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PROFESSOR: We do
have a little bit
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of finishing to do because
we didn't quite finish
00:00:27.041 --> 00:00:29.480
the dynamics of homogeneous
expansion last time.
00:00:29.480 --> 00:00:33.040
So we'll begin by finishing that
after a brief review of where
00:00:33.040 --> 00:00:34.300
we were last time.
00:00:34.300 --> 00:00:38.100
And then we'll move on to
discuss non-euclidean spaces,
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which I hope will be still
the bulk of today's lecture.
00:00:41.930 --> 00:00:42.430
OK.
00:00:42.430 --> 00:00:44.080
In that case, let's get going.
00:00:44.080 --> 00:00:45.610
Again, as I said,
I want to begin
00:00:45.610 --> 00:00:47.270
by just reviewing
some of the things we
00:00:47.270 --> 00:00:48.462
talked about last time.
00:00:48.462 --> 00:00:50.420
And you should consider
this a good opportunity
00:00:50.420 --> 00:00:52.961
to ask questions if you discover
there are things that you're
00:00:52.961 --> 00:00:55.780
not really sure you understood
as well as you'd like.
00:00:55.780 --> 00:00:58.590
We were talking about the
evolution of a closed universe.
00:00:58.590 --> 00:01:02.050
And to summarize that
calculation we first
00:01:02.050 --> 00:01:04.260
re-shuffled the
first order Friedmann
00:01:04.260 --> 00:01:08.370
equation, the equation for a
dot over a quantity squared,
00:01:08.370 --> 00:01:11.310
by bringing all the d t's to
one side and all the d a's
00:01:11.310 --> 00:01:15.210
on the other side after doing
a little bit of rescaling.
00:01:15.210 --> 00:01:17.580
And we got this equation.
00:01:17.580 --> 00:01:20.700
Which we then said
we can integrate.
00:01:20.700 --> 00:01:25.230
And the integral from
time will go from time 0
00:01:25.230 --> 00:01:29.440
to some arbitrary final time
that we called t tilde sub f,
00:01:29.440 --> 00:01:32.310
where the "tilde" indicates
that we multiplied by c.
00:01:32.310 --> 00:01:37.520
And the sub f means it's the
final time of our calculation.
00:01:37.520 --> 00:01:40.200
And on the other side
we have to integrate
00:01:40.200 --> 00:01:42.730
with corresponding
limits of integration.
00:01:42.730 --> 00:01:45.880
Corresponding to t equals
0 is a equals zero.
00:01:45.880 --> 00:01:48.150
So the lower limit
of integration is 0.
00:01:48.150 --> 00:01:50.660
And the final limit
of integration
00:01:50.660 --> 00:01:54.460
is just the value of a tilde
at the final time, whatever
00:01:54.460 --> 00:01:56.680
that is.
00:01:56.680 --> 00:01:59.670
Then we discovered
that we could actually
00:01:59.670 --> 00:02:02.950
do the integral on the right
if we made a substitution.
00:02:02.950 --> 00:02:06.290
And in the lecture last time
we did it in two stages.
00:02:06.290 --> 00:02:09.160
But the combined
substitution is just
00:02:09.160 --> 00:02:11.790
to replace a tilde
by cosine theta,
00:02:11.790 --> 00:02:14.980
according to this formula, if
we combine the two substitutions
00:02:14.980 --> 00:02:17.120
that we made last time.
00:02:17.120 --> 00:02:20.940
And if we do that we could
integrate the right hand side.
00:02:20.940 --> 00:02:22.990
And the integral
then gives us t tilde
00:02:22.990 --> 00:02:25.360
f is equal to the
integral of that,
00:02:25.360 --> 00:02:27.630
which is just this expression.
00:02:27.630 --> 00:02:33.830
And the expression below that
is just the substitution itself.
00:02:33.830 --> 00:02:37.530
How to relate a tilde to theta,
according to substitution
00:02:37.530 --> 00:02:38.790
which gave us that formula.
00:02:38.790 --> 00:02:40.930
So these two formulas
together allow
00:02:40.930 --> 00:02:44.490
us to determine t tilde
sub f, and a tilde
00:02:44.490 --> 00:02:48.020
sub f in terms of
theta and alpha.
00:02:48.020 --> 00:02:50.760
And once we had that we
realized we no longer needed
00:02:50.760 --> 00:02:53.840
to keep those sub f's
because that was really
00:02:53.840 --> 00:02:56.880
just a way of keeping
track of our notation
00:02:56.880 --> 00:02:57.970
during the calculation.
00:02:57.970 --> 00:02:59.800
It holds for any theta sub f.
00:02:59.800 --> 00:03:01.700
So it holds for any theta.
00:03:01.700 --> 00:03:04.650
So then we just rewrote those
same formulas without the sub
00:03:04.650 --> 00:03:05.400
f's.
00:03:05.400 --> 00:03:08.160
And here we wrote it
removing the tildes,
00:03:08.160 --> 00:03:09.970
replacing them by
their definitions.
00:03:09.970 --> 00:03:11.700
And this was the final answer.
00:03:11.700 --> 00:03:16.660
This describes the evolution
of a closed universe expressed
00:03:16.660 --> 00:03:18.540
in this parametric form.
00:03:18.540 --> 00:03:21.180
That is, we were not
able to explicitly write
00:03:21.180 --> 00:03:24.730
a as a function of t, which
is what would have liked
00:03:24.730 --> 00:03:28.460
to determine how the expansion
varied as a function of time.
00:03:28.460 --> 00:03:31.300
But instead we introduced
the auxiliary variable theta,
00:03:31.300 --> 00:03:33.340
often called the
development angle.
00:03:33.340 --> 00:03:36.340
And in terms of theta we
can express both t and a,
00:03:36.340 --> 00:03:40.450
and thereby indirectly have
an unambiguous relationship
00:03:40.450 --> 00:03:43.470
between a and t.
00:03:43.470 --> 00:03:44.760
Any questions about that?
00:03:47.650 --> 00:03:48.150
OK.
00:03:48.150 --> 00:03:51.230
Then we noticed that
those were, in fact,
00:03:51.230 --> 00:03:53.350
the equations of a cycloid.
00:03:53.350 --> 00:03:55.990
And I won't go through
the argument again,
00:03:55.990 --> 00:03:58.830
but the key point is
that the evolution
00:03:58.830 --> 00:04:04.330
of our closed universe scale
factor, as a function of time,
00:04:04.330 --> 00:04:05.990
is described by
what would happen
00:04:05.990 --> 00:04:11.760
if you had a disk rolling
on the horizontal axis
00:04:11.760 --> 00:04:13.940
with a point marked on the
disk which was initially
00:04:13.940 --> 00:04:17.149
straight down, and then as
the disk rolls that point
00:04:17.149 --> 00:04:19.220
traces out a cycloid.
00:04:19.220 --> 00:04:22.440
And that is exactly the
evolution of a closed universe.
00:04:22.440 --> 00:04:26.140
It starts at size zero,
goes to a maximum size,
00:04:26.140 --> 00:04:28.920
and then contracts again in
a completely symmetric way.
00:04:28.920 --> 00:04:31.550
The contraction phase
is the mirror image
00:04:31.550 --> 00:04:32.900
of the expansion phase.
00:04:39.470 --> 00:04:43.060
And then we went on to calculate
the age of a closed universe.
00:04:43.060 --> 00:04:45.880
And that was really more
of an exercise in algebra
00:04:45.880 --> 00:04:47.570
than anything else.
00:04:47.570 --> 00:04:51.480
The age is really given by
this formula to start with.
00:04:51.480 --> 00:04:55.040
This expresses the age, t,
in terms of alpha and theta.
00:04:55.040 --> 00:04:57.570
And the only problem with
that is that nobody really
00:04:57.570 --> 00:04:59.464
knows what alpha and theta mean.
00:04:59.464 --> 00:05:02.130
It's much more useful to have an
expression for the age in terms
00:05:02.130 --> 00:05:05.420
of things that astronomers
directly measure.
00:05:05.420 --> 00:05:10.884
And one needs two things
to replace alpha and theta.
00:05:10.884 --> 00:05:13.050
And the kinds of things
astronomers directly measure
00:05:13.050 --> 00:05:14.640
are things like the
Hubble parameter
00:05:14.640 --> 00:05:19.010
and the mass density, where they
often express the mass density
00:05:19.010 --> 00:05:22.190
in terms of omega where omega
is the mass density divided
00:05:22.190 --> 00:05:23.440
by the critical density.
00:05:23.440 --> 00:05:25.850
And that's what I chose
to do in the formulas
00:05:25.850 --> 00:05:28.070
that we went on to derive.
00:05:28.070 --> 00:05:30.664
So our goal is simply to
take that formula for t
00:05:30.664 --> 00:05:33.080
and figure out how to express
the alpha and the theta that
00:05:33.080 --> 00:05:36.310
appear in it so that we
can get an expression
00:05:36.310 --> 00:05:39.090
in terms of h and omega.
00:05:39.090 --> 00:05:42.040
And to do that, we'll
go one step at a time.
00:05:42.040 --> 00:05:45.590
We started by just writing
down the Friedmann equation.
00:05:45.590 --> 00:05:48.560
And the Friedmann
equation has an a in it.
00:05:48.560 --> 00:05:50.299
And everything
else in it is rho,
00:05:50.299 --> 00:05:52.840
which can be expressed in terms
of omega without any trouble,
00:05:52.840 --> 00:05:54.670
and h itself.
00:05:54.670 --> 00:05:56.970
So everything else in
it consists of variables
00:05:56.970 --> 00:06:00.700
that we're accepting to be
part of our final answer.
00:06:00.700 --> 00:06:04.080
So we could solve the Friedmann
equations for a or a tilde
00:06:04.080 --> 00:06:07.440
and that will allow
us to eliminate
00:06:07.440 --> 00:06:11.810
any a tilde's from
our expressions.
00:06:11.810 --> 00:06:14.270
Next, alpha was
originally defined
00:06:14.270 --> 00:06:17.520
in terms of this formula which
only involves rho and a tilde.
00:06:17.520 --> 00:06:19.260
We now know what
to do with a tilde.
00:06:19.260 --> 00:06:21.314
We could substitute
that formula.
00:06:21.314 --> 00:06:22.980
And rho, we always
knew what to do with.
00:06:22.980 --> 00:06:26.030
We could express that
in terms of omega.
00:06:26.030 --> 00:06:28.730
So we can instantly
solve that equation
00:06:28.730 --> 00:06:31.090
and get an equation for alpha
in terms of the quantities
00:06:31.090 --> 00:06:33.840
that we want to appear
in our final answer.
00:06:33.840 --> 00:06:36.310
And there's still one
more thing we need.
00:06:36.310 --> 00:06:38.410
We need an expression for theta.
00:06:38.410 --> 00:06:41.230
And theta we can get by
looking at the other of those
00:06:41.230 --> 00:06:43.720
two parametric
equations, the one that's
00:06:43.720 --> 00:06:47.820
not the "t equals" equation, but
rather the "a equals" equation.
00:06:47.820 --> 00:06:51.730
And in this equation we know
everything except theta itself.
00:06:51.730 --> 00:06:55.766
So we could substitute for a
over root k from up here- that.
00:06:55.766 --> 00:06:57.140
And on the right
hand side we can
00:06:57.140 --> 00:07:00.060
place alpha by that
expression and then the 1
00:07:00.060 --> 00:07:01.730
minus cosine theta stays.
00:07:01.730 --> 00:07:04.730
And now we could solve
this equation for theta.
00:07:04.730 --> 00:07:07.600
I might just mention that,
in lecture last time I
00:07:07.600 --> 00:07:09.640
actually mis-copied
this equation.
00:07:09.640 --> 00:07:11.864
I forgot the factor of
omega in the numerator.
00:07:11.864 --> 00:07:13.280
So if any of you
were taking notes
00:07:13.280 --> 00:07:15.220
you can go back and
correct your notes.
00:07:15.220 --> 00:07:19.420
But it's correct in
the printed notes.
00:07:19.420 --> 00:07:21.739
And now it's
correct on a screen.
00:07:21.739 --> 00:07:23.780
So this you could either
solve for cosine theta--
00:07:23.780 --> 00:07:26.066
initially what I did was
to solve for cosine theta.
00:07:26.066 --> 00:07:27.440
But then it turned
out to be more
00:07:27.440 --> 00:07:29.110
useful to know
what sine theta is,
00:07:29.110 --> 00:07:31.870
because sine theta appears
in that parametric expression
00:07:31.870 --> 00:07:33.540
for the age.
00:07:33.540 --> 00:07:35.880
So if you know cosine
theta you can, of course,
00:07:35.880 --> 00:07:38.070
get sine theta because
sine theta is just
00:07:38.070 --> 00:07:41.330
the square root of 1 minus
cosine squared theta.
00:07:41.330 --> 00:07:42.482
And that's what we did.
00:07:42.482 --> 00:07:44.190
And that's how we got
a square root here.
00:07:44.190 --> 00:07:46.023
And since square roots
can have either sign,
00:07:46.023 --> 00:07:47.710
there's a plus or
minus there which
00:07:47.710 --> 00:07:52.660
depends on where you are in
the evolution of our universe.
00:07:52.660 --> 00:07:54.770
The right hand side
here is always positive
00:07:54.770 --> 00:07:57.100
because we define that
square root symbol
00:07:57.100 --> 00:07:59.600
to mean the positive
square root,
00:07:59.600 --> 00:08:03.790
where the plus or minus, in
the end, tells you the sine.
00:08:03.790 --> 00:08:07.470
And sine theta itself--
we know what theta does.
00:08:07.470 --> 00:08:09.120
It goes from 0 to 2 pi.
00:08:09.120 --> 00:08:11.070
So sine theta
starts out positive
00:08:11.070 --> 00:08:14.640
and after theta crosses
pi, sine theta is
00:08:14.640 --> 00:08:18.100
negative for the second
half of the evolution.
00:08:18.100 --> 00:08:20.780
Meaning, sine theta is negative
for the contracting phase
00:08:20.780 --> 00:08:23.005
and positive for
the expanding phase.
00:08:26.480 --> 00:08:29.780
We then put all that together
into the formula for CT,
00:08:29.780 --> 00:08:32.150
which is alpha theta
minus sine theta.
00:08:32.150 --> 00:08:34.950
The alpha becomes this factor.
00:08:34.950 --> 00:08:37.970
And the theta minus sine
theta becomes the arc sine
00:08:37.970 --> 00:08:41.570
of this expression, minus
or plus that expression,
00:08:41.570 --> 00:08:44.770
where this is just sine
theta from that formula.
00:08:44.770 --> 00:08:48.620
And it's minus sine theta, which
is why the plus minus becomes
00:08:48.620 --> 00:08:50.480
minus or plus.
00:08:50.480 --> 00:08:52.830
The signs get a little tricky,
but it's, in principle,
00:08:52.830 --> 00:08:53.910
straightforward.
00:08:53.910 --> 00:08:55.670
It all just follows
from this formula.
00:08:55.670 --> 00:08:57.045
And if you know
the sine of theta
00:08:57.045 --> 00:09:00.550
you have all your
problems solved.
00:09:00.550 --> 00:09:02.580
So we put all that
together into a table.
00:09:02.580 --> 00:09:04.260
This is just a copy
of the same formula
00:09:04.260 --> 00:09:07.600
that we had on the
previous slide.
00:09:07.600 --> 00:09:09.840
Theta is what appears in
this right hand column.
00:09:09.840 --> 00:09:12.705
It's indicated as the sine
inverse of some expression.
00:09:12.705 --> 00:09:16.265
It refers to that
expression-- an abbreviation.
00:09:16.265 --> 00:09:17.390
So we know what theta does.
00:09:17.390 --> 00:09:21.690
Theta just goes from 0
to 2 pi in quadrants.
00:09:21.690 --> 00:09:24.470
At least, it pays to divide
it up into quadrants.
00:09:24.470 --> 00:09:29.280
Sine theta is always
positive for the first half
00:09:29.280 --> 00:09:30.860
and negative for
the second half.
00:09:30.860 --> 00:09:34.530
And that means that we have the
plus sign, or the upper sign,
00:09:34.530 --> 00:09:36.900
for the first half of
the motion and the minus
00:09:36.900 --> 00:09:39.830
sign for the second
half of the motion.
00:09:39.830 --> 00:09:42.480
Omega we could just
calculate in terms of theta.
00:09:42.480 --> 00:09:44.846
No problem about filling
in the omega column.
00:09:44.846 --> 00:09:46.970
And we know that we're
expanding for the first half
00:09:46.970 --> 00:09:48.590
and contracting for
the second half,
00:09:48.590 --> 00:09:50.447
so that really
completes the table.
00:09:50.447 --> 00:09:52.280
And the important thing
when you're actually
00:09:52.280 --> 00:09:55.830
using this formula is
to decide what theta is.
00:09:55.830 --> 00:09:57.910
And once you have that,
the sine of that--
00:09:57.910 --> 00:09:59.510
the inverse sine
of that-- well, no.
00:09:59.510 --> 00:10:00.010
Excuse me.
00:10:00.010 --> 00:10:02.080
Theta itself appears there,
and the sine of theta
00:10:02.080 --> 00:10:03.130
appears there.
00:10:03.130 --> 00:10:06.550
And theta itself you
have to figure out
00:10:06.550 --> 00:10:08.910
which is the relevant
value in terms
00:10:08.910 --> 00:10:11.200
of where you are
in the evolution.
00:10:11.200 --> 00:10:14.830
The point is that sine
theta itself goes--
00:10:14.830 --> 00:10:18.930
does not uniquely
determine what theta is.
00:10:18.930 --> 00:10:20.600
OK.
00:10:20.600 --> 00:10:22.240
That, I think,
completes our discussion
00:10:22.240 --> 00:10:25.900
of the evolution of
closed universes.
00:10:25.900 --> 00:10:28.650
I think it completes everything
that we did last time.
00:10:28.650 --> 00:10:29.775
So are there any questions?
00:10:33.830 --> 00:10:34.790
OK.
00:10:34.790 --> 00:10:36.300
Good.
00:10:36.300 --> 00:10:38.820
To finish our discussion
of the evolution
00:10:38.820 --> 00:10:41.360
of matter-dominated
universes, we
00:10:41.360 --> 00:10:44.310
go on to discuss open universes.
00:10:44.310 --> 00:10:47.450
And open universes are
really the same algebra
00:10:47.450 --> 00:10:48.620
as closed universes.
00:10:48.620 --> 00:10:51.760
They just differ
by the sine of k.
00:10:51.760 --> 00:10:55.210
Because one doesn't like to
deal with negative numbers,
00:10:55.210 --> 00:10:58.170
I defined kappa to
be equal to minus k,
00:10:58.170 --> 00:11:00.220
so that for our
open universes kappa
00:11:00.220 --> 00:11:03.730
was positive while
k was negative.
00:11:03.730 --> 00:11:07.200
Then I used a different
substitution for a tilde.
00:11:07.200 --> 00:11:09.500
Instead of a tilde
being a divided
00:11:09.500 --> 00:11:11.800
by the square root of
k, which in this case
00:11:11.800 --> 00:11:14.410
would be the square root of a
negative number-- which, one
00:11:14.410 --> 00:11:16.201
doesn't like to deal
with imaginary numbers
00:11:16.201 --> 00:11:18.410
if one doesn't have any need to.
00:11:18.410 --> 00:11:21.000
So instead, for
the open universe
00:11:21.000 --> 00:11:23.870
I'm defining a tilde to
be a divided by the square
00:11:23.870 --> 00:11:28.310
root of kappa, so that
a tilde is again real.
00:11:28.310 --> 00:11:30.341
Then I'm going to skip
all the algebra here.
00:11:30.341 --> 00:11:32.840
There's a little bit more of
it shown in the printed lecture
00:11:32.840 --> 00:11:34.880
notes.
00:11:34.880 --> 00:11:38.020
But there are no
new concepts here.
00:11:38.020 --> 00:11:40.280
Everything is really
the same, conceptually,
00:11:40.280 --> 00:11:42.880
as it was for the
closed universe.
00:11:42.880 --> 00:11:44.460
One difference is
that this time,
00:11:44.460 --> 00:11:46.350
instead of getting
trigonometric functions,
00:11:46.350 --> 00:11:49.900
we find that we get
hypergeometric functions.
00:11:49.900 --> 00:11:51.360
Hyper-- yeah.
00:11:51.360 --> 00:11:54.770
Hyper trigonometric functions,
I think, is the right word.
00:11:54.770 --> 00:11:59.110
That is, sinhs and coshs
instead of sines and cosines.
00:11:59.110 --> 00:12:00.360
The formulas are very similar.
00:12:00.360 --> 00:12:04.060
These are the formulas we get
for the open universe case,
00:12:04.060 --> 00:12:06.780
compared to those formulas
for the closed universe case.
00:12:09.330 --> 00:12:12.200
We get a change in
the order of-- instead
00:12:12.200 --> 00:12:15.106
of theta minus sine theta, we
get sinh theta minus theta.
00:12:15.106 --> 00:12:16.730
But that's what you
have to get if it's
00:12:16.730 --> 00:12:18.460
going to turn out
to be positive.
00:12:18.460 --> 00:12:20.140
Sine theta is always
less than theta.
00:12:20.140 --> 00:12:22.080
So this is a positive quantity.
00:12:22.080 --> 00:12:23.900
Sinh theta is always
greater than theta,
00:12:23.900 --> 00:12:26.160
so this is a positive quantity.
00:12:26.160 --> 00:12:27.840
And same for the second lines.
00:12:27.840 --> 00:12:30.090
Cosh theta minus 1
is always positive,
00:12:30.090 --> 00:12:32.870
and one minus cosine
theta is always positive.
00:12:32.870 --> 00:12:35.120
So you really know which
order to write them in just
00:12:35.120 --> 00:12:36.720
by knowing that you
want to write down
00:12:36.720 --> 00:12:39.910
something that's positive.
00:12:39.910 --> 00:12:43.810
So these formulas describe the
evolution of the open universe
00:12:43.810 --> 00:12:47.810
exactly the same way as those
formulas describe the evolution
00:12:47.810 --> 00:12:51.390
of a closed,
matter-dominated universe.
00:12:51.390 --> 00:12:52.620
So any questions?
00:12:55.610 --> 00:12:56.110
OK.
00:12:56.110 --> 00:12:58.780
Next step, just repeating what
we did for the closed universe,
00:12:58.780 --> 00:13:02.271
we can derive a formula for
the age of an open universe.
00:13:02.271 --> 00:13:03.770
And again, it's
really just a matter
00:13:03.770 --> 00:13:06.380
of substituting into
the formula we already
00:13:06.380 --> 00:13:10.500
have to be able to re-express it
in terms of useful quantities.
00:13:10.500 --> 00:13:12.940
Which, again, I choose to
be the Hubble expansion
00:13:12.940 --> 00:13:15.252
rate and omega.
00:13:15.252 --> 00:13:19.380
And here I've put together all
three formulas for the age.
00:13:19.380 --> 00:13:20.870
The flat universe,
the first one we
00:13:20.870 --> 00:13:24.110
did, where t is just h
inverse if we bring it
00:13:24.110 --> 00:13:25.570
to the other side, times 2/3.
00:13:28.200 --> 00:13:33.300
And the open universe on the
top, and the closed universe
00:13:33.300 --> 00:13:36.260
on the bottom.
00:13:36.260 --> 00:13:39.000
Now one of the, perhaps
surprising, things
00:13:39.000 --> 00:13:43.910
that one finds here is that
all three of these expressions
00:13:43.910 --> 00:13:46.752
look fairly different
from each other.
00:13:46.752 --> 00:13:48.210
And you might think
that that would
00:13:48.210 --> 00:13:51.730
give some kind of a jagged,
discontinuous curve.
00:13:51.730 --> 00:13:55.310
But you can go ahead and
plot it, which I did.
00:13:55.310 --> 00:13:56.690
And there's the plot.
00:13:56.690 --> 00:13:59.850
It's one nice, smooth curve.
00:13:59.850 --> 00:14:04.050
And we won't go into this
in detail, but many of you
00:14:04.050 --> 00:14:07.570
have had courses in complex
functions, functions
00:14:07.570 --> 00:14:09.140
of a complex variable.
00:14:09.140 --> 00:14:11.810
If you know about functions
of a complex variable
00:14:11.810 --> 00:14:16.870
you can tell that these are,
in fact, all the same function.
00:14:16.870 --> 00:14:20.320
That is, if omega is,
say, bigger than 1,
00:14:20.320 --> 00:14:22.512
this formula can be
evaluated straightforwardly.
00:14:22.512 --> 00:14:24.220
It involves only things
like square roots
00:14:24.220 --> 00:14:26.280
of positive numbers.
00:14:26.280 --> 00:14:28.410
But you could also try
evaluating this formula
00:14:28.410 --> 00:14:30.420
for omega bigger than 1.
00:14:30.420 --> 00:14:33.020
And then you have square roots
of negative numbers appearing.
00:14:33.020 --> 00:14:34.520
But square roots
of negative numbers
00:14:34.520 --> 00:14:36.910
are OK if you know
about complex numbers.
00:14:36.910 --> 00:14:38.727
They're just purely imaginary.
00:14:38.727 --> 00:14:40.810
And then you get trigonometric
functions, and even
00:14:40.810 --> 00:14:43.710
inverse trigonometric functions,
or inverse hyperbolic trig
00:14:43.710 --> 00:14:46.580
functions of
imaginary arguments.
00:14:46.580 --> 00:14:48.090
But all those are well-defined.
00:14:48.090 --> 00:14:50.380
And if you work through
what the definitions are,
00:14:50.380 --> 00:14:52.750
the top line really is
identical to the bottom line.
00:14:52.750 --> 00:14:55.300
Those really are
the same function.
00:14:55.300 --> 00:14:58.720
And that's why one expects
that, when you plot them
00:14:58.720 --> 00:15:05.380
they will join together
smoothly, as they clearly do.
00:15:05.380 --> 00:15:08.090
The point in the middle
here is the first age
00:15:08.090 --> 00:15:10.190
that we derived for
the flat universe
00:15:10.190 --> 00:15:14.200
where omega is equal to 1
and h t is just equal to 2/3.
00:15:14.200 --> 00:15:16.290
That is, t is equal
to 2/3 h inverse
00:15:16.290 --> 00:15:19.450
for a flat universe,
which is the middle dot.
00:15:19.450 --> 00:15:22.770
And the closed universes
are on the right
00:15:22.770 --> 00:15:24.470
and open universes
are on the left.
00:15:26.821 --> 00:15:27.320
OK.
00:15:27.320 --> 00:15:31.310
Any questions about
these age calculations?
00:15:31.310 --> 00:15:32.650
Yes?
00:15:32.650 --> 00:15:38.530
AUDIENCE: I noticed that,
for the open solution,
00:15:38.530 --> 00:15:42.450
there's a case where you
get some imaginary numbers?
00:15:46.940 --> 00:15:50.610
PROFESSOR: If you
use these formulas
00:15:50.610 --> 00:15:52.950
you don't get any
imaginary numbers.
00:15:52.950 --> 00:15:55.000
But if you tried to use,
say, the bottom formula
00:15:55.000 --> 00:15:56.970
for a value of
omega less than 1 it
00:15:56.970 --> 00:15:58.930
would give you
imaginary numbers.
00:15:58.930 --> 00:16:00.570
And it would, in
fact, if you trace
00:16:00.570 --> 00:16:03.310
through what those
imaginary numbers do,
00:16:03.310 --> 00:16:06.200
it would give you the
formula on the top.
00:16:06.200 --> 00:16:08.407
So it's all consistent.
00:16:08.407 --> 00:16:10.240
It's most straightforward
to use the formula
00:16:10.240 --> 00:16:12.410
on the top for omega less
than 1 and the formula
00:16:12.410 --> 00:16:14.250
on the bottom for
omega greater than 1.
00:16:14.250 --> 00:16:16.310
And then one never
confronts imaginary numbers.
00:16:16.310 --> 00:16:17.300
AUDIENCE: OK.
00:16:23.219 --> 00:16:24.510
PROFESSOR: Any other questions?
00:16:27.970 --> 00:16:29.900
OK.
00:16:29.900 --> 00:16:31.270
Where are we going next?
00:16:31.270 --> 00:16:33.580
Finally, just actually
one last graph
00:16:33.580 --> 00:16:36.490
on the evolution of
matter-dominated universes,
00:16:36.490 --> 00:16:39.280
which is the final
form of a of t
00:16:39.280 --> 00:16:41.790
for a matter-dominated universe.
00:16:41.790 --> 00:16:45.320
If you re-scale things
by dividing a over root
00:16:45.320 --> 00:16:49.860
k by alpha, and c t over
alpha-- if you look back
00:16:49.860 --> 00:16:51.524
at our equations--
let me go back
00:16:51.524 --> 00:16:52.690
to where the equations were.
00:16:52.690 --> 00:16:54.810
We're really just
graphing these equations.
00:16:54.810 --> 00:16:56.900
If I divide this
equation by alpha,
00:16:56.900 --> 00:16:58.670
I just get a pure
function of theta
00:16:58.670 --> 00:17:00.320
with dimensionless variables.
00:17:00.320 --> 00:17:03.820
And similarly here, if I
divide a over root k by alpha
00:17:03.820 --> 00:17:07.666
I just get 1 minus cosine
theta, which is a pure number.
00:17:07.666 --> 00:17:09.040
So that's what
I've chosen to do.
00:17:09.040 --> 00:17:13.560
And the dimensionality works the
same for the open case as well.
00:17:13.560 --> 00:17:15.720
So it allows you to
draw a plot which
00:17:15.720 --> 00:17:17.300
is just independent
of parameters.
00:17:17.300 --> 00:17:19.410
All the parameters are
absorbed into the way
00:17:19.410 --> 00:17:21.550
the axes are defined,
which are both
00:17:21.550 --> 00:17:24.109
defined as
dimensionless numbers.
00:17:24.109 --> 00:17:26.980
And in that case
the closed universe
00:17:26.980 --> 00:17:29.310
survives for a duration of 2 pi.
00:17:29.310 --> 00:17:32.710
The axis here, at least, has
the same duration as theta.
00:17:32.710 --> 00:17:36.530
It's not actually theta, because
t is not linear in theta.
00:17:36.530 --> 00:17:38.640
But this point
does change by 2 pi
00:17:38.640 --> 00:17:41.440
as theta goes from 0 to 2 pi.
00:17:41.440 --> 00:17:45.100
And one can see here, the
three possible curves.
00:17:45.100 --> 00:17:49.220
A closed universe which
starts and then falls back,
00:17:49.220 --> 00:17:52.250
a flat universe which
goes off to the right
00:17:52.250 --> 00:17:55.390
and has actually a constant
slope as you go out here,
00:17:55.390 --> 00:17:59.584
and an open universe which
behaves slightly differently.
00:17:59.584 --> 00:18:01.000
Actually, I think
I was wrong when
00:18:01.000 --> 00:18:03.090
I said the flat universe
has a constant slope.
00:18:03.090 --> 00:18:07.590
But the open and the flat case
are similar to each other.
00:18:07.590 --> 00:18:09.880
They both go off to infinity,
but in different ways.
00:18:12.460 --> 00:18:16.250
And all three of them merge
as you go backwards in time.
00:18:16.250 --> 00:18:19.110
That's not something that
might have been obvious
00:18:19.110 --> 00:18:21.030
before we wrote
down the equations.
00:18:21.030 --> 00:18:23.470
But in very early
times all universes
00:18:23.470 --> 00:18:26.510
look like they're
flat universes,
00:18:26.510 --> 00:18:28.850
if you go to early enough times.
00:18:28.850 --> 00:18:30.590
And that actually is
an important point
00:18:30.590 --> 00:18:33.380
which we'll talk
about later in terms
00:18:33.380 --> 00:18:36.170
of what's called the flatness
problem of cosmology.
00:18:36.170 --> 00:18:41.550
That basically is the
consequences of that fact.
00:18:41.550 --> 00:18:42.130
Yes?
00:18:42.130 --> 00:18:43.558
AUDIENCE: Why
didn't you just say
00:18:43.558 --> 00:18:45.938
all of them looks like
open universes [INAUDIBLE]?
00:18:45.938 --> 00:18:46.890
PROFESSOR: [LAUGHS]
00:18:46.890 --> 00:18:49.280
AUDIENCE: I mean, what's
special about flat universes?
00:18:49.280 --> 00:18:50.820
PROFESSOR: Well,
actually, there is
00:18:50.820 --> 00:18:52.028
something special about flat.
00:18:52.028 --> 00:18:55.340
Which is that, if we plotted
omega as the function of time
00:18:55.340 --> 00:18:58.430
all of them approach omega
equals 1 as time goes to 0.
00:18:58.430 --> 00:19:00.320
So there is a very
definite meaning
00:19:00.320 --> 00:19:02.290
of saying that they're
all approaching flat,
00:19:02.290 --> 00:19:06.660
rather than they're all
approaching open or closed.
00:19:06.660 --> 00:19:08.394
Good question, though.
00:19:08.394 --> 00:19:09.810
Because from this
graph you really
00:19:09.810 --> 00:19:13.260
can't tell anything
special about the flat.
00:19:13.260 --> 00:19:15.880
Any other questions?
00:19:15.880 --> 00:19:16.618
Yes?
00:19:16.618 --> 00:19:21.100
AUDIENCE: So does this mean
that the open and flat solutions
00:19:21.100 --> 00:19:24.274
will extend indefinitely?
00:19:24.274 --> 00:19:24.940
PROFESSOR: Yeah.
00:19:24.940 --> 00:19:27.290
The open and flat solutions
extend indefinitely in time.
00:19:27.290 --> 00:19:29.094
That's right, they do.
00:19:29.094 --> 00:19:31.260
And one can see that from
the formulas or the graph.
00:19:34.040 --> 00:19:34.961
Yes?
00:19:34.961 --> 00:19:37.847
AUDIENCE: So for plotting
omega as a function of time,
00:19:37.847 --> 00:19:39.530
I'm a little bit
confused as to how
00:19:39.530 --> 00:19:44.750
it changes as-- for example,
an open universe expands,
00:19:44.750 --> 00:19:47.867
or a closed universe--
because it seemed
00:19:47.867 --> 00:19:50.600
like, from our calculations,
that when the universe was
00:19:50.600 --> 00:19:53.600
expanding, omega was increasing?
00:19:53.600 --> 00:19:57.390
Is that-- at least
for a closed universe?
00:19:57.390 --> 00:19:58.890
PROFESSOR: That is
true for a closed
00:19:58.890 --> 00:20:00.570
universe during the
expanding phase.
00:20:00.570 --> 00:20:02.611
For a closed universe
during the expanding phase,
00:20:02.611 --> 00:20:03.950
omega does increase.
00:20:03.950 --> 00:20:05.880
It starts out as 1
and then it rises
00:20:05.880 --> 00:20:08.910
to infinity when the universe
reaches its maximum size
00:20:08.910 --> 00:20:10.675
and is about to turn
around and go back.
00:20:10.675 --> 00:20:12.800
Because it doesn't mean
the mass density increases.
00:20:12.800 --> 00:20:14.290
That's maybe what's
confusing you.
00:20:14.290 --> 00:20:17.690
The actual mass density always
decreases as it expands,
00:20:17.690 --> 00:20:21.540
but the critical density
decreases even faster.
00:20:21.540 --> 00:20:26.400
So the ratio, omega, actually
rises for a closed universe
00:20:26.400 --> 00:20:27.015
as it expands.
00:20:30.420 --> 00:20:32.850
For an open universe
it's the reverse.
00:20:32.850 --> 00:20:35.100
For an open universe
omega starts
00:20:35.100 --> 00:20:37.660
out being 1 at
early times, as it
00:20:37.660 --> 00:20:41.040
does for any
matter-dominated universe.
00:20:41.040 --> 00:20:42.900
And as the universe
expands, omega
00:20:42.900 --> 00:20:47.710
gets smaller and smaller
for an open universe.
00:20:47.710 --> 00:20:51.410
And it follows-- I don't
have them on slides here,
00:20:51.410 --> 00:20:52.950
but we do have in
the notes formulas
00:20:52.950 --> 00:20:56.280
that we derived, that we did
on the blackboard, that give us
00:20:56.280 --> 00:20:58.920
omega as a function of theta.
00:20:58.920 --> 00:21:01.040
And those formulas, you
can just look at them
00:21:01.040 --> 00:21:03.180
and see how omega behaves
as the universe evolves.
00:21:03.180 --> 00:21:06.400
Because as the universe
evolves, theta just increases.
00:21:06.400 --> 00:21:09.210
And those formulas
do trivially show
00:21:09.210 --> 00:21:11.080
what I said about
how omega evolves.
00:21:14.970 --> 00:21:17.270
Any other questions?
00:21:17.270 --> 00:21:17.954
Yes?
00:21:17.954 --> 00:21:21.356
AUDIENCE: Why a over alpha root
k, but for a flat universe k
00:21:21.356 --> 00:21:22.390
is 0?
00:21:22.390 --> 00:21:23.390
PROFESSOR: That's right.
00:21:23.390 --> 00:21:25.570
I didn't really say that,
but for the flat universe
00:21:25.570 --> 00:21:26.986
there's an arbitrary
normalization
00:21:26.986 --> 00:21:30.150
that one chooses in
drawing this graph.
00:21:30.150 --> 00:21:32.180
And it was really an
arbitrary choice for me
00:21:32.180 --> 00:21:35.660
to draw the flat case
to join on smoothly
00:21:35.660 --> 00:21:37.300
with the open and closed cases.
00:21:37.300 --> 00:21:40.040
I could have put any constant
in front of t to the 2/3.
00:21:40.040 --> 00:21:42.270
And you're right, then
they would not necessarily
00:21:42.270 --> 00:21:45.480
mesh unless I chose that
constant in the right way.
00:21:51.882 --> 00:21:52.382
Yes?
00:21:52.382 --> 00:21:54.847
AUDIENCE: Based on that, is
there a particular reason
00:21:54.847 --> 00:21:58.291
you decided to chose [INAUDIBLE]
a flat universe looks
00:21:58.291 --> 00:21:58.791
like this?
00:21:58.791 --> 00:22:00.434
Is there a particular
thing you're
00:22:00.434 --> 00:22:02.270
trying to show in
choosing [INAUDIBLE]?
00:22:02.270 --> 00:22:02.570
PROFESSOR: OK.
00:22:02.570 --> 00:22:04.445
The question is, is
there a particular reason
00:22:04.445 --> 00:22:07.550
why I chose the
normalization that I chose?
00:22:07.550 --> 00:22:09.440
If I did not choose
it, it would only
00:22:09.440 --> 00:22:12.500
differ by an overall factor.
00:22:12.500 --> 00:22:14.420
So it would still
look-- the flat line
00:22:14.420 --> 00:22:16.610
by itself would look
indistinguishable, really.
00:22:16.610 --> 00:22:18.740
It would just be
higher or lower.
00:22:18.740 --> 00:22:21.180
So the only question is how
it meshes with the others.
00:22:21.180 --> 00:22:22.620
And since the flat
case really is
00:22:22.620 --> 00:22:24.590
the borderline between
the other two cases,
00:22:24.590 --> 00:22:27.380
and since this constant
that appears in front of t
00:22:27.380 --> 00:22:30.370
to the 2/3 has no
physical meaning whatever,
00:22:30.370 --> 00:22:32.300
it seems the
sensible thing to do
00:22:32.300 --> 00:22:36.470
is to plot it so that
it looks like the limit
00:22:36.470 --> 00:22:37.780
of an open or closed universe.
00:22:37.780 --> 00:22:40.300
Because physically it is the
same as the limit of an open
00:22:40.300 --> 00:22:42.175
or a closed universe
coming from either side.
00:22:49.930 --> 00:22:52.029
Any other questions?
00:22:52.029 --> 00:22:53.070
Those are good questions.
00:22:56.410 --> 00:22:56.910
OK.
00:22:56.910 --> 00:23:00.450
In that case, we are
finished with the evolution
00:23:00.450 --> 00:23:05.330
of matter-dominated
universes, and ready to start
00:23:05.330 --> 00:23:09.560
talking about
non-euclidean spaces.
00:23:09.560 --> 00:23:14.640
So what we'll be doing next
is kind of a mini introduction
00:23:14.640 --> 00:23:19.350
to general relativity, which
how non-euclidean spaces
00:23:19.350 --> 00:23:21.790
enter physics.
00:23:21.790 --> 00:23:24.530
Now, needless to say,
general relativity
00:23:24.530 --> 00:23:29.234
is an entire course
separate from this course.
00:23:29.234 --> 00:23:31.150
And of course that even
has more prerequisites
00:23:31.150 --> 00:23:33.510
that this course
has, so we're not
00:23:33.510 --> 00:23:36.070
going to duplicate
what would be taught
00:23:36.070 --> 00:23:38.760
in the general relativity class.
00:23:38.760 --> 00:23:40.440
But it turns out
that the discussion
00:23:40.440 --> 00:23:42.690
of general relativity does,
in fact, divide pretty
00:23:42.690 --> 00:23:46.769
cleanly into two major issues.
00:23:46.769 --> 00:23:49.060
And we will be dealing with
one of those issues but not
00:23:49.060 --> 00:23:50.360
the other.
00:23:50.360 --> 00:23:53.710
In particular, what we
will be doing in this class
00:23:53.710 --> 00:23:58.790
is learning how the formulas
of general relativity
00:23:58.790 --> 00:24:01.680
is used to describe
curved spaces.
00:24:01.680 --> 00:24:06.960
And we will learn how particles
move in curved spaces.
00:24:06.960 --> 00:24:10.630
So we'll be able to
analyze trajectories
00:24:10.630 --> 00:24:12.880
in any curbed space
if somebody tells you
00:24:12.880 --> 00:24:16.080
what the curved space
itself looks like.
00:24:16.080 --> 00:24:19.810
What we will not be
doing is we won't even
00:24:19.810 --> 00:24:26.010
attempt to describe how general
relativity predicts that matter
00:24:26.010 --> 00:24:28.740
should cause space to curve.
00:24:28.740 --> 00:24:31.200
That would be left
entirely for the GR course
00:24:31.200 --> 00:24:34.260
that you may or may not take.
00:24:34.260 --> 00:24:35.800
But it will not
be discussed here.
00:24:35.800 --> 00:24:37.299
There's only one
point where we will
00:24:37.299 --> 00:24:38.560
need a result of that sort.
00:24:38.560 --> 00:24:41.860
We'll want to know how the
matter in our Friedman Roberson
00:24:41.860 --> 00:24:48.120
Walker universe
affects the curvature.
00:24:48.120 --> 00:24:50.090
And there I would just
give you the result.
00:24:50.090 --> 00:24:51.590
I'll try to make
it sound plausible.
00:24:51.590 --> 00:24:53.770
But I won't make
any pretense being
00:24:53.770 --> 00:24:55.260
able to drive that result.
00:24:55.260 --> 00:24:59.260
That is, we will not be able
to drive how much space curves
00:24:59.260 --> 00:25:02.445
as a consequence of the
matter that's in it.
00:25:02.445 --> 00:25:03.820
But we will write
down the answer
00:25:03.820 --> 00:25:04.840
so you at least know
what the answer is
00:25:04.840 --> 00:25:06.555
for a homogeneous
isotropic universe.
00:25:09.600 --> 00:25:14.100
So here's a picturesque
slide about curve space.
00:25:14.100 --> 00:25:17.050
Four years ago, I
think it was, a postdoc
00:25:17.050 --> 00:25:19.900
here named Mustafa Amin
gave this lecture for me
00:25:19.900 --> 00:25:21.110
because I was out of town.
00:25:21.110 --> 00:25:24.249
And he had much more colorful
transparencies than I ever do.
00:25:24.249 --> 00:25:26.290
So I'll be using some of
his transparencies here.
00:25:26.290 --> 00:25:30.840
And this is one of
his opening slides.
00:25:30.840 --> 00:25:33.380
So this is what we want to
talk about-- curved space
00:25:33.380 --> 00:25:38.590
as illustrated in
that nifty picture.
00:25:38.590 --> 00:25:42.410
So I want to begin with a kind
of an historical introduction.
00:25:42.410 --> 00:25:45.740
To be honest, I'm pretty
much following the logic
00:25:45.740 --> 00:25:48.620
of the first chapter of Steve
Weinberg's General Relativity
00:25:48.620 --> 00:25:50.370
book.
00:25:50.370 --> 00:25:52.496
Non-Euclidean geometry
of course starts
00:25:52.496 --> 00:25:54.370
by thinking about
Euclidean geometry and then
00:25:54.370 --> 00:25:58.060
how one might be
move away from it.
00:25:58.060 --> 00:26:00.100
And historically, there's
kind of a clear cut
00:26:00.100 --> 00:26:03.130
path, which was followed.
00:26:03.130 --> 00:26:09.110
Euclid based his geometry, as
described in Euclid's elements,
00:26:09.110 --> 00:26:11.587
in terms of five postulates.
00:26:11.587 --> 00:26:13.670
The first of which is that
a straight line segment
00:26:13.670 --> 00:26:15.960
can be drawn joining
any two points.
00:26:15.960 --> 00:26:17.460
Sounds clear enough.
00:26:17.460 --> 00:26:19.460
Second, any straight
line segment
00:26:19.460 --> 00:26:22.610
can be extended indefinitely
in a straight line.
00:26:22.610 --> 00:26:26.670
Also sounds obvious, which is
what Euclid was banking on.
00:26:26.670 --> 00:26:28.900
Third, given a
straight line segment,
00:26:28.900 --> 00:26:32.980
a circle can be drawn having
the segment as a radius and one
00:26:32.980 --> 00:26:35.389
endpoint as the center.
00:26:35.389 --> 00:26:37.430
That also-- you can imagine
yourself doing that--
00:26:37.430 --> 00:26:39.920
seems straightforward enough.
00:26:39.920 --> 00:26:41.600
But then we come to
the fifth postulate,
00:26:41.600 --> 00:26:43.770
which still sounds
pretty obvious.
00:26:43.770 --> 00:26:48.430
But it's certainly much more
complicated than the others.
00:26:48.430 --> 00:26:52.010
The fifth postulate says that
if a straight line falling
00:26:52.010 --> 00:26:57.640
on two straight lines makes the
interior angle on the same side
00:26:57.640 --> 00:27:02.010
are less than two right angles,
the two straight lines--
00:27:02.010 --> 00:27:03.860
if produced
indefinitely-- meet on
00:27:03.860 --> 00:27:08.240
that-- I think
this is mis-typed.
00:27:08.240 --> 00:27:09.110
Mustafa typed it.
00:27:09.110 --> 00:27:11.316
I guess he's typed
this one, too.
00:27:11.316 --> 00:27:12.440
This one shows the picture.
00:27:18.240 --> 00:27:19.990
Yeah, this should be
on that side in which
00:27:19.990 --> 00:27:22.146
the angles are less
than two right angles.
00:27:22.146 --> 00:27:24.270
So let me just explain it,
independent of the text.
00:27:27.060 --> 00:27:29.600
The question is,
what happens if you
00:27:29.600 --> 00:27:32.530
have one line-- the line that's
shown more or less vertical
00:27:32.530 --> 00:27:35.800
here-- and two lines
that cross it such
00:27:35.800 --> 00:27:38.820
that the interior angles--
here shown as a and b--
00:27:38.820 --> 00:27:41.490
are on one side less than pi.
00:27:41.490 --> 00:27:44.757
Less than two right angles is
the way you could describe it.
00:27:44.757 --> 00:27:46.340
Then, as you can see
from the picture,
00:27:46.340 --> 00:27:47.940
these lines will
meet on this side
00:27:47.940 --> 00:27:50.500
and will not meet on that side.
00:27:50.500 --> 00:27:52.920
And that's what
the postulate says
00:27:52.920 --> 00:27:55.590
that under those circumstances
where two lines cross a given
00:27:55.590 --> 00:27:58.030
line such that the
sum of the two angles
00:27:58.030 --> 00:28:01.127
adds up to less
than pi on one side
00:28:01.127 --> 00:28:02.585
that the lines will
be on that side
00:28:02.585 --> 00:28:04.681
and will not be
on the other side.
00:28:04.681 --> 00:28:05.180
Yes?
00:28:05.180 --> 00:28:06.928
AUDIENCE: What
was his motivation
00:28:06.928 --> 00:28:08.428
for making this the
fifth postulate?
00:28:08.428 --> 00:28:10.284
It seems kind of arbitrary.
00:28:10.284 --> 00:28:13.000
PROFESSOR: OK, the question is
what was Euclid's motivation
00:28:13.000 --> 00:28:16.120
for making this the
fifth postulate?
00:28:16.120 --> 00:28:19.080
Well, I have to admit I
haven't had many conversations
00:28:19.080 --> 00:28:20.920
with Euclid so I'm
not sure I know
00:28:20.920 --> 00:28:22.430
the answer to that question.
00:28:22.430 --> 00:28:25.170
But it was what was needed to
basically complete geometry
00:28:25.170 --> 00:28:26.680
as we know it.
00:28:26.680 --> 00:28:29.140
So much of what you've
learned in geometry
00:28:29.140 --> 00:28:31.850
would not be there if there
was not something equivalent
00:28:31.850 --> 00:28:32.624
to this postulate.
00:28:32.624 --> 00:28:34.790
But actually what I'm going
to be talking about next
00:28:34.790 --> 00:28:36.960
is there has been
discovered there
00:28:36.960 --> 00:28:41.620
are a number of substitutes
for this fifth postulate.
00:28:41.620 --> 00:28:43.332
Mathematicians studied
for a long time
00:28:43.332 --> 00:28:44.790
whether or not this
postulate could
00:28:44.790 --> 00:28:47.840
be derived from the others
since it seems so much more
00:28:47.840 --> 00:28:49.630
complicated than the others.
00:28:49.630 --> 00:28:53.730
And there was a strong
desire during thousands
00:28:53.730 --> 00:28:56.620
of years really-- at
least over 1,000 years--
00:28:56.620 --> 00:29:00.830
among mathematicians to try
to prove the fifth postulate
00:29:00.830 --> 00:29:02.930
from the first four postulates.
00:29:02.930 --> 00:29:05.250
And nobody ever
succeeded in doing that.
00:29:05.250 --> 00:29:09.070
And we now are pretty clear that
it's not possible to do that.
00:29:09.070 --> 00:29:10.660
That the postulate
is independent
00:29:10.660 --> 00:29:13.710
of the other postulates.
00:29:13.710 --> 00:29:15.770
It was discovered along
the way that there
00:29:15.770 --> 00:29:18.750
are a number of equivalent
statements to fifth postulate.
00:29:18.750 --> 00:29:22.457
And you could equally well have
chosen any one of these four
00:29:22.457 --> 00:29:24.540
statements that are
illustrated in these pictures.
00:29:24.540 --> 00:29:27.264
And I'll give you
words one by one
00:29:27.264 --> 00:29:29.680
for what these alternative
versions of the fifth postulate
00:29:29.680 --> 00:29:31.490
would be.
00:29:31.490 --> 00:29:34.860
A, up here, illustrated there,
says that if a straight line
00:29:34.860 --> 00:29:38.660
intersects one of two parallels,
meaning two parallel lines--
00:29:38.660 --> 00:29:40.160
so this is the one
line intersecting
00:29:40.160 --> 00:29:41.334
two parallel lines.
00:29:41.334 --> 00:29:43.750
If it intersects one of them
as the heavy part of the line
00:29:43.750 --> 00:29:46.900
shows, then the theorem says
that if you continue that line
00:29:46.900 --> 00:29:49.740
it will always
intersect the other.
00:29:49.740 --> 00:29:52.880
And certainly obvious from
the picture, that's the way
00:29:52.880 --> 00:29:54.240
it works.
00:29:54.240 --> 00:29:56.850
But that's equivalent
to the fifth postulate
00:29:56.850 --> 00:30:01.280
and not provable from the
other four postulates.
00:30:01.280 --> 00:30:03.700
A second statement--
b is the one
00:30:03.700 --> 00:30:05.960
that's illustrated there--
is the one that I remember
00:30:05.960 --> 00:30:08.170
learning when I was in
high school, I think,
00:30:08.170 --> 00:30:12.330
which says that if you have one
line and another line parallel
00:30:12.330 --> 00:30:16.550
to it-- or rather, I'm sorry
another point off the line
00:30:16.550 --> 00:30:19.200
that there's one and only
one line through that point
00:30:19.200 --> 00:30:22.990
parallel to the original line.
00:30:22.990 --> 00:30:29.110
So that is yet another statement
of this famous fifth postulate.
00:30:29.110 --> 00:30:31.710
Number c is less obvious.
00:30:31.710 --> 00:30:33.590
But it turns out
that once you go away
00:30:33.590 --> 00:30:35.580
from Euclidean geometry,
your space always
00:30:35.580 --> 00:30:37.760
has a built in scale.
00:30:37.760 --> 00:30:41.210
So things are not scalable.
00:30:41.210 --> 00:30:43.785
One example I might mention at
this point of occurred space
00:30:43.785 --> 00:30:45.930
is, say, the
surface of a sphere.
00:30:45.930 --> 00:30:49.720
And the service of a
sphere has some fixed size.
00:30:49.720 --> 00:30:52.114
So if you have a
figure of one size,
00:30:52.114 --> 00:30:54.030
and you wanted, on the
surface of this sphere,
00:30:54.030 --> 00:30:57.060
to make a figure 5 times bigger,
it might not fit on the sphere
00:30:57.060 --> 00:30:57.560
anymore.
00:30:57.560 --> 00:31:00.722
So you can't always
make a figure bigger
00:31:00.722 --> 00:31:01.930
on the surface of the sphere.
00:31:01.930 --> 00:31:05.000
In fact, you could
never make a figure
00:31:05.000 --> 00:31:07.960
bigger without
bending in some way
00:31:07.960 --> 00:31:10.110
on the surface of the sphere.
00:31:10.110 --> 00:31:14.100
So that gives rise to
this third statement
00:31:14.100 --> 00:31:17.960
of the fifth postulate, which
is that, given any figure,
00:31:17.960 --> 00:31:21.440
there exists a figure
similar to it of any size.
00:31:21.440 --> 00:31:24.040
And by similar it means
that for polygons they're
00:31:24.040 --> 00:31:27.250
similar if the corresponding
angles are equal to each other
00:31:27.250 --> 00:31:30.130
as they're supposed to
be on those two images.
00:31:30.130 --> 00:31:33.310
And the corresponding sides
are proportional to each other.
00:31:33.310 --> 00:31:37.390
So a similar figure
is just a blow up--
00:31:37.390 --> 00:31:39.750
a rescaling-- of
the original figure.
00:31:39.750 --> 00:31:42.940
And you can only do it if
the fifth postulate holds.
00:31:42.940 --> 00:31:46.032
You can do it on a flat space
but not on a curved space.
00:31:46.032 --> 00:31:50.910
And I think that is a
less obvious version
00:31:50.910 --> 00:31:52.950
of the fifth postulate.
00:31:52.950 --> 00:31:56.540
And finally, the
fifth postulate is
00:31:56.540 --> 00:31:58.870
linked to the
behavior of triangles.
00:31:58.870 --> 00:32:00.760
We all learned in
Euclidean geometry
00:32:00.760 --> 00:32:05.500
that the sum of the three angles
of a triangle is 180 degrees.
00:32:05.500 --> 00:32:08.760
That is a crucial theorem
of Euclidean geometry that
00:32:08.760 --> 00:32:11.440
depends directly on the fifth
postulate and is, in fact,
00:32:11.440 --> 00:32:12.912
equivalent to the
fifth postulate.
00:32:12.912 --> 00:32:15.120
So you could assume it and
forget the fifth postulate
00:32:15.120 --> 00:32:17.720
and still prove everything.
00:32:17.720 --> 00:32:21.570
So the fact that--
actually, I'm sorry.
00:32:24.470 --> 00:32:26.136
It is equivalent to
the fifth postulate.
00:32:26.136 --> 00:32:27.719
But you don't need
to assume that it's
00:32:27.719 --> 00:32:30.630
true for every triangle to
prove the fifth postulate.
00:32:30.630 --> 00:32:33.880
It turns out that sufficient
and mathematicians always
00:32:33.880 --> 00:32:36.400
look for the minimum
axiom to be able to prove
00:32:36.400 --> 00:32:37.560
what they want to prove.
00:32:37.560 --> 00:32:39.930
The minimum version of
the axiom for triangles
00:32:39.930 --> 00:32:43.220
is to simply assume that
there's just one triangle who's
00:32:43.220 --> 00:32:45.690
angles add up 180 degrees.
00:32:45.690 --> 00:32:48.500
And if there exist just one
triangle whose angles add up
00:32:48.500 --> 00:32:50.210
to 180 degrees, then
the fifth postulate
00:32:50.210 --> 00:32:54.490
has to hold-- turns out--
which is not so obvious, again.
00:32:54.490 --> 00:32:56.410
So these are all
different, equivalent ways
00:32:56.410 --> 00:32:58.370
of staying in the
fifth postulate.
00:32:58.370 --> 00:33:02.730
But it's pretty clear
now that it's not
00:33:02.730 --> 00:33:08.500
possible to prove the fifth
postulate from the first four.
00:33:08.500 --> 00:33:09.640
Any questions about that?
00:33:14.200 --> 00:33:17.070
OK, so a little
bit of history now.
00:33:17.070 --> 00:33:19.670
The first person, apparently,
to seriously explore
00:33:19.670 --> 00:33:23.980
the possibility that the
fifth postulate might be wrong
00:33:23.980 --> 00:33:29.062
was a Jesuit priest named
Giovanni Geralamo Saccheri.
00:33:29.062 --> 00:33:30.520
I'm sure other
people can pronounce
00:33:30.520 --> 00:33:32.740
it much better than I can.
00:33:32.740 --> 00:33:36.230
And in 1733, which is
the same year he died,
00:33:36.230 --> 00:33:41.190
he published a study of
what geometry would be like
00:33:41.190 --> 00:33:44.660
if the fifth postulate
did not hold.
00:33:44.660 --> 00:33:47.469
And he titled it- this
is a lot title, which
00:33:47.469 --> 00:33:49.010
I don't know how to
pronounce really,
00:33:49.010 --> 00:33:50.790
but in English it's
apparently translated
00:33:50.790 --> 00:33:55.620
as Euclid Freed of Every Flaw,
treating the fifth postulate
00:33:55.620 --> 00:34:03.010
as kind of a flaw in Euclid's
axiomization of geometry.
00:34:03.010 --> 00:34:06.150
Saccheri was, in fact, convinced
that the fifth postulate
00:34:06.150 --> 00:34:07.330
was true.
00:34:07.330 --> 00:34:09.489
He didn't really want to
consider the possibility
00:34:09.489 --> 00:34:11.449
that it was false.
00:34:11.449 --> 00:34:13.600
But he was nonetheless
exploring the possibility
00:34:13.600 --> 00:34:15.440
that it was false
because he understood
00:34:15.440 --> 00:34:17.409
the concept of a proof
by contradiction.
00:34:17.409 --> 00:34:19.159
He was looking for
a contradiction
00:34:19.159 --> 00:34:23.570
to be able to prove that
mathematics would not
00:34:23.570 --> 00:34:26.290
be consistent if you assume
the fifth postulate was false.
00:34:26.290 --> 00:34:27.706
And therefore, the
fifth postulate
00:34:27.706 --> 00:34:29.540
would be proven to be true.
00:34:29.540 --> 00:34:31.110
So he was exploring
what would happen
00:34:31.110 --> 00:34:32.929
if the fifth
postulate was false,
00:34:32.929 --> 00:34:36.530
looking all the time to
find some inconsistency,
00:34:36.530 --> 00:34:37.886
and was not able to find any.
00:34:37.886 --> 00:34:39.510
So he considered all
of this a failure.
00:34:39.510 --> 00:34:41.440
But he published it anyway.
00:34:41.440 --> 00:34:45.380
And that's the
publication front page.
00:34:47.889 --> 00:34:50.316
The next person to enter
the stage-- or actually
00:34:50.316 --> 00:34:51.690
three people
together, but I only
00:34:51.690 --> 00:34:54.580
have a nice
transparency on Gauss.
00:34:54.580 --> 00:34:57.300
Gauss, Bolyai, and
Lobachevsky went on
00:34:57.300 --> 00:35:00.460
to seriously explore the
possibility of geometry
00:35:00.460 --> 00:35:03.450
without the fifth
postulate, actually assuming
00:35:03.450 --> 00:35:05.180
that the fifth
postulate is false,
00:35:05.180 --> 00:35:09.950
developing what we call GBL,
Gauss-Bolyai-Lobachevsky
00:35:09.950 --> 00:35:11.440
geometry.
00:35:11.440 --> 00:35:13.420
Gauss was a German
mathematician.
00:35:13.420 --> 00:35:14.920
These were the years he lived.
00:35:14.920 --> 00:35:17.170
He, in fact, was born the
son of a poor, working class
00:35:17.170 --> 00:35:19.700
parents, which I found
a little surprising.
00:35:19.700 --> 00:35:23.120
We kind of think of scholars
in those early years
00:35:23.120 --> 00:35:26.580
as being gentleman who
were part of the nobility.
00:35:26.580 --> 00:35:28.180
But Gauss was not
but, nonetheless,
00:35:28.180 --> 00:35:30.570
went on to be one of the
most important mathematicians
00:35:30.570 --> 00:35:31.842
of his age.
00:35:31.842 --> 00:35:33.550
One of the other things
that surprised me
00:35:33.550 --> 00:35:35.883
and to be honest I just learned
all this from Wikipedia.
00:35:35.883 --> 00:35:37.670
I'm no real expert
on the history.
00:35:37.670 --> 00:35:40.430
But they gave a list
of Gauss' students.
00:35:40.430 --> 00:35:43.870
And they included that Richard
Dedekind, Bernhard Riemann,
00:35:43.870 --> 00:35:47.930
Peter Gustav Lejeune Dirichlet,
which is the name I remember,
00:35:47.930 --> 00:35:50.160
Kirchhoff, and Mobius.
00:35:50.160 --> 00:35:52.920
So quite a list of
famous mathematicians.
00:35:52.920 --> 00:35:55.770
So I have to admit, when
I read that, I was just
00:35:55.770 --> 00:35:59.450
about to send off an email
to all of my former students
00:35:59.450 --> 00:36:01.470
saying, look, what's
happening here?
00:36:01.470 --> 00:36:05.680
You're not competing
at all. [LAUGHING]
00:36:05.680 --> 00:36:07.470
But I decided not to do that.
00:36:07.470 --> 00:36:08.530
It would be impolite.
00:36:08.530 --> 00:36:09.220
And who knows?
00:36:09.220 --> 00:36:10.720
Maybe 100 years
from now my students
00:36:10.720 --> 00:36:12.053
will be as famous as these guys.
00:36:12.053 --> 00:36:13.480
You never know.
00:36:13.480 --> 00:36:17.250
We can plan, I hope.
00:36:17.250 --> 00:36:20.170
OK, so the other guys involved
in this and they were all
00:36:20.170 --> 00:36:25.030
working independently were
Bolyai who was, I think,
00:36:25.030 --> 00:36:29.370
a Prussian military man,
actually, and Lobachevsky
00:36:29.370 --> 00:36:31.610
who was also a professional
mathematician working
00:36:31.610 --> 00:36:33.500
in the university.
00:36:33.500 --> 00:36:39.910
The three of them independently
developed a geometry
00:36:39.910 --> 00:36:42.954
in which the fifth postulate
was assumed to be false.
00:36:42.954 --> 00:36:44.495
There are two ways
it could be false.
00:36:48.530 --> 00:36:51.070
In version with the triangles,
for example, a triangle
00:36:51.070 --> 00:36:53.960
could have more than or
less than 180 degrees.
00:37:01.150 --> 00:37:04.660
Since there were assuming that
the fifth postulate was false,
00:37:04.660 --> 00:37:07.710
it meant they had to be assuming
that every version that we just
00:37:07.710 --> 00:37:09.240
talked about of
the fifth postulate
00:37:09.240 --> 00:37:12.035
is false because they are
all equivalent to each other
00:37:12.035 --> 00:37:15.130
and these people realize that.
00:37:15.130 --> 00:37:32.780
So in particular in the
Gauss-Bolyai-Lobachevsky
00:37:32.780 --> 00:37:54.420
geometry, there are
infinitely many lines parallel
00:37:54.420 --> 00:37:55.195
to a given line.
00:38:02.210 --> 00:38:11.980
And no figures of
different size are similar.
00:38:24.960 --> 00:38:42.205
And the sum of the
angles of a triangle
00:38:42.205 --> 00:38:47.890
are always less than
180 degrees or pi,
00:38:47.890 --> 00:38:50.488
depending on whether you're
a radian person or a degree
00:38:50.488 --> 00:38:50.988
person.
00:38:53.860 --> 00:38:58.030
Now I should mention here
that the surface of the sphere
00:38:58.030 --> 00:38:59.690
is, in fact, a
perfectly good example
00:38:59.690 --> 00:39:02.420
of the non-Euclidean geometry.
00:39:02.420 --> 00:39:04.520
But for some reason it
was not taken seriously
00:39:04.520 --> 00:39:08.970
by mathematicians until
long after these guys
00:39:08.970 --> 00:39:10.620
were doing their work.
00:39:10.620 --> 00:39:14.130
And part of the reason, I guess,
is that the surfaces three
00:39:14.130 --> 00:39:17.350
evaluates not just
one of Euclid's axioms
00:39:17.350 --> 00:39:19.925
but two if we go back
to Euclid's axioms.
00:39:25.516 --> 00:39:28.670
The second of Euclid's axioms
said that any straight line
00:39:28.670 --> 00:39:30.480
segment can be
extended indefinitely
00:39:30.480 --> 00:39:31.697
in a straight line.
00:39:31.697 --> 00:39:33.530
And the surface of the
sphere and the analog
00:39:33.530 --> 00:39:35.950
of the straight line
is a great circle.
00:39:35.950 --> 00:39:37.540
And if you extend
the great circle,
00:39:37.540 --> 00:39:39.670
it comes back on itself.
00:39:39.670 --> 00:39:43.090
So the surface of the sphere
violates the fifth postulate.
00:39:43.090 --> 00:39:45.660
But it also violates
the second postulate.
00:39:45.660 --> 00:39:47.670
But still perfectly
consistent geometry,
00:39:47.670 --> 00:39:50.120
and it is a
non-Euclidean geometry.
00:39:50.120 --> 00:39:52.820
And on the surface
of this sphere,
00:39:52.820 --> 00:39:56.330
these statements
all kind of reverse.
00:39:56.330 --> 00:39:59.050
Instead of having infinitely
many lines parallel to a given
00:39:59.050 --> 00:40:03.450
line, you have no lines that
are parallel to a given line.
00:40:03.450 --> 00:40:06.620
Remember, lines are great
circles and lines are parallel
00:40:06.620 --> 00:40:07.780
if they never meet.
00:40:07.780 --> 00:40:10.170
But any two great circles meet.
00:40:10.170 --> 00:40:13.460
So there are no lines
parallel to a given line
00:40:13.460 --> 00:40:17.730
in the geometry of the
surface of the sphere.
00:40:17.730 --> 00:40:19.990
It's, again, true that no
figures of different size
00:40:19.990 --> 00:40:20.490
are similar.
00:40:20.490 --> 00:40:25.210
That has to be true
for any any geometry
00:40:25.210 --> 00:40:27.300
where the fifth
postulate was false.
00:40:27.300 --> 00:40:29.370
The last one,
again, has a choice.
00:40:29.370 --> 00:40:31.080
And it's the opposite
choice for a sphere
00:40:31.080 --> 00:40:34.920
as for the
Gauss-Bolyai-Lobachevsky
00:40:34.920 --> 00:40:36.550
geometry.
00:40:36.550 --> 00:40:38.610
In the Gauss-Bolyai-Lobachevsky
Bolyai- geometry,
00:40:38.610 --> 00:40:41.586
the sum of the angles is
always less than 180 degrees.
00:40:41.586 --> 00:40:43.210
If you picture a
triangle and a sphere,
00:40:43.210 --> 00:40:47.010
you can imagine that the
edges look like they bulge.
00:40:47.010 --> 00:40:49.350
And the sum of the
angles on the surface
00:40:49.350 --> 00:40:50.960
of the sphere of a
triangle are always
00:40:50.960 --> 00:40:53.640
greater than 180 degrees.
00:40:53.640 --> 00:40:55.140
The easiest way to
convince yourself
00:40:55.140 --> 00:40:58.390
that that's true in at
least one important case
00:40:58.390 --> 00:40:59.870
is to imagine a triangle.
00:40:59.870 --> 00:41:02.130
Here's a sphere.
00:41:02.130 --> 00:41:04.372
Everybody see the sphere?
00:41:04.372 --> 00:41:05.330
There's the North Pole.
00:41:05.330 --> 00:41:08.440
There's an equator.
00:41:08.440 --> 00:41:10.240
Imagine a triangle
where one vertex
00:41:10.240 --> 00:41:14.330
is at the North Pole and the
liver disease or on the equator
00:41:14.330 --> 00:41:16.300
and the triangle
looks like this.
00:41:16.300 --> 00:41:18.420
And these are both 90
degree angles down here.
00:41:18.420 --> 00:41:20.660
So you already have 180 degrees.
00:41:20.660 --> 00:41:22.430
And any angle you
have on top just
00:41:22.430 --> 00:41:24.410
adds to the 180
degrees putting you
00:41:24.410 --> 00:41:29.430
above the Euclidean
value of 180 degrees.
00:41:29.430 --> 00:41:31.810
So for a sphere it's
always the opposite
00:41:31.810 --> 00:41:33.465
of this greater
than 180 degrees.
00:41:37.250 --> 00:41:41.390
OK, continuing with Gauss,
Bolyai and Lobachevsky,
00:41:41.390 --> 00:41:47.650
their work was based on
exploring the axioms of Euclid,
00:41:47.650 --> 00:41:51.890
assuming the reverse
of the fifth postulate
00:41:51.890 --> 00:41:55.000
in any one of its forms.
00:41:55.000 --> 00:41:58.870
And they proved
theorems and wrote,
00:41:58.870 --> 00:42:02.810
sort of, their own versions
of Euclid's elements.
00:42:02.810 --> 00:42:05.940
But that still left
open the question
00:42:05.940 --> 00:42:08.940
whether or not all of this
was really consistent.
00:42:08.940 --> 00:42:12.109
That is, from the
thinking of Saccheri
00:42:12.109 --> 00:42:14.400
that we already talked about,
there's always the chance
00:42:14.400 --> 00:42:16.441
that you might find the
contradiction even if you
00:42:16.441 --> 00:42:18.490
haven't found a
contradiction yet.
00:42:18.490 --> 00:42:20.910
And Gauss, Bolyai
and Lobachevsky
00:42:20.910 --> 00:42:23.340
didn't really have any
way of answering that.
00:42:23.340 --> 00:42:25.140
They didn't really
have any way of knowing
00:42:25.140 --> 00:42:26.514
whether if they
continued further
00:42:26.514 --> 00:42:28.970
they might find
some contradiction.
00:42:28.970 --> 00:42:32.690
So what they did certainly
extracted the properties
00:42:32.690 --> 00:42:34.570
of this non-Euclidean geometry.
00:42:34.570 --> 00:42:37.860
But it did not really prove
that the non-Euclidean geometry
00:42:37.860 --> 00:42:40.320
was consistent.
00:42:40.320 --> 00:42:45.880
That happened slightly
later in an argument
00:42:45.880 --> 00:42:49.580
by Felix Klein, who was the
same Klein as the Klein bottle,
00:42:49.580 --> 00:42:50.080
by the way.
00:43:00.710 --> 00:43:02.720
And that was his
famous paper on this
00:43:02.720 --> 00:43:05.850
was published in 1870 somewhat
later than the early work
00:43:05.850 --> 00:43:07.320
that we're talking about.
00:43:07.320 --> 00:43:12.640
And what he did is he gave an
actual construction of the GBL
00:43:12.640 --> 00:43:14.250
geometry.
00:43:14.250 --> 00:43:15.780
And by construction
I mean in terms
00:43:15.780 --> 00:43:19.530
of coordinates using coordinate
geometry ideas, which
00:43:19.530 --> 00:43:22.740
were originally
developed by Descartes.
00:43:22.740 --> 00:43:25.780
That's why we call them
Cartesian coordinate systems.
00:43:25.780 --> 00:43:31.580
So what he gave as a description
of the Gauss-Bolyai-Lobachevsky
00:43:31.580 --> 00:43:38.880
geometry was a space
that consisted of a disk
00:43:38.880 --> 00:43:46.340
with coordinates x and y just
as Descartes would have done.
00:43:46.340 --> 00:43:49.011
He restricts himself to
the inside of the disk.
00:43:49.011 --> 00:43:50.510
So he restricts
himself to x squared
00:43:50.510 --> 00:43:55.130
plus y squared less than 1.
00:43:55.130 --> 00:43:59.540
And what he gave was a
function of two points
00:43:59.540 --> 00:44:02.390
in this disk where the
function represents
00:44:02.390 --> 00:44:04.690
the distance between
those two points.
00:44:04.690 --> 00:44:06.600
He decided that
distances don't have
00:44:06.600 --> 00:44:08.680
to be Euclidean
distances if we're
00:44:08.680 --> 00:44:12.060
trying to explore
non-Euclidean geometry.
00:44:12.060 --> 00:44:15.230
So he invented his
own distance function.
00:44:15.230 --> 00:44:18.290
And it's pretty
complicated looking.
00:44:18.290 --> 00:44:24.160
The function he came up with was
that the cosh of the distance
00:44:24.160 --> 00:44:33.400
between points 1 and 2 divided
by sum number a-- and a
00:44:33.400 --> 00:44:43.470
could be any number-- is
equal to 1 minus x1 times x2--
00:44:43.470 --> 00:44:45.620
these are the two
x-coordinates for points 1
00:44:45.620 --> 00:44:59.680
and 2-- minus y1 y2 over
the square root of 1 minus
00:44:59.680 --> 00:45:03.890
x1 squared minus y1 squared.
00:45:03.890 --> 00:45:06.530
Sorry, it's the product of two
square roots in the denominator
00:45:06.530 --> 00:45:07.030
here.
00:45:09.547 --> 00:45:11.380
And the second square
root is the same thing
00:45:11.380 --> 00:45:18.245
for the two coordinates 1 minus
x2 squared minus y2 squared.
00:45:24.610 --> 00:45:29.500
So this formula is certainly
not obvious to anybody.
00:45:29.500 --> 00:45:34.030
But Klein figured out
that this actually
00:45:34.030 --> 00:45:38.520
does-- the geometry
described by these formulas--
00:45:38.520 --> 00:45:42.070
reproduce completely
the postulates
00:45:42.070 --> 00:45:44.440
of the Gauss-Bolyai-Lobachevsky
geometry,
00:45:44.440 --> 00:45:47.010
including the failure of
Euclid's fifth postulate.
00:45:50.050 --> 00:45:52.175
And since this boils
down to just algebra,
00:45:52.175 --> 00:45:54.340
if algebra is
consistent, it proves
00:45:54.340 --> 00:45:55.760
that the
Gauss-Bolyai-Lobachevsky
00:45:55.760 --> 00:45:57.914
geometry is consistent.
00:45:57.914 --> 00:46:00.330
Now, as I understand it, nobody
can prove that any of this
00:46:00.330 --> 00:46:01.288
actually is consistent.
00:46:01.288 --> 00:46:03.660
People prove
relative consistency.
00:46:03.660 --> 00:46:08.170
So in the assumption that
algebra is consistent,
00:46:08.170 --> 00:46:10.304
theorems about the real
numbers are consistent.
00:46:10.304 --> 00:46:11.720
Felix Klein was
able to prove that
00:46:11.720 --> 00:46:13.950
the Gauss-Bolyai-Lobachevsky
geometry is consistent.
00:46:16.115 --> 00:46:17.490
And this was really
the beginning
00:46:17.490 --> 00:46:18.740
of coordinate geometry.
00:46:18.740 --> 00:46:19.930
I'm sure all of you
are rather familiar
00:46:19.930 --> 00:46:20.971
with coordinate geometry.
00:46:20.971 --> 00:46:22.840
It's a standard topic
now in math courses
00:46:22.840 --> 00:46:24.520
and even in high school.
00:46:24.520 --> 00:46:26.510
And this is really
where it began.
00:46:26.510 --> 00:46:29.610
Euclid had no idea
that it was any value
00:46:29.610 --> 00:46:33.070
in trying to represent geometric
quantities as equations.
00:46:33.070 --> 00:46:37.250
Euclid did everything
in terms of theorems.
00:46:37.250 --> 00:46:39.300
But this opened up
a whole new door
00:46:39.300 --> 00:46:40.610
for how to discuss geometry.
00:46:46.800 --> 00:46:52.840
So the geometry is a slide that
just shows the same formula.
00:46:52.840 --> 00:46:56.562
And I guess this is supposed
to be an image of the disk.
00:46:56.562 --> 00:46:58.020
One thing I should
point out, which
00:46:58.020 --> 00:47:01.290
I forgot to point out, which is
that although this disk looks
00:47:01.290 --> 00:47:06.230
finite, it really is an infinite
space that's being described.
00:47:06.230 --> 00:47:08.360
And one can see that
by looking carefully
00:47:08.360 --> 00:47:09.880
at the distance functions.
00:47:09.880 --> 00:47:12.540
As either one of
these two points--
00:47:12.540 --> 00:47:17.470
point 1 or point 2--
approaches the boundary
00:47:17.470 --> 00:47:20.730
x square plus y square
equals 1, this square
00:47:20.730 --> 00:47:23.080
root denominator blows up.
00:47:23.080 --> 00:47:27.680
So the distance between a
point and another point that's
00:47:27.680 --> 00:47:30.400
approaching the boundary
goes to infinity
00:47:30.400 --> 00:47:32.960
as that point
approaches the boundary.
00:47:32.960 --> 00:47:35.160
So boundary's actually
infinitely far away
00:47:35.160 --> 00:47:37.900
even though in
coordinates they are still
00:47:37.900 --> 00:47:40.970
x squared plus y
squared equals 1.
00:47:40.970 --> 00:47:43.405
So this introduces
another important concept,
00:47:43.405 --> 00:47:45.446
which we'll be using in
general relativity, which
00:47:45.446 --> 00:47:47.665
is the coordinates don't
represent distances.
00:47:47.665 --> 00:47:49.040
Distances could
be very different
00:47:49.040 --> 00:47:50.737
from the way the
coordinates look.
00:47:50.737 --> 00:47:52.320
So that boundary,
even though it looks
00:47:52.320 --> 00:47:55.580
like it's right there
on the blackboard,
00:47:55.580 --> 00:47:57.580
is actually very far away
from the other points.
00:48:07.300 --> 00:48:10.850
OK, so after Klein,
the important new idea
00:48:10.850 --> 00:48:12.690
that Klein introduced
was, first of all,
00:48:12.690 --> 00:48:15.290
the explicit construction
but also the general idea
00:48:15.290 --> 00:48:17.960
that you can describe geometry
not by giving postulates
00:48:17.960 --> 00:48:20.270
but instead by actually
doing a construction where
00:48:20.270 --> 00:48:23.550
you've given names to the
points in terms of coordinates
00:48:23.550 --> 00:48:26.060
and you describe what
happens geometrically
00:48:26.060 --> 00:48:28.950
in terms of some distance
function which describes
00:48:28.950 --> 00:48:30.925
the distance between
two arbitrary points.
00:48:37.780 --> 00:48:44.940
And Gauss went on to make
two other very important
00:48:44.940 --> 00:48:46.610
observations about
geometry, which
00:48:46.610 --> 00:48:49.210
has become essential to
our current understanding
00:48:49.210 --> 00:48:51.660
of non-Euclidean geometry.
00:48:51.660 --> 00:48:54.550
So let me mention two other
ideas that Gauss introduced.
00:49:00.560 --> 00:49:03.180
The first one was
the distinction
00:49:03.180 --> 00:49:07.160
between what he called
inner and outer properties
00:49:07.160 --> 00:49:09.470
of a curved surface.
00:49:09.470 --> 00:49:11.510
His curved spaces were
all two dimensional,
00:49:11.510 --> 00:49:12.515
so they were surfaces.
00:49:24.800 --> 00:49:26.610
So this is most easily
described for say
00:49:26.610 --> 00:49:28.470
a surface of a sphere
where we can all
00:49:28.470 --> 00:49:30.770
visualize what
we're talking about.
00:49:30.770 --> 00:49:33.780
The surface of a sphere we
visualize in three Euclidean
00:49:33.780 --> 00:49:36.910
dimensions and we
think of its properties
00:49:36.910 --> 00:49:39.650
as being determined by that
three dimensional space.
00:49:39.650 --> 00:49:42.280
And the geometry of that
three dimensional space.
00:49:42.280 --> 00:49:43.988
And of course the
three dimensional space
00:49:43.988 --> 00:49:47.000
is Euclidean that we're
embedding our sphere into.
00:49:47.000 --> 00:49:50.780
But the non-Euclidean
aspects are all
00:49:50.780 --> 00:49:54.830
seen in the geometry of the
two dimensional surface.
00:49:54.830 --> 00:49:57.690
Figures drawn on the surface
as if the rest of the three
00:49:57.690 --> 00:50:00.640
dimensional space did not exist.
00:50:00.640 --> 00:50:05.580
And that is this key idea of
inner versus outer properties.
00:50:05.580 --> 00:50:07.330
The outer properties
of the sphere
00:50:07.330 --> 00:50:10.000
are properties that relate
to the three dimensional
00:50:10.000 --> 00:50:12.992
space in which the
sphere is embedded.
00:50:12.992 --> 00:50:14.910
But what Gauss realized
is that there's
00:50:14.910 --> 00:50:18.650
a perfectly well defined
mathematics contained entirely
00:50:18.650 --> 00:50:21.620
in the two dimensional space
of the surface of the sphere.
00:50:21.620 --> 00:50:24.214
You could discuss it
without making any reference
00:50:24.214 --> 00:50:26.630
to the three dimensional space
around it if you wanted to,
00:50:26.630 --> 00:50:28.530
it's just a little
more complicated to be
00:50:28.530 --> 00:50:29.770
able to do that.
00:50:29.770 --> 00:50:34.860
But we will in fact be doing
it explicitly very shortly.
00:50:34.860 --> 00:50:37.442
And all it amounts to
from our point of view
00:50:37.442 --> 00:50:39.650
is assigning coordinates to
the points on the surface
00:50:39.650 --> 00:50:43.770
of the sphere and the distance
function for those coordinates.
00:50:43.770 --> 00:50:45.900
And then one has
a full description
00:50:45.900 --> 00:50:48.150
of this two dimensional
space consisting
00:50:48.150 --> 00:50:51.110
of the surface of the sphere
which no longer needs make
00:50:51.110 --> 00:50:53.650
any reference to
the third dimension
00:50:53.650 --> 00:50:57.270
that you imagined the
sphere embedded in.
00:50:57.270 --> 00:51:01.260
So the study of the properties
of that two dimensional space
00:51:01.260 --> 00:51:03.990
is the study of the inner
properties of the space.
00:51:03.990 --> 00:51:06.300
And if you care about
how it's embedded,
00:51:06.300 --> 00:51:08.300
that's called the
outer properties.
00:51:08.300 --> 00:51:14.130
And Gauss made it clear in
the way he described things
00:51:14.130 --> 00:51:17.240
that from his point of view, the
real thing that mathematicians
00:51:17.240 --> 00:51:19.710
should be doing is studying
the inner properties.
00:51:19.710 --> 00:51:23.636
The outer properties are
not that interesting.
00:51:23.636 --> 00:51:25.510
So that's one key idea
that Gauss introduced.
00:51:29.160 --> 00:51:33.660
And the second is the idea
of what we now call a metric.
00:51:36.480 --> 00:51:40.030
And there are really
two pieces to this.
00:51:40.030 --> 00:51:43.410
The first of which
is that instead
00:51:43.410 --> 00:51:47.020
of giving macroscopic distances,
which is what Klein did,
00:51:47.020 --> 00:51:49.420
he told you how to write down
a formula for the distance
00:51:49.420 --> 00:51:52.550
between any two
points, Gauss realized
00:51:52.550 --> 00:51:56.290
that things could
become more interesting
00:51:56.290 --> 00:51:59.450
if instead of trying to
immediately write down
00:51:59.450 --> 00:52:02.860
a function for the distance
between two arbitrary points,
00:52:02.860 --> 00:52:06.070
you can find your attention
to very nearby points.
00:52:06.070 --> 00:52:09.030
And consider the
distance between two
00:52:09.030 --> 00:52:11.070
far away points
as just the length
00:52:11.070 --> 00:52:14.577
of a line that joins them
where every little segment
00:52:14.577 --> 00:52:17.160
in the line is a short distance
which is governed by the rules
00:52:17.160 --> 00:52:18.969
that you've made up
for short distances.
00:52:18.969 --> 00:52:20.510
So if you understand
short distances,
00:52:20.510 --> 00:52:23.124
long distances are
obtained just by adding.
00:52:23.124 --> 00:52:24.040
That's the basic idea.
00:52:26.760 --> 00:52:28.230
So only short
distances are needed.
00:52:28.230 --> 00:52:30.770
And then there's
another important idea
00:52:30.770 --> 00:52:33.910
which is that the short
distances themselves
00:52:33.910 --> 00:52:36.490
for an interesting
class of spaces
00:52:36.490 --> 00:52:38.930
should not be some arbitrary
function of the coordinates
00:52:38.930 --> 00:52:44.549
of the two points but should in
fact be a quadratic function.
00:52:44.549 --> 00:52:46.340
And in two dimensions,
a quadratic function
00:52:46.340 --> 00:52:49.190
mean something like
this, where gxx is just
00:52:49.190 --> 00:52:52.410
any function of x and y, x
and y are the two coordinates
00:52:52.410 --> 00:52:53.800
of the space.
00:52:53.800 --> 00:52:57.164
gxy is also just any
function of x and u.
00:52:57.164 --> 00:53:02.830
gyy is any function and gxx
multiplies the x squared,
00:53:02.830 --> 00:53:09.310
xy multiplies the x times dy and
g sub yy multiplies dy squared.
00:53:09.310 --> 00:53:13.260
And there's no terms portion
for dx by itself or dy by itself
00:53:13.260 --> 00:53:15.075
or dz by itself.
00:53:15.075 --> 00:53:16.450
And there's no
terms proportional
00:53:16.450 --> 00:53:19.760
to the cubes of those
quantities, it's all quadratic.
00:53:19.760 --> 00:53:21.850
That's the assumption.
00:53:21.850 --> 00:53:24.020
Now what underlies
that assumption is not
00:53:24.020 --> 00:53:27.160
that all spaces have
to have this property.
00:53:27.160 --> 00:53:31.110
This does narrow one down to
a particular class of spaces.
00:53:31.110 --> 00:53:33.810
But one way of
characterizing that class
00:53:33.810 --> 00:53:38.240
is that what Gauss understood
is that that class, the class
00:53:38.240 --> 00:53:40.710
of spaces described
by a metric like this,
00:53:40.710 --> 00:53:43.040
are precisely the class of
spaces which are locally
00:53:43.040 --> 00:53:48.630
Euclidean, meaning that even
though the surface is curved,
00:53:48.630 --> 00:53:53.890
any tiny little patch of it
looks flat and can be covered
00:53:53.890 --> 00:53:56.110
by Euclidean coordinates,
redefining coordinates
00:53:56.110 --> 00:53:59.020
in that one little patch only
where the metric in the one
00:53:59.020 --> 00:54:01.620
little patch would just be
the Euclidean metric given
00:54:01.620 --> 00:54:04.750
by the Pythagorean theorem
of Euclidean geometry,
00:54:04.750 --> 00:54:08.070
dx prime squared
plus dy squared,
00:54:08.070 --> 00:54:10.870
which would be the distance
function for a flat space
00:54:10.870 --> 00:54:13.720
and ordinary
Cartesian coordinates.
00:54:13.720 --> 00:54:16.360
And it turns out
that saying that this
00:54:16.360 --> 00:54:19.270
is true in every
microscopic region
00:54:19.270 --> 00:54:22.760
is equivalent to saying that
the metric over the entire space
00:54:22.760 --> 00:54:27.000
can be written as
a quadratic form,
00:54:27.000 --> 00:54:29.710
meaning one that looks
exactly like that.
00:54:29.710 --> 00:54:31.580
And spaces that
have this property
00:54:31.580 --> 00:54:34.242
are now usually called
Riemannian spaces,
00:54:34.242 --> 00:54:36.840
even though it was Gauss
who first identified them.
00:54:39.390 --> 00:54:41.990
And all the spaces that
we deal with in physics,
00:54:41.990 --> 00:54:44.840
in particular the spaces we
deal with general relativity,
00:54:44.840 --> 00:54:48.860
will be Riemannian spaces.
00:54:48.860 --> 00:54:51.110
Or sometimes they're called
pseudo Riemannian spaces
00:54:51.110 --> 00:54:53.840
because time is treated a little
bit differently in physics.
00:54:53.840 --> 00:54:55.460
And the word
"Riemannian" was really
00:54:55.460 --> 00:54:58.430
built on spatial geometry.
00:54:58.430 --> 00:55:01.117
But the word
"pseudo" only changes
00:55:01.117 --> 00:55:03.700
the fact that this becomes the
Lorenz metric, if you know what
00:55:03.700 --> 00:55:06.750
that means, instead of
the Euclidean metric.
00:55:06.750 --> 00:55:08.910
But the same idea
holds, which is
00:55:08.910 --> 00:55:10.900
that the spaces that
we're interested in
00:55:10.900 --> 00:55:16.060
are spaces which locally
look exactly like flat space.
00:55:16.060 --> 00:55:20.570
And that implies that globally,
you can always write down
00:55:20.570 --> 00:55:27.680
a metric function which is
quadratic, as Gauss said
00:55:27.680 --> 00:55:37.540
So the metric should be
a quadratic function that
00:55:37.540 --> 00:55:39.910
specifies the distance only
between infinitesimally
00:55:39.910 --> 00:55:44.030
separated points, not
finitely separated points
00:55:44.030 --> 00:55:47.610
and should have the
form of ds squared
00:55:47.610 --> 00:55:52.560
is equal to some
function of x and y times
00:55:52.560 --> 00:56:02.200
dx squared plus a different
function of x and y times dx
00:56:02.200 --> 00:56:13.520
times dy plus a different
function of x and y times dy
00:56:13.520 --> 00:56:14.020
squared.
00:56:14.020 --> 00:56:17.946
So I'm just writing the same
form that was on the side.
00:56:17.946 --> 00:56:19.320
This is a very
important formula.
00:56:22.310 --> 00:56:24.920
Incidentally, the
two that appears here
00:56:24.920 --> 00:56:29.190
is only because when we write
this in a different notation,
00:56:29.190 --> 00:56:32.400
this term will occur
twice, once as dx times dy,
00:56:32.400 --> 00:56:34.560
and once as dy times dx.
00:56:34.560 --> 00:56:37.060
And the two times it occurs
are equal to each other,
00:56:37.060 --> 00:56:39.560
so here they're just collected
together with a factor of two
00:56:39.560 --> 00:56:40.700
in front.
00:56:40.700 --> 00:56:41.200
Yes.
00:56:41.200 --> 00:56:44.155
STUDENT: Just to make sure,
those three different functions
00:56:44.155 --> 00:56:45.997
you wrote, those
subscripts aren't
00:56:45.997 --> 00:56:47.830
supposed to mean partial
derivatives, right?
00:56:47.830 --> 00:56:48.996
PROFESSOR: No, that's right.
00:56:48.996 --> 00:56:50.430
The subscripts and
this expression
00:56:50.430 --> 00:56:53.390
only mean that this
multiplies dx squared,
00:56:53.390 --> 00:56:55.370
so it has subscripts xx.
00:56:55.370 --> 00:56:59.020
And this multiplies dx times
dy, so it has subscripts xy.
00:56:59.020 --> 00:57:02.510
That's what the subscripts
mean and nothing more.
00:57:02.510 --> 00:57:04.185
So the subscripts
only mean that they're
00:57:04.185 --> 00:57:06.511
the things that appear
in that equation.
00:57:06.511 --> 00:57:07.010
Yes.
00:57:07.010 --> 00:57:09.495
STUDENT: It seems like
the metric is giving us
00:57:09.495 --> 00:57:15.460
distance in terms of an
infintesimal displacement,
00:57:15.460 --> 00:57:20.560
but then a locally Euclidean
space is already tangent
00:57:20.560 --> 00:57:24.560
infintesimally, so how are
we relating the local metric
00:57:24.560 --> 00:57:26.774
with the global metric?
00:57:26.774 --> 00:57:28.190
PROFESSOR: OK, the
question is how
00:57:28.190 --> 00:57:30.170
do we relate the local
metric which I say
00:57:30.170 --> 00:57:33.810
is Euclidean to
the global metric?
00:57:33.810 --> 00:57:36.770
And the answer I
think for now I will
00:57:36.770 --> 00:57:39.380
stick to just giving kind
of a pictorial answer based
00:57:39.380 --> 00:57:41.830
on the picture here.
00:57:41.830 --> 00:57:46.760
That is once you know the
distances between any two
00:57:46.760 --> 00:57:49.950
points in a tiny
little patch here,
00:57:49.950 --> 00:57:53.350
it's then always possible
to construct coordinates,
00:57:53.350 --> 00:57:56.290
here called x prime
and y prime, such
00:57:56.290 --> 00:57:59.530
that the distance between
any two points as calculated
00:57:59.530 --> 00:58:02.190
from the original metric,
which is the one here,
00:58:02.190 --> 00:58:03.820
is exactly the same
as the distance you
00:58:03.820 --> 00:58:08.110
get using this metric.
00:58:08.110 --> 00:58:09.720
And the claim is
that you can always
00:58:09.720 --> 00:58:14.600
define coordinates x prime and
y prime which make that true.
00:58:14.600 --> 00:58:17.180
That claim is not
absolutely obvious.
00:58:17.180 --> 00:58:19.850
But it's something you can
probably visualize if you just
00:58:19.850 --> 00:58:23.730
imagine that every little tiny
piece of this curved surface
00:58:23.730 --> 00:58:28.040
looks like it was just
a flat surface and then
00:58:28.040 --> 00:58:30.901
a flat surface you know
that you can write down
00:58:30.901 --> 00:58:33.400
a Cartesian coordinate system,
which will have the Euclidean
00:58:33.400 --> 00:58:35.936
metric.
00:58:35.936 --> 00:58:37.850
But it's only an
intuitive statement,
00:58:37.850 --> 00:58:39.820
proving it is actually harder.
00:58:39.820 --> 00:58:40.320
Yes.
00:58:40.320 --> 00:58:44.920
STUDENT: [INAUDIBLE]
bottom formula [INAUDIBLE]
00:58:44.920 --> 00:58:50.900
with positive curvature if we
analogize to second derivative.
00:58:50.900 --> 00:58:52.441
PROFESSOR: I'm sorry,
say that again.
00:58:52.441 --> 00:58:54.796
STUDENT: If we
analogize gxx, gyy,
00:58:54.796 --> 00:59:00.889
in the bottom formula there,
[INAUDIBLE] positive curvature
00:59:00.889 --> 00:59:02.180
[INAUDIBLE] second derivative--
00:59:02.180 --> 00:59:03.260
PROFESSOR: Yeah OK.
00:59:03.260 --> 00:59:07.060
So you're asking does
this mean that we
00:59:07.060 --> 00:59:09.097
have two states of
positive curvature?
00:59:09.097 --> 00:59:10.230
STUDENT: Bottom right.
00:59:10.230 --> 00:59:11.604
PROFESSOR: Bottom
right, oh this.
00:59:14.280 --> 00:59:15.450
These are Mustafa's slides.
00:59:15.450 --> 00:59:16.462
I forgot to say that.
00:59:16.462 --> 00:59:18.670
You can tell from the style,
these are not my slides,
00:59:18.670 --> 00:59:19.980
these are Mustafa's slides.
00:59:19.980 --> 00:59:23.040
And I don't know what
he meant by this.
00:59:23.040 --> 00:59:29.080
You're right, this
does-- well you
00:59:29.080 --> 00:59:30.510
do want the metric
to be positive
00:59:30.510 --> 00:59:34.120
definite, which is not the
same as saying the curvature
00:59:34.120 --> 00:59:35.087
is positive.
00:59:35.087 --> 00:59:36.920
And I think this might
just be the condition
00:59:36.920 --> 00:59:39.280
that the metric is
positive definite,
00:59:39.280 --> 00:59:43.490
that this expression
will always be positive.
00:59:43.490 --> 00:59:44.082
Yeah.
00:59:44.082 --> 00:59:46.165
I'll bet that's what it
is, I don't know for sure.
00:59:46.165 --> 00:59:48.439
I'll bet that's what
that condition is about.
00:59:48.439 --> 00:59:50.230
And you do want that,
the metric had better
00:59:50.230 --> 00:59:54.660
be positive definite
for spatial geometries.
00:59:54.660 --> 00:59:57.930
In fact, in general relativity,
where it's a space time metric,
00:59:57.930 --> 01:00:00.054
it will not be
positive definite.
01:00:00.054 --> 01:00:01.720
For reasons that we'll
talk about later.
01:00:04.800 --> 01:00:07.400
But for geometry, the metric
should be positive definite.
01:00:07.400 --> 01:00:08.775
All distances
should be positive.
01:00:16.260 --> 01:00:18.320
OK.
01:00:18.320 --> 01:00:20.390
So that ends my slides.
01:00:20.390 --> 01:00:22.470
So now I'll continue
on the blackboard.
01:00:41.300 --> 01:00:44.700
OK next I wanted to say
a few general comments
01:00:44.700 --> 01:00:46.540
about general relativity.
01:00:53.470 --> 01:00:56.660
General relativity was of
course invented by the Einstein
01:00:56.660 --> 01:01:00.190
in 1916.
01:01:00.190 --> 01:01:02.690
It's a theory that he was
working on for about 10 years
01:01:02.690 --> 01:01:06.760
after he invented the theory
of special relativity.
01:01:06.760 --> 01:01:09.260
To understand what's
going on there,
01:01:09.260 --> 01:01:11.750
you want to recognize
that special relativity is
01:01:11.750 --> 01:01:15.950
a theory of mechanics
and electrodynamics
01:01:15.950 --> 01:01:19.390
which was designed to be
consistent with the principle
01:01:19.390 --> 01:01:21.270
that the speed of
light is always
01:01:21.270 --> 01:01:24.880
the same speed of light
independent of the speed
01:01:24.880 --> 01:01:28.090
of the source of light or
the speed of the observer
01:01:28.090 --> 01:01:30.120
of the light.
01:01:30.120 --> 01:01:32.117
And it's of course
not easy to do that,
01:01:32.117 --> 01:01:34.450
because you think that if you
move relative to something
01:01:34.450 --> 01:01:38.110
else that's moving, that you
would see its velocity change.
01:01:38.110 --> 01:01:43.940
So in order to make a theory
that where the speed of light
01:01:43.940 --> 01:01:46.320
was an absolute
invariant, Einstein
01:01:46.320 --> 01:01:47.860
had introduced a
number of things.
01:01:47.860 --> 01:01:49.645
And we talked about
those at the beginning
01:01:49.645 --> 01:01:54.170
of the course, the three
primary effects built
01:01:54.170 --> 01:01:56.090
in to special relativity.
01:01:56.090 --> 01:01:58.690
Time dilation,
length contraction,
01:01:58.690 --> 01:02:02.310
and new rules
about simultaneity,
01:02:02.310 --> 01:02:05.670
and how things that look
simultaneous to one observer
01:02:05.670 --> 01:02:09.000
will not look simultaneous
to other observers in a very
01:02:09.000 --> 01:02:10.830
definite, well defined way.
01:02:10.830 --> 01:02:13.320
So by inventing
these rules, Einstein
01:02:13.320 --> 01:02:16.164
was able to devise
a system which
01:02:16.164 --> 01:02:18.330
was consistent with the
idea that the speed of light
01:02:18.330 --> 01:02:20.980
always looked the
same to all observers.
01:02:20.980 --> 01:02:24.750
And at similar times, the
Michelson Morley experiment
01:02:24.750 --> 01:02:26.990
seemed to show that that
was in fact the case
01:02:26.990 --> 01:02:29.290
and ultimately there's a
tremendous amount of evidence
01:02:29.290 --> 01:02:33.040
verifying that what the
hypothesis that Einstein was
01:02:33.040 --> 01:02:36.000
pursuing is the right one
for the way nature behaves.
01:02:36.000 --> 01:02:38.990
The speed of light is invariant.
01:02:38.990 --> 01:02:43.150
But what was lacking in
Einstein's formulation
01:02:43.150 --> 01:02:46.420
of special relativity
was any version
01:02:46.420 --> 01:02:49.730
of a consistent
theory of gravity.
01:02:49.730 --> 01:02:55.810
Gravity had a well defined
description known since Newton.
01:02:55.810 --> 01:02:58.170
But Newton's description
was a description
01:02:58.170 --> 01:03:00.630
of a force at a distance.
01:03:00.630 --> 01:03:05.470
And that is intrinsically
inconsistent with a theory
01:03:05.470 --> 01:03:08.272
like special relativity,
which holds that simultaneity
01:03:08.272 --> 01:03:11.100
is itself relative.
01:03:11.100 --> 01:03:23.960
In Newtonian physics,
the force of gravity
01:03:23.960 --> 01:03:27.980
is equal to Newton's
constant times
01:03:27.980 --> 01:03:29.960
the product of the
two masses that
01:03:29.960 --> 01:03:32.810
are interacting with
each other divided
01:03:32.810 --> 01:03:35.720
by the square of the distance
and then times some union
01:03:35.720 --> 01:03:40.630
vector pointing along the line
that joins the two particles.
01:03:40.630 --> 01:03:43.600
But for that formula to
make any sense at all,
01:03:43.600 --> 01:03:45.850
you have to know where
the two particles are
01:03:45.850 --> 01:03:48.100
at the same instant of time.
01:03:48.100 --> 01:03:50.680
And this r is the distance
between the locations
01:03:50.680 --> 01:03:53.470
of the two particles at
some instant of time.
01:03:53.470 --> 01:03:55.240
And this r hat is
a unit vector that
01:03:55.240 --> 01:03:58.110
points along the line joining
those two particles where
01:03:58.110 --> 01:04:00.350
you've pinned down where
those particles are
01:04:00.350 --> 01:04:02.960
at one instant of time.
01:04:02.960 --> 01:04:05.240
But we know from
the very beginning
01:04:05.240 --> 01:04:09.250
that in special relativity the
notion of two things happening
01:04:09.250 --> 01:04:13.280
at distant points at one
instant of time is ambiguous.
01:04:13.280 --> 01:04:16.350
Different observers will
see different notions
01:04:16.350 --> 01:04:20.470
of what it means for
two events to happen
01:04:20.470 --> 01:04:24.402
at the same time
across this distance.
01:04:24.402 --> 01:04:26.360
And that means there's
no way to make sense out
01:04:26.360 --> 01:04:29.250
of Newton's Law of
gravity in the context
01:04:29.250 --> 01:04:31.250
of special relativity.
01:04:31.250 --> 01:04:33.000
You can't modify
it by just changing
01:04:33.000 --> 01:04:35.110
the way the force
depends on distance.
01:04:35.110 --> 01:04:36.780
You really have to
change the variables
01:04:36.780 --> 01:04:39.340
that it depends on
from the beginning.
01:04:39.340 --> 01:04:39.840
Yes.
01:04:39.840 --> 01:04:43.067
STUDENT: Is that equivalent to
saying the other particle can't
01:04:43.067 --> 01:04:45.225
know how much mass
of one is [INAUDIBLE]
01:04:45.225 --> 01:04:48.355
limit of the information
of the speed of light?
01:04:48.355 --> 01:04:50.980
PROFESSOR: Is that equivalent to
saying that one particle can't
01:04:50.980 --> 01:04:52.646
know what the mass
of the other particle
01:04:52.646 --> 01:04:54.310
is because of limitation.
01:04:54.310 --> 01:04:56.060
Yeah it's the distance,
it's not the mass.
01:04:56.060 --> 01:04:58.530
Because the mass is
preserved in time.
01:04:58.530 --> 01:05:00.140
So one particle
could've measured
01:05:00.140 --> 01:05:02.181
the mass of the other
particle at an earlier time
01:05:02.181 --> 01:05:05.074
and it would be
reasonable to infer,
01:05:05.074 --> 01:05:06.490
given the laws of
physics, that we
01:05:06.490 --> 01:05:07.700
know that would stay the same.
01:05:07.700 --> 01:05:09.199
But one particle
has no way of where
01:05:09.199 --> 01:05:11.630
the other particle
is at the same time.
01:05:11.630 --> 01:05:14.190
And not only does the particle
not have any way of knowing,
01:05:14.190 --> 01:05:16.770
but even an external
observer can't know.
01:05:16.770 --> 01:05:18.270
Because different
external observers
01:05:18.270 --> 01:05:20.130
will have different
definitions of what
01:05:20.130 --> 01:05:23.070
it means to be at the same time.
01:05:23.070 --> 01:05:26.850
So there really is no
way that this could work.
01:05:26.850 --> 01:05:29.200
Now I should mention
that it is still
01:05:29.200 --> 01:05:33.880
possible to have action
at a distance theories
01:05:33.880 --> 01:05:36.520
which are consistent
with special relativity.
01:05:36.520 --> 01:05:45.002
And in fact, Maxwell's equations
can be reformulated that way.
01:05:45.002 --> 01:05:46.710
The easiest way to
make things consistent
01:05:46.710 --> 01:05:49.910
with special relativity is
to describe interactions
01:05:49.910 --> 01:05:51.290
in terms of fields.
01:05:51.290 --> 01:05:54.510
And that is really what
Einstein did originally
01:05:54.510 --> 01:05:55.427
in special relativity.
01:05:55.427 --> 01:05:56.884
He was thinking
about light, he was
01:05:56.884 --> 01:05:58.370
thinking about
Maxwell's equations
01:05:58.370 --> 01:06:00.494
and he was thinking very
explicitly about Maxwell's
01:06:00.494 --> 01:06:01.630
equations.
01:06:01.630 --> 01:06:06.020
And in the Maxwell description
of electromagnetism,
01:06:06.020 --> 01:06:08.230
particles at a
distance don't directly
01:06:08.230 --> 01:06:09.710
interact with each other.
01:06:09.710 --> 01:06:12.130
But rather, each
particle interacts
01:06:12.130 --> 01:06:15.140
with the fields around it and
that is a local interaction.
01:06:15.140 --> 01:06:17.660
A particle interacts
directly only
01:06:17.660 --> 01:06:20.080
with the fields
at the same point.
01:06:20.080 --> 01:06:22.340
But then those fields
obey wave equations
01:06:22.340 --> 01:06:24.830
that can propagate information.
01:06:24.830 --> 01:06:27.150
And the fact that
you have an electron
01:06:27.150 --> 01:06:29.910
here can create a
field which then
01:06:29.910 --> 01:06:32.490
exerts a force on
an electron there.
01:06:32.490 --> 01:06:33.924
And the force on
the electron here
01:06:33.924 --> 01:06:35.590
depends only on the
electric fields here
01:06:35.590 --> 01:06:37.280
or the magnetic fields as well.
01:06:37.280 --> 01:06:40.190
The particle is moving, but
everything is completely local
01:06:40.190 --> 01:06:43.640
and the description
of electromagnetism as
01:06:43.640 --> 01:06:46.490
given in the form of
Maxwell's equations.
01:06:46.490 --> 01:06:48.800
And Maxwell's equations are
completely relativistically
01:06:48.800 --> 01:06:49.299
invariant.
01:06:49.299 --> 01:06:53.200
And that was part of
Einstein's-- it was really
01:06:53.200 --> 01:06:55.580
the key part of Einstein's
motivation in constructing
01:06:55.580 --> 01:06:58.710
the theory of special relativity
in the first place to make
01:06:58.710 --> 01:07:00.710
Maxwell's equations invariant.
01:07:00.710 --> 01:07:03.640
That they held in every frame.
01:07:03.640 --> 01:07:07.240
It is still possible
though and worth
01:07:07.240 --> 01:07:12.100
recognizing that it's possible
to reformulate electromagnetism
01:07:12.100 --> 01:07:15.320
as an action at a
distance theory.
01:07:15.320 --> 01:07:17.470
And it is in fact
described that way
01:07:17.470 --> 01:07:19.590
in volume one of
the Feynman lectures
01:07:19.590 --> 01:07:23.110
for those of you who've looked
at the Feynman lectures.
01:07:23.110 --> 01:07:27.840
In order to make
that work, you have
01:07:27.840 --> 01:07:31.340
to complicate things in
a pretty significant way.
01:07:31.340 --> 01:07:34.710
So I'm going to draw here
just a space time diagram.
01:07:34.710 --> 01:07:38.729
X and CT, CT going up that
way, x going that way.
01:07:38.729 --> 01:07:40.270
So in this diagram
the speed of light
01:07:40.270 --> 01:07:42.570
would be a 45 degree line.
01:07:42.570 --> 01:07:46.620
And let's suppose we have
two particles traveling
01:07:46.620 --> 01:07:48.480
in this space.
01:07:48.480 --> 01:07:52.765
A particle that I will
call a and a particle
01:07:52.765 --> 01:07:55.840
that I will call b.
01:07:55.840 --> 01:07:57.925
Being very original
with these names.
01:08:00.810 --> 01:08:05.100
If we wanted to know
the force on particle a
01:08:05.100 --> 01:08:09.430
at a certain time t indicated
by this dotted line,
01:08:09.430 --> 01:08:14.790
I guess I'll label t to make
it as clear as possible.
01:08:14.790 --> 01:08:17.680
Feynman gives us a formula
where we can determine it
01:08:17.680 --> 01:08:20.100
solely in terms of the
motion of particle B
01:08:20.100 --> 01:08:23.660
without talking
about fields at all.
01:08:23.660 --> 01:08:28.005
But the formula does not depend
on where b is at the same time.
01:08:28.005 --> 01:08:29.380
And it could not
if this is going
01:08:29.380 --> 01:08:31.450
to be a relativistic
description.
01:08:31.450 --> 01:08:34.760
But instead, the way this action
and the distance formulation
01:08:34.760 --> 01:08:39.609
works is when imagine drawing
a 45 degree line backwards,
01:08:39.609 --> 01:08:42.710
meaning a line that
light could travel on,
01:08:42.710 --> 01:08:44.850
and one sees where
that intersects
01:08:44.850 --> 01:08:50.210
the trajectory of particle
B. And that time is t prime.
01:08:50.210 --> 01:08:53.840
And the word that's used for
that symbol is retarded time.
01:08:53.840 --> 01:08:55.060
It's an earlier time.
01:08:55.060 --> 01:08:59.479
It's exactly that time which has
the property that if particle
01:08:59.479 --> 01:09:01.609
b emitted a light
beam at that time,
01:09:01.609 --> 01:09:03.779
it would be arriving at
particle a at just the time
01:09:03.779 --> 01:09:05.750
t that we're interested
in the time when we're
01:09:05.750 --> 01:09:08.840
trying to calculate the
force on particle a.
01:09:08.840 --> 01:09:12.300
And what Feynman gives you in
volume one if you look at it
01:09:12.300 --> 01:09:13.970
is a very complicated
formula that
01:09:13.970 --> 01:09:15.745
determines the
force on particle a
01:09:15.745 --> 01:09:18.910
in terms of not only the
position of this particle
01:09:18.910 --> 01:09:21.620
at time t prime but also
its velocity and even
01:09:21.620 --> 01:09:23.870
its acceleration.
01:09:23.870 --> 01:09:26.470
But if you do know the
position, the velocity,
01:09:26.470 --> 01:09:28.109
and the acceleration
of this particle,
01:09:28.109 --> 01:09:30.395
and of course the
velocity of particle a,
01:09:30.395 --> 01:09:31.770
you can determine
the force on a.
01:09:34.290 --> 01:09:38.534
Not obvious, but it's true.
01:09:38.534 --> 01:09:40.200
But that's certainly
not the easiest way
01:09:40.200 --> 01:09:41.990
to formulate electromagnetism.
01:09:41.990 --> 01:09:44.660
And that's not the way
most of us have learned,
01:09:44.660 --> 01:09:46.160
unless you've
learned by starting
01:09:46.160 --> 01:09:47.870
by reading volume
one of Feynman.
01:09:47.870 --> 01:09:49.870
But most of us learn
Maxwell's equations
01:09:49.870 --> 01:09:52.040
as differential equations.
01:09:52.040 --> 01:09:54.970
Where information is propagated
by the field from one
01:09:54.970 --> 01:09:57.070
point to another.
01:09:57.070 --> 01:09:59.590
In the case of
general relativity,
01:09:59.590 --> 01:10:01.810
one has the same problem.
01:10:01.810 --> 01:10:05.077
How can you describe
something which--
01:10:05.077 --> 01:10:06.660
and the only approximate
you initially
01:10:06.660 --> 01:10:09.525
know is an action
and the distance,
01:10:09.525 --> 01:10:11.150
how can you describe
it in a way that's
01:10:11.150 --> 01:10:13.590
consistent with relativity?
01:10:13.590 --> 01:10:18.050
And the idea that simultaneity
is not a well defined concept.
01:10:18.050 --> 01:10:19.550
So that was the
problem was Einstein
01:10:19.550 --> 01:10:22.040
was wrestling with
for 10 years, how
01:10:22.040 --> 01:10:24.146
to build a theory
of gravity that
01:10:24.146 --> 01:10:26.020
would be consistent with
the basic principles
01:10:26.020 --> 01:10:27.960
of special relativity.
01:10:27.960 --> 01:10:31.440
And the result of those
10 years of cogitating
01:10:31.440 --> 01:10:34.560
is the theory that we
call general relativity.
01:10:34.560 --> 01:10:36.880
And it's essentially
a theory which
01:10:36.880 --> 01:10:40.670
describes gravity
as a field theory
01:10:40.670 --> 01:10:43.800
similar to Maxwell's
field theory
01:10:43.800 --> 01:10:45.990
where all interactions
are local.
01:10:45.990 --> 01:10:49.640
Nothing interacts at a
distance, but particles interact
01:10:49.640 --> 01:10:52.130
with fields at the
same point, the fields
01:10:52.130 --> 01:10:55.960
can propagate information
by obeying wave equations,
01:10:55.960 --> 01:10:59.765
and the fields then
at a distant point
01:10:59.765 --> 01:11:02.820
can exert forces
on other particles.
01:11:02.820 --> 01:11:06.130
But in the case of general
relativity, what Einstein
01:11:06.130 --> 01:11:08.900
concluded was that the
fields that were relevant,
01:11:08.900 --> 01:11:11.000
the fields that
described gravity,
01:11:11.000 --> 01:11:15.170
were in fact the metric
of space and time.
01:11:15.170 --> 01:11:17.780
So general relativity
is the field theory
01:11:17.780 --> 01:11:20.280
of the metric of space and time.
01:11:20.280 --> 01:11:23.880
And gravity is described
solely as a distortion
01:11:23.880 --> 01:11:25.400
of space and time.
01:11:25.400 --> 01:11:27.310
And that's what general
relativity is about
01:11:27.310 --> 01:11:30.970
and that's what we will
be learning more about.
01:11:30.970 --> 01:11:33.770
Now as I said earlier, now
I can say it perhaps more
01:11:33.770 --> 01:11:36.970
explicitly, what we
will be learning about
01:11:36.970 --> 01:11:39.620
is how to describe the
curvature of space time
01:11:39.620 --> 01:11:42.350
as general relativity
describes it.
01:11:42.350 --> 01:11:44.850
We will learn how that
curvature affects things
01:11:44.850 --> 01:11:47.370
like the motions of particles.
01:11:47.370 --> 01:11:49.670
But we will not in
this course learn
01:11:49.670 --> 01:11:53.840
how the presence of
particles and masses
01:11:53.840 --> 01:11:55.540
affects the curvature
of space time.
01:11:55.540 --> 01:11:59.460
That you'll have to take it in
a relativity course to learn.
01:11:59.460 --> 01:12:01.807
OK, I think that's where
we'll be stopping today.
01:12:01.807 --> 01:12:04.390
Just doesn't pay to start a new
topic with a minute and a half
01:12:04.390 --> 01:12:05.075
left.
01:12:05.075 --> 01:12:07.033
But let me just ask if
there are any questions.
01:12:11.010 --> 01:12:12.740
OK, well class
over, I will see you
01:12:12.740 --> 01:12:16.150
folks in a week, because
there's no class next Tuesday.