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PROFESSOR: I'd like
to begin by reviewing

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the last lecture,
where we introduced

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00:00:30,879 --> 00:00:32,253
the idea of
non-Euclidean spaces.

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00:00:35,570 --> 00:00:39,610
The important idea here is that
general relativity describes

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gravity as a distortion
of space and time.

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And that idea becomes
crucial in cosmology,

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00:00:46,880 --> 00:00:49,550
so we want to understand
what that's about.

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00:00:49,550 --> 00:00:53,070
As I explained last
time, there are really

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00:00:53,070 --> 00:00:56,280
two aspects to this
general subject.

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00:00:56,280 --> 00:01:00,210
There's the question of, how
do we treat curved spacetimes

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00:01:00,210 --> 00:01:02,960
and how do particles,
for example, behave

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00:01:02,960 --> 00:01:05,349
when they move through
curved spacetimes.

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00:01:05,349 --> 00:01:07,280
That we will be doing.

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00:01:07,280 --> 00:01:09,110
There's also the
important question

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00:01:09,110 --> 00:01:13,420
of how does matter distort
the spacetime, which

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00:01:13,420 --> 00:01:15,410
is the Einstein field equations.

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00:01:15,410 --> 00:01:16,790
That we will not do.

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00:01:16,790 --> 00:01:20,420
That we will leave for the
general relativity course

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00:01:20,420 --> 00:01:23,040
that you may or may not want
to take at some point, or maybe

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00:01:23,040 --> 00:01:25,710
you already have.

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00:01:25,710 --> 00:01:27,720
The idea of non
Euclidean geometry

29
00:01:27,720 --> 00:01:29,740
really goes back
to Euclid himself,

30
00:01:29,740 --> 00:01:31,900
and the fifth postulate.

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00:01:31,900 --> 00:01:35,640
The distinction between
Euclidean geometry and what

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00:01:35,640 --> 00:01:38,570
is generally called non
Euclidean geometry is entirely

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00:01:38,570 --> 00:01:39,870
in the fifth postulate.

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00:01:39,870 --> 00:01:43,534
We never bothered changing
the first four postulates.

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00:01:43,534 --> 00:01:45,200
This fifth postulate
is explained better

36
00:01:45,200 --> 00:01:47,680
in the next slide,
where we have a diagram.

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00:01:47,680 --> 00:01:52,767
The fifth postulate says that
if a straight line falls on

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00:01:52,767 --> 00:01:57,420
to other straight lines,
such that the sum of the two

39
00:01:57,420 --> 00:02:01,240
included angles is
less than pi, then

40
00:02:01,240 --> 00:02:04,010
these two lines will
meet on the same side

41
00:02:04,010 --> 00:02:06,450
where the sum of the
angles is less than pi,

42
00:02:06,450 --> 00:02:09,509
and will not meet
on the other side.

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00:02:09,509 --> 00:02:13,890
This postulate was something
that attracted attention,

44
00:02:13,890 --> 00:02:15,640
really from the very
beginning, because it

45
00:02:15,640 --> 00:02:18,420
seems like a much more
complicated postulate

46
00:02:18,420 --> 00:02:20,690
than the other four.

47
00:02:20,690 --> 00:02:22,960
And for many years,
mathematicians

48
00:02:22,960 --> 00:02:26,830
tried to derive this
postulate from the others,

49
00:02:26,830 --> 00:02:30,250
thinking that it couldn't really
be a fundamental postulate.

50
00:02:30,250 --> 00:02:36,130
But that never worked, and
eventually in the 17 and 1800s,

51
00:02:36,130 --> 00:02:40,932
mathematicians realized that
the postulate was independent.

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00:02:40,932 --> 00:02:43,390
And you could either assume
it's true or assume it's false.

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00:02:43,390 --> 00:02:48,240
And you get different versions
of geometry in the two cases.

54
00:02:48,240 --> 00:02:51,570
We learned last time that there
are four other ways, at least,

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00:02:51,570 --> 00:02:54,220
of stating the things that
are equivalent to the fifth

56
00:02:54,220 --> 00:02:57,560
postulate, and they're
diagrammed here.

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00:02:57,560 --> 00:03:00,340
If a straight line intersects
one of two parallel lines,

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00:03:00,340 --> 00:03:03,920
it will also
intersect the other.

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00:03:03,920 --> 00:03:06,840
If one has a straight
line and another point,

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00:03:06,840 --> 00:03:10,600
there is one and only one line
through that point, parallel

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00:03:10,600 --> 00:03:13,330
to the original line.

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00:03:13,330 --> 00:03:16,430
If one has a figure,
one can construct

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00:03:16,430 --> 00:03:20,400
a figure which is similar
to it, of any size.

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00:03:20,400 --> 00:03:22,630
And finally, the
famous statement

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00:03:22,630 --> 00:03:25,110
about sum of the
angles that make up

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00:03:25,110 --> 00:03:27,420
the vertices of a triangle.

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00:03:27,420 --> 00:03:29,640
If there exists just one
triangle for which it's

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00:03:29,640 --> 00:03:32,822
180 degrees, then
that's equivalent

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00:03:32,822 --> 00:03:34,280
to the fifth
postulate, and you can

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00:03:34,280 --> 00:03:39,350
prove that every
triangle has 180 degrees.

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00:03:39,350 --> 00:03:44,280
The fifth postulate was
questioned in a serious way

72
00:03:44,280 --> 00:03:51,180
by Giovanni Geralamo
Saccheri in the 16, 1700s,

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00:03:51,180 --> 00:03:54,660
who wrote a detailed study of
what geometry would be like

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00:03:54,660 --> 00:03:57,180
if the fifth
postulate were false.

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00:03:57,180 --> 00:04:01,080
He wrote this believing that the
fifth postulate must be true.

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00:04:01,080 --> 00:04:04,600
And he was looking for a
contradiction, which he never

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00:04:04,600 --> 00:04:07,240
found.

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00:04:07,240 --> 00:04:14,360
Things went further in the later
1700s, with Gauss and Bolyai

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00:04:14,360 --> 00:04:17,980
and Lobachevski, who
independently developed

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00:04:17,980 --> 00:04:21,170
the geometry that we call
Gauss Bolyai Lobachevski

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00:04:21,170 --> 00:04:25,530
geometry, which is a two
dimensional geometry,

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00:04:25,530 --> 00:04:27,246
non Euclidean.

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00:04:27,246 --> 00:04:30,365
It corresponds to what we
now call an open universe,

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00:04:30,365 --> 00:04:35,600
that we'll be learning about
in more detail later today.

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00:04:35,600 --> 00:04:37,450
The Gauss Bolyai
Lobachevski geometry

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00:04:37,450 --> 00:04:40,930
was treated purely axiomatically
by the three authors

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00:04:40,930 --> 00:04:42,580
that I just mentioned.

88
00:04:42,580 --> 00:04:44,740
But it was given a
coordinate representation

89
00:04:44,740 --> 00:04:48,730
by Felix Klein in
1870, which was really

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00:04:48,730 --> 00:04:51,520
the first demonstration
that it really existed.

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00:04:51,520 --> 00:04:53,190
When one treats
it axiomatically,

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00:04:53,190 --> 00:04:55,050
one still always
has the possibility

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00:04:55,050 --> 00:04:58,060
that some contradiction
could be found someplace.

94
00:04:58,060 --> 00:05:01,360
But by the time you put it
into algebraic equations,

95
00:05:01,360 --> 00:05:04,900
then it becomes as consistent
as our understanding

96
00:05:04,900 --> 00:05:07,480
of the real numbers, which
we have a lot of confidence

97
00:05:07,480 --> 00:05:09,690
in, even though I don't
think mathematicians really

98
00:05:09,690 --> 00:05:12,530
know how to prove the
consistency of anything.

99
00:05:12,530 --> 00:05:17,870
But we have a lot of confidence
in this kind of mathematics.

100
00:05:17,870 --> 00:05:19,920
So by 1870, it was
absolutely clear

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00:05:19,920 --> 00:05:23,600
that this open geometry,
this non-Euclidean geometry,

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00:05:23,600 --> 00:05:27,400
was a perfectly consistent,
is a perfectly consistent,

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00:05:27,400 --> 00:05:28,845
formulation of geometry.

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00:05:33,000 --> 00:05:35,390
An important development
coming out of Kline's work

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00:05:35,390 --> 00:05:39,630
is that the idea about
how one describes geometry

106
00:05:39,630 --> 00:05:41,400
changed dramatically.

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00:05:41,400 --> 00:05:45,040
Prior to Kline,
essentially all of geometry

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00:05:45,040 --> 00:05:47,500
was done in the same
way that Euclid did it,

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00:05:47,500 --> 00:05:51,210
by writing down axioms
and then proving theorems.

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00:05:51,210 --> 00:05:54,420
Kline realized that you
could gain a lot of mileage

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00:05:54,420 --> 00:05:57,370
by taking advantage of our
understanding of algebra

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00:05:57,370 --> 00:06:01,270
and calculus by describing
things in terms of functions.

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00:06:01,270 --> 00:06:03,580
And in particular,
geometry is described

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00:06:03,580 --> 00:06:07,590
by giving a distance
function between points.

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00:06:07,590 --> 00:06:11,970
This was further
developed by Gauss,

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00:06:11,970 --> 00:06:16,835
who realized that
distances are additive.

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00:06:16,835 --> 00:06:18,460
So if distances mean
anything like what

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00:06:18,460 --> 00:06:20,080
we think distances
mean, it would

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00:06:20,080 --> 00:06:23,100
be sufficient to describe
the distance between any two

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00:06:23,100 --> 00:06:24,980
arbitrarily close points.

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00:06:24,980 --> 00:06:27,230
And then if you want to know
the distance between two

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00:06:27,230 --> 00:06:29,780
distant points, you draw
a line between them,

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00:06:29,780 --> 00:06:31,560
and measure the
length of that line

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00:06:31,560 --> 00:06:36,250
by adding up an infinite number
of infinitesimal segments.

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00:06:36,250 --> 00:06:40,180
So the idea that distances need
only be defined infinitesimally

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00:06:40,180 --> 00:06:44,810
was very crucial to our current
understanding of geometry.

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00:06:44,810 --> 00:06:47,390
Gauss also introduced
another important idea,

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00:06:47,390 --> 00:06:51,480
which is a restriction on what
that infinitesimal distance

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00:06:51,480 --> 00:06:53,850
function should look like.

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00:06:53,850 --> 00:06:55,850
Gauss proposed that
it should always

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00:06:55,850 --> 00:06:58,120
have the same
quadratic form that it

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00:06:58,120 --> 00:07:00,170
has for Euclidean distances.

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00:07:00,170 --> 00:07:02,780
For Euclidean distances,
the Pythagorean theorem

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00:07:02,780 --> 00:07:05,640
tells us that the distance
between any two points

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00:07:05,640 --> 00:07:09,640
is the sum of the squares
of the coordinate distances.

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00:07:09,640 --> 00:07:14,140
And for non-Euclidean
geometry, we generalize that

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00:07:14,140 --> 00:07:17,850
by allowing each term in
this quadratic expansion

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00:07:17,850 --> 00:07:23,040
to have its own prefactor,
and those prefactors

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00:07:23,040 --> 00:07:24,910
could be functions of position.

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00:07:24,910 --> 00:07:28,800
So g sub xx of xy is just
a function of x and y.

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00:07:28,800 --> 00:07:32,040
And g sub xy is another
function of x and y.

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00:07:32,040 --> 00:07:34,970
And g sub yy is another
function of x and y.

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00:07:34,970 --> 00:07:36,570
And the distance
function is taken

144
00:07:36,570 --> 00:07:40,400
as the sum of those three terms.

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00:07:40,400 --> 00:07:46,050
The important feature that that
quadratic form corresponds to,

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00:07:46,050 --> 00:07:52,570
which was noticed by Gauss, is
that if the distance function

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00:07:52,570 --> 00:07:55,570
has that form, it means that
even though the space is not

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00:07:55,570 --> 00:07:57,420
Euclidean and will
not obey, in general,

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00:07:57,420 --> 00:07:59,105
the axioms of
Euclidean geometry--

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00:07:59,105 --> 00:08:01,340
and in particular
the fifth postulate--

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00:08:01,340 --> 00:08:06,270
it is still true that in
a very tiny neighborhood,

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00:08:06,270 --> 00:08:09,010
it will resemble
Euclidean geometry, where

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00:08:09,010 --> 00:08:11,050
the resemblance will
become more and more exact

154
00:08:11,050 --> 00:08:13,770
as you confine yourself
to tinier and tinier

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00:08:13,770 --> 00:08:15,170
neighborhoods.

156
00:08:15,170 --> 00:08:20,430
And we're kind of aware
of this in everyday life.

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00:08:20,430 --> 00:08:24,200
The surface of the earth is
approximately spherical-- we'll

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00:08:24,200 --> 00:08:26,780
ignore little things like
mountains and roads and bumps,

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00:08:26,780 --> 00:08:29,420
and pretend the surface
of the Earth is spherical.

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00:08:29,420 --> 00:08:31,805
Nonetheless, the surface of
the Earth looks flat to us.

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00:08:31,805 --> 00:08:33,429
And the reason it
looks flat is that we

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00:08:33,429 --> 00:08:35,140
see only a tiny
little bit of it.

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00:08:35,140 --> 00:08:39,370
And a tiny piece of a curved
surface always looks flat.

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00:08:39,370 --> 00:08:41,929
And mathematically
speaking, the way

165
00:08:41,929 --> 00:08:47,000
to introduce enough assumptions
to validate that conclusion

166
00:08:47,000 --> 00:08:49,570
is to assume that the
local distance function is

167
00:08:49,570 --> 00:08:51,840
a quadratic function
of this form.

168
00:08:51,840 --> 00:08:53,840
And what Gauss
originally proved is

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00:08:53,840 --> 00:08:56,840
that if the distance
function is of that form,

170
00:08:56,840 --> 00:08:59,260
it is always true that
in a tiny neighborhood,

171
00:08:59,260 --> 00:09:03,160
to an arbitrary accuracy, you
could define new coordinates--

172
00:09:03,160 --> 00:09:07,310
x prime and y prime in the
notation of this diagram--

173
00:09:07,310 --> 00:09:09,040
where in terms of
the new coordinates

174
00:09:09,040 --> 00:09:11,510
in the tiny neighborhood,
the distances are just

175
00:09:11,510 --> 00:09:13,030
the Euclidean distances.

176
00:09:13,030 --> 00:09:17,870
ds squared equals dx prime
squared plus dy prime squared.

177
00:09:17,870 --> 00:09:21,460
And that's a very crucial fact
that we will be making use of,

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00:09:21,460 --> 00:09:23,919
Einstein made use
of, in the context

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00:09:23,919 --> 00:09:25,960
of general relativity,
which we'll be getting to.

180
00:09:28,470 --> 00:09:31,360
OK, that finishes the review.

181
00:09:31,360 --> 00:09:34,150
Any questions about anything
that we talked about last time?

182
00:09:37,790 --> 00:09:39,100
OK, great.

183
00:09:39,100 --> 00:09:41,330
Now what we want
to do is to go on

184
00:09:41,330 --> 00:09:43,400
to apply these ideas in detail.

185
00:09:43,400 --> 00:09:45,800
In one of them in particular,
build up a full description

186
00:09:45,800 --> 00:09:48,670
of closed and open
universes today.

187
00:09:48,670 --> 00:09:52,720
And we are going
to begin by giving

188
00:09:52,720 --> 00:09:55,260
a mathematical description
of the simplest

189
00:09:55,260 --> 00:09:58,650
non-Euclidean geometry
that we have available,

190
00:09:58,650 --> 00:10:01,510
which is just the surface of
a sphere-- a two dimensional

191
00:10:01,510 --> 00:10:03,320
sphere embedded in
three dimensions.

192
00:10:03,320 --> 00:10:04,850
That is, a two
dimensional surface

193
00:10:04,850 --> 00:10:06,800
embedded in a three
dimensional space,

194
00:10:06,800 --> 00:10:10,770
as is intended to be
shown in that diagram.

195
00:10:10,770 --> 00:10:13,170
The sphere is described
simply by x squared plus y

196
00:10:13,170 --> 00:10:16,160
squared equals plus z
squared equals r squared,

197
00:10:16,160 --> 00:10:19,700
where x, y, and z are just
Euclidean coordinates.

198
00:10:19,700 --> 00:10:21,530
So in this case,
our curved space,

199
00:10:21,530 --> 00:10:23,450
which is the surface
of the sphere,

200
00:10:23,450 --> 00:10:27,830
can be embedded in a Euclidean
space of one higher dimension.

201
00:10:27,830 --> 00:10:29,050
That's not always the case.

202
00:10:29,050 --> 00:10:31,490
We should not pretend that
that will always be the case.

203
00:10:31,490 --> 00:10:33,420
But when it is the
case, it allows

204
00:10:33,420 --> 00:10:36,460
us to study that curved surface
in a very straightforward way,

205
00:10:36,460 --> 00:10:39,010
because everything
is really determined

206
00:10:39,010 --> 00:10:42,960
by the Euclidean geometry of the
space in which this sphere is

207
00:10:42,960 --> 00:10:44,630
embedded.

208
00:10:44,630 --> 00:10:48,400
Nonetheless, when
we're done formalizing

209
00:10:48,400 --> 00:10:51,070
our description of the
surface of the sphere,

210
00:10:51,070 --> 00:10:54,650
the goal will be to concentrate
on what Gauss called

211
00:10:54,650 --> 00:10:58,110
the "inner properties," namely
the properties of the surface

212
00:10:58,110 --> 00:10:58,672
itself.

213
00:10:58,672 --> 00:11:00,880
And we will try to pretend
that the three dimensional

214
00:11:00,880 --> 00:11:02,420
space never even existed.

215
00:11:02,420 --> 00:11:03,890
It won't be required
for anything

216
00:11:03,890 --> 00:11:06,100
that we'll be left
with, once we have

217
00:11:06,100 --> 00:11:09,060
a solid description
of the surface itself.

218
00:11:09,060 --> 00:11:11,900
And this will be very important
for what we'll be doing later.

219
00:11:11,900 --> 00:11:16,240
So it is important to get
in touch with the idea

220
00:11:16,240 --> 00:11:19,010
that we're going to study this
sphere, making use of the fact

221
00:11:19,010 --> 00:11:21,610
that it could be embedded in
three Euclidean dimensions.

222
00:11:21,610 --> 00:11:26,710
But in the end, we want to
think of it as a two dimensional

223
00:11:26,710 --> 00:11:30,950
geometry, which
is non Euclidean.

224
00:11:30,950 --> 00:11:32,790
OK, so our goal will
be to write down

225
00:11:32,790 --> 00:11:36,020
the distance function
for some coordinization

226
00:11:36,020 --> 00:11:39,000
of the surface of the sphere.

227
00:11:39,000 --> 00:11:41,790
I should say at the beginning,
when this picture makes

228
00:11:41,790 --> 00:11:44,320
it most obvious, that one
of the reasons we might

229
00:11:44,320 --> 00:11:46,630
be interested in the surface
of a sphere, if we're

230
00:11:46,630 --> 00:11:49,520
interested in cosmology, is
that we know that we're trying

231
00:11:49,520 --> 00:11:51,540
to build cosmological
models that

232
00:11:51,540 --> 00:11:54,284
are consistent with
homogeneity and isotropy.

233
00:11:54,284 --> 00:11:55,700
Because we discussed
earlier those

234
00:11:55,700 --> 00:11:59,380
are, to a very good
approximation, valid features

235
00:11:59,380 --> 00:12:01,950
of the universe that
we're living in.

236
00:12:01,950 --> 00:12:04,984
So the surface of a sphere
has those properties.

237
00:12:04,984 --> 00:12:06,650
It's certainly
homogeneous, in the sense

238
00:12:06,650 --> 00:12:08,780
that any point on the
surface of a sphere

239
00:12:08,780 --> 00:12:11,110
will look exactly
like any other point.

240
00:12:11,110 --> 00:12:13,620
If you were living
on that sphere,

241
00:12:13,620 --> 00:12:15,790
and you didn't have
any other landmarks,

242
00:12:15,790 --> 00:12:19,620
you'd have no way of knowing
where on the sphere you were.

243
00:12:19,620 --> 00:12:23,390
Furthermore, it's isotropic--
same in all directions.

244
00:12:23,390 --> 00:12:25,180
And when I say
that, it's important

245
00:12:25,180 --> 00:12:27,410
that I really mean it in
the context of the two

246
00:12:27,410 --> 00:12:31,050
dimensional surface, not the
three dimensional geometry.

247
00:12:31,050 --> 00:12:33,050
So the three dimensional
geometry is isotropic.

248
00:12:33,050 --> 00:12:34,674
If you sat at the
center of that sphere

249
00:12:34,674 --> 00:12:36,979
and looked any direction
in three dimensions,

250
00:12:36,979 --> 00:12:38,270
everything would look the same.

251
00:12:38,270 --> 00:12:41,650
But that's not the isotropy
that's important for us.

252
00:12:41,650 --> 00:12:44,200
We want to imagine ourselves
as two dimensional creatures

253
00:12:44,200 --> 00:12:46,002
living on the surface.

254
00:12:46,002 --> 00:12:48,460
And then you can imagine that
if you were a two dimensional

255
00:12:48,460 --> 00:12:50,640
creature living on the
surface-- so you happen

256
00:12:50,640 --> 00:12:52,765
to be at the North Pole,
because that's the easiest

257
00:12:52,765 --> 00:12:55,690
to describe-- you could
imagine looking around

258
00:12:55,690 --> 00:13:00,715
in a circle, 360%
available, and the world

259
00:13:00,715 --> 00:13:02,840
that you'd be living in
would look exactly the same

260
00:13:02,840 --> 00:13:06,120
in all directions
on the surface.

261
00:13:06,120 --> 00:13:08,890
And that's the isotropy
that's important to us,

262
00:13:08,890 --> 00:13:11,280
because the two
dimensional surface here

263
00:13:11,280 --> 00:13:12,900
is what we're soon
going to generalize

264
00:13:12,900 --> 00:13:15,150
to be our three
dimensional world.

265
00:13:15,150 --> 00:13:22,140
And it's isotropy within that
world that we're talking about.

266
00:13:22,140 --> 00:13:25,900
OK, so, first thing
we wanted to do

267
00:13:25,900 --> 00:13:29,070
is to put coordinates on
our two dimensional surface.

268
00:13:29,070 --> 00:13:31,950
If we want to ultimately
forget the third dimension

269
00:13:31,950 --> 00:13:34,230
and live in the
surface, we want to have

270
00:13:34,230 --> 00:13:36,070
coordinates to use
in the surface.

271
00:13:36,070 --> 00:13:37,645
It's a two dimensional
surface, so it

272
00:13:37,645 --> 00:13:39,960
should have two coordinates.

273
00:13:39,960 --> 00:13:42,790
And when we use the
usual coordinization

274
00:13:42,790 --> 00:13:46,406
of a sphere, polar
coordinates, well, it

275
00:13:46,406 --> 00:13:49,230
will be two angles,
theta and phi.

276
00:13:49,230 --> 00:13:52,726
And there are some
different dimensions

277
00:13:52,726 --> 00:13:54,100
that are used in
different books,

278
00:13:54,100 --> 00:13:57,600
but I think almost all physics
book use these conventions.

279
00:13:57,600 --> 00:14:01,250
Theta is an angle
measured from the z-axis,

280
00:14:01,250 --> 00:14:05,500
and phi is an angle measured
by taking the point that you're

281
00:14:05,500 --> 00:14:08,810
trying to describe, which is
that dot there, projecting it

282
00:14:08,810 --> 00:14:12,030
down into the xy plane,
and in the xy plane,

283
00:14:12,030 --> 00:14:15,460
measuring the angle
from the x-axis.

284
00:14:15,460 --> 00:14:16,230
So that's phi.

285
00:14:16,230 --> 00:14:21,690
And theta and phi are the polar
coordinates describing a point

286
00:14:21,690 --> 00:14:23,065
on the surface of the sphere.

287
00:14:23,065 --> 00:14:25,630
And what we want to do is
describe the distance function

288
00:14:25,630 --> 00:14:27,880
in terms of those polar
coordinates-- that's our goal.

289
00:14:34,680 --> 00:14:37,695
OK, to describe the
distance function, what

290
00:14:37,695 --> 00:14:43,670
we want to imagine is two
infinitesimally nearby points--

291
00:14:43,670 --> 00:14:47,390
one described by
coordinates theta and phi,

292
00:14:47,390 --> 00:14:53,500
and one described by theta plus
d theta and phi plus d phi.

293
00:14:56,230 --> 00:15:00,120
So we have one point
described by theta and phi,

294
00:15:00,120 --> 00:15:06,230
and another point described by
theta plus d theta, phi plus d

295
00:15:06,230 --> 00:15:07,850
phi.

296
00:15:07,850 --> 00:15:10,532
So the coordinate changes
are just d theta and d phi.

297
00:15:10,532 --> 00:15:12,240
What we want to know
is how much distance

298
00:15:12,240 --> 00:15:16,720
is undergone by moving
from the first point

299
00:15:16,720 --> 00:15:18,920
to the second point.

300
00:15:18,920 --> 00:15:21,710
And the easiest way to see
it is to make the changes one

301
00:15:21,710 --> 00:15:23,710
at a time.

302
00:15:23,710 --> 00:15:27,080
So first, we can
just vary theta.

303
00:15:27,080 --> 00:15:29,290
And if we just
vary theta, we see

304
00:15:29,290 --> 00:15:33,840
that the point described
by theta and phi moves

305
00:15:33,840 --> 00:15:39,640
along a great circle, which
goes through the z-axis.

306
00:15:39,640 --> 00:15:42,570
And the distance
that the point goes

307
00:15:42,570 --> 00:15:47,170
is just an arc length as a
piece of that great circle.

308
00:15:47,170 --> 00:15:53,090
And since this attended angle
is d theta, and the radius is r,

309
00:15:53,090 --> 00:15:54,750
the distance of that
arc length really

310
00:15:54,750 --> 00:15:58,270
just follows from the definition
of an angle in radians.

311
00:15:58,270 --> 00:16:00,870
The arc length is
r times d theta,

312
00:16:00,870 --> 00:16:04,390
and that's really the definition
of d theta in radians.

313
00:16:04,390 --> 00:16:14,870
So if we vary theta
only, the distance ds

314
00:16:14,870 --> 00:16:17,710
is just equal to
r times d theta.

315
00:16:20,440 --> 00:16:22,960
Everybody happy with that?

316
00:16:22,960 --> 00:16:23,850
OK.

317
00:16:23,850 --> 00:16:27,010
Now if we vary phi, it's
slightly more complicated,

318
00:16:27,010 --> 00:16:28,400
but not much.

319
00:16:28,400 --> 00:16:33,117
If we vary phi, the
point being described

320
00:16:33,117 --> 00:16:34,950
would-- if you vary phi
all the way around--

321
00:16:34,950 --> 00:16:38,590
make a circle around the z-axis.

322
00:16:38,590 --> 00:16:40,790
That circle does
not have radius r.

323
00:16:40,790 --> 00:16:43,600
That's the one thing that may
be a little bit surprising,

324
00:16:43,600 --> 00:16:46,710
until you look at the picture
and see that it's true.

325
00:16:46,710 --> 00:16:50,350
The radius of that circle
is r times sine theta.

326
00:16:50,350 --> 00:16:52,540
So in particular,
if theta were 0,

327
00:16:52,540 --> 00:16:55,060
if you're up around the
North Pole, going around

328
00:16:55,060 --> 00:16:57,720
that circle would just be
going around the point.

329
00:16:57,720 --> 00:16:59,230
0 radius.

330
00:16:59,230 --> 00:17:02,180
And you have maximum radius
when you're at the equator,

331
00:17:02,180 --> 00:17:04,660
going all the way around.

332
00:17:04,660 --> 00:17:06,650
So again, we're
going in a circle

333
00:17:06,650 --> 00:17:10,740
through an angle--
in this case d phi.

334
00:17:10,740 --> 00:17:13,680
So the arc length is just
the angle times the radius.

335
00:17:13,680 --> 00:17:19,270
But the radius is r times
sine theta, not r itself.

336
00:17:19,270 --> 00:17:35,450
So when we vary phi only, ds
is equal to r times sine theta

337
00:17:35,450 --> 00:17:36,160
times d phi.

338
00:17:40,597 --> 00:17:43,380
Any questions?

339
00:17:43,380 --> 00:17:46,100
OK, now, the next
important thing to notice

340
00:17:46,100 --> 00:17:48,820
is that these two
variations that we made

341
00:17:48,820 --> 00:17:50,730
are orthogonal to each other.

342
00:17:50,730 --> 00:17:54,000
When we varied phi, we moved
in the horizontal plane.

343
00:17:54,000 --> 00:17:58,030
There's only motion in the x and
y directions when we vary phi.

344
00:17:58,030 --> 00:18:04,450
When we vary theta, we move
in the vertical direction.

345
00:18:04,450 --> 00:18:07,170
And those two vectors
are orthogonal,

346
00:18:07,170 --> 00:18:08,970
as you can see from the diagram.

347
00:18:08,970 --> 00:18:11,270
And because we have two
orthogonal distances

348
00:18:11,270 --> 00:18:14,490
that we're adding
up, and because we're

349
00:18:14,490 --> 00:18:16,660
in underlying Euclidean
space here-- we

350
00:18:16,660 --> 00:18:19,360
can think of those distances
as being distances in the three

351
00:18:19,360 --> 00:18:21,280
dimensional Euclidean
space that we're

352
00:18:21,280 --> 00:18:25,440
embedded in-- we get to use
the Pythagorean theorem.

353
00:18:25,440 --> 00:18:33,670
So putting together
these two variations,

354
00:18:33,670 --> 00:18:39,050
we get ds squared is equal to
an overall factor of r squared,

355
00:18:39,050 --> 00:18:46,530
times d theta squared plus sine
squared theta, d phi squared.

356
00:18:51,520 --> 00:18:54,340
And that formula then
describes the metric

357
00:18:54,340 --> 00:18:56,450
on the surface of a sphere.

358
00:18:56,450 --> 00:18:58,243
And it describes a
non-Euclidean geometry.

359
00:19:02,690 --> 00:19:04,250
And once we have
that metric, we can

360
00:19:04,250 --> 00:19:05,940
forget the three
dimensional picture

361
00:19:05,940 --> 00:19:08,270
that we've been
drawing, and just

362
00:19:08,270 --> 00:19:09,800
think of a world
in which there are

363
00:19:09,800 --> 00:19:12,460
two coordinates-- theta and
phi-- with that distance

364
00:19:12,460 --> 00:19:13,020
function.

365
00:19:13,020 --> 00:19:15,353
And that's the way we want
to be able to think about it.

366
00:19:18,730 --> 00:19:19,680
OK, everybody happy?

367
00:19:23,550 --> 00:19:25,930
OK, I want to mention,
because it is perhaps

368
00:19:25,930 --> 00:19:29,110
useful in other cases, depending
on your taste of how you like

369
00:19:29,110 --> 00:19:32,900
to solve problems-- the
description I just gave

370
00:19:32,900 --> 00:19:38,015
of deriving this formula
was geometric, that

371
00:19:38,015 --> 00:19:40,190
is, we drew pictures and
wrote down the answer

372
00:19:40,190 --> 00:19:42,890
based on visualizing
the pictures.

373
00:19:42,890 --> 00:19:46,280
But it can also be done
purely algebraically.

374
00:19:46,280 --> 00:19:49,470
To do it purely
algebraically, one

375
00:19:49,470 --> 00:19:54,330
would first write
down formulas that

376
00:19:54,330 --> 00:19:57,900
relayed the angular
coordinates--

377
00:19:57,900 --> 00:19:59,075
I should go back to slides.

378
00:20:06,395 --> 00:20:07,530
What's going on here?

379
00:20:07,530 --> 00:20:08,495
Is my computer frozen?

380
00:20:23,120 --> 00:20:27,610
[INAUDIBLE] Yes?

381
00:20:27,610 --> 00:20:29,068
AUDIENCE: For that
formula, are you

382
00:20:29,068 --> 00:20:32,719
assuming that d theta and d phi
are really small so that you

383
00:20:32,719 --> 00:20:35,490
can [INAUDIBLE] the triangle?

384
00:20:35,490 --> 00:20:38,830
PROFESSOR: Yes, these are
infinitesimal separations only.

385
00:20:38,830 --> 00:20:41,490
That's the key idea of Gauss.

386
00:20:41,490 --> 00:20:43,100
And yes, we're
making use of that.

387
00:20:43,100 --> 00:20:45,100
This formula will not
hold if d theta and d phi

388
00:20:45,100 --> 00:20:47,640
were large angles.

389
00:20:47,640 --> 00:20:51,125
Holds only when
they're infinitesimal.

390
00:20:51,125 --> 00:20:51,625
Yes?

391
00:20:51,625 --> 00:20:53,000
AUDIENCE: And then
similiarly, we

392
00:20:53,000 --> 00:20:55,633
can use the line integral for
calculating the distances based

393
00:20:55,633 --> 00:20:56,382
on this metric?

394
00:20:56,382 --> 00:20:57,590
PROFESSOR: Yes, that's right.

395
00:20:57,590 --> 00:21:00,490
If we wanted to know the
distances between two finitely

396
00:21:00,490 --> 00:21:04,260
separated points, we would
construct a line between them,

397
00:21:04,260 --> 00:21:06,470
and then integrate
along that line.

398
00:21:06,470 --> 00:21:10,310
And by line what we mean is
the path of shortest distance,

399
00:21:10,310 --> 00:21:14,760
which we'll be learning about
more next time, probably.

400
00:21:14,760 --> 00:21:17,080
Those are not necessarily
easy to calculate.

401
00:21:17,080 --> 00:21:19,560
In this case, they're
calculable, but not really

402
00:21:19,560 --> 00:21:20,060
easy.

403
00:21:24,592 --> 00:21:25,425
Any other questions?

404
00:21:28,920 --> 00:21:33,300
OK, so what I wanted
to do was to look

405
00:21:33,300 --> 00:21:35,690
at the definition of the
coordinate system, as now

406
00:21:35,690 --> 00:21:36,769
shown on the screen.

407
00:21:36,769 --> 00:21:38,810
And from that, we can
write down the relationship

408
00:21:38,810 --> 00:21:42,000
between x, y, and z,
and theta and phi.

409
00:21:42,000 --> 00:21:44,400
And those relationships
are that x

410
00:21:44,400 --> 00:21:50,060
is equal to r times sine
theta times cosine phi.

411
00:21:50,060 --> 00:21:56,850
y is equal to r times
sine theta times sine phi.

412
00:21:56,850 --> 00:22:01,070
And z is equal to r
times cosine theta.

413
00:22:05,130 --> 00:22:08,750
And once one writes
those formulas,

414
00:22:08,750 --> 00:22:13,370
then one can just use
straightforward calculus

415
00:22:13,370 --> 00:22:15,740
to get the metric,
without needing

416
00:22:15,740 --> 00:22:17,070
to draw any pictures at all.

417
00:22:17,070 --> 00:22:20,050
You may have wanted to draw
pictures to get these formulas,

418
00:22:20,050 --> 00:22:21,720
but once you have
these formulas,

419
00:22:21,720 --> 00:22:25,390
you can get that by
straightforward calculus.

420
00:22:25,390 --> 00:22:29,280
I'll sketch the calculation
without writing it out in full.

421
00:22:29,280 --> 00:22:35,790
But given this formula
for x, we can write down

422
00:22:35,790 --> 00:22:40,780
what dx is by calculus,
by chain rule.

423
00:22:40,780 --> 00:22:43,860
So dx-- it's a function
of two variables.

424
00:22:43,860 --> 00:22:48,100
So it would be the partial of x,
with respect to theta, times d

425
00:22:48,100 --> 00:22:50,620
theta, plus the
partial derivative

426
00:22:50,620 --> 00:22:54,315
of x, with respect
to phi, times d phi.

427
00:22:58,800 --> 00:23:02,230
And we'll go work out what
these partial derivatives are

428
00:23:02,230 --> 00:23:04,460
if I differentiate this
with respect to theta.

429
00:23:04,460 --> 00:23:07,970
The sine theta turns
into cosine theta.

430
00:23:07,970 --> 00:23:18,330
So the first term becomes
r cosine theta, cosine phi,

431
00:23:18,330 --> 00:23:20,350
d theta.

432
00:23:20,350 --> 00:23:23,170
And then plus, from here we have
the partial of x with respect

433
00:23:23,170 --> 00:23:24,070
to phi.

434
00:23:24,070 --> 00:23:26,970
We just differentiate this
expression with respect to phi.

435
00:23:26,970 --> 00:23:31,000
The derivative of cosine
phi is minus sine phi.

436
00:23:31,000 --> 00:23:46,560
So the plus sign becomes a minus
sign-- r sine theta, sine phi d

437
00:23:46,560 --> 00:23:49,320
phi.

438
00:23:49,320 --> 00:23:51,800
And then I won't
continue, but we

439
00:23:51,800 --> 00:23:56,520
could do the same
thing for dy and dz.

440
00:23:56,520 --> 00:24:01,450
And once we have expressions
for dx, dy, and dz, we

441
00:24:01,450 --> 00:24:06,750
can calculate ds squared, using
the fact again that all of this

442
00:24:06,750 --> 00:24:08,309
is embedded in Euclidean space.

443
00:24:08,309 --> 00:24:10,350
That's where we're starting,
although in the end,

444
00:24:10,350 --> 00:24:13,140
we want to forget
that Euclidean space.

445
00:24:13,140 --> 00:24:14,730
But we could still
make use of it

446
00:24:14,730 --> 00:24:19,300
here, and write ds squared is
equal to dx squared, plus dy

447
00:24:19,300 --> 00:24:21,875
squared, plus dz squared.

448
00:24:25,210 --> 00:24:28,840
One can then plug in the
expression that we have for dx,

449
00:24:28,840 --> 00:24:32,500
in terms of d theta and d phi,
and the analogous expressions

450
00:24:32,500 --> 00:24:38,080
that I'm not writing
down for dy and dz.

451
00:24:38,080 --> 00:24:39,690
And when one puts
them in here, one

452
00:24:39,690 --> 00:24:42,060
makes lots of use
of the identity

453
00:24:42,060 --> 00:24:45,700
that cosine squared plus
sine squared equals 1

454
00:24:45,700 --> 00:24:49,900
And after using that identity
a number of times, what you get

455
00:24:49,900 --> 00:24:52,230
when you just put
together this algebra

456
00:24:52,230 --> 00:24:58,030
is exactly what we had
before-- r squared times

457
00:24:58,030 --> 00:25:05,230
d theta squared, plus sine
squared theta, d phi squared.

458
00:25:10,570 --> 00:25:12,620
So the important
point is that once you

459
00:25:12,620 --> 00:25:15,400
have the identities that relate
these two different coordinate

460
00:25:15,400 --> 00:25:19,420
systems, and if you know the
distance function in the xyz

461
00:25:19,420 --> 00:25:22,120
coordinate system,
you're home free,

462
00:25:22,120 --> 00:25:25,030
as far as geometry is concerned.

463
00:25:25,030 --> 00:25:28,390
One could just use calculus
from there on if one wants to.

464
00:25:28,390 --> 00:25:32,712
Although usually the geometric
pictures make things easier.

465
00:25:32,712 --> 00:25:33,920
OK, any questions about that?

466
00:25:45,590 --> 00:25:51,840
OK in that case, we are
now ready to move on,

467
00:25:51,840 --> 00:25:53,640
having discussed
the two dimensional

468
00:25:53,640 --> 00:25:57,802
surface of a sphere embedded
in three Euclidean dimensions.

469
00:25:57,802 --> 00:25:59,760
The next thing I'd like
to do-- and this really

470
00:25:59,760 --> 00:26:02,727
will be our closed
universe cosmology--

471
00:26:02,727 --> 00:26:04,560
we're going to discuss
the three dimensional

472
00:26:04,560 --> 00:26:08,850
surface of a sphere in
four Euclidean dimensions

473
00:26:08,850 --> 00:26:11,740
by analogy.

474
00:26:11,740 --> 00:26:14,370
The previous exercise
was a warm- up.

475
00:26:14,370 --> 00:26:19,290
Now we want to introduce a four
dimensional Euclidean space.

476
00:26:19,290 --> 00:26:45,000
So our sphere now
will obey the equation

477
00:26:45,000 --> 00:26:50,990
x squared plus y squared
plus z squared plus-- we need

478
00:26:50,990 --> 00:26:53,910
a new letter for our new
dimension in our four

479
00:26:53,910 --> 00:26:58,800
dimensional space, and I'm
using w. x, y, z, and w.

480
00:27:04,810 --> 00:27:09,550
So that equation describes
a three dimensional sphere

481
00:27:09,550 --> 00:27:11,260
in a four dimensional
Euclidean space.

482
00:27:11,260 --> 00:27:14,700
And our goal is to do the
same thing to that equation

483
00:27:14,700 --> 00:27:17,500
that we just did to the two
dimensional sphere embedded

484
00:27:17,500 --> 00:27:19,510
in a three dimensional space.

485
00:27:19,510 --> 00:27:23,850
Now of course it becomes much
harder to visualize anything.

486
00:27:23,850 --> 00:27:26,650
If you ask how do we do
visualize things in four

487
00:27:26,650 --> 00:27:29,670
dimensions, I think probably the
best answer is that we usually

488
00:27:29,670 --> 00:27:31,310
don't.

489
00:27:31,310 --> 00:27:33,852
And what we're going to
take advantage of mainly

490
00:27:33,852 --> 00:27:35,560
is that once you know
how to write things

491
00:27:35,560 --> 00:27:38,810
in terms of equations,
you don't usually

492
00:27:38,810 --> 00:27:40,470
have to visualize things.

493
00:27:40,470 --> 00:27:42,100
Or if you do, you
can usually get by,

494
00:27:42,100 --> 00:27:47,580
by visualizing subspaces
of the full space.

495
00:27:47,580 --> 00:27:51,250
If you want to visualize
the full sphere

496
00:27:51,250 --> 00:27:56,560
in some rational way, I think
the crutch that I usually use,

497
00:27:56,560 --> 00:27:59,550
and that most people
use, is if you

498
00:27:59,550 --> 00:28:02,120
have just one extra
dimension, in this case w,

499
00:28:02,120 --> 00:28:04,010
try to think of w as
a time coordinate.

500
00:28:04,010 --> 00:28:05,926
Even though it's not
really a time coordinate,

501
00:28:05,926 --> 00:28:08,440
it still gives you a way
of visualizing things.

502
00:28:08,440 --> 00:28:11,200
So if we think of w as
a time coordinate here,

503
00:28:11,200 --> 00:28:14,500
the smallest possible value
of w would be minus r.

504
00:28:14,500 --> 00:28:16,240
The maximum possible
value of w would

505
00:28:16,240 --> 00:28:19,440
be plus r, consistent
with this constraint,

506
00:28:19,440 --> 00:28:21,810
consistent with
being on the sphere.

507
00:28:21,810 --> 00:28:24,520
So when w is equal to minus
r, the other coordinates

508
00:28:24,520 --> 00:28:27,789
have to be 0 to be on the
surface of the sphere.

509
00:28:27,789 --> 00:28:29,580
So you can think of
that as a sphere that's

510
00:28:29,580 --> 00:28:33,660
just appearing at time minus
1, with initially 0 radius.

511
00:28:33,660 --> 00:28:37,900
Then as w increases, x
squared plus y squared plus z

512
00:28:37,900 --> 00:28:39,230
squared increases.

513
00:28:39,230 --> 00:28:42,240
So you could think of it
sphere that starts at 0 size,

514
00:28:42,240 --> 00:28:45,620
gets bigger, gets as
big as r in radius

515
00:28:45,620 --> 00:28:49,530
when w equals 0 and then gets
smaller again and disappears.

516
00:28:49,530 --> 00:28:53,036
So that's one way of thinking
of this four dimensional sphere.

517
00:28:53,036 --> 00:28:53,536
Yes?

518
00:28:53,536 --> 00:28:53,921
AUDIENCE: Is that
kind of like looking

519
00:28:53,921 --> 00:28:55,486
at the cross sections
of the sphere?

520
00:28:55,486 --> 00:28:57,110
PROFESSOR: Is that
kind of like looking

521
00:28:57,110 --> 00:28:58,170
at the cross sections
of the sphere?

522
00:28:58,170 --> 00:28:58,841
Exactly, yes.

523
00:28:58,841 --> 00:29:00,840
One is looking at cross
sections of the sphere--

524
00:29:00,840 --> 00:29:05,160
successive cross sections
at successive values of w.

525
00:29:05,160 --> 00:29:07,595
And if you do it in succession,
it makes w act like a time

526
00:29:07,595 --> 00:29:07,910
coordinate.

527
00:29:07,910 --> 00:29:08,330
But you're right.

528
00:29:08,330 --> 00:29:09,830
The fact that w is
a time coordinate

529
00:29:09,830 --> 00:29:11,087
is kind of irrelevant.

530
00:29:11,087 --> 00:29:12,545
You could imagine
just drawing them

531
00:29:12,545 --> 00:29:14,380
on a piece of
paper in any order.

532
00:29:14,380 --> 00:29:16,630
And for any fixed
value of w, you're

533
00:29:16,630 --> 00:29:19,970
seeing a cross section of
what this sphere looks like.

534
00:29:19,970 --> 00:29:22,994
That is how the xyz coordinates
behave for a fixed value of w.

535
00:29:25,900 --> 00:29:30,800
Now to coordinatize the
surface of our sphere.

536
00:29:30,800 --> 00:29:32,232
Last time we used
two coordinates,

537
00:29:32,232 --> 00:29:33,940
because we had a two
dimensional surface.

538
00:29:33,940 --> 00:29:35,350
This time we're going
to want to use three

539
00:29:35,350 --> 00:29:37,350
coordinates, because this
is a three dimensional

540
00:29:37,350 --> 00:29:39,370
surface that we're describing.

541
00:29:39,370 --> 00:29:44,087
And that means that we need
at least one new coordinate.

542
00:29:44,087 --> 00:29:46,170
And the new coordinate
that I'm going to introduce

543
00:29:46,170 --> 00:29:49,880
will be another angle,
which I'm going to call psi.

544
00:29:49,880 --> 00:29:53,160
And the angle I'm going to
define, as in this diagram,

545
00:29:53,160 --> 00:29:57,250
as the angle from the
new axis, the w axis.

546
00:29:57,250 --> 00:30:04,880
So psi is the angle of any
arbitrary point to the w axis.

547
00:30:04,880 --> 00:30:06,810
And therefore the
w coordinate itself

548
00:30:06,810 --> 00:30:13,390
is going to be r times cosine
psi, just by projecting.

549
00:30:13,390 --> 00:30:17,810
And the square root of the sum
of x squared plus y squared

550
00:30:17,810 --> 00:30:23,250
plus z squared is then the
other component of that vector.

551
00:30:23,250 --> 00:30:24,980
And beyond the sphere,
it's easy to see

552
00:30:24,980 --> 00:30:26,980
that the square root of
x squared plus y squared

553
00:30:26,980 --> 00:30:29,870
plus z squared has to
be r times sine psi.

554
00:30:34,259 --> 00:30:36,050
OK, now we still need
two more coordinates.

555
00:30:36,050 --> 00:30:39,000
This is only one coordinate--
we want to have three.

556
00:30:39,000 --> 00:30:42,030
But the two other coordinates
will just be our old friends,

557
00:30:42,030 --> 00:30:46,930
theta and phi We're going
to keep theta and phi.

558
00:30:46,930 --> 00:30:48,580
And in order to keep
them, what we'll

559
00:30:48,580 --> 00:30:51,640
imagine doing is that for
any point on the surface

560
00:30:51,640 --> 00:30:55,890
of this three dimensional sphere
of the four dimensional space,

561
00:30:55,890 --> 00:30:58,840
we could imagine just
ignoring the w coordinate.

562
00:30:58,840 --> 00:31:00,627
And then we have x,
y, and z-coordinates,

563
00:31:00,627 --> 00:31:02,960
and we can just ask what are
the values of theta and phi

564
00:31:02,960 --> 00:31:05,790
that would go with those
x, y, and z-coordinates.

565
00:31:05,790 --> 00:31:09,230
So theta and phi are just
defined by quote, "projecting"

566
00:31:09,230 --> 00:31:13,280
the original point into the
three dimensional xyz space,

567
00:31:13,280 --> 00:31:16,800
which just means ignore the
w coordinates, look at xyz,

568
00:31:16,800 --> 00:31:18,930
and ask what would be
the angular coordinates,

569
00:31:18,930 --> 00:31:24,010
theta and phi, for those
values of x,y, and z.

570
00:31:24,010 --> 00:31:26,285
And it's easy to take
those words I just said

571
00:31:26,285 --> 00:31:27,493
and turn them into equations.

572
00:31:34,920 --> 00:31:38,497
We like to write down the
analog of these equations.

573
00:31:38,497 --> 00:31:40,080
But we want to have
four equations now

574
00:31:40,080 --> 00:31:43,540
that will specify x, y, z, and
w as a function of our three

575
00:31:43,540 --> 00:31:45,870
angles, psi, theta, and phi.

576
00:31:49,020 --> 00:31:49,860
But that's not hard.

577
00:31:49,860 --> 00:31:54,201
I'll show you at
the bottom with w.

578
00:31:56,847 --> 00:32:04,900
w we already said is just
r times the cosine of psi.

579
00:32:04,900 --> 00:32:06,615
Just coming from
the fact that psi

580
00:32:06,615 --> 00:32:13,316
is defined as the angle from
the point to the w-axis,

581
00:32:13,316 --> 00:32:15,190
and that's enough as
you see from the picture

582
00:32:15,190 --> 00:32:19,726
to imply that w is equal
to r times cosine psi.

583
00:32:19,726 --> 00:32:24,810
The other point, x,y, and z,
really just follow by induction

584
00:32:24,810 --> 00:32:27,810
from what we already know.

585
00:32:27,810 --> 00:32:30,120
Each of these formulas
will hold in the three

586
00:32:30,120 --> 00:32:32,840
dimensional subspace,
except that r,

587
00:32:32,840 --> 00:32:35,520
the radius of a sphere in the
three dimensional subspace,

588
00:32:35,520 --> 00:32:41,070
is not r anymore but
is r times sine psi.

589
00:32:41,070 --> 00:32:47,850
So x is equal to r
times sine psi times

590
00:32:47,850 --> 00:32:51,880
what it was already--
sine theta cosine phi.

591
00:32:54,530 --> 00:33:06,290
y is equal to 4 times sine
psi times sine theta sine phi.

592
00:33:06,290 --> 00:33:15,157
And z is equal to r times
sine psi times cosine theta.

593
00:33:15,157 --> 00:33:16,740
So I just take each
of these equations

594
00:33:16,740 --> 00:33:19,590
and multiply them by sine
psi to get the new equations.

595
00:33:22,150 --> 00:33:24,520
And you can
straightforwardly check--

596
00:33:24,520 --> 00:33:27,160
if we take x squared plus y
squared plus z squared plus w

597
00:33:27,160 --> 00:33:29,970
squared here, it makes
successive use of the identity

598
00:33:29,970 --> 00:33:33,484
that sine squared plus
cosine squared equals 1.

599
00:33:33,484 --> 00:33:35,900
We'll be able to show that x
squared plus y squared plus z

600
00:33:35,900 --> 00:33:41,572
squared plus w squared equals r
squared, like it's supposed to.

601
00:33:41,572 --> 00:33:43,780
OK, so is everybody happy
with this coordinatization?

602
00:33:47,000 --> 00:33:48,730
OK, I should
mention, by the way,

603
00:33:48,730 --> 00:33:53,980
that if you ever have the need
to describe a sphere in 26

604
00:33:53,980 --> 00:33:57,510
dimensions or whatever, this
process easily iterates,

605
00:33:57,510 --> 00:33:59,640
once you've known
how to do it once.

606
00:33:59,640 --> 00:34:02,120
That is, every time you
add a new dimension,

607
00:34:02,120 --> 00:34:04,870
you invent a new letter
for the new axis.

608
00:34:04,870 --> 00:34:09,139
You define a new angle, which
is the angle from that access.

609
00:34:09,139 --> 00:34:11,670
And then the new
coordinatization

610
00:34:11,670 --> 00:34:13,760
is just to set
the new coordinate

611
00:34:13,760 --> 00:34:16,889
equal to r times the
cosine of the new angle.

612
00:34:16,889 --> 00:34:19,070
And then take all
the old equations

613
00:34:19,070 --> 00:34:22,080
and put in an extra factor
of the sine of the new angle.

614
00:34:22,080 --> 00:34:24,362
And you got it.

615
00:34:24,362 --> 00:34:26,320
So you could do that as
many times as you want,

616
00:34:26,320 --> 00:34:29,817
if you want to describe a
very high dimensional sphere.

617
00:34:29,817 --> 00:34:32,150
We should say something about
the range of these angles.

618
00:34:35,630 --> 00:34:39,330
The original angle phi--
maybe I should go back

619
00:34:39,330 --> 00:34:41,120
a few slides now.

620
00:34:41,120 --> 00:34:47,650
The original angle phi, as
you can see from the slide,

621
00:34:47,650 --> 00:34:53,610
goes around the xy plane, so
it has a range of 0 to 2 pi.

622
00:34:53,610 --> 00:34:56,932
The original angle theta is
an angle from the z-axis,

623
00:34:56,932 --> 00:34:58,390
and the furthest
you could never be

624
00:34:58,390 --> 00:35:01,390
away from pointing towards an
axis is pointing away from it.

625
00:35:01,390 --> 00:35:04,010
And that's pi, not 2 pi.

626
00:35:04,010 --> 00:35:07,100
So theta has a range of 0 to pi.

627
00:35:07,100 --> 00:35:12,840
And similarly, psi
is also defined

628
00:35:12,840 --> 00:35:15,830
as an angle from an axis.

629
00:35:15,830 --> 00:35:17,610
So again, the furthest
you could ever

630
00:35:17,610 --> 00:35:21,200
be away from pointing towards an
axis is pointing away from it.

631
00:35:21,200 --> 00:35:25,260
So psi, like theta, will
have a range of 0 to pi.

632
00:35:25,260 --> 00:35:29,800
And if you ever need to
coordinatize a 26 dimensional

633
00:35:29,800 --> 00:35:32,470
sphere, as I just mentioned,
you keep adding new angles.

634
00:35:32,470 --> 00:35:34,790
Each of the new angles
goes from 0 to pi.

635
00:35:34,790 --> 00:35:36,650
Thus each new angle
was introduced

636
00:35:36,650 --> 00:35:40,790
as an angle between the point
that you're trying to describe

637
00:35:40,790 --> 00:35:42,730
and the new axis.

638
00:35:42,730 --> 00:35:47,000
So, they're all angles
like theta and psi.

639
00:35:47,000 --> 00:35:57,460
So 0 is less than phi,
is less than 2 pi.

640
00:35:57,460 --> 00:36:02,700
But 0 is-- these should be less
than or equal to's-- 0 is less

641
00:36:02,700 --> 00:36:07,130
than or equal to theta, is
less than or equal to pi.

642
00:36:07,130 --> 00:36:11,910
And 0 is less than
or equal to psi,

643
00:36:11,910 --> 00:36:13,250
is less than or equal to pi.

644
00:36:24,210 --> 00:36:29,030
OK, next we want to get
the metric of our three

645
00:36:29,030 --> 00:36:32,540
dimensional spherical
surface embedded in our four

646
00:36:32,540 --> 00:36:33,920
dimensional Euclidean space.

647
00:36:36,680 --> 00:36:40,039
And I'm going to do it by
the geometric sort of way.

648
00:36:40,039 --> 00:36:41,580
I'll try to just
motivate the pieces.

649
00:36:44,794 --> 00:36:46,960
There'll be some cross here
between actually algebra

650
00:36:46,960 --> 00:36:47,980
and geometry.

651
00:36:50,585 --> 00:36:51,960
Let's see, first
I should mention

652
00:36:51,960 --> 00:36:54,240
that once you have this,
you can get the answer

653
00:36:54,240 --> 00:36:57,380
by the same brute force process
that we describe, but didn't

654
00:36:57,380 --> 00:36:58,360
really carry out here.

655
00:36:58,360 --> 00:36:59,860
That's tedious, but
it's pretty well

656
00:36:59,860 --> 00:37:01,860
guaranteed to get you the
right answer if you're

657
00:37:01,860 --> 00:37:04,700
careful enough, and does not
require drawing any pictures

658
00:37:04,700 --> 00:37:07,300
or having any the
visualization of the geometry,

659
00:37:07,300 --> 00:37:09,880
so it does have some advantages.

660
00:37:09,880 --> 00:37:11,210
But I will not do it that way.

661
00:37:11,210 --> 00:37:14,540
I will do it in a
geometric sort of way,

662
00:37:14,540 --> 00:37:16,370
because I think it's
easier to understand

663
00:37:16,370 --> 00:37:17,480
the geometric sort of way.

664
00:37:20,740 --> 00:37:22,720
So here goes.

665
00:37:22,720 --> 00:37:25,370
As we did over there, we
will vary our coordinates one

666
00:37:25,370 --> 00:37:26,980
at a time, and
then see how we can

667
00:37:26,980 --> 00:37:31,189
combine the
different variations.

668
00:37:31,189 --> 00:37:33,480
Again we'll find that they're
orthogonal to each other.

669
00:37:33,480 --> 00:37:36,090
So I'll be able to combine them
just by adding the squares.

670
00:37:36,090 --> 00:37:39,130
But we don't necessarily
know that the beginning.

671
00:37:39,130 --> 00:37:45,800
So let's start by varying
psi, the new coordinate.

672
00:37:45,800 --> 00:37:49,260
And there, things
are very simple

673
00:37:49,260 --> 00:37:51,070
because our new
coordinate is just

674
00:37:51,070 --> 00:37:54,990
defined as the
angle from an axis.

675
00:37:54,990 --> 00:37:58,130
So if we vary psi,
the point in question

676
00:37:58,130 --> 00:38:01,320
just makes a circle
around the origin,

677
00:38:01,320 --> 00:38:05,610
and the circle has radius,
capital R. So the variation

678
00:38:05,610 --> 00:38:09,110
if I vary psi is
just r times d psi.

679
00:38:12,860 --> 00:38:19,300
So ds is equal to r times d psi.

680
00:38:19,300 --> 00:38:20,420
What could be simpler?

681
00:38:24,650 --> 00:38:28,500
OK, now we'll imagine varying
either theta or phi or both.

682
00:38:28,500 --> 00:38:30,630
And since we already
pretty well understand

683
00:38:30,630 --> 00:38:32,820
this three
dimensional subspace--

684
00:38:32,820 --> 00:38:34,350
this is the previous
problem-- I'm

685
00:38:34,350 --> 00:38:36,350
going to talk about varying
them simultaneously.

686
00:38:40,290 --> 00:38:47,030
So vary theta and phi.

687
00:38:47,030 --> 00:38:51,650
Well then we know that
ds squared is really

688
00:38:51,650 --> 00:38:53,780
given by this formula.

689
00:38:53,780 --> 00:38:56,040
We are just varying
theta and phi

690
00:38:56,040 --> 00:38:57,250
in a three dimensional space.

691
00:38:57,250 --> 00:39:01,450
The fourth direction
doesn't change in this case.

692
00:39:01,450 --> 00:39:05,530
But the radius involved is not
what we originally called r.

693
00:39:05,530 --> 00:39:07,620
But the radius in the
three dimensional space

694
00:39:07,620 --> 00:39:10,192
is r times sine psi.

695
00:39:10,192 --> 00:39:11,650
That's the square
root of x squared

696
00:39:11,650 --> 00:39:13,100
plus y squared plus z squared.

697
00:39:16,430 --> 00:39:23,280
So what we get is ds squared
is equal to R squared

698
00:39:23,280 --> 00:39:30,200
times sine squared psi
times d theta squared

699
00:39:30,200 --> 00:39:34,540
plus sine squared
theta, d phi squared.

700
00:39:41,722 --> 00:39:42,805
Everybody happy with that?

701
00:39:45,530 --> 00:39:50,140
OK, now I'm going to
first jump ahead and then

702
00:39:50,140 --> 00:39:52,360
come back and justify
what we're doing.

703
00:39:52,360 --> 00:40:04,090
But if these variations
are orthogonal-- which

704
00:40:04,090 --> 00:40:06,650
I will argue shortly
that they are,

705
00:40:06,650 --> 00:40:10,220
so I'm not doing
this for nothing--

706
00:40:10,220 --> 00:40:11,970
if the variations
are orthogonal,

707
00:40:11,970 --> 00:40:14,620
then we just add
the sum of squares

708
00:40:14,620 --> 00:40:16,610
using the generalized
Pythagorean theorem.

709
00:40:16,610 --> 00:40:18,860
In this case, Pythagorean
theorem in four dimensions--

710
00:40:18,860 --> 00:40:20,510
that's four
Euclidean dimensions,

711
00:40:20,510 --> 00:40:23,170
so we should be able to use it.

712
00:40:23,170 --> 00:40:27,070
So what we get for
our final answer is

713
00:40:27,070 --> 00:40:33,270
ds squared is an overall
factor of r squared,

714
00:40:33,270 --> 00:40:45,040
times d psi squared plus
sine squared psi, times d

715
00:40:45,040 --> 00:40:53,800
theta squared, plus sine
squared theta, d phi squared.

716
00:41:07,940 --> 00:41:09,740
I just added the
sum of the squares.

717
00:41:09,740 --> 00:41:11,675
Now I need to justify
this orthogonality.

718
00:41:39,180 --> 00:41:43,970
OK, to do that, let me introduce
a vector notation in the four

719
00:41:43,970 --> 00:41:46,690
dimensional space
of x,y z, and w.

720
00:41:46,690 --> 00:41:51,064
OK, we're justifying this in
the Euclidean embedding space.

721
00:41:51,064 --> 00:41:53,480
And the Euclidean embedding
space of these transformations

722
00:41:53,480 --> 00:41:55,640
are orthogonal.

723
00:41:55,640 --> 00:41:57,730
So let me imagine varying psi.

724
00:42:03,850 --> 00:42:07,400
And then I could construct
a four dimensional vector

725
00:42:07,400 --> 00:42:11,230
dr-- I'll call it
sub psi because it

726
00:42:11,230 --> 00:42:12,537
arises from varying psi.

727
00:42:12,537 --> 00:42:14,120
So this is the four
dimensional vector

728
00:42:14,120 --> 00:42:18,500
that describes the motion of
this point r as psi is varied.

729
00:42:22,130 --> 00:42:25,590
And first let me just give
these components names.

730
00:42:25,590 --> 00:42:29,510
I'll call it-- rather
than repeat the r,

731
00:42:29,510 --> 00:42:35,291
I'm just going to call this
d psi sub x, d psi sub y,

732
00:42:35,291 --> 00:42:39,570
d psi sub z, and d psi sub w.

733
00:42:43,640 --> 00:42:45,270
This is just by definition.

734
00:42:45,270 --> 00:42:47,446
I'm just naming the
components of that vector.

735
00:42:47,446 --> 00:42:49,320
And since the vector
already has a subscript,

736
00:42:49,320 --> 00:42:50,820
I don't want to
have two subscripts.

737
00:42:50,820 --> 00:42:54,407
So I've changed the
name of the vector

738
00:42:54,407 --> 00:42:55,490
for writing as components.

739
00:42:58,040 --> 00:43:01,900
And similarly, here
I will just vary

740
00:43:01,900 --> 00:43:05,470
one of these two angles, theta
and phi I'll just vary theta,

741
00:43:05,470 --> 00:43:10,340
and let you know that
varying psi is no different,

742
00:43:10,340 --> 00:43:12,470
and you can easily see that.

743
00:43:12,470 --> 00:43:17,590
So if I vary theta,
the variation

744
00:43:17,590 --> 00:43:21,800
of r when I vary theta
will be dr sub theta.

745
00:43:21,800 --> 00:43:26,820
And its components I
will just call d theta

746
00:43:26,820 --> 00:43:36,570
x, d theta y, d theta
z, and d theta w.

747
00:43:36,570 --> 00:43:37,820
So these are just definitions.

748
00:43:37,820 --> 00:43:41,460
I haven't said any
actual facts yet.

749
00:43:41,460 --> 00:43:42,990
But I've defined
these two vectors,

750
00:43:42,990 --> 00:43:45,732
and given names to
their components.

751
00:43:45,732 --> 00:43:48,190
And now we want to look at them
and take their dot product.

752
00:43:48,190 --> 00:43:50,606
Their dot product is just a
four dimensional Euclidean dot

753
00:43:50,606 --> 00:43:51,270
product.

754
00:43:51,270 --> 00:43:56,560
So the dot product is just dx
times d theta d psi y d theta y

755
00:43:56,560 --> 00:44:01,355
plus d psi z times d
theta z plus d-- blah.

756
00:44:01,355 --> 00:44:02,480
I think we should write it.

757
00:44:05,120 --> 00:44:08,890
So this actually is now a
fact about Euclidean geometry

758
00:44:08,890 --> 00:44:11,170
in four dimensions.

759
00:44:11,170 --> 00:44:14,390
The dot product of
these two vectors

760
00:44:14,390 --> 00:44:21,710
is just equal to the
product of the x components,

761
00:44:21,710 --> 00:44:27,340
plus the product of
their y components,

762
00:44:27,340 --> 00:44:33,870
plus the product of
their z components,

763
00:44:33,870 --> 00:44:36,230
plus the product of
their w components.

764
00:44:47,750 --> 00:44:49,915
And now what we want to
do is to look at this sum

765
00:44:49,915 --> 00:44:53,540
and argue they're 0, because
if they're orthogonal,

766
00:44:53,540 --> 00:44:55,350
the dot product of two
vectors should be 0.

767
00:44:58,590 --> 00:45:03,150
OK, so let me first look
at the dr sub theta vector.

768
00:45:11,460 --> 00:45:14,720
OK, what do we know about it?

769
00:45:14,720 --> 00:45:20,290
Well, we know that when we vary
theta, from these formulas,

770
00:45:20,290 --> 00:45:21,980
w does not change--
and we can easily

771
00:45:21,980 --> 00:45:24,550
see that from the
picture as well.

772
00:45:24,550 --> 00:45:30,280
So d theta sub w equals 0.

773
00:45:33,360 --> 00:45:36,652
And since these are
all products, that

774
00:45:36,652 --> 00:45:38,110
means that this
last term vanishes,

775
00:45:38,110 --> 00:45:41,327
no matter what d psi w is.

776
00:45:41,327 --> 00:45:43,410
So we know we don't need
to worry about that term.

777
00:45:43,410 --> 00:45:45,326
We only need to worry
about these three terms.

778
00:45:47,810 --> 00:45:50,800
Now what do we know
about those three terms?

779
00:45:50,800 --> 00:45:54,390
If you look at dr sub theta,
and look at its three spacial

780
00:45:54,390 --> 00:45:58,540
components-- x,y,
and z-- from here,

781
00:45:58,540 --> 00:46:03,190
we could see that varying theta
does the same thing to x, y,

782
00:46:03,190 --> 00:46:07,910
and z as it did over here,
except a different overall

783
00:46:07,910 --> 00:46:09,590
factor out front.

784
00:46:09,590 --> 00:46:11,840
So in particular, what
I want to point out

785
00:46:11,840 --> 00:46:14,240
is that varying
theta does not change

786
00:46:14,240 --> 00:46:16,980
x squared plus y
squared plus z squared.

787
00:46:16,980 --> 00:46:18,900
It leaves it constant.

788
00:46:18,900 --> 00:46:21,010
So if we think of
the xyz space, we

789
00:46:21,010 --> 00:46:26,070
could imagine a sphere
in the xyz space,

790
00:46:26,070 --> 00:46:29,540
and varying theta always
causes a variation

791
00:46:29,540 --> 00:46:31,336
that's tangential
to that sphere.

792
00:46:31,336 --> 00:46:32,960
It never moves in
the radial direction.

793
00:46:38,770 --> 00:46:47,704
So the three vector, defined
by d theta x, d theta y,

794
00:46:47,704 --> 00:47:01,560
d theta z, is tangential in the
three dimensional subspace xyz.

795
00:47:01,560 --> 00:47:04,724
Just as it was when we
didn't have a w coordinate.

796
00:47:04,724 --> 00:47:06,640
The w coordinate doesn't
change anything here.

797
00:47:09,447 --> 00:47:11,030
So that's a little
bit [? soluable. ?]

798
00:47:11,030 --> 00:47:12,010
Are people happy with that?

799
00:47:12,010 --> 00:47:13,468
Do you know what
I'm talking about?

800
00:47:18,670 --> 00:47:20,370
OK.

801
00:47:20,370 --> 00:47:31,680
Now we want to
look at dr sub psi,

802
00:47:31,680 --> 00:47:36,622
and it will have a w
component-- d psi sub w--

803
00:47:36,622 --> 00:47:37,830
but we don't care about that.

804
00:47:37,830 --> 00:47:40,204
We know we don't care about
it because that piece already

805
00:47:40,204 --> 00:47:41,970
dropped out of our expression.

806
00:47:41,970 --> 00:47:44,100
So we want to know
what this vector looks

807
00:47:44,100 --> 00:47:47,040
like in the xyz space.

808
00:47:47,040 --> 00:47:50,060
We don't care about what it
looks like in the w space.

809
00:47:50,060 --> 00:47:54,120
So in the xyz space, we can
look at these formulas here.

810
00:47:54,120 --> 00:48:00,160
As we vary psi, x, y,
and z could change.

811
00:48:00,160 --> 00:48:02,180
But they all change
by the same factor--

812
00:48:02,180 --> 00:48:05,030
whatever factor psi changes by.

813
00:48:05,030 --> 00:48:10,330
The same sign psi appears
in all three lines.

814
00:48:10,330 --> 00:48:13,840
So changing psi can only
multiply x, y, and z all

815
00:48:13,840 --> 00:48:14,944
by the same factor.

816
00:48:14,944 --> 00:48:16,360
And what that means
is that if you

817
00:48:16,360 --> 00:48:18,850
think of this geometrically
in the xyz space,

818
00:48:18,850 --> 00:48:21,632
varying psi moves the point
only in the radial direction.

819
00:48:21,632 --> 00:48:23,840
If you multiply all of the
coordinates by a constant,

820
00:48:23,840 --> 00:48:27,480
you are just moving in
the radial direction.

821
00:48:27,480 --> 00:48:32,900
So dr psi has the property that
when we look at only its x, y,

822
00:48:32,900 --> 00:48:42,840
and z components-- d
psi x d psi y, d psi z,

823
00:48:42,840 --> 00:48:44,960
it is radial in the three
dimensional subspace.

824
00:48:48,900 --> 00:48:52,430
So, the sum of
these three terms--

825
00:48:52,430 --> 00:48:55,590
this is what we're
trying to evaluate--

826
00:48:55,590 --> 00:49:00,760
is the dot product of a radial
vector and a tangential vector.

827
00:49:00,760 --> 00:49:02,349
And the dot product
of a radial vector

828
00:49:02,349 --> 00:49:03,890
and the tangential
vector will always

829
00:49:03,890 --> 00:49:05,931
be 0, because there are
orthogonal to each other.

830
00:49:10,380 --> 00:49:16,760
Sorry for the overlap
here, but equal 0,

831
00:49:16,760 --> 00:49:22,020
and that's because radial is
perpendicular to tangential.

832
00:49:36,010 --> 00:49:38,770
OK, everybody happy with that?

833
00:49:38,770 --> 00:49:40,660
OK if so, we have
important result now.

834
00:49:40,660 --> 00:49:45,550
We have derived the metric for
the three dimensional surface

835
00:49:45,550 --> 00:49:50,360
of a four dimensional sphere
embedded in four Euclidean

836
00:49:50,360 --> 00:49:51,130
dimensions.

837
00:49:51,130 --> 00:49:53,590
And that, in fact, is
precisely the closed universe

838
00:49:53,590 --> 00:49:54,800
of cosmology.

839
00:49:54,800 --> 00:49:58,350
It's the homogeneous
isotropic description

840
00:49:58,350 --> 00:49:59,435
of a closed universe.

841
00:50:15,312 --> 00:50:17,645
OK, next thing I want to point
out is just a definition.

842
00:50:25,430 --> 00:50:28,800
An important feature of
non Euclidean geometry

843
00:50:28,800 --> 00:50:31,040
and general relativity--
because they're

844
00:50:31,040 --> 00:50:33,800
connected to each other--
is that there never

845
00:50:33,800 --> 00:50:38,420
is a unique, useful
coordinate system.

846
00:50:38,420 --> 00:50:41,911
And Euclidean spaces,
there is a unique, useful

847
00:50:41,911 --> 00:50:42,660
coordinate system.

848
00:50:42,660 --> 00:50:44,290
It's the Cartesian system.

849
00:50:44,290 --> 00:50:47,011
Sometimes it's also useful
to use polar coordinates

850
00:50:47,011 --> 00:50:49,510
or something else, but by and
large the Cartesian coordinate

851
00:50:49,510 --> 00:50:51,960
system is the
natural description

852
00:50:51,960 --> 00:50:54,590
of Euclidean spaces.

853
00:50:54,590 --> 00:50:56,770
And the coordinates of a
Cartesian coordinate system

854
00:50:56,770 --> 00:50:59,510
really are distances.

855
00:50:59,510 --> 00:51:02,720
Once, however, you go
from Euclidean geometry

856
00:51:02,720 --> 00:51:08,494
to non Euclidean geometry-- from
flat spaces to curved spaces--

857
00:51:08,494 --> 00:51:10,410
you're usually in a
situation where there just

858
00:51:10,410 --> 00:51:13,410
is no natural coordinate system.

859
00:51:13,410 --> 00:51:16,716
When we invented
this psi, theta, phi,

860
00:51:16,716 --> 00:51:18,840
we really made a number of
arbitrary choices there.

861
00:51:18,840 --> 00:51:20,756
We could have defined
things quite differently

862
00:51:20,756 --> 00:51:22,710
if we wanted to.

863
00:51:22,710 --> 00:51:26,710
So in general, one has
to deal with the fact

864
00:51:26,710 --> 00:51:30,640
that the coordinates no
longer represent distances,

865
00:51:30,640 --> 00:51:34,150
and therefore there's
a lot of arbitrariness

866
00:51:34,150 --> 00:51:36,960
in the way you choose the
coordinates in the first place.

867
00:51:36,960 --> 00:51:45,250
So in particular, we
could think of psi

868
00:51:45,250 --> 00:51:48,890
equals zero as the center of
our new coordinate system,

869
00:51:48,890 --> 00:51:50,885
with coordinates
psi, theta, and phi.

870
00:51:53,560 --> 00:51:56,900
psi equals 0 corresponds
to being along the w axis,

871
00:51:56,900 --> 00:51:58,490
so it's a unique point.

872
00:51:58,490 --> 00:52:00,180
When you say that
psi is equal to 0,

873
00:52:00,180 --> 00:52:03,300
it no longer matters
what theta and phi are.

874
00:52:03,300 --> 00:52:08,379
You're at the point w equals
r and x, y, and z equals 0.

875
00:52:08,379 --> 00:52:09,920
So we can think of
that as the origin

876
00:52:09,920 --> 00:52:11,060
of our new coordinate system.

877
00:52:11,060 --> 00:52:12,768
And we can think of
then the value of psi

878
00:52:12,768 --> 00:52:15,620
as measuring how far we
are from that origin.

879
00:52:15,620 --> 00:52:17,610
So psi will become
our radial coordinate.

880
00:52:25,160 --> 00:52:27,745
So thinking of it
as our sphere, psi

881
00:52:27,745 --> 00:52:32,680
equals 0 we might think of as
the North Pole this sphere.

882
00:52:32,680 --> 00:52:37,600
But we're also going to think of
it as the origin of psi, theta,

883
00:52:37,600 --> 00:52:38,300
phi space.

884
00:52:43,770 --> 00:52:46,510
And we will sometimes use
other radial variables--

885
00:52:46,510 --> 00:52:50,660
other variables for the
distance from the origin.

886
00:52:50,660 --> 00:52:52,490
So in particular,
another coordinate

887
00:52:52,490 --> 00:52:55,430
that's very commonly
used is u, which

888
00:52:55,430 --> 00:52:58,655
is just defined to be the
sine of the angle psi.

889
00:53:01,600 --> 00:53:03,450
And notice that the
sine of the angle psi

890
00:53:03,450 --> 00:53:04,810
shows up in a lot of equations.

891
00:53:04,810 --> 00:53:07,140
So taking that as
our natural variable

892
00:53:07,140 --> 00:53:10,130
is a reasonable thing
to do on occasion.

893
00:53:10,130 --> 00:53:11,820
Both are useful.

894
00:53:11,820 --> 00:53:16,130
If we do use this, then
we can rewrite the metric

895
00:53:16,130 --> 00:53:19,009
in terms of u, instead of psi.

896
00:53:19,009 --> 00:53:20,550
And in order to do
that, we just have

897
00:53:20,550 --> 00:53:24,014
to know how du relates to
d psi, because the metric

898
00:53:24,014 --> 00:53:25,680
is written in terms
of the differentials

899
00:53:25,680 --> 00:53:27,680
of the coordinates.

900
00:53:27,680 --> 00:53:32,490
So that's easy to calculate.

901
00:53:32,490 --> 00:53:38,510
du would be equal to
cosine of psi d psi.

902
00:53:42,030 --> 00:53:44,850
But if we're trying to
rewrite the metric solely

903
00:53:44,850 --> 00:53:48,340
in terms of u, we don't want
to have to divide or multiply

904
00:53:48,340 --> 00:53:52,580
by cosine psi, because that's
written in terms of psi.

905
00:53:52,580 --> 00:53:54,930
But we could express
cosine psi in terms of u,

906
00:53:54,930 --> 00:53:58,750
because if u equals sine
psi, then cosine psi

907
00:53:58,750 --> 00:54:01,720
is the square root
of 1 minus u squared.

908
00:54:01,720 --> 00:54:05,990
So I can rewrite this as
the square root of 1 minus u

909
00:54:05,990 --> 00:54:08,150
squared times d psi.

910
00:54:13,200 --> 00:54:20,030
And then d psi squared, which
is what appears in our metric,

911
00:54:20,030 --> 00:54:27,930
can be rewritten just
using that as du squared

912
00:54:27,930 --> 00:54:29,765
divided by 1 minus u squared.

913
00:54:35,590 --> 00:54:40,450
And the full metric now, in
terms of the u theta and psi

914
00:54:40,450 --> 00:54:49,470
coordinates, can be
written as r squared

915
00:54:49,470 --> 00:54:55,170
times du squared over
1 minus u squared,

916
00:54:55,170 --> 00:55:01,720
plus u squared times
d theta squared

917
00:55:01,720 --> 00:55:06,670
plus sine squared
theta d phi squared.

918
00:55:10,510 --> 00:55:14,320
So this is another way
of writing the metric

919
00:55:14,320 --> 00:55:16,887
for this three dimensional
sphere embedded

920
00:55:16,887 --> 00:55:18,095
in four Euclidean dimensions.

921
00:55:22,650 --> 00:55:24,226
Any questions?

922
00:55:24,226 --> 00:55:25,112
Yes?

923
00:55:25,112 --> 00:55:26,820
AUDIENCE: The value
of u doesn't uniquely

924
00:55:26,820 --> 00:55:28,700
determine a point on
the sphere, right?

925
00:55:28,700 --> 00:55:31,530
Because [INAUDIBLE].

926
00:55:31,530 --> 00:55:32,910
PROFESSOR: Very good point.

927
00:55:32,910 --> 00:55:35,375
In case you didn't
hear the question,

928
00:55:35,375 --> 00:55:37,470
it was pointed out
that the value of u,

929
00:55:37,470 --> 00:55:41,090
unlike the value of psi, is
not uniquely indicate a point,

930
00:55:41,090 --> 00:55:47,560
because on the entire sphere,
there are two points u

931
00:55:47,560 --> 00:55:50,660
for every value of sine psi--
one in the northern hemisphere

932
00:55:50,660 --> 00:55:53,240
and one in the
southern hemisphere,

933
00:55:53,240 --> 00:55:57,910
if we think of
hemispheres as dividing

934
00:55:57,910 --> 00:56:00,959
whether w is
positive or negative.

935
00:56:00,959 --> 00:56:02,000
So in fact, that's right.

936
00:56:02,000 --> 00:56:03,541
If we use the u
coordinate, we should

937
00:56:03,541 --> 00:56:05,890
remember that if we want to
talk about the whole sphere,

938
00:56:05,890 --> 00:56:07,931
we should remember that
the whole sphere is twice

939
00:56:07,931 --> 00:56:09,690
as big as what we
see if we just let

940
00:56:09,690 --> 00:56:15,310
u vary between 0 and
its maximum value, 1.

941
00:56:15,310 --> 00:56:18,620
u equals 1 corresponds to
the equator of this sphere.

942
00:56:27,400 --> 00:56:30,130
Another point which I'd like to
make now that we've written it

943
00:56:30,130 --> 00:56:33,021
this way is that writing it this
way is the easiest way to see--

944
00:56:33,021 --> 00:56:34,520
although we can see
it in other ways

945
00:56:34,520 --> 00:56:38,430
as well-- that if u is very
small, if we look right

946
00:56:38,430 --> 00:56:42,210
in the vicinity of the origin
of our new coordinate system,

947
00:56:42,210 --> 00:56:47,470
if u is very small, 1 minus
u squared is very close to 1.

948
00:56:47,470 --> 00:56:49,860
The square of a small
quantity is extra small,

949
00:56:49,860 --> 00:56:52,790
so this denominator
is extra close to 1.

950
00:56:52,790 --> 00:56:55,190
And that means that for
very small values of u,

951
00:56:55,190 --> 00:56:59,570
what we have is du squared plus
u squared times this quantity.

952
00:56:59,570 --> 00:57:05,470
And this is just polar
coordinates in Euclidean space.

953
00:57:05,470 --> 00:57:09,040
So for u very small, we do see
that we have a local Euclidean

954
00:57:09,040 --> 00:57:09,741
space.

955
00:57:09,741 --> 00:57:11,240
And that, if you
might remember, was

956
00:57:11,240 --> 00:57:13,900
one of the key points
about writing the metric

957
00:57:13,900 --> 00:57:17,300
as the sum of squares
in the first place.

958
00:57:17,300 --> 00:57:20,080
And it's true about every point,
although the coordinate system

959
00:57:20,080 --> 00:57:21,940
here only makes it
obvious about the origin.

960
00:57:21,940 --> 00:57:24,100
But we know that the space
actually is homogeneous,

961
00:57:24,100 --> 00:57:25,610
from the way we constructed it.

962
00:57:25,610 --> 00:57:30,600
So what's true about the
origin is true about any point.

963
00:57:30,600 --> 00:57:32,860
So the coordinate
system, the metric

964
00:57:32,860 --> 00:57:34,690
around any point, if
you look close enough

965
00:57:34,690 --> 00:57:38,710
in the vicinity of the point,
looks like a Euclidean space,

966
00:57:38,710 --> 00:57:41,196
which is what we expected
from the very beginning,

967
00:57:41,196 --> 00:57:42,820
but we can see it
very explicitly here.

968
00:57:50,330 --> 00:57:51,870
OK.

969
00:57:51,870 --> 00:57:54,270
So far, this is just geometry.

970
00:57:54,270 --> 00:57:57,530
But this will be a model
for a homogeneous, isotropic

971
00:57:57,530 --> 00:57:59,376
universe.

972
00:57:59,376 --> 00:58:00,750
We know it's
homogeneous, we know

973
00:58:00,750 --> 00:58:03,770
it's isotropic from the
way we discussed it.

974
00:58:03,770 --> 00:58:05,420
When we write the
metric this way,

975
00:58:05,420 --> 00:58:07,840
it's obvious that it's
isotropic about the origin,

976
00:58:07,840 --> 00:58:10,910
because this construction we
know is just polar coordinates,

977
00:58:10,910 --> 00:58:12,390
and we know polar
coordinates don't

978
00:58:12,390 --> 00:58:16,340
really single out any direction,
even though manifestly they

979
00:58:16,340 --> 00:58:17,970
look like they
do, because you're

980
00:58:17,970 --> 00:58:19,720
measuring angles
from the z-axis.

981
00:58:19,720 --> 00:58:24,290
But we know that really
describes an isotropic sphere.

982
00:58:24,290 --> 00:58:26,350
It's not obvious that
this formula describes

983
00:58:26,350 --> 00:58:28,160
a homogeneous space,
because it makes

984
00:58:28,160 --> 00:58:31,220
it look like u equals
0 is a special point.

985
00:58:31,220 --> 00:58:33,450
But we do know that
it did correspond

986
00:58:33,450 --> 00:58:35,820
to a homogeneous space from
the way we constructed it.

987
00:58:35,820 --> 00:58:38,660
It really is just a
three dimensional sphere

988
00:58:38,660 --> 00:58:40,880
embedded in four
Euclidean dimensions.

989
00:58:40,880 --> 00:58:43,300
And if you think of
it as the sphere,

990
00:58:43,300 --> 00:58:46,620
there clearly is no special
point on the sphere.

991
00:58:46,620 --> 00:58:49,950
So the homogeneity is a
feature of this metric,

992
00:58:49,950 --> 00:58:51,240
but a hidden feature.

993
00:58:51,240 --> 00:58:55,070
It's hard to see how
you would transform

994
00:58:55,070 --> 00:58:57,220
coordinates to, for example,
put a different point

995
00:58:57,220 --> 00:58:58,240
at the origin.

996
00:58:58,240 --> 00:59:00,500
But it is possible, and
we know it's possible,

997
00:59:00,500 --> 00:59:02,850
because of the way we
originally constructed.

998
00:59:02,850 --> 00:59:04,820
And if we had to
do it, we go back

999
00:59:04,820 --> 00:59:07,400
to the original construction
and actually do it.

1000
00:59:07,400 --> 00:59:10,820
That is, if I told you I wanted
some other point in that system

1001
00:59:10,820 --> 00:59:13,870
to be the origin, you
could trace it all way back

1002
00:59:13,870 --> 00:59:17,506
to the four dimensional
Euclidean space,

1003
00:59:17,506 --> 00:59:19,880
and figure out how you have
to do a rotation in that four

1004
00:59:19,880 --> 00:59:22,590
dimensional Euclidean space to
make the point that I told you

1005
00:59:22,590 --> 00:59:25,510
I wanted to be the origin
to actually be the origin.

1006
00:59:25,510 --> 00:59:27,520
So you'd be able to do that.

1007
00:59:27,520 --> 00:59:29,610
It would be some work,
but you would in fact

1008
00:59:29,610 --> 00:59:30,960
know how to do that.

1009
00:59:30,960 --> 00:59:34,934
We really do understand that
the space is homogeneous, which

1010
00:59:34,934 --> 00:59:36,725
is guaranteed by our
original construction.

1011
00:59:39,830 --> 00:59:40,990
OK.

1012
00:59:40,990 --> 00:59:43,510
Finish with the basic
geometry-- one more

1013
00:59:43,510 --> 00:59:46,332
chance to ask any
questions about it.

1014
00:59:46,332 --> 00:59:48,310
OK, next I want to
go on now to talk

1015
00:59:48,310 --> 00:59:51,230
about how this fits
into general relativity.

1016
00:59:51,230 --> 00:59:53,890
And here we are going to
be confronting-- actually

1017
00:59:53,890 --> 00:59:57,150
the only place we ever will
confront-- the issue of how

1018
00:59:57,150 --> 01:00:00,750
matter causes space
to curve, which

1019
01:00:00,750 --> 01:00:02,470
is the aspect of
general relativity

1020
01:00:02,470 --> 01:00:04,690
that we're not really
going to do it all.

1021
01:00:04,690 --> 01:00:06,690
So I'll basically just
be giving you the answer.

1022
01:00:06,690 --> 01:00:11,110
Although we will in fact
know enough to narrow down

1023
01:00:11,110 --> 01:00:13,290
the range of possible
answers, pretty much.

1024
01:00:13,290 --> 01:00:15,669
But there will be a
fudge factor that I'll

1025
01:00:15,669 --> 01:00:17,460
have to just tell you
the right answer for.

1026
01:00:38,662 --> 01:00:40,120
So what I want to
do now is to make

1027
01:00:40,120 --> 01:00:46,370
a connection between this
formalism and the model

1028
01:00:46,370 --> 01:00:49,750
that we already discussed of
the expanding universe whose

1029
01:00:49,750 --> 01:00:54,570
dynamics we derived using
Newtonian mechanics.

1030
01:00:54,570 --> 01:00:56,930
So using internet
Newtonian mechanics,

1031
01:00:56,930 --> 01:01:00,380
we introduced a
scale factor a of t.

1032
01:01:00,380 --> 01:01:06,600
And convinced ourselves
that a dot over a squared

1033
01:01:06,600 --> 01:01:13,870
is equal to 8 pi over
3 g rho, minus kc

1034
01:01:13,870 --> 01:01:17,580
squared over a squared.

1035
01:01:17,580 --> 01:01:20,230
And furthermore,
that this a of t

1036
01:01:20,230 --> 01:01:23,900
describes the relationship
between physical distances

1037
01:01:23,900 --> 01:01:24,976
and coordinate distances.

1038
01:01:30,790 --> 01:01:35,180
Namely, if we have objects that
are at rest in this expanding

1039
01:01:35,180 --> 01:01:38,930
universe, comoving objects
is the phrase usually used

1040
01:01:38,930 --> 01:01:40,300
to describe that.

1041
01:01:40,300 --> 01:01:43,170
If we have comoving objects,
the comoving objects

1042
01:01:43,170 --> 01:01:47,254
will sit at fixed coordinates
in our coordinate system.

1043
01:01:47,254 --> 01:01:48,920
And the distance
between any two of them

1044
01:01:48,920 --> 01:01:53,470
will be some fixed distance-- l
sub c-- a coordinate distance.

1045
01:01:53,470 --> 01:01:56,310
But the physical distance will
vary with time, proportional

1046
01:01:56,310 --> 01:01:57,810
to this scale factor.

1047
01:01:57,810 --> 01:02:00,896
So the physical distance
between any two points

1048
01:02:00,896 --> 01:02:02,270
will be a function
of time, which

1049
01:02:02,270 --> 01:02:05,285
is the scale factor times the
time-independent coordinate

1050
01:02:05,285 --> 01:02:05,785
distance.

1051
01:02:09,520 --> 01:02:13,040
OK, if we look at our metric
for this sphere, and say

1052
01:02:13,040 --> 01:02:14,550
we're going to
assume that this is

1053
01:02:14,550 --> 01:02:18,270
going to be the metric that
describes the space that we're

1054
01:02:18,270 --> 01:02:25,415
describing over here, then
clearly this r that sits out

1055
01:02:25,415 --> 01:02:29,540
front rescales all
of the distances.

1056
01:02:29,540 --> 01:02:32,240
All the distances are
proportional to r,

1057
01:02:32,240 --> 01:02:35,590
just as all the distances
here are proportional to a.

1058
01:02:35,590 --> 01:02:39,770
So that could only work
if r is proportional to a.

1059
01:02:39,770 --> 01:02:43,890
So that's our key
conclusion here--

1060
01:02:43,890 --> 01:02:48,910
that r is going to
have to vary with time,

1061
01:02:48,910 --> 01:02:51,230
and be proportional to a.

1062
01:02:56,219 --> 01:02:57,760
But we can even say
a little bit more

1063
01:02:57,760 --> 01:03:01,510
than that, because we can
look at dimensionality.

1064
01:03:01,510 --> 01:03:05,390
And here comes in handy that
I insisted from the beginning

1065
01:03:05,390 --> 01:03:07,050
in introducing this
idea of a notch.

1066
01:03:07,050 --> 01:03:10,400
The notch helps us here
to get this formula right.

1067
01:03:10,400 --> 01:03:18,195
The units of r-- r is just
distance-- so the units of r

1068
01:03:18,195 --> 01:03:20,800
are just distance
units-- and I'll

1069
01:03:20,800 --> 01:03:22,550
pretend that we're using meters.

1070
01:03:22,550 --> 01:03:25,370
It doesn't really matter what
actual units we're using,

1071
01:03:25,370 --> 01:03:29,100
so I'll call this m for meters.

1072
01:03:29,100 --> 01:03:39,440
On the other hand, a of t
comes from this formula, where

1073
01:03:39,440 --> 01:03:41,240
physical distances are
measured in meters,

1074
01:03:41,240 --> 01:03:44,000
but coordinate distances
are measured in notches,

1075
01:03:44,000 --> 01:03:47,790
so a is meters per notch, as
we've said many times before.

1076
01:03:47,790 --> 01:03:50,795
So a of t is meters per notch.

1077
01:03:54,934 --> 01:03:56,850
So that tells us something
about this constant

1078
01:03:56,850 --> 01:03:59,710
of proportionality-- it has to
have the right units to turn

1079
01:03:59,710 --> 01:04:02,170
meters per notch into meters.

1080
01:04:02,170 --> 01:04:06,360
That is, it has to
have units of notches.

1081
01:04:06,360 --> 01:04:07,800
Where did we get notches from?

1082
01:04:07,800 --> 01:04:16,530
The other thing that we know
is that the little k that

1083
01:04:16,530 --> 01:04:19,570
appears in the Friedman
equation, which we know

1084
01:04:19,570 --> 01:04:22,500
is a constant-- we also
know it's a constant that

1085
01:04:22,500 --> 01:04:27,090
has units of 1 over notches
squared, as we worked out

1086
01:04:27,090 --> 01:04:27,860
some time ago.

1087
01:04:43,240 --> 01:04:48,165
Units of k is 1 over
notches squared.

1088
01:04:50,549 --> 01:04:52,090
And that's the only
thing around that

1089
01:04:52,090 --> 01:04:54,040
we can find that has
units of notches,

1090
01:04:54,040 --> 01:04:56,810
so we're going to use that
to make the units turn out

1091
01:04:56,810 --> 01:04:59,500
right in this equation.

1092
01:04:59,500 --> 01:05:05,650
So to get the units right,
we can write it as r of t

1093
01:05:05,650 --> 01:05:09,535
is equal to some
constant-- actually, it's

1094
01:05:09,535 --> 01:05:13,050
more convenient to square
this-- r squared of t

1095
01:05:13,050 --> 01:05:22,050
is equal to some constant times
a squared of t, divided by k.

1096
01:05:22,050 --> 01:05:24,210
And now that constant
is dimensionless.

1097
01:05:24,210 --> 01:05:28,330
The units are all built into the
a's and the k's and everything

1098
01:05:28,330 --> 01:05:28,830
else.

1099
01:05:28,830 --> 01:05:32,245
AUDIENCE: Is the constant
multiplied by that, or--

1100
01:05:32,245 --> 01:05:32,870
PROFESSOR: Yes.

1101
01:05:37,110 --> 01:05:40,030
That equals sign
was a big mistake.

1102
01:05:40,030 --> 01:05:43,070
Constant times a squared over k.

1103
01:05:43,070 --> 01:05:44,990
And that constant is
now dimensionless.

1104
01:05:44,990 --> 01:05:47,690
Because this is
meters squared, this

1105
01:05:47,690 --> 01:05:49,260
is meters squared
per notch squared,

1106
01:05:49,260 --> 01:05:50,540
and this is just
per notch squared,

1107
01:05:50,540 --> 01:05:51,950
so the notches cancel over here.

1108
01:05:56,060 --> 01:06:01,210
OK, now what this
constant is really

1109
01:06:01,210 --> 01:06:03,705
is the statement
of how curved is

1110
01:06:03,705 --> 01:06:05,100
our space-- r is
really a measure

1111
01:06:05,100 --> 01:06:08,320
the curvature of our space-- how
curved is our space for a given

1112
01:06:08,320 --> 01:06:11,140
description of what
the matter is doing?

1113
01:06:11,140 --> 01:06:14,590
a of t is directly related to
the [INAUDIBLE] calculations

1114
01:06:14,590 --> 01:06:16,220
you've already done.

1115
01:06:16,220 --> 01:06:19,700
So this clearly is a
formula of exactly the type

1116
01:06:19,700 --> 01:06:24,290
that I told you we weren't
going to learn how to deal with.

1117
01:06:24,290 --> 01:06:28,550
We're not capable in this course
of describing the Einstein

1118
01:06:28,550 --> 01:06:31,940
field equations,
which determine how

1119
01:06:31,940 --> 01:06:35,660
matter causes a space to curve.

1120
01:06:35,660 --> 01:06:38,080
So I'll just tell
you the answer.

1121
01:06:38,080 --> 01:06:41,590
The answer is that this
constant is equal to 1.

1122
01:06:41,590 --> 01:06:44,600
So it's certainly
a simple answer,

1123
01:06:44,600 --> 01:06:46,330
but we won't be
able to derive it.

1124
01:06:49,440 --> 01:06:54,010
So we end up with
just r squared of t

1125
01:06:54,010 --> 01:06:58,234
is equal to a squared
of t, divided by k.

1126
01:07:32,830 --> 01:07:35,540
Now it may be useful at this
point to remind ourselves

1127
01:07:35,540 --> 01:07:37,830
what little k meant
in the first place.

1128
01:07:37,830 --> 01:07:40,100
We introduced little
k in the context

1129
01:07:40,100 --> 01:07:42,790
of describing a purely
Newtonian model of an expanding

1130
01:07:42,790 --> 01:07:46,460
universe, where we imagined
just a finite sphere of matter

1131
01:07:46,460 --> 01:07:47,760
expanding.

1132
01:07:47,760 --> 01:07:53,430
And in doing that,
we defined k to be

1133
01:07:53,430 --> 01:07:59,750
equal to minus 2 e
divided by c squared,

1134
01:07:59,750 --> 01:08:05,040
where e itself was not a
quantity that we proved

1135
01:08:05,040 --> 01:08:07,360
was conserved.

1136
01:08:07,360 --> 01:08:12,042
And it's related to an
energy, but as we discussed,

1137
01:08:12,042 --> 01:08:14,000
there are various ways
that you could relate it

1138
01:08:14,000 --> 01:08:16,140
to the energies of different
pieces of the system.

1139
01:08:23,040 --> 01:08:24,880
But e was given by
that expression.

1140
01:08:30,290 --> 01:08:36,340
And if I put this
into that, just

1141
01:08:36,340 --> 01:08:42,590
to see more clearly how
our discussion relates

1142
01:08:42,590 --> 01:08:45,950
to our Newtonian
discussion, we get

1143
01:08:45,950 --> 01:08:53,227
r squared is equal to a squared
of t times c squared over 2 e.

1144
01:08:57,520 --> 01:09:01,870
And I wrote it this way mainly
to illustrate, or demonstrate,

1145
01:09:01,870 --> 01:09:06,010
an important point, which
is that our calculation was

1146
01:09:06,010 --> 01:09:09,715
non-relativistic,
but there's a c

1147
01:09:09,715 --> 01:09:12,290
squared appearing
in this formula.

1148
01:09:12,290 --> 01:09:15,819
This c squared really just
arose from our definitions.

1149
01:09:15,819 --> 01:09:20,479
And if we had some
other quantity here

1150
01:09:20,479 --> 01:09:23,301
whose units were--
we have to sub

1151
01:09:23,301 --> 01:09:25,884
units of meters per second for
this formula to come out right.

1152
01:09:25,884 --> 01:09:28,220
If we put some
other velocity here,

1153
01:09:28,220 --> 01:09:32,279
then this constant would not be
1 anymore, but something else.

1154
01:09:32,279 --> 01:09:34,044
So saying that
this constant is 1

1155
01:09:34,044 --> 01:09:37,149
is saying that this
formula is meaningful.

1156
01:09:37,149 --> 01:09:39,790
Putting the c squared
there simplifies things.

1157
01:09:39,790 --> 01:09:43,740
And that in turn means
that the curvature really

1158
01:09:43,740 --> 01:09:46,479
is a relativistic effect.

1159
01:09:46,479 --> 01:09:48,010
OK, we think of
relativistic effects

1160
01:09:48,010 --> 01:09:50,229
as effects that disappear
as the speed of light

1161
01:09:50,229 --> 01:09:51,975
goes to infinity.

1162
01:09:51,975 --> 01:09:54,100
So this formula tells us
that as the speed of light

1163
01:09:54,100 --> 01:09:56,720
goes to infinity-- for
fixed values of things

1164
01:09:56,720 --> 01:09:59,971
like the mass density, which
are buried in a squared of t--

1165
01:09:59,971 --> 01:10:01,970
as the speed of light
goes to infinity for fixed

1166
01:10:01,970 --> 01:10:07,950
values of the mass density,
r squared goes to infinity.

1167
01:10:07,950 --> 01:10:10,590
Now infinity may sound
like it's backwards,

1168
01:10:10,590 --> 01:10:12,230
but it's the right way.

1169
01:10:12,230 --> 01:10:13,880
1 over r is really
the curvature.

1170
01:10:13,880 --> 01:10:16,070
r is the radius of
curvature of the space.

1171
01:10:16,070 --> 01:10:18,610
As r goes to infinity,
our curved space

1172
01:10:18,610 --> 01:10:20,210
looks more and more flat.

1173
01:10:20,210 --> 01:10:22,170
So we're saying that
if you could imagine

1174
01:10:22,170 --> 01:10:24,602
varying the speed of light,
as you made the speed of light

1175
01:10:24,602 --> 01:10:26,060
larger and larger,
this space would

1176
01:10:26,060 --> 01:10:28,180
become flatter and flatter.

1177
01:10:28,180 --> 01:10:30,320
So this curvature
of the space really

1178
01:10:30,320 --> 01:10:33,350
is a relativistic effect,
which is related to the fact

1179
01:10:33,350 --> 01:10:36,830
that the speed of light is
finite and not infinite.

1180
01:10:36,830 --> 01:10:37,695
Yes?

1181
01:10:37,695 --> 01:10:39,570
AUDIENCE: Sorry, when
we replace a with that,

1182
01:10:39,570 --> 01:10:41,140
are we missing a minus sign?

1183
01:10:41,140 --> 01:10:42,473
PROFESSOR: Oh we might be, yeah.

1184
01:10:50,460 --> 01:10:51,595
Minus sign now fixed.

1185
01:10:51,595 --> 01:10:53,720
The point is that for the
case we're talking about,

1186
01:10:53,720 --> 01:10:56,650
e would be negative and
k would be positive.

1187
01:10:56,650 --> 01:11:01,180
So this formula needs an
absolute value sign in it.

1188
01:11:01,180 --> 01:11:01,680
Thank you.

1189
01:11:12,760 --> 01:11:17,400
OK, it may also be
useful to relate r

1190
01:11:17,400 --> 01:11:22,400
more directly to astronomical
observables, which we can do,

1191
01:11:22,400 --> 01:11:26,670
because we have the
Friedman equation up there,

1192
01:11:26,670 --> 01:11:31,770
which relates a to rho.

1193
01:11:31,770 --> 01:11:35,820
And a dot over a is also
the Hubble expansion rate,

1194
01:11:35,820 --> 01:11:37,240
so that's h squared.

1195
01:11:37,240 --> 01:11:38,640
So this formula
tells exactly how

1196
01:11:38,640 --> 01:11:41,115
to write a in terms
of rho and h squared.

1197
01:11:43,409 --> 01:11:44,950
And in fact, it
tells us how to write

1198
01:11:44,950 --> 01:11:48,940
a over the square root of k
in terms of rho and H squared.

1199
01:11:48,940 --> 01:11:53,600
And that's exactly what r
is-- it's a squared over k.

1200
01:11:53,600 --> 01:11:56,960
So putting those
equations together,

1201
01:11:56,960 --> 01:12:02,010
we could write r is equal to c
times the inverse of the Hubble

1202
01:12:02,010 --> 01:12:09,260
expansion rate, over the
square root of omega minus 1.

1203
01:12:09,260 --> 01:12:16,020
Where omega you would
call is equal to rho

1204
01:12:16,020 --> 01:12:18,690
divided by rho sub c.

1205
01:12:18,690 --> 01:12:27,530
And rho sub c is 3 h squared
over 8 pi-- Newton's constant.

1206
01:12:36,087 --> 01:12:37,420
So this formula says two things.

1207
01:12:37,420 --> 01:12:39,730
It says that the
radius of curvature

1208
01:12:39,730 --> 01:12:42,030
becomes infinite
if c were infinite.

1209
01:12:42,030 --> 01:12:43,520
And that says what
I already said,

1210
01:12:43,520 --> 01:12:45,502
that's a relativistic effect.

1211
01:12:45,502 --> 01:12:47,210
It also tells you the
radius of curvature

1212
01:12:47,210 --> 01:12:50,550
goes to 0 as omega goes to 1.

1213
01:12:50,550 --> 01:12:53,870
So omega approaching 1 is
the flat universe case,

1214
01:12:53,870 --> 01:12:55,710
which is what we've
already mumbled about,

1215
01:12:55,710 --> 01:12:57,980
but this formula shows
it very directly.

1216
01:12:57,980 --> 01:13:02,360
As omega approaches 1,
for a fixed value of h

1217
01:13:02,360 --> 01:13:04,650
and a fixed value of
the speed of light,

1218
01:13:04,650 --> 01:13:06,430
the radius of curvature
goes to infinity.

1219
01:13:06,430 --> 01:13:07,971
The space becomes
more and more flat.

1220
01:13:24,100 --> 01:13:27,180
Look, I'm just going to
write one more formula, which

1221
01:13:27,180 --> 01:13:30,090
is really just a redefinition,
but an important redefinition

1222
01:13:30,090 --> 01:13:32,210
as far as where we're
going to be going next.

1223
01:13:32,210 --> 01:13:35,640
And then we'll finish
today's lecture,

1224
01:13:35,640 --> 01:13:36,840
and continue next week.

1225
01:13:46,230 --> 01:13:50,290
What I wanted to do is to
put these definitions back

1226
01:13:50,290 --> 01:13:55,310
into the metric itself.

1227
01:13:55,310 --> 01:14:02,520
So we can write ds squared is
equal to a squared of t divided

1228
01:14:02,520 --> 01:14:07,040
by k-- which is what we
previously called r squared--

1229
01:14:07,040 --> 01:14:13,530
times du squared over
1 minus u squared,

1230
01:14:13,530 --> 01:14:19,510
plus u squared times d theta
squared plus sine squared

1231
01:14:19,510 --> 01:14:23,390
theta d phi squared.

1232
01:14:28,470 --> 01:14:31,350
And now what I want to do is
make one further redefinition

1233
01:14:31,350 --> 01:14:35,090
of this radial variable, which,
remember, initially was psi.

1234
01:14:35,090 --> 01:14:38,050
Then we let u be equal
to the sine of psi.

1235
01:14:38,050 --> 01:14:40,110
Now I'm going to make
one further substitution.

1236
01:14:40,110 --> 01:14:46,920
I'm going to let little
r be equal to u divided

1237
01:14:46,920 --> 01:14:51,900
by the square root of k,
to bring this k inside.

1238
01:14:51,900 --> 01:14:54,170
And that then is
also equal to sine

1239
01:14:54,170 --> 01:14:57,310
of psi, divided by
the square root of k.

1240
01:14:59,930 --> 01:15:03,580
And when we do that, the metric
takes a slightly simpler form.

1241
01:15:03,580 --> 01:15:07,190
ds squared is equal
to a squared of t

1242
01:15:07,190 --> 01:15:08,960
all by itself on
the outside now.

1243
01:15:11,550 --> 01:15:14,710
And then this, after
we factor in the k,

1244
01:15:14,710 --> 01:15:23,840
becomes dr squared over
1 minus k r squared,

1245
01:15:23,840 --> 01:15:29,930
plus r squared times
d theta squared

1246
01:15:29,930 --> 01:15:34,490
plus sine squared
theta, d phi squared.

1247
01:15:43,830 --> 01:15:46,339
And this is the form that the
metric is usually written in.

1248
01:15:46,339 --> 01:15:48,005
It's called the
Robertson-Walker metric.

1249
01:16:17,600 --> 01:16:19,390
So we've only discussed
closed universes.

1250
01:16:19,390 --> 01:16:21,580
I had hoped to discuss
closed and open,

1251
01:16:21,580 --> 01:16:23,300
but open will in fact
follow very quickly

1252
01:16:23,300 --> 01:16:24,870
from what we already have.

1253
01:16:24,870 --> 01:16:29,310
So we'll begin next time by
discussing open universes.