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PROFESSOR: OK.
00:00:21.810 --> 00:00:23.160
Good morning, everybody.
00:00:23.160 --> 00:00:26.870
Welcome to lecture 12 of A286.
00:00:26.870 --> 00:00:29.360
I can't think of any
announcements for today,
00:00:29.360 --> 00:00:31.140
but let me begin
by asking if there
00:00:31.140 --> 00:00:35.160
are any questions either about
logistics or about physics.
00:00:39.810 --> 00:00:40.310
OK.
00:00:40.310 --> 00:00:42.030
In that case, let's get started.
00:00:42.030 --> 00:00:46.810
I want to begin by having a
rapid run through of the things
00:00:46.810 --> 00:00:49.690
we talked about last time
just to firm everything
00:00:49.690 --> 00:00:52.880
up in our minds and
get us ready to go on.
00:00:52.880 --> 00:00:57.570
So, last time we were talking
about non-euclidean geometry
00:00:57.570 --> 00:00:59.510
in a serious way.
00:00:59.510 --> 00:01:02.920
We began by considering the
surface of a sphere, just
00:01:02.920 --> 00:01:04.810
a two-dimensional
sphere embedded
00:01:04.810 --> 00:01:08.040
in a three-dimensional
space, described
00:01:08.040 --> 00:01:11.430
by the simple equation x squared
plus y squared plus z squared
00:01:11.430 --> 00:01:14.090
equals R squared.
00:01:14.090 --> 00:01:17.630
We said that if we wanted to
talk about the surface itself,
00:01:17.630 --> 00:01:20.300
we'd want to have
coordinates for the surface
00:01:20.300 --> 00:01:24.930
and not just speak of things
in terms of x, y, and z.
00:01:24.930 --> 00:01:28.380
So we introduced the
standard polar coordinates--
00:01:28.380 --> 00:01:31.560
theta and phi, which
are related to x, y,
00:01:31.560 --> 00:01:36.160
and z by these fairly
well-known equations.
00:01:36.160 --> 00:01:38.020
Then we wanted to
know the metric
00:01:38.020 --> 00:01:40.280
in terms of our new
variables theta and phi,
00:01:40.280 --> 00:01:44.020
which is the main goal--
to figure out the metric.
00:01:44.020 --> 00:01:49.010
So we first considered
varying the two variables one
00:01:49.010 --> 00:01:50.530
at a time.
00:01:50.530 --> 00:01:53.900
By varying theta, we see
that the point described
00:01:53.900 --> 00:01:58.950
by theta phi would sweep out
a circle whose radius is R,
00:01:58.950 --> 00:02:02.600
and the angle
subtended is d theta.
00:02:02.600 --> 00:02:06.310
So the arc length is
just R times d theta.
00:02:06.310 --> 00:02:09.289
So for varying
theta, the arc length
00:02:09.289 --> 00:02:12.370
is given by that
simple equation.
00:02:12.370 --> 00:02:15.100
Similarly, we went
on to ask ourselves
00:02:15.100 --> 00:02:17.560
what happens when we vary phi.
00:02:17.560 --> 00:02:20.970
As we vary phi, the
point described by theta
00:02:20.970 --> 00:02:24.130
comma phi again
sweeps out a circle,
00:02:24.130 --> 00:02:27.350
but this time it's a circle
in the horizontal plane whose
00:02:27.350 --> 00:02:31.090
radius is not R but whose
radius has this projection
00:02:31.090 --> 00:02:34.610
factor that's R
times sine theta.
00:02:34.610 --> 00:02:37.770
So the angle is
again-- excuse me,
00:02:37.770 --> 00:02:42.030
the arc length is again d phi,
the angles, times the radius,
00:02:42.030 --> 00:02:44.760
but the radius is R sine theta.
00:02:44.760 --> 00:02:51.050
So ds, the total arc length,
is R times sine theta d phi.
00:02:51.050 --> 00:02:52.789
Then, to put them
together, we notice
00:02:52.789 --> 00:02:55.330
that these two variations are
orthogonal to each other, which
00:02:55.330 --> 00:02:57.882
you could see pretty
directly from the diagram.
00:02:57.882 --> 00:02:59.590
So if we do both of
them at the same time
00:02:59.590 --> 00:03:03.320
and ask what's the total
length of the displacement,
00:03:03.320 --> 00:03:06.290
it's just a simple application
of the Pythagorean theorem.
00:03:06.290 --> 00:03:09.600
And we get the sum
of the squares.
00:03:09.600 --> 00:03:12.500
So, varying theta
gives us Rd theta.
00:03:12.500 --> 00:03:16.054
Varying phi gives us
R sine theta d phi.
00:03:16.054 --> 00:03:17.470
And putting them
together, we just
00:03:17.470 --> 00:03:20.380
get ds squared is the sum of
the squares of those, which
00:03:20.380 --> 00:03:24.400
is R squared times d theta
squared plus sine squared
00:03:24.400 --> 00:03:26.310
theta d phi squared.
00:03:26.310 --> 00:03:28.770
And that's the standard
metric for the service
00:03:28.770 --> 00:03:31.720
of a sphere in
polar coordinates.
00:03:31.720 --> 00:03:32.900
So that was a warm up.
00:03:32.900 --> 00:03:35.520
What we really want to do is to
elevate that problem one more
00:03:35.520 --> 00:03:38.370
dimension, and then we have
a model for the universe.
00:03:38.370 --> 00:03:40.830
We can use the same
method to construct
00:03:40.830 --> 00:03:44.870
a three-dimensional space, which
is a three-dimensional surface
00:03:44.870 --> 00:03:47.920
of a sphere embedded in
four euclidean dimensions,
00:03:47.920 --> 00:03:50.590
and that becomes a perfectly
viable homogeneous,
00:03:50.590 --> 00:03:55.690
isotropic, non-euclidean metric
that can describe a universe
00:03:55.690 --> 00:03:57.310
and, in particular,
describes the type
00:03:57.310 --> 00:04:00.720
of universe called
a closed universe.
00:04:00.720 --> 00:04:04.760
So to do that, we
introduce one more axis, w.
00:04:04.760 --> 00:04:07.930
And we consider the sphere
described by x squared plus y
00:04:07.930 --> 00:04:11.862
squared plus z squared plus
w squared equals R squared.
00:04:11.862 --> 00:04:13.320
So it's a
three-dimensional surface
00:04:13.320 --> 00:04:15.960
of a sphere in four dimensions.
00:04:15.960 --> 00:04:18.130
We then need to introduce
one more variable
00:04:18.130 --> 00:04:20.120
to describe points
on the surface,
00:04:20.120 --> 00:04:23.540
and we introduce this in
the form of a new angle.
00:04:23.540 --> 00:04:26.960
The new angle I
chose to call psi.
00:04:26.960 --> 00:04:31.520
And we measure that angle from
the new axis, from the w-axis.
00:04:31.520 --> 00:04:33.680
So the new angle psi
is simply the angle
00:04:33.680 --> 00:04:38.060
from the w-axis, which
means that the projection
00:04:38.060 --> 00:04:43.600
of our vector from the origin
to the point in the w direction
00:04:43.600 --> 00:04:49.050
is just the R cosine psi and
the projection into the x, y, z
00:04:49.050 --> 00:04:53.180
subspace is R times sine psi.
00:04:53.180 --> 00:04:57.260
And the four equations that
describe x, y, z, and w
00:04:57.260 --> 00:04:59.590
are shown there.
00:04:59.590 --> 00:05:03.360
And all we did is we set the
w-coordinate equal to R times
00:05:03.360 --> 00:05:06.510
cosine psi, which is
just the statement
00:05:06.510 --> 00:05:08.840
that psi measures the
angle from the w-axis.
00:05:08.840 --> 00:05:10.680
Nothing more.
00:05:10.680 --> 00:05:16.370
And then we multiplied x, y,
and z by a factor of sine psi,
00:05:16.370 --> 00:05:19.330
so that now we still have
maintained the condition that x
00:05:19.330 --> 00:05:22.030
squared plus y squared plus
z squared plus w squared
00:05:22.030 --> 00:05:25.180
is equal to R squared,
which you can prove directly
00:05:25.180 --> 00:05:28.750
by manipulating this using
the famous identity sine
00:05:28.750 --> 00:05:30.960
squared plus cosine
squared equals 1.
00:05:30.960 --> 00:05:34.240
Nothing more profound than that.
00:05:34.240 --> 00:05:40.249
So, we're now ready to go
ahead and find the new metric,
00:05:40.249 --> 00:05:42.790
and this time it'll really be
something nontrivial, something
00:05:42.790 --> 00:05:47.510
you didn't already
know from high school.
00:05:47.510 --> 00:05:51.420
The new displacement
is to vary psi.
00:05:51.420 --> 00:05:53.885
If we vary psi, it's
really the same story
00:05:53.885 --> 00:05:56.460
as we've seen before except
in a different plane.
00:05:56.460 --> 00:05:59.330
ds is just equal
to R times d psi.
00:05:59.330 --> 00:06:01.500
I guess, as we
vary psi, the point
00:06:01.500 --> 00:06:04.450
described by these coordinates
makes a full circle of radius
00:06:04.450 --> 00:06:06.610
R.
00:06:06.610 --> 00:06:10.550
OK, now what we want to do
is put all this together.
00:06:10.550 --> 00:06:14.760
If we vary psi, we know that ds
is equal to R times deep psi.
00:06:14.760 --> 00:06:17.340
If we just vary
theta or phi, it's
00:06:17.340 --> 00:06:18.700
the same thing we had before.
00:06:18.700 --> 00:06:20.039
We don't need to rethink it.
00:06:20.039 --> 00:06:21.580
All we need to do
is remember there's
00:06:21.580 --> 00:06:24.880
an extra factor of sine psi
in front of all [INAUDIBLE]
00:06:24.880 --> 00:06:28.060
in the XYZ subspace.
00:06:28.060 --> 00:06:31.080
So if you vary theta
or phi, ds squared
00:06:31.080 --> 00:06:34.590
is just equal to what we
had before for the metric
00:06:34.590 --> 00:06:40.340
multiplied by the extra
factor of sine squared of psi.
00:06:40.340 --> 00:06:44.620
Then to put them together,
if we assume for the moment
00:06:44.620 --> 00:06:47.690
that they are orthogonal
to each other,
00:06:47.690 --> 00:06:49.692
then we just add the
sum of the squares.
00:06:49.692 --> 00:06:50.900
And that is the right answer.
00:06:50.900 --> 00:06:53.970
But I'll justify it in a minute.
00:06:53.970 --> 00:06:56.990
But jumping ahead and
making the assumption
00:06:56.990 --> 00:07:00.440
that these separate
displacements are always
00:07:00.440 --> 00:07:03.130
orthogonal to each
other, ds squared
00:07:03.130 --> 00:07:04.860
is then just the
sum of the squares,
00:07:04.860 --> 00:07:10.360
and we get this matrix to
describe our closed universe
00:07:10.360 --> 00:07:15.790
in terms of the variables
psi, theta, and phi.
00:07:15.790 --> 00:07:18.090
To prove this
orthogonality, which
00:07:18.090 --> 00:07:22.770
is crucial for
believing that result,
00:07:22.770 --> 00:07:24.730
I gave an argument
last time, and I'll
00:07:24.730 --> 00:07:27.740
outline again on
the slides here.
00:07:27.740 --> 00:07:30.830
We can consider the two
displacement vectors
00:07:30.830 --> 00:07:33.970
that we're trying to
show to be orthogonal.
00:07:33.970 --> 00:07:37.910
dR sub psi is a
four-dimensional vector,
00:07:37.910 --> 00:07:40.200
which represents the
displacement of the point being
00:07:40.200 --> 00:07:43.920
described by these coordinates
when psi is changed
00:07:43.920 --> 00:07:47.600
to psi plus deep psi,
infinitesimal change in the psi
00:07:47.600 --> 00:07:49.490
coordinate.
00:07:49.490 --> 00:07:53.650
Similarly, I'm going
to let dR sub theta be
00:07:53.650 --> 00:07:56.535
the displacement vector
that the point described
00:07:56.535 --> 00:08:00.030
by these coordinates
undergoes when theta is varied
00:08:00.030 --> 00:08:02.397
by an infinitesimal
amount, d theta.
00:08:02.397 --> 00:08:04.355
And what we're trying to
show is that these two
00:08:04.355 --> 00:08:05.563
are orthogonal to each other.
00:08:05.563 --> 00:08:08.960
So if we do them both, the
magnitude of the change
00:08:08.960 --> 00:08:11.310
is just the sum of
squares, the square root
00:08:11.310 --> 00:08:13.050
of the sum of squares.
00:08:13.050 --> 00:08:17.220
So, first looking
at dR sub theta,
00:08:17.220 --> 00:08:21.240
we notice that dR sub
theta has no w-component.
00:08:21.240 --> 00:08:23.470
And to make that clear,
we should go back a couple
00:08:23.470 --> 00:08:26.900
slides and look at
how w is defined.
00:08:26.900 --> 00:08:29.370
w is defined as R
times cosine psi.
00:08:29.370 --> 00:08:31.550
So if we vary theta,
w doesn't change.
00:08:31.550 --> 00:08:34.740
It doesn't depend on theta.
00:08:34.740 --> 00:08:38.159
So if dR sub theta
has no w-component,
00:08:38.159 --> 00:08:41.460
it means that when we take
the dot product of dR psi
00:08:41.460 --> 00:08:43.260
with the dR theta,
we want to show
00:08:43.260 --> 00:08:45.630
that this is 0 to show
that they're orthogonal.
00:08:45.630 --> 00:08:48.540
The w-components won't
enter, because one of the two
00:08:48.540 --> 00:08:50.930
w-components is 0,
and the dot product
00:08:50.930 --> 00:08:53.881
is a sum of the product
of the x-components
00:08:53.881 --> 00:08:55.380
plus the product
of the y-components
00:08:55.380 --> 00:08:57.160
plus the product
of the z-components
00:08:57.160 --> 00:08:59.360
plus the product of
the w-components.
00:08:59.360 --> 00:09:01.120
So w-compontents only
enter as a product
00:09:01.120 --> 00:09:02.610
of the two w-components.
00:09:02.610 --> 00:09:06.270
So as long as one of them is 0,
there's no contribution there.
00:09:06.270 --> 00:09:08.710
So the four-dimensional
dot product
00:09:08.710 --> 00:09:11.510
reduces to a
three-dimensional dot product.
00:09:11.510 --> 00:09:14.420
And here I'm introducing
a peculiar notation
00:09:14.420 --> 00:09:18.060
when I put a subscript--
a superscript, rather-- 3
00:09:18.060 --> 00:09:18.729
in a vector.
00:09:18.729 --> 00:09:20.520
I just mean take the
first three components
00:09:20.520 --> 00:09:24.622
and ignore the fourth and
think of it as a 3 vector.
00:09:24.622 --> 00:09:26.830
So the dot product that
we're trying to calculate now
00:09:26.830 --> 00:09:29.470
is just the dot product
of two 3 vectors--
00:09:29.470 --> 00:09:31.650
the one that we get when
we vary psi and the one
00:09:31.650 --> 00:09:35.160
that we get when we vary theta.
00:09:35.160 --> 00:09:37.790
Next thing to notice
is that we can
00:09:37.790 --> 00:09:40.690
look at the properties
of these two vectors.
00:09:40.690 --> 00:09:46.560
And dR psi, the vector we
get when we vary psi, I claim
00:09:46.560 --> 00:09:49.480
is in the radial direction
in this three-dimensional
00:09:49.480 --> 00:09:50.620
subspace.
00:09:50.620 --> 00:09:52.410
And we can see that
by looking again
00:09:52.410 --> 00:09:55.630
at these formulas
that relate the angles
00:09:55.630 --> 00:09:57.670
to the Cartesian coordinates.
00:09:57.670 --> 00:10:02.880
When we vary psi,
sine psi changes,
00:10:02.880 --> 00:10:08.110
but sine psi multiplies x, y,
and z all by the same amount.
00:10:08.110 --> 00:10:09.870
So sine psi changes.
00:10:09.870 --> 00:10:12.775
It changes x, y, and
z proportionally.
00:10:12.775 --> 00:10:14.650
And if you change x, y,
and z proportionally,
00:10:14.650 --> 00:10:16.750
it means you're moving
in the radial direction
00:10:16.750 --> 00:10:18.425
in this three-dimensional
subspace.
00:10:21.110 --> 00:10:26.560
On the other hand, dR theta is
what we get when we vary theta.
00:10:26.560 --> 00:10:28.830
And from the
beginning, theta was
00:10:28.830 --> 00:10:32.450
defined in a way that
parametrized the sphere.
00:10:32.450 --> 00:10:34.745
So varying theta only
moves you along the sphere.
00:10:34.745 --> 00:10:38.330
It does not change your
distance from the origin.
00:10:38.330 --> 00:10:43.200
So varying theta is
purely tangential.
00:10:43.200 --> 00:10:45.470
So we have a dot product
between a radial vector
00:10:45.470 --> 00:10:47.360
and a tangential vector,
and those are always
00:10:47.360 --> 00:10:49.400
orthogonal to each other.
00:10:49.400 --> 00:10:53.650
So we get a dot product
of 0, as claimed.
00:10:53.650 --> 00:10:56.410
So the two original
four vectors are
00:10:56.410 --> 00:10:59.552
orthogonal to each other, which
is what we're trying to prove.
00:10:59.552 --> 00:11:01.940
OK, everybody happy with that?
00:11:01.940 --> 00:11:02.890
It is a crucial step.
00:11:02.890 --> 00:11:04.460
You haven't really
gotten the results
00:11:04.460 --> 00:11:08.230
unless you know these
vectors are orthogonal.
00:11:08.230 --> 00:11:11.090
OK, almost done now.
00:11:11.090 --> 00:11:14.380
We then later in lecture
talked about the implications
00:11:14.380 --> 00:11:16.520
of general relativity,
and here we
00:11:16.520 --> 00:11:19.209
didn't prove what
we were claiming.
00:11:19.209 --> 00:11:21.000
We just admitted that
there are some things
00:11:21.000 --> 00:11:23.920
in general relativity that we're
just going to have to assume,
00:11:23.920 --> 00:11:26.860
and this really is
almost the only one.
00:11:26.860 --> 00:11:32.990
General relativity tells us how
matter causes space to curve.
00:11:32.990 --> 00:11:35.720
And it does that in the
form of what are called
00:11:35.720 --> 00:11:37.730
the Einstein field equations.
00:11:37.730 --> 00:11:40.280
And we're not going to learn
the Einstein field equations.
00:11:40.280 --> 00:11:43.660
That's the subject of a
general relativity course.
00:11:43.660 --> 00:11:46.520
So we're just going
to have to assume
00:11:46.520 --> 00:11:50.250
what general relativity tells
us about how space curves,
00:11:50.250 --> 00:11:53.320
and in particular in this
instance, what it tells
00:11:53.320 --> 00:11:57.950
us is that the radius of
curvature R-- this R that we've
00:11:57.950 --> 00:12:00.320
introduced into our metric--
is the radius of curvature
00:12:00.320 --> 00:12:07.140
of the space, is related to the
matter and motion by R squared
00:12:07.140 --> 00:12:10.320
being equal to a squared
of t divided by k.
00:12:10.320 --> 00:12:13.880
And we did argue last time
that, that k in the denominator
00:12:13.880 --> 00:12:17.150
really is necessary just to
make the units turn out right.
00:12:17.150 --> 00:12:19.360
So we really know by
dimensional analysis
00:12:19.360 --> 00:12:23.200
that this formula has to
hold up to some factor.
00:12:23.200 --> 00:12:26.870
The fact that the
factor is 1 is a fact
00:12:26.870 --> 00:12:29.000
about general relativity,
which we're not
00:12:29.000 --> 00:12:31.520
showing at this point.
00:12:31.520 --> 00:12:34.770
When one puts this
back into the metric
00:12:34.770 --> 00:12:38.620
to express the metric
in terms of a of t,
00:12:38.620 --> 00:12:42.250
we find finally
that the metric can
00:12:42.250 --> 00:12:44.950
be written as shown
in the box here,
00:12:44.950 --> 00:12:47.370
and this is the
last equation, where
00:12:47.370 --> 00:12:49.460
I've made a substitution
of variables.
00:12:49.460 --> 00:12:54.430
I replaced the angle psi by
a radial coordinate little r,
00:12:54.430 --> 00:12:56.940
which is defined
to be sine of psi
00:12:56.940 --> 00:12:59.640
divided by the square root of k.
00:12:59.640 --> 00:13:01.560
And this form of
the metric is what's
00:13:01.560 --> 00:13:03.660
called the
Robertson-Walker metric.
00:13:03.660 --> 00:13:05.760
And it's a famous
form of the metric.
00:13:05.760 --> 00:13:08.840
This is what people
normally use.
00:13:08.840 --> 00:13:12.700
So that finishes everything
we said last time, I think.
00:13:12.700 --> 00:13:13.325
Any questions?
00:13:16.230 --> 00:13:17.860
Yes.
00:13:17.860 --> 00:13:20.935
AUDIENCE: What is the motivation
for saying that the-- where
00:13:20.935 --> 00:13:23.480
you can describe space as
a three-dimensional sphere
00:13:23.480 --> 00:13:24.790
in a four space.
00:13:24.790 --> 00:13:27.663
Is it because it's only
real geometry that,
00:13:27.663 --> 00:13:29.532
where there's
isotropian [INAUDIBLE]?
00:13:29.532 --> 00:13:30.740
PROFESSOR: Yes, that's right.
00:13:30.740 --> 00:13:34.330
I was going to be saying
that shortly, but yes.
00:13:34.330 --> 00:13:37.270
This metric and its open
universe counterpart
00:13:37.270 --> 00:13:41.240
and flat space together
make up the most general
00:13:41.240 --> 00:13:45.262
possible metric, which is
homogeneous and isotropic.
00:13:45.262 --> 00:13:46.720
At this stage, I'm
really not going
00:13:46.720 --> 00:13:48.110
to claim that anyway,
but at this stage,
00:13:48.110 --> 00:13:49.890
what we do know is
that this metric is
00:13:49.890 --> 00:13:51.667
homogeneous and isotropic.
00:13:51.667 --> 00:13:53.500
And certainly what we're
trying to construct
00:13:53.500 --> 00:13:55.670
is metrics, which are
homogeneous and isotropic.
00:13:55.670 --> 00:13:59.330
But this also is actually
the only possibility
00:13:59.330 --> 00:14:02.040
within that small class.
00:14:02.040 --> 00:14:03.065
Any other questions?
00:14:08.890 --> 00:14:09.514
OK.
00:14:09.514 --> 00:14:11.555
In that case, we will
continue on the blackboard.
00:14:24.420 --> 00:14:26.380
So what we have
derived so far is
00:14:26.380 --> 00:14:28.405
the metric for a
closed universe.
00:14:31.460 --> 00:14:33.487
Maybe I'll start by
getting on the blackboard
00:14:33.487 --> 00:14:35.070
the same formula
that's up there, just
00:14:35.070 --> 00:14:36.480
so I can see it better.
00:14:36.480 --> 00:14:39.910
Even though you can probably
see equally well either way.
00:14:39.910 --> 00:14:46.610
A closed universe is
described by ds squared
00:14:46.610 --> 00:14:58.870
is equal to a squared of t
times dr squared over 1 minus kr
00:14:58.870 --> 00:15:14.000
squared plus r squared d theta
squared plus sine squared theta
00:15:14.000 --> 00:15:14.650
d phi squared.
00:15:21.360 --> 00:15:28.070
And to relate this variable r
to our previous definitions,
00:15:28.070 --> 00:15:34.150
little r is equal
to the sine of psi
00:15:34.150 --> 00:15:35.910
divided by the square root of k.
00:15:40.850 --> 00:15:41.860
OK.
00:15:41.860 --> 00:15:42.900
Question back there?
00:15:42.900 --> 00:15:48.804
AUDIENCE: Psi is-- sorry this
is just [INAUDIBLE] think.
00:15:48.804 --> 00:15:51.267
Psi is which angle?
00:15:51.267 --> 00:15:53.350
PROFESSOR: OK, the question
is psi is which angle.
00:15:53.350 --> 00:15:55.150
Psi is the angle
we introduced when
00:15:55.150 --> 00:15:59.210
we went from two-dimension
sphere embedded in three
00:15:59.210 --> 00:16:01.010
dimensions to one
dimension higher.
00:16:01.010 --> 00:16:02.950
Psi is the angle
from the new axis.
00:16:02.950 --> 00:16:04.540
The angle from the w axis.
00:16:04.540 --> 00:16:05.081
AUDIENCE: OK.
00:16:09.230 --> 00:16:09.880
PROFESSOR: OK.
00:16:09.880 --> 00:16:13.200
So, we've covered a
lot of ground here.
00:16:13.200 --> 00:16:15.440
We have our first
non-euclidean metric
00:16:15.440 --> 00:16:17.670
that's visibly important.
00:16:17.670 --> 00:16:22.200
But we know from our work on the
Newtonian model of the universe
00:16:22.200 --> 00:16:25.280
that this little k doesn't
have to be positive.
00:16:25.280 --> 00:16:29.210
It can be positive,
negative, or 0.
00:16:29.210 --> 00:16:32.810
If k is 0, the
metric is actually
00:16:32.810 --> 00:16:35.140
just the metric of a flat space.
00:16:35.140 --> 00:16:36.830
But when k is
negative, it's a case
00:16:36.830 --> 00:16:38.920
that we haven't
talked about yet.
00:16:38.920 --> 00:16:42.750
So we want to know, what
will we write for a metric
00:16:42.750 --> 00:16:44.760
if k were negative?
00:16:44.760 --> 00:16:49.820
And the answer to that turns
out to be perfectly simple.
00:16:49.820 --> 00:16:53.440
This formula has a k in it.
00:16:53.440 --> 00:16:55.500
Lots of times in
our experience--
00:16:55.500 --> 00:16:59.280
I'm sure we all know-- when
we write an equation for one
00:16:59.280 --> 00:17:01.710
sine of a variable, we
find that the same equation
00:17:01.710 --> 00:17:04.510
works even if the variable
has the other sine.
00:17:04.510 --> 00:17:07.960
So if you're buying and selling
stocks, if the stocks go up
00:17:07.960 --> 00:17:11.880
and the stocks go down, you
can use the same equations.
00:17:11.880 --> 00:17:14.620
The price today is the price
yesterday plus the increment,
00:17:14.620 --> 00:17:16.619
and the increment could
be positive or negative.
00:17:16.619 --> 00:17:19.690
But that equation-- price
today equals price yesterday
00:17:19.690 --> 00:17:21.930
plus income-- it still works.
00:17:21.930 --> 00:17:24.030
And same thing here.
00:17:24.030 --> 00:17:27.430
If k it happens to
be negative, there's
00:17:27.430 --> 00:17:30.766
nothing wrong with this formula.
00:17:30.766 --> 00:17:35.070
It, in fact, describes an
open universe just as well
00:17:35.070 --> 00:17:38.300
as it describes a
closed universe.
00:17:38.300 --> 00:17:42.090
Now notice, however, that
things are a little bit tricky.
00:17:42.090 --> 00:17:45.800
If you look at the equation that
I wrote immediately below here,
00:17:45.800 --> 00:17:47.760
if k is negative, we
have the square root
00:17:47.760 --> 00:17:51.480
of a negative number here,
so the denominator would
00:17:51.480 --> 00:17:55.250
be imaginary, and what would
that say about r and psi?
00:17:55.250 --> 00:17:57.710
It would obviously confuse us.
00:17:57.710 --> 00:18:02.020
So you really have to write
the metric and the correct form
00:18:02.020 --> 00:18:08.370
before you can just
change the sign of k.
00:18:08.370 --> 00:18:11.090
If we had written the
metric in terms of psi
00:18:11.090 --> 00:18:14.650
and not made the substitution,
we could just as well
00:18:14.650 --> 00:18:17.760
have written the metric
for a closed universe
00:18:17.760 --> 00:18:25.370
as a squared of t divided
by k times d psi squared
00:18:25.370 --> 00:18:34.080
plus sine squared psi d theta
squared plus sine squared
00:18:34.080 --> 00:18:39.240
theta d phi squared.
00:18:39.240 --> 00:18:41.640
This is an alternative metric
for the closed universe.
00:18:41.640 --> 00:18:43.160
It's, in fact, where we started.
00:18:43.160 --> 00:18:48.170
We then made a substitution,
replacing sine psi by little r.
00:18:48.170 --> 00:18:50.830
If we had the metric in
this form and we said,
00:18:50.830 --> 00:18:54.540
well, let k be negative
instead of positive,
00:18:54.540 --> 00:18:58.200
then notice that a squared
is certainly positive,
00:18:58.200 --> 00:19:01.150
so we'd have a negative number
out here times things which
00:19:01.150 --> 00:19:02.970
are also manifestly positive.
00:19:02.970 --> 00:19:04.640
We would have a
negative definite metric
00:19:04.640 --> 00:19:07.340
instead of a positive
definite metric.
00:19:07.340 --> 00:19:11.420
So we could not change the sign
of little k in this formula
00:19:11.420 --> 00:19:14.190
and get what we want.
00:19:14.190 --> 00:19:16.160
So you have to careful.
00:19:16.160 --> 00:19:17.670
It doesn't always work.
00:19:17.670 --> 00:19:22.110
But it does work when you
write the metric in this form.
00:19:22.110 --> 00:19:24.370
Now since it
doesn't always work,
00:19:24.370 --> 00:19:27.620
and since we haven't really
made any sound arguments yet,
00:19:27.620 --> 00:19:30.440
I'd like to spend a
little time describing--
00:19:30.440 --> 00:19:33.100
I'm not going to do the
calculation because it's
00:19:33.100 --> 00:19:34.950
too messy, but I'd
like spend little time
00:19:34.950 --> 00:19:40.700
describing how you would
show that this metric works
00:19:40.700 --> 00:19:43.960
for an open universe.
00:19:43.960 --> 00:19:47.730
So first thing to
recognize is-- what
00:19:47.730 --> 00:19:51.230
do we actually mean
when we say it "works"?
00:19:51.230 --> 00:19:54.310
Can somebody tell me what I
probably mean when I say that?
00:19:58.140 --> 00:19:58.640
Yes.
00:19:58.640 --> 00:20:00.890
AUDIENCE: It doesn't have
any glaring contradictions?
00:20:00.890 --> 00:20:03.240
PROFESSOR: Doesn't have
any glaring contradictions.
00:20:03.240 --> 00:20:07.547
Yeah, that's good, but we can
be more specific, especially
00:20:07.547 --> 00:20:09.880
since I'm going to try to
describe how we would actually
00:20:09.880 --> 00:20:12.130
show it, and it's a little hard
to show that something doesn't
00:20:12.130 --> 00:20:13.296
have glaring contradictions.
00:20:16.980 --> 00:20:19.890
What do we actually care about
in constructing these metrics?
00:20:22.701 --> 00:20:23.200
Yes?
00:20:23.200 --> 00:20:24.991
AUDIENCE: Does the goal
that they hold well
00:20:24.991 --> 00:20:28.619
in limits, i.e.,
the flat universes?
00:20:28.619 --> 00:20:31.160
PROFESSOR: That the physics will
hold well in certain limits,
00:20:31.160 --> 00:20:33.530
like they should approach
the flat universe and limit.
00:20:33.530 --> 00:20:35.960
We certainly do
want that to happen,
00:20:35.960 --> 00:20:37.430
but there is
something else that we
00:20:37.430 --> 00:20:39.430
want that doesn't
involve taking limits,
00:20:39.430 --> 00:20:41.055
because you have
different things which
00:20:41.055 --> 00:20:43.526
all approach the same
limit, of course.
00:20:43.526 --> 00:20:46.150
Making sure an answer approaches
the right limits is a good way
00:20:46.150 --> 00:20:48.960
to test the answer, because
most wrong answers will not
00:20:48.960 --> 00:20:50.110
have the right limits.
00:20:50.110 --> 00:20:52.340
But merely knowing you
have the right limits
00:20:52.340 --> 00:20:54.750
does not prove that you
have the right answer.
00:20:54.750 --> 00:20:55.250
Yes?
00:20:55.250 --> 00:20:57.749
AUDIENCE: It could also reflect
an isotropic and homogeneous
00:20:57.749 --> 00:20:58.570
non-euclidean?
00:20:58.570 --> 00:20:59.370
PROFESSOR: Exactly.
00:20:59.370 --> 00:20:59.890
Exactly.
00:20:59.890 --> 00:21:02.270
What we're looking for is
a homogeneous and isotropic
00:21:02.270 --> 00:21:04.467
non-euclidean space,
because that's
00:21:04.467 --> 00:21:05.800
what we know about our universe.
00:21:05.800 --> 00:21:07.670
It's homogeneous and
isotropic, and we're
00:21:07.670 --> 00:21:11.410
trying to build a mathematical
model of those facts.
00:21:11.410 --> 00:21:14.940
So we want homogeneity
and isotropy.
00:21:14.940 --> 00:21:19.280
If we limit isotropy to
isotropy about the origin, which
00:21:19.280 --> 00:21:21.910
is enough if we're going to
later prove homogeneity, which
00:21:21.910 --> 00:21:24.420
will prove that all
points are equivalent,
00:21:24.420 --> 00:21:27.590
isotropy about the
origin is obvious here,
00:21:27.590 --> 00:21:30.810
because the angular part
is just exactly what we
00:21:30.810 --> 00:21:34.890
had for a sphere in three
euclidean dimensions.
00:21:34.890 --> 00:21:39.790
So it behaves on angles exactly
like a euclidean problem,
00:21:39.790 --> 00:21:42.580
so we know that it's isotropic.
00:21:42.580 --> 00:21:44.500
If you point out
that algebraically
00:21:44.500 --> 00:21:47.570
as you look at it, it's not
obvious that it's isotropic.
00:21:47.570 --> 00:21:50.510
We just know where that
expression came from.
00:21:50.510 --> 00:21:53.100
It came from the sphere.
00:21:53.100 --> 00:22:00.500
In terms of theta and phi,
it's not manifestly isotropic.
00:22:00.500 --> 00:22:02.840
And that's because in
choosing theta and phi,
00:22:02.840 --> 00:22:05.620
we chose a special
point, the North Pole,
00:22:05.620 --> 00:22:08.340
to measure our angle theta from.
00:22:08.340 --> 00:22:13.550
And the choice of that special
point for our coordinate system
00:22:13.550 --> 00:22:16.860
broke the isotropy.
00:22:16.860 --> 00:22:19.890
But we know that deep
down it is isotropic.
00:22:19.890 --> 00:22:24.110
And that idea, that you can have
such isotropy without having
00:22:24.110 --> 00:22:28.740
manifest isotropy is
also crucial to how
00:22:28.740 --> 00:22:31.950
homogeneity plays
out in this metric.
00:22:31.950 --> 00:22:34.650
I claim, and we know
really, that this metric
00:22:34.650 --> 00:22:35.421
is homogeneous.
00:22:35.421 --> 00:22:36.920
At least we know
where it came from.
00:22:36.920 --> 00:22:39.120
It came, again, from
the surface of a sphere
00:22:39.120 --> 00:22:42.560
one dimension higher than
just the angular part.
00:22:42.560 --> 00:22:45.830
And then spherical picture--it's
obviously homogeneous.
00:22:45.830 --> 00:22:48.970
But nonetheless, in building
our coordinate system,
00:22:48.970 --> 00:22:50.690
we had to break the homogeneity.
00:22:50.690 --> 00:22:54.020
We chose a special
point-- again,
00:22:54.020 --> 00:22:55.770
what we might call
the North Pole--
00:22:55.770 --> 00:22:59.330
in this case, the point
where w has its maximum value
00:22:59.330 --> 00:23:00.860
and made that point special.
00:23:00.860 --> 00:23:03.270
The point where w had its
maximum value in the x, y, z, w
00:23:03.270 --> 00:23:06.970
space is the point
which is now the origin
00:23:06.970 --> 00:23:09.940
of this coordinate system.
00:23:09.940 --> 00:23:15.620
So, if we wanted to prove
that this metric really
00:23:15.620 --> 00:23:19.360
is homogeneous, we
would like to prove it
00:23:19.360 --> 00:23:21.720
for the k equals minus 1 case.
00:23:21.720 --> 00:23:24.320
But let's first,
imaging, what would we
00:23:24.320 --> 00:23:26.850
do if we wanted to prove
that it was homogeneous
00:23:26.850 --> 00:23:30.520
for the k equals plus 1
case or k positive case?
00:23:30.520 --> 00:23:32.480
The case that we really
think we do understand.
00:23:32.480 --> 00:23:34.590
The closed universe.
00:23:34.590 --> 00:23:37.370
For the closed universe, this
does not look homogeneous.
00:23:37.370 --> 00:23:38.910
It looks like the
origin is special.
00:23:38.910 --> 00:23:41.140
r equals 0 is special.
00:23:41.140 --> 00:23:43.560
But we know that it
came from the sphere,
00:23:43.560 --> 00:23:50.270
and if somebody asked us to
prove that, that metric was
00:23:50.270 --> 00:23:53.960
homogeneous-- more
particularly, somebody
00:23:53.960 --> 00:23:57.000
might, for example,
challenge us to construct
00:23:57.000 --> 00:24:01.760
a coordinate transformation,
which would preserve the metric
00:24:01.760 --> 00:24:10.330
and map some arbitrary point r
0 theta 0 phi 0 to the origin.
00:24:10.330 --> 00:24:12.959
We might undertake
that challenge.
00:24:12.959 --> 00:24:13.750
It's a lot of work.
00:24:13.750 --> 00:24:14.630
We're not going
to actually do it.
00:24:14.630 --> 00:24:16.406
And I promise I'll
never ask you to do it.
00:24:16.406 --> 00:24:18.780
But I want to talk a little
bit about how we would do it,
00:24:18.780 --> 00:24:22.267
because we do have a method,
which we know will work.
00:24:22.267 --> 00:24:24.100
And knowing that we
have a method that works
00:24:24.100 --> 00:24:26.570
is all we really need to know.
00:24:26.570 --> 00:24:55.360
So suppose we wanted a
coordinate transformation
00:24:55.360 --> 00:25:08.800
that preserves the
form of the metric
00:25:08.800 --> 00:25:19.972
and maps some arbitrary
point, and I'll
00:25:19.972 --> 00:25:24.310
give the coordinates to
this arbitrary point a name.
00:25:24.310 --> 00:25:29.820
I'll call it r 0,
theta 0, and phi 0.
00:25:29.820 --> 00:25:31.480
These are coordinates
of a point.
00:25:31.480 --> 00:25:35.110
So we're going to map this
arbitrary point to the origin.
00:25:39.800 --> 00:25:42.494
Notice that this is a concrete
statement about homogeneity.
00:25:42.494 --> 00:25:44.410
If we can map an arbitrary
point to the origin
00:25:44.410 --> 00:25:45.830
while preserving
the metric, we're
00:25:45.830 --> 00:25:47.530
really proving that
an arbitrary point
00:25:47.530 --> 00:25:49.100
is equivalent to the origin.
00:25:49.100 --> 00:25:51.350
And if an arbitrary point
is equivalent to the origin,
00:25:51.350 --> 00:25:53.312
then all points are
equivalent to the origin
00:25:53.312 --> 00:25:54.520
and equivalent to each other.
00:25:54.520 --> 00:25:55.820
We're done.
00:25:55.820 --> 00:25:58.410
That proves homogeneity.
00:25:58.410 --> 00:26:01.250
So, suppose we
wanted to do this.
00:26:01.250 --> 00:26:03.280
How would we do it?
00:26:03.280 --> 00:26:06.710
The point is that knowing how we
got this metric from the sphere
00:26:06.710 --> 00:26:10.150
allows us to go back to
sphere and rotate the sphere
00:26:10.150 --> 00:26:15.970
and rotate back and derive
the coordinate transformation
00:26:15.970 --> 00:26:17.140
that we want.
00:26:17.140 --> 00:26:20.770
And I'll just describe that
in slightly more detail.
00:26:20.770 --> 00:26:30.930
What we would first
do is-- I claim
00:26:30.930 --> 00:26:32.900
we can do it in three
steps, each of which
00:26:32.900 --> 00:26:36.810
we know how to do
although they're messy.
00:26:36.810 --> 00:26:38.260
I guess I'll start over here.
00:26:46.060 --> 00:26:54.510
So step one is to find
the x, y, z, w-coordinates
00:26:54.510 --> 00:26:56.280
that go with this point.
00:26:56.280 --> 00:26:58.430
Because once we have the
x, y, z, w-coordinates,
00:26:58.430 --> 00:26:59.990
we're in our
four-dimensional space
00:26:59.990 --> 00:27:01.920
where we know how
to do rotations.
00:27:01.920 --> 00:27:08.560
So the first thing
we do is we just find
00:27:08.560 --> 00:27:12.310
the corresponding x, y, z,
w-coordinates where x0 is just
00:27:12.310 --> 00:27:19.065
the x-coordinate that goes
with r0, theta0, and phi0.
00:27:19.065 --> 00:27:26.360
And y0 is the y-coordinate that
goes with r0, theta0, and phi0.
00:27:26.360 --> 00:27:31.984
And z0 is the z-coordinate,
and w0 is the w-coordinate.
00:27:31.984 --> 00:27:33.650
And these are just
points we had before.
00:27:33.650 --> 00:27:35.191
I'm writing them
symbolically, but we
00:27:35.191 --> 00:27:37.740
know how to express the
Cartesian coordinates
00:27:37.740 --> 00:27:39.961
in terms of the angles.
00:27:39.961 --> 00:27:40.460
Yes?
00:27:40.460 --> 00:27:44.150
AUDIENCE: Do we
have a psi0 as well?
00:27:44.150 --> 00:27:46.820
PROFESSOR: r0 replaced psi0.
00:27:46.820 --> 00:27:50.784
r0 is the sine of psi
divided by root k.
00:27:50.784 --> 00:27:52.200
So we only need
three coordinates.
00:27:52.200 --> 00:27:54.720
We could have different
choices of what we call them.
00:27:54.720 --> 00:27:56.580
We could've used psi here.
00:27:56.580 --> 00:28:00.470
The reason I'm using little r is
that I want to, when I'm done,
00:28:00.470 --> 00:28:03.410
describe what we would
do if k were a negative.
00:28:03.410 --> 00:28:05.200
And if k were a
negative, we already
00:28:05.200 --> 00:28:08.730
said that psi does
not actually work.
00:28:08.730 --> 00:28:10.605
We have to use different
coordinates in order
00:28:10.605 --> 00:28:15.750
to smoothly write an open
universe coordinate system.
00:28:18.820 --> 00:28:19.320
Yes?
00:28:19.320 --> 00:28:20.986
AUDIENCE: How is that
not like cheating?
00:28:20.986 --> 00:28:24.442
Because, I mean, you did
define r with the root k,
00:28:24.442 --> 00:28:26.150
and now we're just
kind of ignoring them.
00:28:26.150 --> 00:28:28.025
PROFESSOR: Well that's
exactly-- the question
00:28:28.025 --> 00:28:29.680
is why is this not cheating?
00:28:29.680 --> 00:28:31.190
And the reason it's
not cheating is
00:28:31.190 --> 00:28:34.290
because of what I'm
about to show you.
00:28:34.290 --> 00:28:38.300
What I'm saying really is
that just setting k negative
00:28:38.300 --> 00:28:41.290
is something you might
expect to probably work.
00:28:41.290 --> 00:28:43.790
I think you have good grounds
to expect it to probably work.
00:28:43.790 --> 00:28:44.690
Now what we're
talking about is how
00:28:44.690 --> 00:28:45.981
to actually show that it works.
00:28:48.720 --> 00:28:49.220
Yes?
00:28:49.220 --> 00:28:51.132
AUDIENCE: Professor,
what's the necessity
00:28:51.132 --> 00:28:55.912
of defining w as opposed
to, I don't know, t?
00:28:55.912 --> 00:28:59.367
The traditional, like, t--
00:28:59.367 --> 00:28:59.950
PROFESSOR: OK.
00:28:59.950 --> 00:29:02.840
The question is why did I call
the fourth variable w and not
00:29:02.840 --> 00:29:03.740
t.
00:29:03.740 --> 00:29:06.250
The answer is that the variable
we're talking about here
00:29:06.250 --> 00:29:08.630
is not time.
00:29:08.630 --> 00:29:11.650
It's another spatial coordinate.
00:29:11.650 --> 00:29:13.910
So, for that reason
I think it's better
00:29:13.910 --> 00:29:15.700
to call it w than to call it t.
00:29:15.700 --> 00:29:18.740
Of course, needless to
say, the name of a variable
00:29:18.740 --> 00:29:21.060
doesn't have any
actual significance.
00:29:21.060 --> 00:29:24.040
So I certainly could have
called it t equally well,
00:29:24.040 --> 00:29:27.399
but I think that would have
caused some confusion by people
00:29:27.399 --> 00:29:28.940
thinking it was
time, which it's not.
00:29:31.452 --> 00:29:32.285
Any other questions?
00:29:37.640 --> 00:29:38.160
OK.
00:29:38.160 --> 00:29:40.550
So, what I'm
outlining is the steps
00:29:40.550 --> 00:29:43.430
that you would use to prove
that this is homogeneous.
00:29:43.430 --> 00:29:45.367
I'm doing it for
the closed case.
00:29:45.367 --> 00:29:47.700
Now the point is that if we
can do it for the open case,
00:29:47.700 --> 00:29:49.330
we'll prove that
the open metric is
00:29:49.330 --> 00:29:51.560
what we want it to be--
homogeneous and isotropic.
00:29:51.560 --> 00:29:53.220
And that's the only
criteria that we
00:29:53.220 --> 00:29:56.580
have for goodness of a metric.
00:29:56.580 --> 00:29:59.020
So, continuing with
the closed case
00:29:59.020 --> 00:30:02.520
in mind, the first step
of doing this mapping,
00:30:02.520 --> 00:30:07.140
to map some arbitrary
point to the origin,
00:30:07.140 --> 00:30:11.245
is to first find its
x, y, z, w-coordinates,
00:30:11.245 --> 00:30:14.110
its Cartesian coordinates.
00:30:14.110 --> 00:30:17.220
Once we have the
Cartesian coordinates,
00:30:17.220 --> 00:30:20.020
we know that in the
four-dimensional space
00:30:20.020 --> 00:30:23.262
we can perform ordinary
euclidean rotations.
00:30:23.262 --> 00:30:24.970
Rotations in four
dimensions [INAUDIBLE],
00:30:24.970 --> 00:30:26.040
not three dimensions.
00:30:26.040 --> 00:30:27.650
And even rotations
in three dimensions
00:30:27.650 --> 00:30:30.180
are not all that simple, but
nonetheless in principle,
00:30:30.180 --> 00:30:33.290
we know how to do rotations
in four dimensions.
00:30:33.290 --> 00:30:35.140
And we know what
we're trying to do
00:30:35.140 --> 00:30:37.970
in the four-dimensional picture.
00:30:37.970 --> 00:30:39.540
We're trying to
rotate this point
00:30:39.540 --> 00:30:42.470
to the origin of our
coordinate system.
00:30:42.470 --> 00:30:44.630
And the origin of our
coordinate system-- the way
00:30:44.630 --> 00:30:49.350
we've done this mapping-- is
to make w equals capital R,
00:30:49.350 --> 00:30:52.270
w equal to its maximum
value, the center
00:30:52.270 --> 00:30:53.720
of our coordinate system.
00:30:53.720 --> 00:30:55.970
That's where psi was
equal to 0, and now
00:30:55.970 --> 00:30:59.480
where our new variable
little r is equal to 0.
00:30:59.480 --> 00:31:04.110
So we'd like to map this point,
whatever it is, to the point
00:31:04.110 --> 00:31:07.380
where w has a maximum value,
and the other coordinates all
00:31:07.380 --> 00:31:08.740
vanish.
00:31:08.740 --> 00:31:09.490
So we can do that.
00:31:09.490 --> 00:31:11.320
We can find a rotation
that does that.
00:31:11.320 --> 00:31:13.580
It's not even unique,
because you can always
00:31:13.580 --> 00:31:16.730
rotate about the final axis.
00:31:16.730 --> 00:31:19.100
But in any case, we imagine
that we can do that.
00:31:19.100 --> 00:31:26.345
And that's step two-- is
to find the right rotation.
00:31:30.580 --> 00:31:36.140
And a general rotation is
a linear transformation,
00:31:36.140 --> 00:31:42.440
so you can write it as x prime,
y prime, z prime, w prime,
00:31:42.440 --> 00:31:45.600
as a four vector,
is equal to some 4
00:31:45.600 --> 00:32:06.290
by 4 rotation matrix times
the original four coordinates.
00:32:06.290 --> 00:32:07.710
So an equation of
this form would
00:32:07.710 --> 00:32:10.260
describe a
four-dimensional rotation.
00:32:10.260 --> 00:32:16.850
And we in particular want the
four-dimensional rotation which
00:32:16.850 --> 00:32:28.240
maps-- maybe I'll write it as a
similar matrix equation-- what
00:32:28.240 --> 00:32:35.440
we want is that the matrix,
when it operates on x0, y0, z0,
00:32:35.440 --> 00:32:38.947
and w0, which remember are
just the four coordinates
00:32:38.947 --> 00:32:40.530
that correspond to
our original point,
00:32:40.530 --> 00:32:49.200
r0, theta0, phi0, we want
this to map into 0, 0, 0,
00:32:49.200 --> 00:32:53.910
r, which is the four-dimensional
description of the origin
00:32:53.910 --> 00:32:55.410
of our new coordinate system.
00:33:04.531 --> 00:33:06.530
And then finally, step
three is the obvious one.
00:33:06.530 --> 00:33:09.200
Once we've found the
transformation that
00:33:09.200 --> 00:33:12.550
maps to the origin,
now we just go back
00:33:12.550 --> 00:33:16.220
to our original
angular coordinates.
00:33:16.220 --> 00:33:29.430
So, now we set r prime
equal to the radius function
00:33:29.430 --> 00:33:35.340
of x prime, y prime,
z prime, and w prime.
00:33:35.340 --> 00:33:37.080
And this just means
the r-coordinate--
00:33:37.080 --> 00:33:40.750
that corresponds to
those four euclidean
00:33:40.750 --> 00:33:45.520
coordinates and similarly
for the other variables.
00:33:45.520 --> 00:33:48.570
Theta prime is
the theta function
00:33:48.570 --> 00:33:52.490
of x prime, y prime,
z prime, and w prime.
00:33:52.490 --> 00:33:57.930
And phi prime is equal
to the phi function.
00:33:57.930 --> 00:33:59.940
All these are
functions that we know.
00:33:59.940 --> 00:34:01.898
I just don't want to
write them out explicitly,
00:34:01.898 --> 00:34:03.620
because that's a lot of work.
00:34:03.620 --> 00:34:10.980
So it's phi of x prime, y
prime, z prime, w prime.
00:34:10.980 --> 00:34:13.510
So now with the three
steps, we have our mapping.
00:34:13.510 --> 00:34:18.000
We could start with an arbitrary
point, perform the rotation,
00:34:18.000 --> 00:34:21.324
and then calculate the
angular variables again.
00:34:24.489 --> 00:34:26.860
Ok, do people understand
what I'm talking about here?
00:34:26.860 --> 00:34:27.711
Oh good.
00:34:27.711 --> 00:34:28.210
OK.
00:34:28.210 --> 00:34:30.777
And the good things is I
promise I won't make you do it.
00:34:30.777 --> 00:34:32.610
And I've never done it
either, to be honest.
00:34:32.610 --> 00:34:34.639
But it's obvious
that we can do it.
00:34:34.639 --> 00:34:39.070
And if we can do this, this
would prove and especially
00:34:39.070 --> 00:34:40.908
demonstrate homogeneity.
00:34:40.908 --> 00:34:42.949
It would be a mapping that
would map an arbitrary
00:34:42.949 --> 00:34:46.230
point to the origin, proving
that, that arbitrary point was
00:34:46.230 --> 00:34:50.429
equivalent to the origin as
far as the metric is concerned.
00:34:50.429 --> 00:34:52.921
And now my claim, I
mean, I'm not really
00:34:52.921 --> 00:34:54.420
going to prove this
either, but it's
00:34:54.420 --> 00:34:56.830
a claim that could be verified
by going through things.
00:34:56.830 --> 00:34:58.829
And it also seems
highly plausible--
00:34:58.829 --> 00:35:00.620
that if you looked at
each of these steps--
00:35:00.620 --> 00:35:01.994
these are all just
algebra steps,
00:35:01.994 --> 00:35:04.490
these are all just
algebraic equations--
00:35:04.490 --> 00:35:06.570
that they would work just
as well for negative k
00:35:06.570 --> 00:35:09.440
as they will for positive k.
00:35:09.440 --> 00:35:12.825
And by doing these same
series of manipulations
00:35:12.825 --> 00:35:15.910
for negative k, you
would prove that
00:35:15.910 --> 00:35:17.750
the Robertson-Walker
metric for negative k
00:35:17.750 --> 00:35:19.440
is homogeneous,
which is our goal.
00:35:19.440 --> 00:35:20.731
We already know it's isotropic.
00:35:20.731 --> 00:35:22.980
If we can prove it's
homogeneous, we're home free.
00:35:22.980 --> 00:35:25.120
It's [INAUDIBLE].
00:35:25.120 --> 00:35:25.620
Yes?
00:35:25.620 --> 00:35:30.540
PROFESSOR: Does the necessity
of placing the origin at r,
00:35:30.540 --> 00:35:35.944
like big R, on w is that
just because that thing is
00:35:35.944 --> 00:35:36.444
expanding?
00:35:36.444 --> 00:35:40.380
Or-- I'm kind of having
trouble understanding
00:35:40.380 --> 00:35:43.824
why it wouldn't be
just straight 0s.
00:35:43.824 --> 00:35:46.780
Like, why there's that
[INAUDIBLE] value [INAUDIBLE]?
00:35:46.780 --> 00:35:50.510
PROFESSOR: OK, the question
is why does the origin look
00:35:50.510 --> 00:35:53.410
like this as opposed
to just being all 0s.
00:35:53.410 --> 00:35:55.602
The answer is that all 0s
is not even in our space.
00:35:55.602 --> 00:35:57.560
Because, remember the
space we're interested in
00:35:57.560 --> 00:35:59.072
is the surface of the sphere.
00:35:59.072 --> 00:36:00.530
And the surface of
the sphere obeys
00:36:00.530 --> 00:36:03.370
x squared plus y squared
plus z squared plus w squared
00:36:03.370 --> 00:36:05.730
equals r squared.
00:36:05.730 --> 00:36:08.830
So if all the
coordinates were 0,
00:36:08.830 --> 00:36:10.970
it's not part of
our space at all.
00:36:10.970 --> 00:36:13.090
So we're using this
four-dimensional space,
00:36:13.090 --> 00:36:15.469
the embedding space,
to make things simple.
00:36:15.469 --> 00:36:17.010
But in the end,
we're only interested
00:36:17.010 --> 00:36:18.750
in the three-dimensional
surface.
00:36:18.750 --> 00:36:20.614
So the origin of our
coordinate system
00:36:20.614 --> 00:36:22.030
for that
three-dimensional surface
00:36:22.030 --> 00:36:24.360
had better be in the
three-dimensional surface.
00:36:24.360 --> 00:36:26.360
So of course, other choices
we could have made--
00:36:26.360 --> 00:36:28.568
we could have put it anywhere
we want on the surface.
00:36:28.568 --> 00:36:30.860
Choosing to put it where
w has its maximum value
00:36:30.860 --> 00:36:33.910
is just an arbitrary convention.
00:36:33.910 --> 00:36:34.410
Yes?
00:36:34.410 --> 00:36:36.860
AUDIENCE: So you were saying
that most metrics are not
00:36:36.860 --> 00:36:38.149
homogeneous?
00:36:38.149 --> 00:36:38.940
PROFESSOR: Oh yeah.
00:36:38.940 --> 00:36:39.440
Sure.
00:36:39.440 --> 00:36:41.370
Most metrics are
not homogeneous.
00:36:41.370 --> 00:36:44.640
Most objects are not around.
00:36:44.640 --> 00:36:46.140
AUDIENCE: When we're
doing this math
00:36:46.140 --> 00:36:47.726
to show that the
Robertson-Walker was
00:36:47.726 --> 00:36:50.570
homogeneous, it
didn't seem that we
00:36:50.570 --> 00:36:53.990
use the exact form of the
Robertson-Walker metric
00:36:53.990 --> 00:36:54.700
at all in it.
00:36:54.700 --> 00:36:56.797
We just said, it is
possible to do these--
00:36:56.797 --> 00:36:57.630
PROFESSOR: Well, no.
00:36:57.630 --> 00:37:00.690
We did use the form when
we made this rotation.
00:37:00.690 --> 00:37:03.400
We used the form that, in
the euclidean formulation,
00:37:03.400 --> 00:37:06.040
we knew that it was
rotationally invariant.
00:37:06.040 --> 00:37:08.890
And the rotational invariance
in the euclidean formulation
00:37:08.890 --> 00:37:12.970
is homogeneity and
guarantees homogeneity,
00:37:12.970 --> 00:37:14.180
but it's a special property.
00:37:14.180 --> 00:37:17.510
If it was ellipsoidal
shaped instead of spherical,
00:37:17.510 --> 00:37:20.830
when you rotated it, it
would not be invariant.
00:37:20.830 --> 00:37:23.450
If it had any bumps or
lumps when you rotated it,
00:37:23.450 --> 00:37:26.485
it would not be invariant.
00:37:26.485 --> 00:37:26.985
Yes, Aviv.
00:37:26.985 --> 00:37:29.615
AUDIENCE: I feel like there
should be a fourth step where
00:37:29.615 --> 00:37:33.920
you show that the metric doesn't
change forms [INAUDIBLE].
00:37:33.920 --> 00:37:35.160
PROFESSOR: Yeah.
00:37:35.160 --> 00:37:37.730
We do need to know the
metric does not change form,
00:37:37.730 --> 00:37:39.980
but I think we do
have a guaranteed--
00:37:39.980 --> 00:37:41.564
maybe we should've
said some words.
00:37:41.564 --> 00:37:43.730
I don't think it's really
a fourth step in the sense
00:37:43.730 --> 00:37:46.270
that I don't think it
requires anymore algebra.
00:37:46.270 --> 00:37:50.260
But the point is that we know
that the metric is invariant,
00:37:50.260 --> 00:37:52.780
that the four-dimensional
metric is
00:37:52.780 --> 00:37:55.430
invariant under this
rotation, which was really
00:37:55.430 --> 00:37:57.510
the only non-trivial step.
00:37:57.510 --> 00:38:00.420
And otherwise, besides
the rotation, all we did
00:38:00.420 --> 00:38:07.550
is we went from the
r theta phi variables
00:38:07.550 --> 00:38:09.150
to the utility
euclidean variables,
00:38:09.150 --> 00:38:11.233
and then we went back from
the euclidean variables
00:38:11.233 --> 00:38:13.070
to the r theta phi variables.
00:38:13.070 --> 00:38:16.470
But we already know how to
go from euclidean variables
00:38:16.470 --> 00:38:19.710
to r theta phi variables, and
it results in that metric.
00:38:22.216 --> 00:38:24.090
And it will still result
in that metric when,
00:38:24.090 --> 00:38:25.890
if a prime is on all
of the coordinates.
00:38:25.890 --> 00:38:28.645
AUDIENCE: So you don't
have to say, like,
00:38:28.645 --> 00:38:31.591
suppose we have a
[INAUDIBLE] byproduct.
00:38:31.591 --> 00:38:35.354
And do the mapping,
figure out what
00:38:35.354 --> 00:38:37.728
the ds squared is in
terms of r theta phi
00:38:37.728 --> 00:38:39.447
to show that it's the same?
00:38:39.447 --> 00:38:42.671
Do we have to do
that [INAUDIBLE]?
00:38:42.671 --> 00:38:44.170
PROFESSOR: OK, the
question is do we
00:38:44.170 --> 00:38:46.710
have to explicitly
show that ds squared is
00:38:46.710 --> 00:38:48.210
the same for the
new variables as it
00:38:48.210 --> 00:38:49.910
was for the old variables.
00:38:49.910 --> 00:38:52.480
We certainly want to be
convinced that that's true,
00:38:52.480 --> 00:38:55.592
and we certainly want to have
an argument which convinces us.
00:38:55.592 --> 00:38:57.050
But I would claim
that if you think
00:38:57.050 --> 00:38:59.330
about what underlies
these steps,
00:38:59.330 --> 00:39:02.740
I only gave a schematic
description of them.
00:39:02.740 --> 00:39:05.810
If you think about what
underlies these steps,
00:39:05.810 --> 00:39:09.990
I think it's implicit that
the form of the metric
00:39:09.990 --> 00:39:11.810
is what we had.
00:39:11.810 --> 00:39:15.360
The form of the metric that
we had was completely dictated
00:39:15.360 --> 00:39:17.540
by the transformation,
which expressed
00:39:17.540 --> 00:39:22.240
r theta and phi in
terms of x, y, z, and w.
00:39:22.240 --> 00:39:24.160
And as long as you know
the metric in x, y, z,
00:39:24.160 --> 00:39:27.870
and w, and that's the
euclidean metric both before
00:39:27.870 --> 00:39:31.790
and after our
rotation, then when you
00:39:31.790 --> 00:39:33.568
use the same equations
to go from x, y, z,
00:39:33.568 --> 00:39:38.270
w to r, theta, and phi, you'll
always get the same metric.
00:39:38.270 --> 00:39:41.340
So I think we are
guaranteed by this process
00:39:41.340 --> 00:39:44.170
to get a metric for
our new variables, r
00:39:44.170 --> 00:39:45.750
prime, theta prime,
and phi prime,
00:39:45.750 --> 00:39:49.950
which has exactly the same
form as the original metric.
00:39:49.950 --> 00:39:51.666
Because it's the same
calculation again.
00:39:51.666 --> 00:39:53.790
The only difference is that
this time the variables
00:39:53.790 --> 00:39:57.060
all have primes on them.
00:39:57.060 --> 00:39:59.560
And the crucial step, the
step that was nontrivial,
00:39:59.560 --> 00:40:02.580
is the fact that this rotation
did not change the metric.
00:40:02.580 --> 00:40:04.520
That's where the
homogeneity was built in,
00:40:04.520 --> 00:40:06.220
that we started with
a sphere that we
00:40:06.220 --> 00:40:08.800
knew was rotationally invariant.
00:40:08.800 --> 00:40:14.060
And this whole calculation
just extracts that homogeneity
00:40:14.060 --> 00:40:15.635
that we built in
from the beginning.
00:40:20.070 --> 00:40:20.570
Yes?
00:40:20.570 --> 00:40:24.800
AUDIENCE: For k negative,
it's r squared over k,
00:40:24.800 --> 00:40:26.540
so is r negative?
00:40:26.540 --> 00:40:27.350
PROFESSOR: No.
00:40:27.350 --> 00:40:32.690
r is still positive
when expressed
00:40:32.690 --> 00:40:35.800
in terms of x, y, z, and w.
00:40:35.800 --> 00:40:38.630
Let me think if I can show that.
00:40:42.650 --> 00:40:45.030
I'll show that next time.
00:40:45.030 --> 00:40:49.865
It's a little involved,
but it will be positive.
00:40:49.865 --> 00:40:54.620
I might add that if we
look at this formula
00:40:54.620 --> 00:40:58.070
and ask what's going
on, what's going on--
00:40:58.070 --> 00:41:01.280
we don't necessarily need to
know this-- but what's going on
00:41:01.280 --> 00:41:04.916
is that in going from the
closed case to the open case,
00:41:04.916 --> 00:41:07.540
k goes from positive to negative
and therefore square root of k
00:41:07.540 --> 00:41:09.670
becomes imaginary.
00:41:09.670 --> 00:41:12.780
But psi also becomes imaginary.
00:41:12.780 --> 00:41:15.920
To describe the relationship
between the closed metric
00:41:15.920 --> 00:41:18.740
and the open metric,
if you're using psi,
00:41:18.740 --> 00:41:20.750
you have to say that
for the closed metric,
00:41:20.750 --> 00:41:23.600
you'll use real values of
psi, and for the open metric,
00:41:23.600 --> 00:41:26.710
you'll use imaginary
values of psi.
00:41:26.710 --> 00:41:30.920
And that makes r real and
makes this formula work.
00:41:30.920 --> 00:41:33.330
And you could also then
see how this formula works.
00:41:33.330 --> 00:41:35.970
If psi is assigned
imaginary values,
00:41:35.970 --> 00:41:37.820
then deep psi
squared is negative,
00:41:37.820 --> 00:41:40.070
so this negative sign
cancels that negative sign,
00:41:40.070 --> 00:41:43.680
and you again get a
positive definite metric.
00:41:43.680 --> 00:41:44.180
Yes?
00:41:44.180 --> 00:41:47.474
AUDIENCE: So how can we choose
the imaginary [INAUDIBLE]
00:41:47.474 --> 00:41:48.140
of sine and psi.
00:41:48.140 --> 00:41:52.380
Is that just to reflect or
is that a choice that we're
00:41:52.380 --> 00:41:55.337
making for our model?
00:41:55.337 --> 00:41:55.920
PROFESSOR: OK.
00:41:55.920 --> 00:41:56.130
Yeah.
00:41:56.130 --> 00:41:57.860
The question is why
do we choose to use
00:41:57.860 --> 00:41:59.970
the imaginary value of psi.
00:41:59.970 --> 00:42:06.000
And the answer is
perhaps that I failed
00:42:06.000 --> 00:42:09.380
to state all the conditions
we're interested in when I said
00:42:09.380 --> 00:42:13.000
what properties we want
this metric to have.
00:42:13.000 --> 00:42:15.040
We want the metric
that we're seeking
00:42:15.040 --> 00:42:18.330
to be homogeneous and
isotropic, as we said.
00:42:18.330 --> 00:42:20.777
What we didn't say,
but what I kind of
00:42:20.777 --> 00:42:22.610
had in the back of my
mind as an assumption,
00:42:22.610 --> 00:42:26.150
is that the metric should
also be positive definite.
00:42:26.150 --> 00:42:28.200
You can construct
other metrics which
00:42:28.200 --> 00:42:30.860
are homogeneous and isotropic
but not positive definite.
00:42:30.860 --> 00:42:32.495
In fact, you would
if you let psi
00:42:32.495 --> 00:42:34.507
be real and let k be negative.
00:42:34.507 --> 00:42:36.840
You'd have a negative definite
metric, which would still
00:42:36.840 --> 00:42:39.670
be homogeneous and isotropic.
00:42:39.670 --> 00:42:41.840
So to enforce all
three properties,
00:42:41.840 --> 00:42:46.380
you have to use
some imagination.
00:42:46.380 --> 00:42:49.160
And the easiest way to do
it is to write the metric
00:42:49.160 --> 00:42:53.320
in the magical form where you
can just let k go to minus k,
00:42:53.320 --> 00:42:56.900
and that's the
Robertson-Walker form here.
00:42:56.900 --> 00:43:00.569
And if you write it this way,
when you let k go to minus k,
00:43:00.569 --> 00:43:02.110
it becomes negative
definite, and you
00:43:02.110 --> 00:43:03.194
have to scratch your head.
00:43:03.194 --> 00:43:05.110
And if you really clever,
you might say, well,
00:43:05.110 --> 00:43:06.580
if I assign
negative-- excuse me,
00:43:06.580 --> 00:43:08.410
if I assign imaginary
values to psi,
00:43:08.410 --> 00:43:11.250
it'll become positive
definite again.
00:43:11.250 --> 00:43:15.300
That works, but it's
less straightforward.
00:43:15.300 --> 00:43:15.800
Yes?
00:43:15.800 --> 00:43:17.800
AUDIENCE: Do we want it
to be positive definite
00:43:17.800 --> 00:43:18.800
so that it's visible?
00:43:18.800 --> 00:43:19.300
PROFESSOR: Exactly.
00:43:19.300 --> 00:43:21.424
We want it to be positive
definite so it's visible.
00:43:21.424 --> 00:43:24.250
AUDIENCE: Is there any way we
can have a negative definite
00:43:24.250 --> 00:43:27.066
for a multidimensional
space that's reflects
00:43:27.066 --> 00:43:30.180
a 3D positive definite space?
00:43:30.180 --> 00:43:32.430
PROFESSOR: Is there anyway
in higher dimensional space
00:43:32.430 --> 00:43:36.980
or something that we can have
it maybe have mixed signs
00:43:36.980 --> 00:43:38.004
or be negative.
00:43:38.004 --> 00:43:39.420
Well in fact, we
will see shortly,
00:43:39.420 --> 00:43:41.530
because we're going
to add in time,
00:43:41.530 --> 00:43:43.330
time will occur with
the opposite sign,
00:43:43.330 --> 00:43:45.710
and it will not be
positive definite anymore.
00:43:45.710 --> 00:43:48.060
But that will nonetheless
correspond to real physics.
00:43:48.060 --> 00:43:48.779
AUDIENCE: Right.
00:43:48.779 --> 00:43:53.179
So why do we have to force
that ds squared into spaces?
00:43:53.179 --> 00:43:55.470
PROFESSOR: Because now we
are talking only about space,
00:43:55.470 --> 00:43:59.320
and certainly for our
universe, space is positive.
00:43:59.320 --> 00:44:03.600
Now, I might add since
you brought this up,
00:44:03.600 --> 00:44:08.350
that in relativity, there's no
clear distinction between space
00:44:08.350 --> 00:44:11.190
and time, so you might
wonder why should I
00:44:11.190 --> 00:44:14.420
be saying that space is
positive and time is negative.
00:44:14.420 --> 00:44:17.050
And perhaps I'm
oversimplifying a bit when
00:44:17.050 --> 00:44:20.200
I say that space is positive
and time is negative.
00:44:20.200 --> 00:44:25.404
But what is a requirement
for general relativity
00:44:25.404 --> 00:44:27.820
to match our universe, and
therefore a requirement that we
00:44:27.820 --> 00:44:30.760
impose on the general
relativity theory,
00:44:30.760 --> 00:44:34.710
is that the metric have three
positive eigenvalues and one
00:44:34.710 --> 00:44:35.892
negative eigenvalue.
00:44:35.892 --> 00:44:37.600
And that's how it's
described, and that's
00:44:37.600 --> 00:44:39.830
called the signature of
the metric-- the number
00:44:39.830 --> 00:44:43.190
of positive eigenvalues and the
number of negative eigenvalues.
00:44:43.190 --> 00:44:46.690
And reality clearly has--
well, I shouldn't say clearly.
00:44:46.690 --> 00:44:48.670
String theory is
more dimensions--
00:44:48.670 --> 00:44:52.760
but to describe our
macroscopic world, clearly,
00:44:52.760 --> 00:44:57.070
we have quantities
that we intuitively
00:44:57.070 --> 00:44:59.020
identify as three space
dimensions in one time
00:44:59.020 --> 00:44:59.744
dimension.
00:44:59.744 --> 00:45:01.160
And the metric
that describes that
00:45:01.160 --> 00:45:03.915
is a metric whose signature
is three positive eigenvalues
00:45:03.915 --> 00:45:05.687
and one negative eigenvalue.
00:45:05.687 --> 00:45:07.770
Although, some people
reverse the sign conventions
00:45:07.770 --> 00:45:10.980
and say it's three
negative and 1 positive,
00:45:10.980 --> 00:45:12.339
which works just as well.
00:45:12.339 --> 00:45:14.130
You can define the
metric with either sign.
00:45:14.130 --> 00:45:15.694
But this is the one
that we're using,
00:45:15.694 --> 00:45:17.735
the one that corresponds
to space being positive.
00:45:20.201 --> 00:45:21.200
OK, any other questions?
00:45:24.740 --> 00:45:25.240
OK.
00:45:25.240 --> 00:45:28.686
In that case, let's move onward.
00:45:28.686 --> 00:45:30.338
What did I want to
talk about next?
00:45:35.330 --> 00:45:36.482
OK.
00:45:36.482 --> 00:45:37.940
I wanted to write
on the blackboard
00:45:37.940 --> 00:45:42.080
a statement which came about
earlier due to questioning
00:45:42.080 --> 00:45:44.260
but hasn't been written on
the blackboard yet, which
00:45:44.260 --> 00:45:48.330
is that there's a
theorem which says
00:45:48.330 --> 00:45:52.350
that the most general possible
three-dimensional metric, which
00:45:52.350 --> 00:45:56.475
is homogeneous and
isotropic, is this space.
00:46:01.740 --> 00:46:18.220
So any three-dimensional,
homogeneous, and isotropic
00:46:18.220 --> 00:46:31.480
space can be described by
the Robertson-Walker metric.
00:46:46.100 --> 00:46:50.375
We are not going to prove
this, but it is a theorem.
00:46:50.375 --> 00:46:52.750
If you want to see a proof,
there's a proof, for example,
00:46:52.750 --> 00:46:58.330
in Steve Weinberg's gravitation
and cosmology textbook.
00:46:58.330 --> 00:47:01.600
And in a lot of other
books, I'm sure.
00:47:01.600 --> 00:47:03.370
But we'll take it for granted.
00:47:03.370 --> 00:47:06.050
These are certainly the only
homogeneous and isotopic spaces
00:47:06.050 --> 00:47:07.670
that we know how to construct.
00:47:07.670 --> 00:47:10.100
And in fact, it's the
only ones that exist.
00:47:10.100 --> 00:47:13.490
Now, I emphasize
that if the space is
00:47:13.490 --> 00:47:15.780
homogeneous and
isotropic, obviously
00:47:15.780 --> 00:47:17.840
the metric does not
have to look exactly
00:47:17.840 --> 00:47:21.880
like that, because you can
choose different coordinates.
00:47:21.880 --> 00:47:24.330
We could take these coordinates
and make some arbitrary
00:47:24.330 --> 00:47:26.830
transformation and make the
metric look incredibly ugly.
00:47:26.830 --> 00:47:29.440
It would still be a homogeneous
and isotropic space.
00:47:29.440 --> 00:47:32.270
But the claim is that any
homogeneous and isotropic space
00:47:32.270 --> 00:47:34.530
can be written in metric
that looks exactly
00:47:34.530 --> 00:47:36.470
like this with a proper
choice of coordinates.
00:47:45.800 --> 00:47:46.430
OK.
00:47:46.430 --> 00:47:49.695
Next thing I want to discuss
is the size of these universes.
00:47:52.770 --> 00:47:55.740
Size in the generalized
sense of, actually
00:47:55.740 --> 00:47:57.850
the question of
whether it's positive--
00:47:57.850 --> 00:48:00.630
whether it's infinite or finite.
00:48:00.630 --> 00:48:04.030
Notice that our
closed universe, one
00:48:04.030 --> 00:48:08.340
can see from the embedding
in four euclidean dimensions,
00:48:08.340 --> 00:48:09.390
is finite.
00:48:09.390 --> 00:48:12.390
The surface of a
sphere is finite.
00:48:12.390 --> 00:48:17.207
And one can also see it
from the form of the metric,
00:48:17.207 --> 00:48:19.540
although one has to think a
little bit about how exactly
00:48:19.540 --> 00:48:20.910
it works.
00:48:20.910 --> 00:48:25.370
If we start at the origin
and let r get bigger,
00:48:25.370 --> 00:48:27.190
clearly something
funny's going to happen
00:48:27.190 --> 00:48:30.290
when r is equal to 1 over
the square root of k.
00:48:30.290 --> 00:48:32.320
That is, when kr
squared is equal to 1,
00:48:32.320 --> 00:48:34.720
this metric will
become singular.
00:48:34.720 --> 00:48:38.910
If one goes back to the angular
description in terms of psi,
00:48:38.910 --> 00:48:41.610
it's clearer what's
going on there.
00:48:41.610 --> 00:48:49.484
When kr squared is 1, that's
exactly where sine psi is one.
00:48:49.484 --> 00:48:50.900
And that just means
you've reached
00:48:50.900 --> 00:48:53.140
the equator of your sphere.
00:48:57.902 --> 00:48:58.485
Quick picture.
00:49:06.300 --> 00:49:09.530
Size measured from the w-axis.
00:49:09.530 --> 00:49:13.680
The equator corresponds
to sine psi equals 1.
00:49:13.680 --> 00:49:18.910
And then if you continue,
psi gets bigger up to pi,
00:49:18.910 --> 00:49:21.110
but r starts getting
smaller again.
00:49:21.110 --> 00:49:22.790
So r is double
valued in the sense
00:49:22.790 --> 00:49:26.240
that there are two latitudes
at which r has the same value--
00:49:26.240 --> 00:49:28.000
one of the northern
hemisphere and one
00:49:28.000 --> 00:49:29.124
in the southern hemisphere.
00:49:31.590 --> 00:49:35.030
But in any case, r starts from
0, goes to some maximum value,
00:49:35.030 --> 00:49:36.680
and then goes back to 0.
00:49:36.680 --> 00:49:38.982
Everything is finite.
00:49:38.982 --> 00:49:41.065
And one could integrate
and find the total volume,
00:49:41.065 --> 00:49:42.050
and it's finite.
00:49:42.050 --> 00:49:46.080
And I think it was or will
be a homework problem where
00:49:46.080 --> 00:49:49.220
you do exactly that integral
at the volume of a closed
00:49:49.220 --> 00:49:51.840
universe.
00:49:51.840 --> 00:49:56.720
On the other hand, if k--
let's first set it equal to 0.
00:49:56.720 --> 00:50:01.180
If k is 0, we have here just
the euclidean metric and polar
00:50:01.180 --> 00:50:04.640
coordinates
describing flat space.
00:50:04.640 --> 00:50:09.390
And there's no limit on r. r
can become as large as you want.
00:50:09.390 --> 00:50:14.440
You can hypothesize
that space somehow ends,
00:50:14.440 --> 00:50:19.520
but we don't believe-- we
don't know of any end to space.
00:50:19.520 --> 00:50:23.130
And there are-- in any case, a
precise way you could describe
00:50:23.130 --> 00:50:25.680
what it would mean for space
to end in general relativity
00:50:25.680 --> 00:50:29.000
and the usual postulates of
general relativity is that ,
00:50:29.000 --> 00:50:32.410
that doesn't happen-- that space
doesn't just have an arbitrary
00:50:32.410 --> 00:50:33.600
end.
00:50:33.600 --> 00:50:37.650
So the flat case is an infinite
space when k is equal to 0.
00:50:37.650 --> 00:50:40.260
It's just an infinite
euclidean space.
00:50:40.260 --> 00:50:44.810
Similarly, if k is
negative, then nothing funny
00:50:44.810 --> 00:50:47.770
happens as r increases.
00:50:47.770 --> 00:50:51.120
So there's no reason not to
let r increase to infinity.
00:50:51.120 --> 00:50:53.530
Anything else would just
be putting in an arbitrary
00:50:53.530 --> 00:50:56.010
wall into space
without any motivation
00:50:56.010 --> 00:50:58.980
for believing that
such a wall is there.
00:50:58.980 --> 00:51:03.320
And one has to remember that
r is not physical distance.
00:51:03.320 --> 00:51:05.040
So the fact that r
can go to infinity
00:51:05.040 --> 00:51:09.064
doesn't necessarily
make the space infinite,
00:51:09.064 --> 00:51:10.730
but you can calculate
physical distance.
00:51:25.140 --> 00:51:27.660
We can calculate the
physical distance
00:51:27.660 --> 00:51:31.139
from the origin to the
radius r, and we get that
00:51:31.139 --> 00:51:32.430
just by integrating the metric.
00:51:32.430 --> 00:51:34.610
The metric tells us what
the actual physical length
00:51:34.610 --> 00:51:36.990
is for an infinitesimal segment.
00:51:36.990 --> 00:51:40.140
That's what the metric
meant in the first place.
00:51:40.140 --> 00:51:45.840
So the integral that we'd be
doing of an a of t out front,
00:51:45.840 --> 00:51:51.670
and then we'll be integrating
dr prime over the square root
00:51:51.670 --> 00:52:00.290
of 1 minus kr prime
squared from 0 up to r.
00:52:00.290 --> 00:52:02.562
Remember k is negative.
00:52:02.562 --> 00:52:04.770
So this is a positive quantity
under the square root.
00:52:04.770 --> 00:52:08.760
It's not going to cause any
problems by vanishing on us.
00:52:08.760 --> 00:52:11.900
And this is an integral,
which is in fact, doable.
00:52:11.900 --> 00:52:16.260
And it's just equal
to, still the a
00:52:16.260 --> 00:52:19.780
of t in front, which I think
I left out in the notes,
00:52:19.780 --> 00:52:25.490
times an inverse hyperbolic
cinch of the square root
00:52:25.490 --> 00:52:34.992
of minus kr over
square root of minus k.
00:52:34.992 --> 00:52:35.950
Remember k is negative.
00:52:35.950 --> 00:52:39.210
See these are all square
roots of positive numbers.
00:52:39.210 --> 00:52:41.390
And that cinch function,
the inverse cinch
00:52:41.390 --> 00:52:43.260
can get to be as
large as one wants
00:52:43.260 --> 00:52:46.230
by letting r be as
large as one wants.
00:52:46.230 --> 00:52:47.697
It grows without bound.
00:52:47.697 --> 00:52:50.030
And that means the physical
distance grows without bound
00:52:50.030 --> 00:52:52.960
as r grows to infinity.
00:52:52.960 --> 00:52:54.750
And it grows faster
than linear, I think.
00:53:00.392 --> 00:53:01.100
I take that back.
00:53:01.100 --> 00:53:01.690
I'm not sure.
00:53:07.080 --> 00:53:07.580
Yes?
00:53:07.580 --> 00:53:09.080
AUDIENCE: I guess
I'm still confused
00:53:09.080 --> 00:53:12.400
because r is sine of psi
over square root of k,
00:53:12.400 --> 00:53:15.774
so sine psi is bounded
by 1 and negative 1,
00:53:15.774 --> 00:53:19.640
so if we're letting r
go to infinity, how--
00:53:19.640 --> 00:53:23.330
PROFESSOR: That formula only
works for the closed case.
00:53:23.330 --> 00:53:26.400
r equals sine psi over root k.
00:53:26.400 --> 00:53:31.870
We can apply it to the open case
if we let psi become imaginary.
00:53:31.870 --> 00:53:35.070
But then the bounds that
you said no longer apply.
00:53:37.790 --> 00:53:41.210
The sine of an
imaginary variable
00:53:41.210 --> 00:53:43.400
is, in fact, the cinch
of a real variable.
00:53:58.730 --> 00:54:00.100
OK?
00:54:00.100 --> 00:54:03.160
Next thing I want
to point out is
00:54:03.160 --> 00:54:11.150
that the
Gauss-Bolyai-Lobachevski
00:54:11.150 --> 00:54:15.180
geometry that you did a
homework problem about or it
00:54:15.180 --> 00:54:19.100
was an extra credit problem,
so some of you did not.
00:54:19.100 --> 00:54:21.335
But we talked about the
Gauss-Bolyai-Lobachevski
00:54:21.335 --> 00:54:21.835
geometry.
00:54:29.200 --> 00:54:37.050
That really is just an
open Robertson-Walker, RW
00:54:37.050 --> 00:54:44.725
Robertson-Walker metric,
but in two space dimensions.
00:54:53.220 --> 00:54:56.640
But it's completely analogous
to the Robertson-Walker metric
00:54:56.640 --> 00:55:00.390
in three space dimensions
that we're talking about here.
00:55:00.390 --> 00:55:03.030
So you might recall that
Felix Klein construction
00:55:03.030 --> 00:55:04.880
looked very complicated.
00:55:04.880 --> 00:55:07.210
That's because of the
coordinates that he used.
00:55:07.210 --> 00:55:09.990
Those coordinates might be
simple from some point of view,
00:55:09.990 --> 00:55:12.730
but from the point of view
of illustrating homogeneity,
00:55:12.730 --> 00:55:15.170
they're very
complicated coordinates.
00:55:15.170 --> 00:55:18.630
And anyway, to physicists,
the Robertson-Walker open
00:55:18.630 --> 00:55:21.660
coordinate system is familiar,
and the Felix Klein coordinate
00:55:21.660 --> 00:55:22.470
system is not.
00:55:28.550 --> 00:55:29.170
OK.
00:55:29.170 --> 00:55:30.890
If there are no
further questions
00:55:30.890 --> 00:55:34.830
about these spatial metrics, the
next thing I want to talk about
00:55:34.830 --> 00:55:36.780
is adding time to the picture.
00:55:36.780 --> 00:55:37.570
Because in the end,
we're going to be
00:55:37.570 --> 00:55:38.980
interested in a
spacetime metric,
00:55:38.980 --> 00:55:40.720
because that's what
general relativity is
00:55:40.720 --> 00:55:42.535
all about-- spacetime metrics.
00:56:26.970 --> 00:56:27.470
OK.
00:56:27.470 --> 00:56:30.720
Everything is going to
hinge on an important fact
00:56:30.720 --> 00:56:34.880
from special relativity, which
we are going to assume but not
00:56:34.880 --> 00:56:38.700
prove, because most
you have had courses
00:56:38.700 --> 00:56:40.490
about special
relativity elsewhere.
00:56:40.490 --> 00:56:43.800
And for those you who
have not, you can either,
00:56:43.800 --> 00:56:46.510
if you wish, read an
appendix to lecture
00:56:46.510 --> 00:56:49.540
notes five, in which the fact
that I'm about to show you
00:56:49.540 --> 00:56:52.130
is derived, or you
could just assume it.
00:56:52.130 --> 00:56:55.100
Whichever you prefer, depending
on how much time you have.
00:56:55.100 --> 00:56:58.470
This is not a course
about special relativity.
00:56:58.470 --> 00:57:02.250
You're not required
to learn how to derive
00:57:02.250 --> 00:57:03.950
the fact that I'm
about to write.
00:57:03.950 --> 00:57:06.350
And what I'm about
to write starts
00:57:06.350 --> 00:57:11.030
with a definition
given any two events.
00:57:11.030 --> 00:57:15.750
An event is a
point in spacetime.
00:57:15.750 --> 00:57:18.400
Snapping my finger-- this
clearly speaking event.
00:57:18.400 --> 00:57:21.550
It happens at a certain
place in a certain time.
00:57:21.550 --> 00:57:27.410
And every real events occupies
some small volume of spacetime.
00:57:27.410 --> 00:57:29.870
An ideal event is a
point in spacetime.
00:57:29.870 --> 00:57:31.910
And we'll be talking
about ideal events.
00:57:31.910 --> 00:57:36.190
Which, our model, as I said,
has points in spacetime.
00:57:36.190 --> 00:57:38.230
So given any two
events, one can talk
00:57:38.230 --> 00:57:40.800
about a separation
between those events.
00:57:40.800 --> 00:57:43.910
And they will be separated
in both space and time,
00:57:43.910 --> 00:57:46.330
although either one
of those could be 0.
00:57:46.330 --> 00:57:49.500
But they're not both 0,
or it's the same event.
00:57:49.500 --> 00:57:55.690
And it's possible to define
an interesting quantity, which
00:57:55.690 --> 00:57:59.220
is the difference between
the x-coordinates of the two
00:57:59.220 --> 00:58:00.970
events.
00:58:00.970 --> 00:58:03.500
This will be the separation
between two events, which I'll
00:58:03.500 --> 00:58:08.100
call a and b, and xa and
xb are the x-coordinates
00:58:08.100 --> 00:58:09.055
of those two events.
00:58:11.630 --> 00:58:15.460
And probably you all
have enough imagination
00:58:15.460 --> 00:58:20.016
to guess that ya and yb are
the y-coordinates of those two
00:58:20.016 --> 00:58:20.515
events.
00:58:24.301 --> 00:58:29.540
And za and zb are the
z-coordinates of those events.
00:58:29.540 --> 00:58:33.530
And now here's a surprising one,
if you haven't already seen it.
00:58:33.530 --> 00:58:36.030
We're going to have
minus c squared
00:58:36.030 --> 00:58:43.570
times ta minus tb squared.
00:58:43.570 --> 00:58:45.335
Now, this is all in
special relativity.
00:58:45.335 --> 00:58:47.070
I maybe should clarify.
00:58:47.070 --> 00:58:51.670
We haven't gotten to general
relativity or cosmology yet.
00:58:51.670 --> 00:58:53.140
But we need to
understand something
00:58:53.140 --> 00:58:54.595
from special relativity first.
00:58:57.330 --> 00:58:59.830
So in special relativity,
it's natural to define
00:58:59.830 --> 00:59:02.850
that interval
between two events.
00:59:02.850 --> 00:59:04.910
And the magical
property, which is
00:59:04.910 --> 00:59:07.510
why we define this integral
in the first place,
00:59:07.510 --> 00:59:12.710
is that if we had two
different inertial observers,
00:59:12.710 --> 00:59:16.370
we could calculate
how the coordinates
00:59:16.370 --> 00:59:18.050
as seen by one
observer are related
00:59:18.050 --> 00:59:20.170
to the coordinates as seen
by the other observer.
00:59:20.170 --> 00:59:23.700
And that's called the
Lorentz transformation.
00:59:23.700 --> 00:59:26.950
And one finds that this
particular quantity
00:59:26.950 --> 00:59:31.710
will have exactly the same
value to both observers always.
00:59:31.710 --> 00:59:36.350
The two observers will, in
general, find different values
00:59:36.350 --> 00:59:39.350
for every one of the
four quantities here.
00:59:39.350 --> 00:59:43.020
But when the four quantities
are added up with a minus sign
00:59:43.020 --> 00:59:48.750
in front of the time
term, the calculations
00:59:48.750 --> 00:59:53.330
would show that you get the same
value for inertial observers.
00:59:53.330 --> 00:59:55.640
So this quantity is
called Lorentz invariant,
00:59:55.640 --> 00:59:58.400
meaning it's invariant under
Lorentz transformations.
00:59:58.400 --> 01:00:00.000
And it makes it a
very important thing
01:00:00.000 --> 01:00:03.440
to talk about
because in the end,
01:00:03.440 --> 01:00:06.350
physical things have to
be essentially Lorentz
01:00:06.350 --> 01:00:09.550
invariant because the laws of
physics are Lorentz invariant.
01:00:09.550 --> 01:00:12.710
The laws of physics are the
same in all Lorentz frames,
01:00:12.710 --> 01:00:15.190
so ultimately they have to
involve quantities, which
01:00:15.190 --> 01:00:17.632
have some simple
relationship from one Lorentz
01:00:17.632 --> 01:00:18.740
frame to another.
01:00:22.900 --> 01:00:32.540
OK now, we'd still like to
have a clearer notion, I think,
01:00:32.540 --> 01:00:35.317
of what this quantity means.
01:00:35.317 --> 01:00:37.150
It's defined by that
equation and principle,
01:00:37.150 --> 01:00:39.610
but it would be nice if we
had some understanding of what
01:00:39.610 --> 01:00:40.340
it means.
01:00:40.340 --> 01:00:42.890
And I think the
easiest way to describe
01:00:42.890 --> 01:00:47.860
what it means is to
look at special frames,
01:00:47.860 --> 01:00:51.250
even though the important
feature of this quantity
01:00:51.250 --> 01:00:55.939
is that it has the same
numerical value in all frames.
01:00:55.939 --> 01:00:57.980
So the numerical value is
the same in all frames,
01:00:57.980 --> 01:01:00.730
but some frames make it
easier to interpret it.
01:01:00.730 --> 01:01:02.740
That's what I'm claiming.
01:01:02.740 --> 01:01:07.730
So, what that frame is depends
on the value of s squared,
01:01:07.730 --> 01:01:10.670
or at least the sine of it.
01:01:10.670 --> 01:01:16.910
For sab squared
greater than 0, which
01:01:16.910 --> 01:01:18.840
means it's dominated
by the spatial terms
01:01:18.840 --> 01:01:22.710
there, because those
are the positive ones.
01:01:22.710 --> 01:01:29.470
And therefore, the separation
is called spacelike.
01:01:36.650 --> 01:01:40.500
Some books put a hyphen between
space and like and some don't.
01:01:40.500 --> 01:01:41.045
I don't.
01:01:44.930 --> 01:01:50.270
And there's a theorem that
says that if the separation
01:01:50.270 --> 01:01:54.040
between two events is
spacelike, there always
01:01:54.040 --> 01:01:56.964
exists a Lorentz frame, an
inertial frame, in which
01:01:56.964 --> 01:01:58.505
the two events happen
simultaneously.
01:02:10.140 --> 01:02:12.700
Backwards E is a
There Exists symbol.
01:02:12.700 --> 01:02:26.940
Then there exists
an inertial frame
01:02:26.940 --> 01:02:30.385
in which a and b
are simultaneous.
01:02:41.000 --> 01:02:43.120
In that frame, we
could look at what
01:02:43.120 --> 01:02:46.000
that formula tells
us sab squared is.
01:02:46.000 --> 01:02:48.920
Since they're
simultaneous, ta equals tb,
01:02:48.920 --> 01:02:52.020
and therefore the last
term does not contribute.
01:02:52.020 --> 01:02:57.190
So in that frame, s squared
is just xa minus xb squared
01:02:57.190 --> 01:03:01.329
plus ya minus yb squared
plus za minus zb squared.
01:03:01.329 --> 01:03:02.370
And we know what that is.
01:03:02.370 --> 01:03:05.320
That's just the euclidean
length, the euclidean distance
01:03:05.320 --> 01:03:08.330
between the two points.
01:03:08.330 --> 01:03:20.130
So, in this frame,
sab squared is
01:03:20.130 --> 01:03:35.799
just equal to the distance
between events squared.
01:03:35.799 --> 01:03:37.590
Or you take the square
root because they're
01:03:37.590 --> 01:03:38.400
positive numbers.
01:03:38.400 --> 01:03:41.270
You could say sab is equal to
the distance between the two
01:03:41.270 --> 01:03:42.400
events.
01:03:42.400 --> 01:03:46.490
So when sab squared
is positive, sab
01:03:46.490 --> 01:03:48.740
is just the distance
between the two events
01:03:48.740 --> 01:03:50.531
in the frame in which
they're simultaneous.
01:04:05.160 --> 01:04:11.050
If sab squared is not positive,
it could be negative or 0.
01:04:11.050 --> 01:04:13.535
Let me go to the
negative case first.
01:04:18.840 --> 01:04:23.800
For sab squared less than
0, for it to be less than 0,
01:04:23.800 --> 01:04:26.280
it means that this expression
is dominated by the time term
01:04:26.280 --> 01:04:28.450
because that's
the negative term.
01:04:28.450 --> 01:04:30.700
And therefore the separation
is called timelike.
01:04:46.880 --> 01:04:49.350
And again, there's a theorem.
01:04:49.350 --> 01:04:52.240
The theorem says that if the
separation between two events
01:04:52.240 --> 01:04:54.275
is timelike, there exists
a frame in which it
01:04:54.275 --> 01:04:55.525
happened in the same location.
01:05:31.254 --> 01:05:32.712
If they're at the
same location, we
01:05:32.712 --> 01:05:35.380
could again look at
that equation that
01:05:35.380 --> 01:05:38.490
defines sab squared
and ask what form
01:05:38.490 --> 01:05:40.660
does it take in
this special frame
01:05:40.660 --> 01:05:42.840
where the two events
are the same location.
01:05:42.840 --> 01:05:44.590
That means the first
three terms are all 0
01:05:44.590 --> 01:05:46.620
because-- same location.
01:05:46.620 --> 01:05:49.950
It means in that frame,
sab squared is negative,
01:05:49.950 --> 01:05:52.190
and it's just minus c
squared times the time
01:05:52.190 --> 01:05:53.120
separation squared.
01:06:00.900 --> 01:06:15.330
So in that frame, sab
squared is equal to minus
01:06:15.330 --> 01:06:22.860
c squared tau ab
squared, where tau ab
01:06:22.860 --> 01:06:24.470
is just equal to
the time separation.
01:06:35.750 --> 01:06:39.030
So when sab squared is
negative, its meaning
01:06:39.030 --> 01:06:42.370
is as minus c squared times the
square of the time separation
01:06:42.370 --> 01:06:44.120
in the frame where the
two events happened
01:06:44.120 --> 01:06:46.770
at the same place.
01:06:46.770 --> 01:06:48.880
Now, this notion of the
two events happening
01:06:48.880 --> 01:06:54.620
at the same place has a
particularly simple intuition
01:06:54.620 --> 01:06:57.340
if the two events that
we're talking about happen
01:06:57.340 --> 01:07:02.780
on the same object, like two
flashes of the same strobe
01:07:02.780 --> 01:07:05.360
if that strobe is moving
at a constant velocity.
01:07:05.360 --> 01:07:06.960
Otherwise, all bets are off.
01:07:06.960 --> 01:07:09.150
But if that strobe is moving
at a constant velocity
01:07:09.150 --> 01:07:13.350
so that the frame of the
strobe is an inertial frame,
01:07:13.350 --> 01:07:15.530
then the frame of the
strobe is, in fact,
01:07:15.530 --> 01:07:17.190
the frame in which
the two events
01:07:17.190 --> 01:07:18.590
happened at the same place.
01:07:18.590 --> 01:07:23.100
They both happened at the bulb
of the strobe light, which
01:07:23.100 --> 01:07:27.030
in the frame of the
strobe is just one point.
01:07:27.030 --> 01:07:32.740
So this time interval, which
the sab squared measures,
01:07:32.740 --> 01:07:35.820
is simply the time interval as
measured by the object itself,
01:07:35.820 --> 01:07:38.750
is measured by an observer
following the strobe.
01:07:38.750 --> 01:07:42.780
And if we place the strobe by
a person with a wristwatch,
01:07:42.780 --> 01:07:45.560
this notion of time, which
is called proper time,
01:07:45.560 --> 01:07:48.490
is simply the time measured
by the person's wristwatch.
01:07:48.490 --> 01:07:51.007
A clock that follows the
object so that anything that
01:07:51.007 --> 01:07:53.215
happened to that object
happens at the same location.
01:08:03.790 --> 01:08:19.069
So if events happen
to the same object,
01:08:19.069 --> 01:08:32.675
ta ab is just the time interval
measured by that object.
01:08:52.109 --> 01:08:58.760
And as you give these things
names for the spacelike case,
01:08:58.760 --> 01:09:17.410
sab is often called the proper
distance between the events,
01:09:17.410 --> 01:09:25.379
and ta ab is the proper time
interval between the events.
01:09:39.960 --> 01:09:40.700
OK.
01:09:40.700 --> 01:09:45.069
One more case to do,
which is, if it's not
01:09:45.069 --> 01:09:48.859
positive or negative,
there's only one remaining
01:09:48.859 --> 01:09:51.359
choice, which is
it's got to be 0.
01:09:51.359 --> 01:09:55.340
If sab squared is 0,
then again looking back
01:09:55.340 --> 01:09:58.260
at the original definition, it
means that the spatial piece
01:09:58.260 --> 01:10:02.580
is equal to minus c squared
times the time piece
01:10:02.580 --> 01:10:04.750
so that they all cancel.
01:10:04.750 --> 01:10:07.030
If you think about it, that's
precisely the statement
01:10:07.030 --> 01:10:09.210
that these two
events are located
01:10:09.210 --> 01:10:12.270
in just the right situation so
the light beam that leaves one
01:10:12.270 --> 01:10:14.340
will just arrive at the other.
01:10:14.340 --> 01:10:18.510
Because it says that some
of the first three terms,
01:10:18.510 --> 01:10:19.910
which is the
distance squared, is
01:10:19.910 --> 01:10:22.145
equal to c squared times
the time interval squared.
01:10:22.145 --> 01:10:23.550
And that just says
that something
01:10:23.550 --> 01:10:24.800
travels at the speed of light.
01:10:24.800 --> 01:10:28.370
It could travel that distance
in that time and go from point a
01:10:28.370 --> 01:10:31.670
to point b or vice versa.
01:10:31.670 --> 01:10:33.010
Only one or the other, not both.
01:10:33.010 --> 01:10:35.800
But it's always
one or the other.
01:10:35.800 --> 01:10:41.920
So, for that reason, the
interval is called lightlike.
01:10:53.600 --> 01:11:06.010
And it means that a light
pulse can travel from a to b,
01:11:06.010 --> 01:11:07.974
or I could have
interchanged a or b.
01:11:07.974 --> 01:11:08.890
Everything is squared.
01:11:08.890 --> 01:11:10.265
It doesn't matter
which is which.
01:11:12.800 --> 01:11:14.560
Now, there's a
peculiar thing here.
01:11:14.560 --> 01:11:18.320
You would think that if a light
pulse can travel from a to b,
01:11:18.320 --> 01:11:20.870
there would still be
some relevant measure
01:11:20.870 --> 01:11:24.430
to how far apart a and b are.
01:11:24.430 --> 01:11:28.130
However, what we're
basically seeing here
01:11:28.130 --> 01:11:31.250
is that if a and b are
lightlike separated,
01:11:31.250 --> 01:11:34.050
in any given reference frame you
could talk about what the time
01:11:34.050 --> 01:11:36.450
interval is and that will be
equal to the space interval
01:11:36.450 --> 01:11:38.920
up to a factor of c.
01:11:38.920 --> 01:11:40.520
But if we imagine
looking at this
01:11:40.520 --> 01:11:43.420
at different frames,
different inertial frames,
01:11:43.420 --> 01:11:45.220
these two points
can get arbitrarily
01:11:45.220 --> 01:11:47.130
close together or
arbitrarily far apart,
01:11:47.130 --> 01:11:49.320
depending on what frame
we look at them in.
01:11:49.320 --> 01:11:51.990
There is no Lorentz
invariant measure
01:11:51.990 --> 01:11:54.230
of how far apart they look.
01:11:54.230 --> 01:11:56.707
The Lorentz invariant-- the
only Lorentz invariant measure
01:11:56.707 --> 01:11:58.290
simply tells us that
they're lightlike
01:11:58.290 --> 01:11:59.900
related to each other.
01:11:59.900 --> 01:12:01.800
And this leads to some
very peculiar issues
01:12:01.800 --> 01:12:05.450
when you try to prove rigorous
theorems about relativity.
01:12:05.450 --> 01:12:08.110
You can't really say whether
two lightlike points, two
01:12:08.110 --> 01:12:10.500
lightlike separated
points are close or far.
01:12:10.500 --> 01:12:14.180
Because there's no real meaning
for them to be close or far.
01:12:22.181 --> 01:12:22.680
OK.
01:12:22.680 --> 01:12:25.180
Let me just say one more fact
about special relativity,
01:12:25.180 --> 01:12:28.850
and then we'll quit for today
and come back on Thursday
01:12:28.850 --> 01:12:31.320
and then talk about
how to extend this
01:12:31.320 --> 01:12:32.492
into general relativity.
01:12:32.492 --> 01:12:33.450
OK, there's a question.
01:12:33.450 --> 01:12:33.820
Yes?
01:12:33.820 --> 01:12:34.986
AUDIENCE: Just really quick.
01:12:34.986 --> 01:12:37.271
For the Lorentz
invariant to equal to 0,
01:12:37.271 --> 01:12:41.215
does that mean that
the objects should
01:12:41.215 --> 01:12:44.666
be moving at the speed of
light relative to each other?
01:12:44.666 --> 01:12:46.373
Is it like that?
01:12:46.373 --> 01:12:46.956
PROFESSOR: OK.
01:12:46.956 --> 01:12:49.880
The question is, if the
separation is lightlike,
01:12:49.880 --> 01:12:50.750
s squared is 0.
01:12:50.750 --> 01:12:52.309
Does that mean that
these two objects
01:12:52.309 --> 01:12:54.600
are moving at the speed of
light relative to each other
01:12:54.600 --> 01:12:56.270
or something like that?
01:12:56.270 --> 01:12:57.440
No, it does not.
01:12:57.440 --> 01:12:59.380
It only talks about
their positions.
01:12:59.380 --> 01:13:03.530
It doesn't say anything about
the motion of these objects.
01:13:03.530 --> 01:13:06.340
It's only a statement about
their x and t-coordinates
01:13:06.340 --> 01:13:07.140
at some instant.
01:13:09.850 --> 01:13:11.930
OK, let me still write
one more equation
01:13:11.930 --> 01:13:14.867
on the blackboard to kind of
finish the special relativity
01:13:14.867 --> 01:13:15.825
part of the discussion.
01:13:20.250 --> 01:13:23.190
In the end, we are
interested in the metric.
01:13:23.190 --> 01:13:26.260
And what makes a
metric a little bit
01:13:26.260 --> 01:13:28.650
different from a
distance function
01:13:28.650 --> 01:13:31.630
is that metrics refer to
infinitesimal distances.
01:13:31.630 --> 01:13:34.660
So we're going to want to know
the infinitesimal form of that.
01:13:34.660 --> 01:13:36.640
And it's obvious,
so it's nothing
01:13:36.640 --> 01:13:37.809
to make a big deal about.
01:13:37.809 --> 01:13:39.850
But I think it's worth
writing on the blackboard.
01:13:39.850 --> 01:13:42.080
The infinitesimal
form of that equation
01:13:42.080 --> 01:13:47.590
is that ds squared is equal
to dx squared plus dy squared
01:13:47.590 --> 01:13:54.110
plus dz squared minus c squared
dt squared where dxdy, dz
01:13:54.110 --> 01:13:56.600
and dt are the
infinitesimal coordinate
01:13:56.600 --> 01:14:00.420
differences between two events.
01:14:00.420 --> 01:14:05.150
And it's in that form that we'll
be beginning from and taking
01:14:05.150 --> 01:14:08.510
off into the world
of general relativity
01:14:08.510 --> 01:14:13.200
and the metric of general
relativistic spacetimes.
01:14:13.200 --> 01:14:15.610
So we'll continue
with this on Thursday.