

1130
00:56:17,946 --> 00:56:19,320
This is a very
important formula.

1131
00:56:22,310 --> 00:56:24,920
Incidentally, the
two that appears here

1132
00:56:24,920 --> 00:56:29,190
is only because when we write
this in a different notation,

1133
00:56:29,190 --> 00:56:32,400
this term will occur
twice, once as dx times dy,

1134
00:56:32,400 --> 00:56:34,560
and once as dy times dx.

1135
00:56:34,560 --> 00:56:37,060
And the two times it occurs
are equal to each other,

1136
00:56:37,060 --> 00:56:39,560
so here they're just collected
together with a factor of two

1137
00:56:39,560 --> 00:56:40,700
in front.

1138
00:56:40,700 --> 00:56:41,200
Yes.

1139
00:56:41,200 --> 00:56:44,155
STUDENT: Just to make sure,
those three different functions

1140
00:56:44,155 --> 00:56:45,997
you wrote, those
subscripts aren't

1141
00:56:45,997 --> 00:56:47,830
supposed to mean partial
derivatives, right?

1142
00:56:47,830 --> 00:56:48,996
PROFESSOR: No, that's right.

1143
00:56:48,996 --> 00:56:50,430
The subscripts and
this expression

1144
00:56:50,430 --> 00:56:53,390
only mean that this
multiplies dx squared,

1145
00:56:53,390 --> 00:56:55,370
so it has subscripts xx.

1146
00:56:55,370 --> 00:56:59,020
And this multiplies dx times
dy, so it has subscripts xy.

1147
00:56:59,020 --> 00:57:02,510
That's what the subscripts
mean and nothing more.

1148
00:57:02,510 --> 00:57:04,185
So the subscripts
only mean that they're

1149
00:57:04,185 --> 00:57:06,511
the things that appear
in that equation.

1150
00:57:06,511 --> 00:57:07,010
Yes.

1151
00:57:07,010 --> 00:57:09,495
STUDENT: It seems like
the metric is giving us

1152
00:57:09,495 --> 00:57:15,460
distance in terms of an
infintesimal displacement,

1153
00:57:15,460 --> 00:57:20,560
but then a locally Euclidean
space is already tangent

1154
00:57:20,560 --> 00:57:24,560
infintesimally, so how are
we relating the local metric

1155
00:57:24,560 --> 00:57:26,774
with the global metric?

1156
00:57:26,774 --> 00:57:28,190
PROFESSOR: OK, the
question is how

1157
00:57:28,190 --> 00:57:30,170
do we relate the local
metric which I say

1158
00:57:30,170 --> 00:57:33,810
is Euclidean to
the global metric?

1159
00:57:33,810 --> 00:57:36,770
And the answer I
think for now I will

1160
00:57:36,770 --> 00:57:39,380
stick to just giving kind
of a pictorial answer based

1161
00:57:39,380 --> 00:57:41,830
on the picture here.

1162
00:57:41,830 --> 00:57:46,760
That is once you know the
distances between any two

1163
00:57:46,760 --> 00:57:49,950
points in a tiny
little patch here,

1164
00:57:49,950 --> 00:57:53,350
it's then always possible
to construct coordinates,

1165
00:57:53,350 --> 00:57:56,290
here called x prime
and y prime, such

1166
00:57:56,290 --> 00:57:59,530
that the distance between
any two points as calculated

1167
00:57:59,530 --> 00:58:02,190
from the original metric,
which is the one here,

1168
00:58:02,190 --> 00:58:03,820
is exactly the same
as the distance you

1169
00:58:03,820 --> 00:58:08,110
get using this metric.

1170
00:58:08,110 --> 00:58:09,720
And the claim is
that you can always

1171
00:58:09,720 --> 00:58:14,600
define coordinates x prime and
y prime which make that true.

1172
00:58:14,600 --> 00:58:17,180
That claim is not
absolutely obvious.

1173
00:58:17,180 --> 00:58:19,850
But it's something you can
probably visualize if you just

1174
00:58:19,850 --> 00:58:23,730
imagine that every little tiny
piece of this curved surface

1175
00:58:23,730 --> 00:58:28,040
looks like it was just
a flat surface and then

1176
00:58:28,040 --> 00:58:30,901
a flat surface you know
that you can write down

1177
00:58:30,901 --> 00:58:33,400
a Cartesian coordinate system,
which will have the Euclidean

1178
00:58:33,400 --> 00:58:35,936
metric.

1179
00:58:35,936 --> 00:58:37,850
But it's only an
intuitive statement,

1180
00:58:37,850 --> 00:58:39,820
proving it is actually harder.

1181
00:58:39,820 --> 00:58:40,320
Yes.

1182
00:58:40,320 --> 00:58:44,920
STUDENT: [INAUDIBLE]
bottom formula [INAUDIBLE]

1183
00:58:44,920 --> 00:58:50,900
with positive curvature if we
analogize to second derivative.

1184
00:58:50,900 --> 00:58:52,441
PROFESSOR: I'm sorry,
say that again.

1185
00:58:52,441 --> 00:58:54,796
STUDENT: If we
analogize gxx, gyy,

1186
00:58:54,796 --> 00:59:00,889
in the bottom formula there,
[INAUDIBLE] positive curvature

1187
00:59:00,889 --> 00:59:02,180
[INAUDIBLE] second derivative--

1188
00:59:02,180 --> 00:59:03,260
PROFESSOR: Yeah OK.

1189
00:59:03,260 --> 00:59:07,060
So you're asking does
this mean that we

1190
00:59:07,060 --> 00:59:09,097
have two states of
positive curvature?

1191
00:59:09,097 --> 00:59:10,230
STUDENT: Bottom right.

1192
00:59:10,230 --> 00:59:11,604
PROFESSOR: Bottom
right, oh this.

1193
00:59:14,280 --> 00:59:15,450
These are Mustafa's slides.

1194
00:59:15,450 --> 00:59:16,462
I forgot to say that.

1195
00:59:16,462 --> 00:59:18,670
You can tell from the style,
these are not my slides,

1196
00:59:18,670 --> 00:59:19,980
these are Mustafa's slides.

1197
00:59:19,980 --> 00:59:23,040
And I don't know what
he meant by this.

1198
00:59:23,040 --> 00:59:29,080
You're right, this
does-- well you

1199
00:59:29,080 --> 00:59:30,510
do want the metric
to be positive

1200
00:59:30,510 --> 00:59:34,120
definite, which is not the
same as saying the curvature

1201
00:59:34,120 --> 00:59:35,087
is positive.

1202
00:59:35,087 --> 00:59:36,920
And I think this might
just be the condition

1203
00:59:36,920 --> 00:59:39,280
that the metric is
positive definite,

1204
00:59:39,280 --> 00:59:43,490
that this expression
will always be positive.

1205
00:59:43,490 --> 00:59:44,082
Yeah.

1206
00:59:44,082 --> 00:59:46,165
I'll bet that's what it
is, I don't know for sure.

1207
00:59:46,165 --> 00:59:48,439
I'll bet that's what
that condition is about.

1208
00:59:48,439 --> 00:59:50,230
And you do want that,
the metric had better

1209
00:59:50,230 --> 00:59:54,660
be positive definite
for spatial geometries.

1210
00:59:54,660 --> 00:59:57,930
In fact, in general relativity,
where it's a space time metric,

1211
00:59:57,930 --> 01:00:00,054
it will not be
positive definite.

1212
01:00:00,054 --> 01:00:01,720
For reasons that we'll
talk about later.

1213
01:00:04,800 --> 01:00:07,400
But for geometry, the metric
should be positive definite.

1214
01:00:07,400 --> 01:00:08,775
All distances
should be positive.

1215
01:00:16,260 --> 01:00:18,320
OK.

1216
01:00:18,320 --> 01:00:20,390
So that ends my slides.

1217
01:00:20,390 --> 01:00:22,470
So now I'll continue
on the blackboard.

1218
01:00:41,300 --> 01:00:44,700
OK next I wanted to say
a few general comments

1219
01:00:44,700 --> 01:00:46,540
about general relativity.

1220
01:00:53,470 --> 01:00:56,660
General relativity was of
course invented by the Einstein

1221
01:00:56,660 --> 01:01:00,190
in 1916.

1222
01:01:00,190 --> 01:01:02,690
It's a theory that he was
working on for about 10 years

1223
01:01:02,690 --> 01:01:06,760
after he invented the theory
of special relativity.

1224
01:01:06,760 --> 01:01:09,260
To understand what's
going on there,

1225
01:01:09,260 --> 01:01:11,750
you want to recognize
that special relativity is

1226
01:01:11,750 --> 01:01:15,950
a theory of mechanics
and electrodynamics

1227
01:01:15,950 --> 01:01:19,390
which was designed to be
consistent with the principle

1228
01:01:19,390 --> 01:01:21,270
that the speed of
light is always

1229
01:01:21,270 --> 01:01:24,880
the same speed of light
independent of the speed

1230
01:01:24,880 --> 01:01:28,090
of the source of light or
the speed of the observer

1231
01:01:28,090 --> 01:01:30,120
of the light.

1232
01:01:30,120 --> 01:01:32,117
And it's of course
not easy to do that,

1233
01:01:32,117 --> 01:01:34,450
because you think that if you
move relative to something

1234
01:01:34,450 --> 01:01:38,110
else that's moving, that you
would see its velocity change.

1235
01:01:38,110 --> 01:01:43,940
So in order to make a theory
that where the speed of light

1236
01:01:43,940 --> 01:01:46,320
was an absolute
invariant, Einstein

1237
01:01:46,320 --> 01:01:47,860
had introduced a
number of things.

1238
01:01:47,860 --> 01:01:49,645
And we talked about
those at the beginning

1239
01:01:49,645 --> 01:01:54,170
of the course, the three
primary effects built

1240
01:01:54,170 --> 01:01:56,090
in to special relativity.

1241
01:01:56,090 --> 01:01:58,690
Time dilation,
length contraction,

1242
01:01:58,690 --> 01:02:02,310
and new rules
about simultaneity,

1243
01:02:02,310 --> 01:02:05,670
and how things that look
simultaneous to one observer

1244
01:02:05,670 --> 01:02:09,000
will not look simultaneous
to other observers in a very

1245
01:02:09,000 --> 01:02:10,830
definite, well defined way.

1246
01:02:10,830 --> 01:02:13,320
So by inventing
these rules, Einstein

1247
01:02:13,320 --> 01:02:16,164
was able to devise
a system which

1248
01:02:16,164 --> 01:02:18,330
was consistent with the
idea that the speed of light

1249
01:02:18,330 --> 01:02:20,980
always looked the
same to all observers.

1250
01:02:20,980 --> 01:02:24,750
And at similar times, the
Michelson Morley experiment

1251
01:02:24,750 --> 01:02:26,990
seemed to show that that
was in fact the case

1252
01:02:26,990 --> 01:02:29,290
and ultimately there's a
tremendous amount of evidence

1253
01:02:29,290 --> 01:02:33,040
verifying that what the
hypothesis that Einstein was

1254
01:02:33,040 --> 01:02:36,000
pursuing is the right one
for the way nature behaves.

1255
01:02:36,000 --> 01:02:38,990
The speed of light is invariant.

1256
01:02:38,990 --> 01:02:43,150
But what was lacking in
Einstein's formulation

1257
01:02:43,150 --> 01:02:46,420
of special relativity
was any version

1258
01:02:46,420 --> 01:02:49,730
of a consistent
theory of gravity.

1259
01:02:49,730 --> 01:02:55,810
Gravity had a well defined
description known since Newton.

1260
01:02:55,810 --> 01:02:58,170
But Newton's description
was a description

1261
01:02:58,170 --> 01:03:00,630
of a force at a distance.

1262
01:03:00,630 --> 01:03:05,470
And that is intrinsically
inconsistent with a theory

1263
01:03:05,470 --> 01:03:08,272
like special relativity,
which holds that simultaneity

1264
01:03:08,272 --> 01:03:11,100
is itself relative.

1265
01:03:11,100 --> 01:03:23,960
In Newtonian physics,
the force of gravity

1266
01:03:23,960 --> 01:03:27,980
is equal to Newton's
constant times

1267
01:03:27,980 --> 01:03:29,960
the product of the
two masses that

1268
01:03:29,960 --> 01:03:32,810
are interacting with
each other divided

1269
01:03:32,810 --> 01:03:35,720
by the square of the distance
and then times some union

1270
01:03:35,720 --> 01:03:40,630
vector pointing along the line
that joins the two particles.

1271
01:03:40,630 --> 01:03:43,600
But for that formula to
make any sense at all,

1272
01:03:43,600 --> 01:03:45,850
you have to know where
the two particles are

1273
01:03:45,850 --> 01:03:48,100
at the same instant of time.

1274
01:03:48,100 --> 01:03:50,680
And this r is the distance
between the locations

1275
01:03:50,680 --> 01:03:53,470
of the two particles at
some instant of time.

1276
01:03:53,470 --> 01:03:55,240
And this r hat is
a unit vector that

1277
01:03:55,240 --> 01:03:58,110
points along the line joining
those two particles where

1278
01:03:58,110 --> 01:04:00,350
you've pinned down where
those particles are

1279
01:04:00,350 --> 01:04:02,960
at one instant of time.

1280
01:04:02,960 --> 01:04:05,240
But we know from
the very beginning

1281
01:04:05,240 --> 01:04:09,250
that in special relativity the
notion of two things happening

1282
01:04:09,250 --> 01:04:13,280
at distant points at one
instant of time is ambiguous.

1283
01:04:13,280 --> 01:04:16,350
Different observers will
see different notions

1284
01:04:16,350 --> 01:04:20,470
of what it means for
two events to happen

1285
01:04:20,470 --> 01:04:24,402
at the same time
across this distance.

1286
01:04:24,402 --> 01:04:26,360
And that means there's
no way to make sense out

1287
01:04:26,360 --> 01:04:29,250
of Newton's Law of
gravity in the context

1288
01:04:29,250 --> 01:04:31,250
of special relativity.

1289
01:04:31,250 --> 01:04:33,000
You can't modify
it by just changing

1290
01:04:33,000 --> 01:04:35,110
the way the force
depends on distance.

1291
01:04:35,110 --> 01:04:36,780
You really have to
change the variables

1292
01:04:36,780 --> 01:04:39,340
that it depends on
from the beginning.

1293
01:04:39,340 --> 01:04:39,840
Yes.

1294
01:04:39,840 --> 01:04:43,067
STUDENT: Is that equivalent to
saying the other particle can't

1295
01:04:43,067 --> 01:04:45,225
know how much mass
of one is [INAUDIBLE]

1296
01:04:45,225 --> 01:04:48,355
limit of the information
of the speed of light?

1297
01:04:48,355 --> 01:04:50,980
PROFESSOR: Is that equivalent to
saying that one particle can't

1298
01:04:50,980 --> 01:04:52,646
know what the mass
of the other particle

1299
01:04:52,646 --> 01:04:54,310
is because of limitation.

1300
01:04:54,310 --> 01:04:56,060
Yeah it's the distance,
it's not the mass.

1301
01:04:56,060 --> 01:04:58,530
Because the mass is
preserved in time.

1302
01:04:58,530 --> 01:05:00,140
So one particle
could've measured

1303
01:05:00,140 --> 01:05:02,181
the mass of the other
particle at an earlier time

1304
01:05:02,181 --> 01:05:05,074
and it would be
reasonable to infer,

1305
01:05:05,074 --> 01:05:06,490
given the laws of
physics, that we

1306
01:05:06,490 --> 01:05:07,700
know that would stay the same.

1307
01:05:07,700 --> 01:05:09,199
But one particle
has no way of where

1308
01:05:09,199 --> 01:05:11,630
the other particle
is at the same time.

1309
01:05:11,630 --> 01:05:14,190
And not only does the particle
not have any way of knowing,

1310
01:05:14,190 --> 01:05:16,770
but even an external
observer can't know.

1311
01:05:16,770 --> 01:05:18,270
Because different
external observers

1312
01:05:18,270 --> 01:05:20,130
will have different
definitions of what

1313
01:05:20,130 --> 01:05:23,070
it means to be at the same time.

1314
01:05:23,070 --> 01:05:26,850
So there really is no
way that this could work.

1315
01:05:26,850 --> 01:05:29,200
Now I should mention
that it is still

1316
01:05:29,200 --> 01:05:33,880
possible to have action
at a distance theories

1317
01:05:33,880 --> 01:05:36,520
which are consistent
with special relativity.

1318
01:05:36,520 --> 01:05:45,002
And in fact, Maxwell's equations
can be reformulated that way.

1319
01:05:45,002 --> 01:05:46,710
The easiest way to
make things consistent

1320
01:05:46,710 --> 01:05:49,910
with special relativity is
to describe interactions

1321
01:05:49,910 --> 01:05:51,290
in terms of fields.

1322
01:05:51,290 --> 01:05:54,510
And that is really what
Einstein did originally

1323
01:05:54,510 --> 01:05:55,427
in special relativity.

1324
01:05:55,427 --> 01:05:56,884
He was thinking
about light, he was

1325
01:05:56,884 --> 01:05:58,370
thinking about
Maxwell's equations

1326
01:05:58,370 --> 01:06:00,494
and he was thinking very
explicitly about Maxwell's

1327
01:06:00,494 --> 01:06:01,630
equations.

1328
01:06:01,630 --> 01:06:06,020
And in the Maxwell description
of electromagnetism,

1329
01:06:06,020 --> 01:06:08,230
particles at a
distance don't directly

1330
01:06:08,230 --> 01:06:09,710
interact with each other.

1331
01:06:09,710 --> 01:06:12,130
But rather, each
particle interacts

1332
01:06:12,130 --> 01:06:15,140
with the fields around it and
that is a local interaction.

1333
01:06:15,140 --> 01:06:17,660
A particle interacts
directly only

1334
01:06:17,660 --> 01:06:20,080
with the fields
at the same point.

1335
01:06:20,080 --> 01:06:22,340
But then those fields
obey wave equations

1336
01:06:22,340 --> 01:06:24,830
that can propagate information.

1337
01:06:24,830 --> 01:06:27,150
And the fact that
you have an electron

1338
01:06:27,150 --> 01:06:29,910
here can create a
field which then

1339
01:06:29,910 --> 01:06:32,490
exerts a force on
an electron there.

1340
01:06:32,490 --> 01:06:33,924
And the force on
the electron here

1341
01:06:33,924 --> 01:06:35,590
depends only on the
electric fields here

1342
01:06:35,590 --> 01:06:37,280
or the magnetic fields as well.

1343
01:06:37,280 --> 01:06:40,190
The particle is moving, but
everything is completely local

1344
01:06:40,190 --> 01:06:43,640
and the description
of electromagnetism as

1345
01:06:43,640 --> 01:06:46,490
given in the form of
Maxwell's equations.

1346
01:06:46,490 --> 01:06:48,800
And Maxwell's equations are
completely relativistically

1347
01:06:48,800 --> 01:06:49,299
invariant.

1348
01:06:49,299 --> 01:06:53,200
And that was part of
Einstein's-- it was really

1349
01:06:53,200 --> 01:06:55,580
the key part of Einstein's
motivation in constructing

1350
01:06:55,580 --> 01:06:58,710
the theory of special relativity
in the first place to make

1351
01:06:58,710 --> 01:07:00,710
Maxwell's equations invariant.

1352
01:07:00,710 --> 01:07:03,640
That they held in every frame.

1353
01:07:03,640 --> 01:07:07,240
It is still possible
though and worth

1354
01:07:07,240 --> 01:07:12,100
recognizing that it's possible
to reformulate electromagnetism

1355
01:07:12,100 --> 01:07:15,320
as an action at a
distance theory.

1356
01:07:15,320 --> 01:07:17,470
And it is in fact
described that way

1357
01:07:17,470 --> 01:07:19,590
in volume one of
the Feynman lectures

1358
01:07:19,590 --> 01:07:23,110
for those of you who've looked
at the Feynman lectures.

1359
01:07:23,110 --> 01:07:27,840
In order to make
that work, you have

1360
01:07:27,840 --> 01:07:31,340
to complicate things in
a pretty significant way.

1361
01:07:31,340 --> 01:07:34,710
So I'm going to draw here
just a space time diagram.

1362
01:07:34,710 --> 01:07:38,729
X and CT, CT going up that
way, x going that way.

1363
01:07:38,729 --> 01:07:40,270
So in this diagram
the speed of light

1364
01:07:40,270 --> 01:07:42,570
would be a 45 degree line.

1365
01:07:42,570 --> 01:07:46,620
And let's suppose we have
two particles traveling

1366
01:07:46,620 --> 01:07:48,480
in this space.

1367
01:07:48,480 --> 01:07:52,765
A particle that I will
call a and a particle

1368
01:07:52,765 --> 01:07:55,840
that I will call b.

1369
01:07:55,840 --> 01:07:57,925
Being very original
with these names.

1370
01:08:00,810 --> 01:08:05,100
If we wanted to know
the force on particle a

1371
01:08:05,100 --> 01:08:09,430
at a certain time t indicated
by this dotted line,

1372
01:08:09,430 --> 01:08:14,790
I guess I'll label t to make
it as clear as possible.

1373
01:08:14,790 --> 01:08:17,680
Feynman gives us a formula
where we can determine it

1374
01:08:17,680 --> 01:08:20,100
solely in terms of the
motion of particle B

1375
01:08:20,100 --> 01:08:23,660
without talking
about fields at all.

1376
01:08:23,660 --> 01:08:28,005
But the formula does not depend
on where b is at the same time.

1377
01:08:28,005 --> 01:08:29,380
And it could not
if this is going

1378
01:08:29,380 --> 01:08:31,450
to be a relativistic
description.

1379
01:08:31,450 --> 01:08:34,760
But instead, the way this action
and the distance formulation

1380
01:08:34,760 --> 01:08:39,609
works is when imagine drawing
a 45 degree line backwards,

1381
01:08:39,609 --> 01:08:42,710
meaning a line that
light could travel on,

1382
01:08:42,710 --> 01:08:44,850
and one sees where
that intersects

1383
01:08:44,850 --> 01:08:50,210
the trajectory of particle
B. And that time is t prime.

1384
01:08:50,210 --> 01:08:53,840
And the word that's used for
that symbol is retarded time.

1385
01:08:53,840 --> 01:08:55,060
It's an earlier time.

1386
01:08:55,060 --> 01:08:59,479
It's exactly that time which has
the property that if particle

1387
01:08:59,479 --> 01:09:01,609
b emitted a light
beam at that time,

1388
01:09:01,609 --> 01:09:03,779
it would be arriving at
particle a at just the time

1389
01:09:03,779 --> 01:09:05,750
t that we're interested
in the time when we're

1390
01:09:05,750 --> 01:09:08,840
trying to calculate the
force on particle a.

1391
01:09:08,840 --> 01:09:12,300
And what Feynman gives you in
volume one if you look at it

1392
01:09:12,300 --> 01:09:13,970
is a very complicated
formula that

1393
01:09:13,970 --> 01:09:15,745
determines the
force on particle a

1394
01:09:15,745 --> 01:09:18,910
in terms of not only the
position of this particle

1395
01:09:18,910 --> 01:09:21,620
at time t prime but also
its velocity and even

1396
01:09:21,620 --> 01:09:23,870
its acceleration.

1397
01:09:23,870 --> 01:09:26,470
But if you do know the
position, the velocity,

1398
01:09:26,470 --> 01:09:28,109
and the acceleration
of this particle,

1399
01:09:28,109 --> 01:09:30,395
and of course the
velocity of particle a,

1400
01:09:30,395 --> 01:09:31,770
you can determine
the force on a.

1401
01:09:34,290 --> 01:09:38,534
Not obvious, but it's true.

1402
01:09:38,534 --> 01:09:40,200
But that's certainly
not the easiest way

1403
01:09:40,200 --> 01:09:41,990
to formulate electromagnetism.

1404
01:09:41,990 --> 01:09:44,660
And that's not the way
most of us have learned,

1405
01:09:44,660 --> 01:09:46,160
unless you've
learned by starting

1406
01:09:46,160 --> 01:09:47,870
by reading volume
one of Feynman.

1407
01:09:47,870 --> 01:09:49,870
But most of us learn
Maxwell's equations

1408
01:09:49,870 --> 01:09:52,040
as differential equations.

1409
01:09:52,040 --> 01:09:54,970
Where information is propagated
by the field from one

1410
01:09:54,970 --> 01:09:57,070
point to another.

1411
01:09:57,070 --> 01:09:59,590
In the case of
general relativity,

1412
01:09:59,590 --> 01:10:01,810
one has the same problem.

1413
01:10:01,810 --> 01:10:05,077
How can you describe
something which--

1414
01:10:05,077 --> 01:10:06,660
and the only approximate
you initially

1415
01:10:06,660 --> 01:10:09,525
know is an action
and the distance,

1416
01:10:09,525 --> 01:10:11,150
how can you describe
it in a way that's

1417
01:10:11,150 --> 01:10:13,590
consistent with relativity?

1418
01:10:13,590 --> 01:10:18,050
And the idea that simultaneity
is not a well defined concept.

1419
01:10:18,050 --> 01:10:19,550
So that was the
problem was Einstein

1420
01:10:19,550 --> 01:10:22,040
was wrestling with
for 10 years, how

1421
01:10:22,040 --> 01:10:24,146
to build a theory
of gravity that

1422
01:10:24,146 --> 01:10:26,020
would be consistent with
the basic principles

1423
01:10:26,020 --> 01:10:27,960
of special relativity.

1424
01:10:27,960 --> 01:10:31,440
And the result of those
10 years of cogitating

1425
01:10:31,440 --> 01:10:34,560
is the theory that we
call general relativity.

1426
01:10:34,560 --> 01:10:36,880
And it's essentially
a theory which

1427
01:10:36,880 --> 01:10:40,670
describes gravity
as a field theory

1428
01:10:40,670 --> 01:10:43,800
similar to Maxwell's
field theory

1429
01:10:43,800 --> 01:10:45,990
where all interactions
are local.

1430
01:10:45,990 --> 01:10:49,640
Nothing interacts at a
distance, but particles interact

1431
01:10:49,640 --> 01:10:52,130
with fields at the
same point, the fields

1432
01:10:52,130 --> 01:10:55,960
can propagate information
by obeying wave equations,

1433
01:10:55,960 --> 01:10:59,765
and the fields then
at a distant point

1434
01:10:59,765 --> 01:11:02,820
can exert forces
on other particles.

1435
01:11:02,820 --> 01:11:06,130
But in the case of general
relativity, what Einstein

1436
01:11:06,130 --> 01:11:08,900
concluded was that the
fields that were relevant,

1437
01:11:08,900 --> 01:11:11,000
the fields that
described gravity,

1438
01:11:11,000 --> 01:11:15,170
were in fact the metric
of space and time.

1439
01:11:15,170 --> 01:11:17,780
So general relativity
is the field theory

1440
01:11:17,780 --> 01:11:20,280
of the metric of space and time.

1441
01:11:20,280 --> 01:11:23,880
And gravity is described
solely as a distortion

1442
01:11:23,880 --> 01:11:25,400
of space and time.

1443
01:11:25,400 --> 01:11:27,310
And that's what general
relativity is about

1444
01:11:27,310 --> 01:11:30,970
and that's what we will
be learning more about.

1445
01:11:30,970 --> 01:11:33,770
Now as I said earlier, now
I can say it perhaps more

1446
01:11:33,770 --> 01:11:36,970
explicitly, what we
will be learning about

1447
01:11:36,970 --> 01:11:39,620
is how to describe the
curvature of space time

1448
01:11:39,620 --> 01:11:42,350
as general relativity
describes it.

1449
01:11:42,350 --> 01:11:44,850
We will learn how that
curvature affects things

1450
01:11:44,850 --> 01:11:47,370
like the motions of particles.

1451
01:11:47,370 --> 01:11:49,670
But we will not in
this course learn

1452
01:11:49,670 --> 01:11:53,840
how the presence of
particles and masses

1453
01:11:53,840 --> 01:11:55,540
affects the curvature
of space time.

1454
01:11:55,540 --> 01:11:59,460
That you'll have to take it in
a relativity course to learn.

1455
01:11:59,460 --> 01:12:01,807
OK, I think that's where
we'll be stopping today.

1456
01:12:01,807 --> 01:12:04,390
Just doesn't pay to start a new
topic with a minute and a half

1457
01:12:04,390 --> 01:12:05,075
left.

1458
01:12:05,075 --> 01:12:07,033
But let me just ask if
there are any questions.

1459
01:12:11,010 --> 01:12:12,740
OK, well class
over, I will see you

1460
01:12:12,740 --> 01:12:16,150
folks in a week, because
there's no class next Tuesday.