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PROFESSOR: OK.
00:00:22.420 --> 00:00:23.769
In that case let's get going.
00:00:23.769 --> 00:00:25.310
In today's lecture,
we're going to be
00:00:25.310 --> 00:00:27.320
sort of splitting the
lecture of things,
00:00:27.320 --> 00:00:29.910
if the timing goes as I plan.
00:00:29.910 --> 00:00:31.880
We're going to
start by finishing
00:00:31.880 --> 00:00:34.320
talking about the
geodesic equation.
00:00:34.320 --> 00:00:36.070
And then if all
goes well, we will
00:00:36.070 --> 00:00:39.090
start talking about the
energy of radiation--
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completely changing
topics altogether.
00:00:42.420 --> 00:00:45.890
I want to begin, as usual,
by reviewing quickly
00:00:45.890 --> 00:00:49.960
what we talked about last time,
just to remind us where we are.
00:00:49.960 --> 00:00:53.630
Last time, we at first, at
the beginning of lecture,
00:00:53.630 --> 00:00:56.910
talked about how to add time
into the Robertson-Walker
00:00:56.910 --> 00:00:57.770
Metric.
00:00:57.770 --> 00:01:02.990
And this is the formula that
we claimed was the correct one.
00:01:02.990 --> 00:01:06.850
For a spacetime
metric, ds squared,
00:01:06.850 --> 00:01:09.930
the meaning is closely
analogous to the meaning
00:01:09.930 --> 00:01:12.120
that it would have in
special relativity.
00:01:12.120 --> 00:01:15.320
The main difference being
that in special relativity
00:01:15.320 --> 00:01:17.550
we always talk about
what is observed
00:01:17.550 --> 00:01:22.460
by inertial frames of reference
and inertial observers.
00:01:22.460 --> 00:01:26.130
In general relativity, the
concept of an inertial observer
00:01:26.130 --> 00:01:28.480
is not so clear
cut, but we can talk
00:01:28.480 --> 00:01:31.630
about observers for whom there
is no forces acting on them
00:01:31.630 --> 00:01:34.580
other than possibly
gravitational forces.
00:01:34.580 --> 00:01:36.690
And whether or not there
are gravitational forces
00:01:36.690 --> 00:01:39.630
is always, itself, a
framed dependent question.
00:01:39.630 --> 00:01:42.380
So it does not have
a definite answer.
00:01:42.380 --> 00:01:45.040
So observers for which there
is no forces acting on them
00:01:45.040 --> 00:01:47.500
other than
gravitational forces are
00:01:47.500 --> 00:01:49.600
called free-falling observers.
00:01:49.600 --> 00:01:51.740
And they play the role
of inertial observers
00:01:51.740 --> 00:01:56.890
that the inertial observers
play in special relativity.
00:01:56.890 --> 00:01:58.980
So if ds squared
is positive, it's
00:01:58.980 --> 00:02:01.740
the square of the spatial
separation measured
00:02:01.740 --> 00:02:04.800
by a local free-falling
observer, for whom the two
00:02:04.800 --> 00:02:07.350
events happen at the same time.
00:02:07.350 --> 00:02:08.850
Last time, I think,
I did not really
00:02:08.850 --> 00:02:11.480
mention or emphasize
the word local.
00:02:11.480 --> 00:02:15.140
But the point is that
in general relativity
00:02:15.140 --> 00:02:17.235
we expect in any
small region one
00:02:17.235 --> 00:02:20.190
can construct an accelerating
coordinate system in which
00:02:20.190 --> 00:02:22.250
the effects of gravity
are canceled out,
00:02:22.250 --> 00:02:24.690
as the equivalence principle
tells us we can do.
00:02:24.690 --> 00:02:27.540
And then you essentially see the
effects of special relativity.
00:02:27.540 --> 00:02:29.490
But it's only a small
region, in principle,
00:02:29.490 --> 00:02:31.860
on an infinitesimal region.
00:02:31.860 --> 00:02:34.460
So these measurements
that correspond
00:02:34.460 --> 00:02:36.360
to special relativity
measurements
00:02:36.360 --> 00:02:39.510
are always made locally
by an observer who
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is, in principle, arbitrarily
close to the events being
00:02:42.210 --> 00:02:44.360
measured.
00:02:44.360 --> 00:02:47.080
If ds squared is negative,
then it's equal to minus c
00:02:47.080 --> 00:02:50.160
squared times the square
of the time separation
00:02:50.160 --> 00:02:53.330
that would be measured by a
local free-falling observer
00:02:53.330 --> 00:02:56.440
for whom the two events
happen at the same location.
00:02:56.440 --> 00:02:58.990
I should point out that
a special case of this
00:02:58.990 --> 00:03:01.190
is an observer looking
at his own wristwatch.
00:03:01.190 --> 00:03:04.040
His own wristwatch is
always at the same location
00:03:04.040 --> 00:03:08.630
relevant to him, so it's a
special case of this statement.
00:03:08.630 --> 00:03:12.050
So it says that ds squared is
equal to minus c squared times
00:03:12.050 --> 00:03:16.430
the time that a
free-falling observer would
00:03:16.430 --> 00:03:18.752
read on his own wrist watch.
00:03:18.752 --> 00:03:21.290
And If ds squared is 0, it
means that the two events
00:03:21.290 --> 00:03:25.220
can be joined by a light pulse
going from one to the other.
00:03:25.220 --> 00:03:27.470
Having said this, we can
go back to this formula
00:03:27.470 --> 00:03:31.140
and understand why the
formula is what it is.
00:03:31.140 --> 00:03:33.830
The spatial part is
what it is because
00:03:33.830 --> 00:03:37.230
any homogeneous and
isotropic spatial metric can
00:03:37.230 --> 00:03:39.010
be written in this form.
00:03:39.010 --> 00:03:41.950
And we are assuming that the
universe we're describing
00:03:41.950 --> 00:03:44.620
is homogeneous and isotropic.
00:03:44.620 --> 00:03:49.260
The dc squared piece is really
dictated by item two here.
00:03:49.260 --> 00:03:51.940
We want the t that we
write in this metric
00:03:51.940 --> 00:03:55.510
to be the cosmic time variable
that we've been speaking about.
00:03:55.510 --> 00:03:57.970
And that means that it is
the time variable measured
00:03:57.970 --> 00:04:00.050
on the watches of
observers who are
00:04:00.050 --> 00:04:02.420
at rest in this
coordinate system.
00:04:02.420 --> 00:04:04.620
And that means that
it has to be simply
00:04:04.620 --> 00:04:05.990
minus c squared dt squared.
00:04:05.990 --> 00:04:08.800
Or else dt would not have
the right relationship
00:04:08.800 --> 00:04:11.760
to a ds squared to be
consistent with what
00:04:11.760 --> 00:04:14.460
the s squared is supposed to be.
00:04:14.460 --> 00:04:16.230
And then we also
talked about why
00:04:16.230 --> 00:04:24.510
there are no dt dr terms,
or dt d theta, or dt d phi.
00:04:24.510 --> 00:04:28.350
We said that any such term
would violate isotropy.
00:04:28.350 --> 00:04:31.160
If you had a dt dr
term, for example,
00:04:31.160 --> 00:04:33.090
it would make the
positive dr direction
00:04:33.090 --> 00:04:35.330
different from the
negative dr direction.
00:04:35.330 --> 00:04:38.250
And that can't be
something that happens
00:04:38.250 --> 00:04:39.530
in an isotropic universe.
00:04:42.290 --> 00:04:45.870
That then is our
metric for cosmology,
00:04:45.870 --> 00:04:48.654
the Roberrtson-Walker Metric.
00:04:48.654 --> 00:04:50.945
And another important thing
is what is it good for, now
00:04:50.945 --> 00:04:53.070
that we decided that's
the right metric?
00:04:53.070 --> 00:04:55.150
What use is to us?
00:04:55.150 --> 00:04:57.210
And what we haven't done
yet, but it's actually
00:04:57.210 --> 00:05:00.120
on the homework, we need
the full spacetime metric
00:05:00.120 --> 00:05:02.180
to be able to find
geodesics, to be
00:05:02.180 --> 00:05:04.530
able to learn the
paths of particles
00:05:04.530 --> 00:05:08.570
moving through this
model universe.
00:05:08.570 --> 00:05:12.200
So we will be making important
use of this Roberrtson-Walker
00:05:12.200 --> 00:05:15.725
Metric with its
spacetime contributions.
00:05:18.011 --> 00:05:18.510
OK.
00:05:18.510 --> 00:05:19.530
Any questions about that?
00:05:19.530 --> 00:05:21.405
Now I'm ready to change
gears to some extent.
00:05:21.405 --> 00:05:23.280
Yes, Ani?
00:05:23.280 --> 00:05:26.140
AUDIANCE: So in general, the
spatial part of the metric,
00:05:26.140 --> 00:05:27.850
we can get from the geometry?
00:05:27.850 --> 00:05:29.790
And in general, can
you just add a minus
00:05:29.790 --> 00:05:32.464
c squared dt square
for the temporal part?
00:05:32.464 --> 00:05:33.880
PROFESSOR: It's
not quite general.
00:05:33.880 --> 00:05:37.020
Remember we used an argument
based on isotropy here.
00:05:39.602 --> 00:05:42.900
So I think it's safe to
say that any metric you'll
00:05:42.900 --> 00:05:46.104
find in this class
is likely to have
00:05:46.104 --> 00:05:48.520
the time entering, and nothing
more complicated than minus
00:05:48.520 --> 00:05:50.110
c squared dt squared.
00:05:50.110 --> 00:05:52.820
But it's not a general statement
about general relativity.
00:05:57.042 --> 00:05:57.875
Any other questions?
00:06:02.089 --> 00:06:02.589
Yes.
00:06:02.589 --> 00:06:11.970
AUDIANCE: [INAUDIBLE],
00:06:11.970 --> 00:06:14.520
PROFESSOR: OK, the question is,
what would be a circumstance
00:06:14.520 --> 00:06:17.470
where we would have to deal
with something more complicated?
00:06:17.470 --> 00:06:19.000
The answer would
be, I think, all
00:06:19.000 --> 00:06:26.220
you need is to add to this model
universe perturbations that
00:06:26.220 --> 00:06:27.910
break the uniformity.
00:06:27.910 --> 00:06:30.080
If we tried to describe
the real universe instead
00:06:30.080 --> 00:06:33.350
of this ideal universe, where
our ideal universe is perfectly
00:06:33.350 --> 00:06:35.580
isotropic and
homogeneous, if it said
00:06:35.580 --> 00:06:38.910
we wanted to describe the lumps
and bumps of the real universe,
00:06:38.910 --> 00:06:40.710
then it would become
more complicated.
00:06:40.710 --> 00:06:42.930
And we would probably
need a dt, dr term.
00:06:51.110 --> 00:06:55.090
OK, next we went on to talking
about the geodesic equation.
00:06:55.090 --> 00:06:59.290
According to General Relativity,
the trajectories of particles
00:06:59.290 --> 00:07:02.220
that have no forces acting
on them other than gravity,
00:07:02.220 --> 00:07:05.870
these free-falling
observers, are geodesics
00:07:05.870 --> 00:07:08.250
in the spacetime..
00:07:08.250 --> 00:07:11.260
So that means we want to learn
how to calculate geodesics,
00:07:11.260 --> 00:07:14.280
which means paths whose
length is stationary
00:07:14.280 --> 00:07:17.160
under small variations.
00:07:17.160 --> 00:07:19.370
So we considered first
just simple geodesics
00:07:19.370 --> 00:07:22.380
in the spatial metric, because
that's easier to think about.
00:07:22.380 --> 00:07:24.780
What is the shortest
distance between two points
00:07:24.780 --> 00:07:28.500
in a space that's described
by some arbitrary metric?
00:07:28.500 --> 00:07:30.780
So first we talked about
how we describe the metric.
00:07:30.780 --> 00:07:34.940
And we introduced two features
in this first formula here.
00:07:34.940 --> 00:07:37.520
One is that instead of
calling the coordinates XYZ
00:07:37.520 --> 00:07:38.750
or something like that.
00:07:38.750 --> 00:07:42.190
We called them x1, x2, x3, so
that we could talk about them
00:07:42.190 --> 00:07:45.890
all together in one formula
without writing separate pieces
00:07:45.890 --> 00:07:48.640
for the different coordinates.
00:07:48.640 --> 00:07:51.570
So i and j represent
1, 2, and 3, or just 1,
00:07:51.570 --> 00:07:56.760
and 2, which is the labeling
of the spatial coordinates.
00:07:56.760 --> 00:07:59.830
And the other important
piece of notation
00:07:59.830 --> 00:08:01.820
that is introduced
in that formula
00:08:01.820 --> 00:08:04.930
is the Einstein
summation convention.
00:08:04.930 --> 00:08:08.680
Whenever there's an
index, like i and j here,
00:08:08.680 --> 00:08:11.340
which are repeated
with one index lower
00:08:11.340 --> 00:08:13.590
and one index upper,
they're automatically
00:08:13.590 --> 00:08:17.670
summed over all of the values
that the coordinates take,
00:08:17.670 --> 00:08:20.565
without writing summation sign.
00:08:20.565 --> 00:08:22.230
It saves a lot of writing.
00:08:22.230 --> 00:08:25.930
And it turns out that one always
sums under those circumstances,
00:08:25.930 --> 00:08:30.740
so there's no need to write the
sums with the summation sign.
00:08:30.740 --> 00:08:34.349
Next we want to
ask ourselves, how
00:08:34.349 --> 00:08:36.470
are we going to
describe the path?
00:08:36.470 --> 00:08:37.970
Before we can find
the minimum path,
00:08:37.970 --> 00:08:41.250
we need at least a language
to talk about paths.
00:08:41.250 --> 00:08:43.600
And we could describe a
path going from some point A
00:08:43.600 --> 00:08:48.960
to some point B, by giving a
function x supra i of lambda.
00:08:48.960 --> 00:08:50.890
Well, lambda is an
arbitrary parameter,
00:08:50.890 --> 00:08:53.700
that parametrizes the path.
00:08:53.700 --> 00:08:55.910
x supra i are a
set of coordinates.
00:08:55.910 --> 00:08:58.540
i runs over the values
of all the coordinates
00:08:58.540 --> 00:09:01.430
of whatever system
you're dealing with.
00:09:01.430 --> 00:09:05.290
And you construct such
a function where xi of 0
00:09:05.290 --> 00:09:09.330
is the starting point, which are
the coordinates of the point A.
00:09:09.330 --> 00:09:14.930
And xi of some value lambda f,
where f just stands for final,
00:09:14.930 --> 00:09:16.406
will be the end of the path.
00:09:16.406 --> 00:09:17.780
And it's supposed
to end at point
00:09:17.780 --> 00:09:20.520
B. So the final
coordinates of the path
00:09:20.520 --> 00:09:28.170
should be x supra i sub b, the
coordinates of the point B.
00:09:28.170 --> 00:09:31.310
Then we want to use this
description of the path
00:09:31.310 --> 00:09:33.850
to figure out what the length
is of a segment of the path.
00:09:33.850 --> 00:09:36.820
And then the full length will
be the sum of the segments.
00:09:36.820 --> 00:09:40.060
So for each segment, we
just apply the metric
00:09:40.060 --> 00:09:42.890
to the change in coordinates.
00:09:42.890 --> 00:09:45.680
The change in coordinates,
as lambda is varied,
00:09:45.680 --> 00:09:49.690
is just the derivative of
xi with respect to lambda
00:09:49.690 --> 00:09:52.740
times the change in lambda.
00:09:52.740 --> 00:09:57.810
And putting that in
for both dxi and dxj
00:09:57.810 --> 00:10:00.870
one gets this
formula, relating ds
00:10:00.870 --> 00:10:02.800
squared, the square
of the length
00:10:02.800 --> 00:10:06.370
of an infinitesimal segment
to d lambda squared,
00:10:06.370 --> 00:10:09.338
the square of the parameter
that describes that length.
00:10:12.090 --> 00:10:15.090
Then the full length is gotten
by, first of all, taking
00:10:15.090 --> 00:10:16.720
the square root of
this equation to get
00:10:16.720 --> 00:10:19.580
the infinitesimal length, ds.
00:10:19.580 --> 00:10:22.840
And then taking the integral
of that over the path
00:10:22.840 --> 00:10:24.400
from beginning to end.
00:10:24.400 --> 00:10:27.090
And that, then, gives us
the full length of the path,
00:10:27.090 --> 00:10:29.760
thinking of it as the
sum of the length of each
00:10:29.760 --> 00:10:32.220
of infinitesimal segment.
00:10:32.220 --> 00:10:33.080
OK?
00:10:33.080 --> 00:10:36.180
Fair enough?
00:10:36.180 --> 00:10:38.390
Now that we have this
formula for the length, now
00:10:38.390 --> 00:10:40.250
we have the next
challenge, which
00:10:40.250 --> 00:10:43.750
is to figure out how to
calculate the path which
00:10:43.750 --> 00:10:45.950
minimizes that length.
00:10:45.950 --> 00:10:47.660
And I didn't use
the word last time,
00:10:47.660 --> 00:10:50.310
but that what is called
the calculus of variations.
00:10:50.310 --> 00:10:53.680
And I looked up a little bit of
the history in the Wikipedia.
00:10:53.680 --> 00:10:57.190
The calculus of variations
dates back to 1696,
00:10:57.190 --> 00:10:59.720
when Johann Bernoulli
invented it,
00:10:59.720 --> 00:11:03.840
applied it to the
brachistochrone problem,
00:11:03.840 --> 00:11:08.120
which is the problem of finding
a path for which a frictionless
00:11:08.120 --> 00:11:11.290
object will slide and get to
its destination in the least
00:11:11.290 --> 00:11:12.760
possible time.
00:11:12.760 --> 00:11:14.460
And it turns out
to be a cycloid,
00:11:14.460 --> 00:11:17.590
just like the cycloid that
describes our closed universes,
00:11:17.590 --> 00:11:21.160
closed matter
dominating the universe.
00:11:21.160 --> 00:11:25.390
And the problem was also
solved by-- Johann Bernoulli
00:11:25.390 --> 00:11:28.060
then announced this
problem to the world
00:11:28.060 --> 00:11:31.200
and challenged other
mathematicians to solve it.
00:11:31.200 --> 00:11:37.400
There's a famous story that
Newton noticed this question
00:11:37.400 --> 00:11:39.395
in his mail when he
got home at 4:00 AM,
00:11:39.395 --> 00:11:41.770
or something like that, from
the mint-- he was apparently
00:11:41.770 --> 00:11:45.899
a hardworking guy--
but nonetheless when
00:11:45.899 --> 00:11:47.690
he seen this problem
he couldn't go to bed.
00:11:47.690 --> 00:11:51.780
He went ahead and
solved it by morning,
00:11:51.780 --> 00:11:55.340
which is a good MIT student
kind of thing to do.
00:11:58.710 --> 00:12:03.400
So the technique is to consider
a small variation from whatever
00:12:03.400 --> 00:12:06.460
path you're hoping
to be the minimum.
00:12:06.460 --> 00:12:08.670
And we're going to
calculate the first order
00:12:08.670 --> 00:12:11.340
change in the
length of the path,
00:12:11.340 --> 00:12:14.170
starting from our original
path, x of lambda,
00:12:14.170 --> 00:12:18.230
to some new path,
x tilde of lambda.
00:12:18.230 --> 00:12:23.200
And we parametrize the
new path by writing it
00:12:23.200 --> 00:12:26.360
as the old path,
plus a correction.
00:12:26.360 --> 00:12:28.280
And I've introduced
a factor, alpha,
00:12:28.280 --> 00:12:30.530
multiplying the correction,
because it makes it easier
00:12:30.530 --> 00:12:32.910
to talk about derivatives.
00:12:32.910 --> 00:12:36.950
And wi of lambda is just
some arbitrary deviation
00:12:36.950 --> 00:12:39.577
from the original path.
00:12:39.577 --> 00:12:41.910
But we want to always go
through the same starting point
00:12:41.910 --> 00:12:43.701
to the same endpoint,
because there's never
00:12:43.701 --> 00:12:46.520
going to be a minimum if we're
allowed to move the endpoints.
00:12:46.520 --> 00:12:48.010
So the endpoints are fixed.
00:12:48.010 --> 00:12:50.230
And that means that
this path deviation,
00:12:50.230 --> 00:12:54.640
w super i in my notation, has
to vanish at the two endpoints.
00:12:54.640 --> 00:13:00.750
So we impose these two
equations on the variation wi.
00:13:00.750 --> 00:13:04.930
Then what do I do is take
the derivative of the path
00:13:04.930 --> 00:13:09.730
length of the varied path, x
tilde with respect to alpha,
00:13:09.730 --> 00:13:13.090
and if we had a minimum
length to start with,
00:13:13.090 --> 00:13:15.560
the derivative
should always vanish.
00:13:15.560 --> 00:13:17.250
That is, the minimum
should always
00:13:17.250 --> 00:13:20.110
occur when alpha equals
0, if the original path
00:13:20.110 --> 00:13:23.240
of the true path, the
true minimum path.
00:13:23.240 --> 00:13:26.150
And if alpha equals
0 is the minimum,
00:13:26.150 --> 00:13:29.270
the derivative should always
vanish at alpha equals 0.
00:13:29.270 --> 00:13:30.370
And vice versa.
00:13:30.370 --> 00:13:34.270
If we know that this happens
for every variation wi,
00:13:34.270 --> 00:13:36.940
then we know that our path
is at least an extremum,
00:13:36.940 --> 00:13:40.590
and, presumably, a minimum.
00:13:40.590 --> 00:13:43.800
And the path itself is just
written by the same formulas
00:13:43.800 --> 00:13:49.070
we had before, except for x
tilde instead of x itself.
00:13:49.070 --> 00:13:52.800
And I've introduced an axillary
quantity, a of lambda alpha,
00:13:52.800 --> 00:13:55.404
which is just what appears
inside the square root.
00:13:55.404 --> 00:13:57.070
That just saves some
writing, because it
00:13:57.070 --> 00:13:58.540
has to be written
a number of times
00:13:58.540 --> 00:13:59.998
in the course of
the manipulations.
00:14:02.970 --> 00:14:06.100
So our goal now is to
carry out this derivative.
00:14:06.100 --> 00:14:08.430
And the derivative acts
only on the integrand,
00:14:08.430 --> 00:14:09.920
because the limits
of integration
00:14:09.920 --> 00:14:11.980
do not depend on alpha.
00:14:11.980 --> 00:14:15.410
So just carry the derivative
into the integrand
00:14:15.410 --> 00:14:17.350
and differentiate
this square root
00:14:17.350 --> 00:14:20.340
of a of lambda, which is,
itself, a product of factors
00:14:20.340 --> 00:14:22.360
that we have to use--
product rule and chain
00:14:22.360 --> 00:14:25.030
rule and various manipulations.
00:14:25.030 --> 00:14:28.040
And after we carry out
those manipulations,
00:14:28.040 --> 00:14:31.680
we end up with this expression
in a straightforward
00:14:31.680 --> 00:14:36.980
way involving a few steps,
which I won't show again.
00:14:36.980 --> 00:14:39.910
And the complication is
that what we want to do
00:14:39.910 --> 00:14:43.760
is to figure out for what
paths that expression
00:14:43.760 --> 00:14:45.760
will vanish for all wi.
00:14:45.760 --> 00:14:48.580
We want it to vanish for
all possible variations
00:14:48.580 --> 00:14:50.000
of the path.
00:14:50.000 --> 00:14:55.190
And what's complicated
is that wi appears here
00:14:55.190 --> 00:14:58.490
as a multiplicative
factor in the first term,
00:14:58.490 --> 00:15:03.010
but as a differentiated
factor in the second term.
00:15:03.010 --> 00:15:05.860
And that makes it very
hard to know, initially,
00:15:05.860 --> 00:15:08.400
when those two terms might
cancel each other to give you
00:15:08.400 --> 00:15:11.220
0, which is what
we're looking for.
00:15:11.220 --> 00:15:13.020
But the brilliant
trick that, I guess,
00:15:13.020 --> 00:15:16.780
Newton invented, along
with Bernoulli and others,
00:15:16.780 --> 00:15:18.807
is to integrate by parts.
00:15:18.807 --> 00:15:21.390
Integration by parts, I'm sure,
was not a well-known procedure
00:15:21.390 --> 00:15:23.480
at that time.
00:15:23.480 --> 00:15:25.910
But if we integrate the
second term by parts,
00:15:25.910 --> 00:15:28.760
we could remove the
derivative acting on w,
00:15:28.760 --> 00:15:31.420
and arrange for w to be
a multiplicative factor
00:15:31.420 --> 00:15:33.540
in both terms.
00:15:33.540 --> 00:15:37.570
And a crucial thing that
makes the whole thing useful
00:15:37.570 --> 00:15:40.080
is that when you do
integrate by parts,
00:15:40.080 --> 00:15:43.650
you discover that you don't
get any endpoint contributions,
00:15:43.650 --> 00:15:45.510
because the endpoint
contributions would
00:15:45.510 --> 00:15:48.250
be proportional to
wi at the endpoints.
00:15:48.250 --> 00:15:50.290
And remember, wi has to
vanish at the endpoints,
00:15:50.290 --> 00:15:52.780
because that's the
condition that we're not
00:15:52.780 --> 00:15:55.350
changing the points
A and B. We're always
00:15:55.350 --> 00:15:58.250
talking about paths that
have the same starting point
00:15:58.250 --> 00:16:01.430
and the same ending point.
00:16:01.430 --> 00:16:09.460
So integrating by parts,
we get this expression,
00:16:09.460 --> 00:16:12.940
where now wi multiplies
everything, as just
00:16:12.940 --> 00:16:14.385
simply a multiplicative factor.
00:16:14.385 --> 00:16:15.760
To write it in
this form, you had
00:16:15.760 --> 00:16:17.690
to do a little bit of
juggling of indices.
00:16:17.690 --> 00:16:20.240
The other important trick
in these manipulations
00:16:20.240 --> 00:16:23.530
is to juggle indices, which
I'll not show you explicitly.
00:16:23.530 --> 00:16:25.850
But the thing to remember
is that these indices that
00:16:25.850 --> 00:16:28.190
are being summed over
can be called anything
00:16:28.190 --> 00:16:31.010
and it's still the same sum.
00:16:31.010 --> 00:16:33.660
So when you want to get
terms to cancel each other,
00:16:33.660 --> 00:16:35.820
you may have to change
the names of indices
00:16:35.820 --> 00:16:38.145
to get them to just
cancel identically.
00:16:38.145 --> 00:16:41.370
But that's straightforward.
00:16:41.370 --> 00:16:43.822
So we get this expression.
00:16:43.822 --> 00:16:45.530
And now we want this
expression to vanish
00:16:45.530 --> 00:16:48.550
for every possible wi of lambda.
00:16:48.550 --> 00:16:50.140
And we argued that
the only way it
00:16:50.140 --> 00:16:53.290
could vanish for every
possible wi of lambda
00:16:53.290 --> 00:16:57.180
is if the expression in curly
brackets, itself, vanishes.
00:16:57.180 --> 00:16:59.867
Yeah, if we only know the
values for some particular wi
00:16:59.867 --> 00:17:01.450
of lambda, then there
are lots of ways
00:17:01.450 --> 00:17:02.824
it could vanish,
because it could
00:17:02.824 --> 00:17:05.609
be positive in some places
and negative in others.
00:17:05.609 --> 00:17:08.069
But the only way it
could vanish for all wi
00:17:08.069 --> 00:17:10.990
is for the quantity in
curly brackets to vanish.
00:17:10.990 --> 00:17:12.940
So that gives us our
final, or at least,
00:17:12.940 --> 00:17:15.445
almost final expression
of the geodesic equation.
00:17:15.445 --> 00:17:21.010
And that's where we left off
last time, with that equation.
00:17:21.010 --> 00:17:23.650
So note that this is
just an equation that
00:17:23.650 --> 00:17:27.269
would either be obeyed or not
obeyed by the function x super
00:17:27.269 --> 00:17:28.780
i of lambda.
00:17:28.780 --> 00:17:30.330
It's just a
differential equation
00:17:30.330 --> 00:17:33.270
involving x super i of
lambda and the metric, which
00:17:33.270 --> 00:17:35.890
we assume is given.
00:17:35.890 --> 00:17:36.390
OK.
00:17:36.390 --> 00:17:37.973
So are there any
questions about that?
00:17:41.600 --> 00:17:43.670
Everybody happy?
00:17:43.670 --> 00:17:44.542
Great.
00:17:44.542 --> 00:17:46.375
OK, now we'll continue
on on the blackboard.
00:17:57.722 --> 00:17:59.180
OK, the first thing
I want to do is
00:17:59.180 --> 00:18:03.020
to simplify the equation a bit.
00:18:03.020 --> 00:18:05.120
This equation is
fairly complicated,
00:18:05.120 --> 00:18:08.467
because of those square roots
of A's in the denominators.
00:18:08.467 --> 00:18:10.550
The square root of A is a
pretty complicated thing
00:18:10.550 --> 00:18:12.332
to start with, and
the square root of A
00:18:12.332 --> 00:18:14.040
here is even
differentiated, because it's
00:18:14.040 --> 00:18:16.460
got the lambda making
an incredible mess,
00:18:16.460 --> 00:18:18.610
if you understand all that.
00:18:18.610 --> 00:18:20.430
So it would be nice
to simplify that.
00:18:20.430 --> 00:18:23.440
And we do have one trick
which we can still do,
00:18:23.440 --> 00:18:25.420
which we haven't done yet.
00:18:25.420 --> 00:18:29.800
We originally constructed
our path, xi of lambda,
00:18:29.800 --> 00:18:33.300
as a function of some
arbitrary parameter, lambda.
00:18:33.300 --> 00:18:37.630
Lambda just measures arbitrary
points along the path.
00:18:37.630 --> 00:18:39.460
There are many, many
ways to do that,
00:18:39.460 --> 00:18:42.169
an infinite number of
ways that you can do that.
00:18:42.169 --> 00:18:43.960
And this formula will
work for all of them,
00:18:43.960 --> 00:18:45.250
it's completely general.
00:18:45.250 --> 00:18:47.120
The formula, when
we derived it, we
00:18:47.120 --> 00:18:52.170
didn't make any assumptions
about how lambda was chosen.
00:18:52.170 --> 00:18:56.290
But we can simplify the formula
by making a particular choice
00:18:56.290 --> 00:18:57.390
for lambda.
00:18:57.390 --> 00:18:59.410
And the choice that
simplifies things
00:18:59.410 --> 00:19:02.780
is to choose lambda to
be the arc length itself.
00:19:02.780 --> 00:19:05.160
Lambda should be the
distance along the path.
00:19:05.160 --> 00:19:07.460
And then we're
trying to express xi
00:19:07.460 --> 00:19:11.700
as a function of how
far you've already gone.
00:19:11.700 --> 00:19:22.290
And that has the effect, if
we go back to what Ai was,
00:19:22.290 --> 00:19:31.150
A of lambda really is just
the path length per lambda.
00:19:31.150 --> 00:19:36.370
So if lambda is the path length
itself, A is just equal to 1.
00:19:40.528 --> 00:19:43.080
I'm trying to get a formula
that shows that more clearly.
00:19:43.080 --> 00:19:43.680
Here.
00:19:43.680 --> 00:19:45.880
If we remember that
this quantity is A,
00:19:45.880 --> 00:19:48.670
this tells us that ds
squared is equal to A times d
00:19:48.670 --> 00:19:49.780
lambda squared.
00:19:49.780 --> 00:19:52.600
So if ds is the
same as d lambda,
00:19:52.600 --> 00:19:55.530
as you've chosen your parameter
to be the path length,
00:19:55.530 --> 00:19:57.840
this formula makes
it clear that that's
00:19:57.840 --> 00:20:01.510
equivalent to A equal to 1.
00:20:01.510 --> 00:20:04.780
So going back to the
formula, if A is 1,
00:20:04.780 --> 00:20:08.121
we would just drop it from
both sides of the equation.
00:20:08.121 --> 00:20:10.620
And all that really matters, I
should point out here, maybe,
00:20:10.620 --> 00:20:13.480
because we'll be using it
later, is that A is a constant.
00:20:13.480 --> 00:20:16.700
As long as A is a constant,
it will not be differentiated,
00:20:16.700 --> 00:20:19.350
and then it will cancel on the
left side and the right side.
00:20:19.350 --> 00:20:20.780
So we don't necessarily
care that it is 1,
00:20:20.780 --> 00:20:22.279
but we do care that
it's a constant.
00:20:22.279 --> 00:20:24.590
And then it just disappears
from the formula.
00:20:24.590 --> 00:20:27.350
And then we get the
simpler formula.
00:20:27.350 --> 00:20:33.040
And now we'll continue
on the blackboard.
00:20:33.040 --> 00:20:51.900
The simpler formula is
just dds of gij dxj ds
00:20:51.900 --> 00:21:01.530
is equal to 1/2 times the
derivative of gjk, with respect
00:21:01.530 --> 00:21:15.020
to xi, times dxj
ds dxk ds, where
00:21:15.020 --> 00:21:16.870
s is equal to the path length.
00:21:22.840 --> 00:21:24.750
So I've replaced
lambda by s, because we
00:21:24.750 --> 00:21:26.095
set lambda equal to s.
00:21:26.095 --> 00:21:29.290
And s has a more specific
meaning than lambda did.
00:21:29.290 --> 00:21:31.740
Lambda was a completely
arbitrary parametrization
00:21:31.740 --> 00:21:32.320
of the path.
00:21:35.940 --> 00:21:40.680
So this one deserves a
big box, because it really
00:21:40.680 --> 00:21:43.072
is the final formula
for geodesics.
00:21:43.072 --> 00:21:45.030
Once we write it in terms
of different letters,
00:21:45.030 --> 00:21:49.260
we will later, but this
actually is the formula.
00:21:49.260 --> 00:21:51.770
Now I should
mention just largely
00:21:51.770 --> 00:21:55.360
for the sake of your
knowing what's going on,
00:21:55.360 --> 00:21:58.580
if you ever look at some other
general relativity books,
00:21:58.580 --> 00:22:01.630
this is not the formula that
the geodesic equation is usually
00:22:01.630 --> 00:22:03.070
written in.
00:22:03.070 --> 00:22:04.360
Frankly, it is the best form.
00:22:04.360 --> 00:22:06.130
If you want to
find the geodesic,
00:22:06.130 --> 00:22:09.560
usually this form of writing
the equation is the easiest.
00:22:09.560 --> 00:22:12.030
But most general
relativity books
00:22:12.030 --> 00:22:14.160
prefer instead to
just give a formula
00:22:14.160 --> 00:22:16.560
for the second derivative, here.
00:22:16.560 --> 00:22:18.820
Which involves just
expanding this term,
00:22:18.820 --> 00:22:21.390
and then when we
shuffle things, to try
00:22:21.390 --> 00:22:24.410
to simplify the expressions.
00:22:24.410 --> 00:22:40.110
So one can write, to
start, d ds of gij dxj ds.
00:22:40.110 --> 00:22:41.760
We're just going to expand it.
00:22:41.760 --> 00:22:43.680
Now we're going to be making
use of all the rules of calculus
00:22:43.680 --> 00:22:44.350
that we've learn.
00:22:44.350 --> 00:22:46.183
Every rule you've ever
learned will probably
00:22:46.183 --> 00:22:47.980
get used in this calculation.
00:22:47.980 --> 00:22:51.140
So it will be
using product rule,
00:22:51.140 --> 00:22:53.760
of course, because we have a
product of two things here.
00:22:53.760 --> 00:22:55.850
But we also have
the complication
00:22:55.850 --> 00:22:59.530
that gij is not explicitly
a function of s.
00:22:59.530 --> 00:23:03.270
But gij is a
function of position.
00:23:03.270 --> 00:23:07.040
And the position that one is
that for any given value of s
00:23:07.040 --> 00:23:10.240
depends on s, because we're
moving along the path,
00:23:10.240 --> 00:23:12.350
x super i of s.
00:23:12.350 --> 00:23:19.970
So the gij here, is
evaluated at x super i of s.
00:23:22.510 --> 00:23:24.240
I should give this a new letter.
00:23:24.240 --> 00:23:27.730
x super k of s.
00:23:27.730 --> 00:23:32.750
So it depends on s, through
the argument of its argument.
00:23:32.750 --> 00:23:36.280
So that's a chain
rule situation.
00:23:36.280 --> 00:23:41.560
And what we get here is,
from just differentiating
00:23:41.560 --> 00:23:43.820
the second factor, that's easy.
00:23:43.820 --> 00:23:54.320
We get gij DC d
squared xj ds squared.
00:23:54.320 --> 00:23:58.380
And then, from the derivative
of the derivative chain rule
00:23:58.380 --> 00:24:11.600
piece, we get the partial of
gij, with respect to xk times
00:24:11.600 --> 00:24:19.030
the dxj ds times dxk ds.
00:24:26.100 --> 00:24:29.840
And then to continue, this
piece gets brought over
00:24:29.840 --> 00:24:32.500
to the other side, because we're
trying to get an equation just
00:24:32.500 --> 00:24:34.145
for the second
derivative of the path.
00:24:54.420 --> 00:25:04.690
So then we get g sub ij d
squared x super j ds squared is
00:25:04.690 --> 00:25:14.470
equal to 1/2 di-- I'll define
that in a second-- g sub jk
00:25:14.470 --> 00:25:29.120
minus 2 dk gij dxi ds dxj ds.
00:25:34.910 --> 00:25:38.970
where this partial
derivative with the subscript
00:25:38.970 --> 00:25:43.270
is just an abbreviation for
the derivative with respect
00:25:43.270 --> 00:25:44.910
to the coordinate
with that index.
00:25:47.910 --> 00:25:49.213
So that's just an abbreviation.
00:25:54.690 --> 00:25:59.890
Now you could think of this
as a matrix times a vector
00:25:59.890 --> 00:26:01.820
is equal to an expression.
00:26:01.820 --> 00:26:05.230
What we like to do is just get
an expression for this vector.
00:26:05.230 --> 00:26:08.210
So if we think of it as
a matrix times a vector,
00:26:08.210 --> 00:26:10.080
all we have to do
is invert the matrix
00:26:10.080 --> 00:26:12.671
to be able to get an expression
for the vector itself.
00:26:12.671 --> 00:26:13.170
Yes!
00:26:13.170 --> 00:26:14.878
AUDIANCE: Should that
closing parenthesis
00:26:14.878 --> 00:26:18.840
be more [INAUDIBLE]?
00:26:18.840 --> 00:26:20.799
PROFESSOR: Oh, Yeah,
I think you're right,
00:26:20.799 --> 00:26:21.715
it doesn't look right.
00:26:26.770 --> 00:26:27.750
Yeah.
00:26:27.750 --> 00:26:37.960
Thank you This has to
multiply everything.
00:26:41.820 --> 00:26:44.600
Oops!
00:26:44.600 --> 00:26:47.295
OK, OK.
00:26:47.295 --> 00:26:48.920
Given enough chances
I'll get it right.
00:26:53.205 --> 00:26:58.710
OK, now everybody
happy this time?
00:26:58.710 --> 00:27:03.100
Thank you very much for
getting it straight.
00:27:03.100 --> 00:27:06.280
OK, So as I was saying,
we want to isolate
00:27:06.280 --> 00:27:07.450
this second derivative.
00:27:07.450 --> 00:27:09.033
We're hoping to get
just an expression
00:27:09.033 --> 00:27:10.460
for the second derivative.
00:27:10.460 --> 00:27:14.482
And this can be interpreted
as a matrix times a vector
00:27:14.482 --> 00:27:15.190
equals something.
00:27:15.190 --> 00:27:17.090
We want to just
invert that matrix.
00:27:17.090 --> 00:27:18.392
Yes?
00:27:18.392 --> 00:27:21.679
AUDIANCE: Isn't the ds and
[? the idx ?] [INAUDIBLE]?
00:27:21.679 --> 00:27:23.720
PROFESSOR: Oh, do I have
that wrong too, perhaps?
00:27:32.360 --> 00:27:35.630
I think we want j and
k there, that don't we?
00:27:35.630 --> 00:27:42.370
OK, attempt number four, or
did I lose count as well?
00:27:42.370 --> 00:27:45.770
j and k are the
indices and the i
00:27:45.770 --> 00:27:48.520
matches the free i on the left.
00:27:48.520 --> 00:27:51.350
And all the other
indices are sound.
00:27:51.350 --> 00:27:55.470
I think, probably, I finally
achieved the right formula.
00:27:59.460 --> 00:28:00.460
Thanks for all the help.
00:28:04.220 --> 00:28:08.223
So inverting a
matrix, the principal
00:28:08.223 --> 00:28:10.890
is a straightforward
mathematical operation.
00:28:10.890 --> 00:28:14.690
In general relativity, we give
a name to the inverse metric,
00:28:14.690 --> 00:28:17.890
and it's the same
letter g with indices,
00:28:17.890 --> 00:28:20.140
with superscripts
instead of subscripts.
00:28:20.140 --> 00:28:23.410
And that's defined to
be the matrix inverse.
00:28:23.410 --> 00:28:38.440
So g super ij is defined to be
the matrix inverse of g sub ij.
00:28:41.020 --> 00:28:44.790
And to put that
into an equation,
00:28:44.790 --> 00:28:48.870
we could say that if we
take g with upper indices--
00:28:48.870 --> 00:28:52.720
and I'll write those
upper indices as i and l--
00:28:52.720 --> 00:29:00.100
and multiply it by a g
with lower indices l and j,
00:29:00.100 --> 00:29:04.680
when you sum over adjacent
indices in this index
00:29:04.680 --> 00:29:07.060
notation, that's
exactly what corresponds
00:29:07.060 --> 00:29:10.660
to the definition of
matrix multiplication.
00:29:10.660 --> 00:29:13.980
So this is just the matrix
g with upper indices times
00:29:13.980 --> 00:29:18.260
the matrix g with lower indices,
and it's the i j'th element
00:29:18.260 --> 00:29:19.820
of that matrix.
00:29:19.820 --> 00:29:22.290
And we're saying it should
be the identity matrix,
00:29:22.290 --> 00:29:26.620
and that means that the i j'th
element should be 0 if it's off
00:29:26.620 --> 00:29:31.180
diagonal, and 1 if it's
diagonal, if i equals j.
00:29:31.180 --> 00:29:32.650
And that's exactly
the definition
00:29:32.650 --> 00:29:34.680
of a chronic or a delta.
00:29:34.680 --> 00:29:39.060
So this is equal to delta ij.
00:29:39.060 --> 00:29:43.670
We remember that delta ij is
0 if i is not equal to j, 1
00:29:43.670 --> 00:29:44.860
if i is equal to j.
00:29:44.860 --> 00:29:46.280
That's the definition.
00:29:46.280 --> 00:29:48.120
And it corresponds to
that identity matrix
00:29:48.120 --> 00:29:48.925
in matrix language.
00:29:52.100 --> 00:29:57.680
So this is the relationship that
actually defines g super il.
00:29:57.680 --> 00:30:01.790
And it is just the statement
that g with upper indices
00:30:01.790 --> 00:30:04.130
is the matrix inverse
of g with lower indices.
00:30:06.940 --> 00:30:11.560
Using this, we can bring
this g to the other side
00:30:11.560 --> 00:30:14.940
essentially by
multiplying by g inverse.
00:30:14.940 --> 00:30:19.660
And I will save a little
time by not writing that out
00:30:19.660 --> 00:30:25.700
in gory detail, but rather
I'll just write the result.
00:30:25.700 --> 00:30:28.110
And the result is written in
terms of a new symbol that
00:30:28.110 --> 00:30:29.770
gets defined, which
is an absolutely
00:30:29.770 --> 00:30:32.910
standard symbol in
General Relativity.
00:30:32.910 --> 00:30:38.820
The formula is d
squared x i, ds squared
00:30:38.820 --> 00:30:42.590
is equal to-- we know it's
going to be equal to stuff
00:30:42.590 --> 00:30:46.860
times the product
of two derivatives.
00:30:46.860 --> 00:30:49.330
And the stuff that
appears is just
00:30:49.330 --> 00:30:54.050
given a name, capital gamma,
which has an upper index
00:30:54.050 --> 00:30:57.670
i, which matches the left
hand side of the equation.
00:30:57.670 --> 00:31:01.230
And two lower indices, which
I'm calling j and k, which
00:31:01.230 --> 00:31:04.810
will get summed with the
derivatives that follow,
00:31:04.810 --> 00:31:10.755
d x j ds, dx k, ds.
00:31:16.100 --> 00:31:20.080
And this quantity,
gamma super i sub jk
00:31:20.080 --> 00:31:21.720
are just the terms
that would appear
00:31:21.720 --> 00:31:24.130
when we do these manipulations.
00:31:24.130 --> 00:31:28.070
And I'll write what they are.
00:31:28.070 --> 00:31:39.155
Gamma super i sub jk is
equal to 1/2 g super il
00:31:39.155 --> 00:31:42.900
times the derivative
with respect
00:31:42.900 --> 00:31:52.290
to j of g sub lk,
plus the derivative
00:31:52.290 --> 00:31:58.740
with respect to k of g sub lj.
00:31:58.740 --> 00:32:05.530
And then minus the derivative
with respect to l of g sub jk.
00:32:14.840 --> 00:32:20.240
And this quantity has
several different names.
00:32:20.240 --> 00:32:22.430
Everybody agrees how to
define it up to the sign.
00:32:22.430 --> 00:32:23.929
There are different
sign conventions
00:32:23.929 --> 00:32:26.620
that are used in
different books.
00:32:26.620 --> 00:32:28.680
And there are also
different names for it.
00:32:28.680 --> 00:32:31.070
It's often called the
affine connection.
00:32:34.694 --> 00:32:36.860
If you look, for example
in Steve Weinberg's General
00:32:36.860 --> 00:32:39.990
Relativity book, he calls
it the affine connection.
00:32:39.990 --> 00:32:43.090
It's also very often called
the Christofel connection,
00:32:43.090 --> 00:32:44.300
or the Christofel symbol.
00:33:00.610 --> 00:33:03.660
And frankly those are
the only names for it
00:33:03.660 --> 00:33:07.340
that I've seen, personally.
00:33:07.340 --> 00:33:09.400
But there's a book
about [INAUDIBLE]
00:33:09.400 --> 00:33:12.170
by Sean Carroll which
is a very good book.
00:33:12.170 --> 00:33:14.900
And he claims that it's
sometimes also called
00:33:14.900 --> 00:33:17.550
the Riemann connection And
it's also sometimes called
00:33:17.550 --> 00:33:19.930
the Levi-Civita connection.
00:33:19.930 --> 00:33:23.250
So it's got lots of names, which
I guess means lots of people's
00:33:23.250 --> 00:33:24.540
independently invented it.
00:33:27.840 --> 00:33:30.050
But in any case,
that's the answer.
00:33:30.050 --> 00:33:33.380
And it's just a way of rewriting
the formula we have up there.
00:33:33.380 --> 00:33:34.970
And for solving
problems, the formula,
00:33:34.970 --> 00:33:36.540
the way we wrote up
there, is almost always
00:33:36.540 --> 00:33:37.560
the best way to do it.
00:33:37.560 --> 00:33:40.160
So this is really just
window dressing, largely
00:33:40.160 --> 00:33:42.556
for the purpose of making
contact with other books
00:33:42.556 --> 00:33:43.680
that you might come across.
00:33:47.360 --> 00:33:48.860
OK, so that finishes
the derivation
00:33:48.860 --> 00:33:50.330
of the geodesic equation.
00:33:50.330 --> 00:33:52.760
Now I'd like to give
an example of its use.
00:33:52.760 --> 00:33:54.399
But before I do
that, let me just
00:33:54.399 --> 00:33:56.940
pause to ask if there are any
questions about the derivation?
00:34:02.350 --> 00:34:03.490
OK.
00:34:03.490 --> 00:34:05.460
So on your homework,
you will, in fact,
00:34:05.460 --> 00:34:10.900
be applying this formalism to
the Robertson-Walker metric.
00:34:10.900 --> 00:34:15.210
And you'll learn how
moving particles slow down
00:34:15.210 --> 00:34:19.650
as they move through
an expanding universe,
00:34:19.650 --> 00:34:23.520
completely in an analogy to
the way photons, which we've
00:34:23.520 --> 00:34:26.000
already learned,
lose energy as they
00:34:26.000 --> 00:34:28.489
travel through an
expanding universe.
00:34:28.489 --> 00:34:31.770
So particles with
mass also lose energy
00:34:31.770 --> 00:34:33.799
in a well-defined
way, which you'll
00:34:33.799 --> 00:34:36.780
be calculating on the homework.
00:34:36.780 --> 00:34:39.879
For example, though, I'll
do something different.
00:34:39.879 --> 00:34:44.330
A fun metric to talk about
is the Schwarzschild metric,
00:34:44.330 --> 00:34:47.730
which describes, among
other things, black holes.
00:34:47.730 --> 00:34:49.310
It in principle,
describes anything
00:34:49.310 --> 00:34:52.250
which is spherically symmetric
and has a gravitational field.
00:34:52.250 --> 00:34:54.760
But black holes are the
most interesting example,
00:34:54.760 --> 00:34:58.930
because it's where the
most surprises lie.
00:34:58.930 --> 00:35:10.140
So the Schwarzschild
metric has the form
00:35:10.140 --> 00:35:18.170
ds squared is equal to minus
c squared d tau squared, which
00:35:18.170 --> 00:35:24.810
is equal to-- this is just a
definition, it defines d tau--
00:35:24.810 --> 00:35:27.560
but in terms of the
coordinates, it's
00:35:27.560 --> 00:35:32.100
minus 1 minus 2 G,
Newton's constant, M,
00:35:32.100 --> 00:35:35.035
the mass of the object
we're discussing--
00:35:35.035 --> 00:35:38.910
the mass of the black hole, if
it is a black hole-- divided
00:35:38.910 --> 00:35:43.270
by r times c squared, r
is the radial coordinate,
00:35:43.270 --> 00:35:57.520
times c squared dt squared,
plus 1 minus 2 GM over rc
00:35:57.520 --> 00:36:07.970
squared times dr squared
plus r squared times d theta
00:36:07.970 --> 00:36:14.705
squared plus sine squared
theta d phi squared.
00:36:20.120 --> 00:36:24.280
Now here, theta and phi
are the usual polar angles.
00:36:24.280 --> 00:36:26.940
We're using a polar
coordinate system.
00:36:26.940 --> 00:36:33.880
So as usual, theta
lies between 0 and pi.
00:36:33.880 --> 00:36:36.300
0, what we might call
the North Pole, and pi
00:36:36.300 --> 00:36:39.760
what we might call
the South Pole.
00:36:39.760 --> 00:36:44.792
And phi is what is often
called an azimuthal angle,
00:36:44.792 --> 00:36:46.840
it goes around.
00:36:46.840 --> 00:36:48.740
And the way one
describes coordinates
00:36:48.740 --> 00:36:50.360
on the surface of
the Earth, phi would
00:36:50.360 --> 00:36:54.860
be the longitude variable.
00:36:54.860 --> 00:36:59.460
So 0 is less than or
equal to phi is less than
00:36:59.460 --> 00:37:04.690
or equal to 2 pi
where phi equals
00:37:04.690 --> 00:37:08.380
2 pi is identified
with phi equals 0.
00:37:08.380 --> 00:37:12.150
And you can go around and come
back to where you started.
00:37:17.650 --> 00:37:20.670
Now notice that if we set
capital M, the mass of this
00:37:20.670 --> 00:37:24.540
object equal to 0, the metric
becomes the trivial metric
00:37:24.540 --> 00:37:29.050
of Special Relativity written
in spherical polar coordinates.
00:37:29.050 --> 00:37:32.080
So all complications go
away if there's no mass.
00:37:32.080 --> 00:37:34.930
The object disappears.
00:37:34.930 --> 00:37:36.800
But as long as the
mass is non-zero
00:37:36.800 --> 00:37:40.300
there are factors that
multiply the dr squared term
00:37:40.300 --> 00:37:42.080
and the c squared
dt squared term.
00:37:45.330 --> 00:37:49.314
Notice that the factors that
do that multiplying-- now one
00:37:49.314 --> 00:37:50.480
of these should be inverted.
00:37:54.630 --> 00:37:58.420
Important inverse, it's a
minus 1 power for that factor.
00:38:01.240 --> 00:38:06.029
Notice that r can
be small enough so
00:38:06.029 --> 00:38:07.320
that these factors will vanish.
00:38:10.110 --> 00:38:12.940
And the place where that happens
is called the Schwarzschild
00:38:12.940 --> 00:38:17.620
radius after the same person
who invented the metric.
00:38:17.620 --> 00:38:26.400
So r sub Schwarzschild is equal
to 2 GM divided by c squared.
00:38:26.400 --> 00:38:31.460
When little r is equal to that,
this quantity in parentheses
00:38:31.460 --> 00:38:34.090
vanishes, which means
we get infinity here,
00:38:34.090 --> 00:38:38.760
because it's inverted,
and we get a 0 there.
00:38:38.760 --> 00:38:44.860
Now when a term in the metric
is either 0 or infinite,
00:38:44.860 --> 00:38:48.300
one calls that a singularity.
00:38:48.300 --> 00:38:51.820
In this case, it's a
removable singularity.
00:38:51.820 --> 00:38:53.850
That is, the
Schwarzschild singularity
00:38:53.850 --> 00:38:56.590
is only there
because Schwarzschild
00:38:56.590 --> 00:38:59.402
chose to use these
particular coordinates.
00:38:59.402 --> 00:39:01.110
These are simpler than
other coordinates.
00:39:01.110 --> 00:39:03.530
He wasn't foolish to use them.
00:39:03.530 --> 00:39:05.250
But the appearance
of that singularity
00:39:05.250 --> 00:39:08.600
is really caused solely by
the choice of coordinates.
00:39:08.600 --> 00:39:15.070
There really is no singularity
at the Schwarzschild horizon.
00:39:15.070 --> 00:39:17.520
And that was shown
some years later
00:39:17.520 --> 00:39:21.090
by other people constructing
other coordinate systems.
00:39:21.090 --> 00:39:23.760
The coordinate system
is best known today
00:39:23.760 --> 00:39:27.370
that avoids the
Schwarzschild singularity is
00:39:27.370 --> 00:39:29.840
a coordinate system called
the Kruskal coordinate system.
00:39:56.404 --> 00:39:58.570
But we will not be looking
at the Kruskal coordinate
00:39:58.570 --> 00:40:00.820
system in this class.
00:40:00.820 --> 00:40:03.260
Leave that for the GR class
that you'll take some time.
00:40:08.080 --> 00:40:14.320
OK, now the masses
sum parameter,
00:40:14.320 --> 00:40:15.820
notice that the
metric is completely
00:40:15.820 --> 00:40:17.400
determined by the mass.
00:40:17.400 --> 00:40:21.450
And that's the same situation as
we found in Newtonian gravity.
00:40:21.450 --> 00:40:25.460
The metric outside of the
spherically symmetric object,
00:40:25.460 --> 00:40:28.220
by the gravitational field
in Newtonian Physics outside
00:40:28.220 --> 00:40:30.295
of a spherical symmetric
object, depends only
00:40:30.295 --> 00:40:31.890
on the total mass,
which does not
00:40:31.890 --> 00:40:33.770
depend, at all, on
how it's distributed
00:40:33.770 --> 00:40:35.630
as long as it's
spherically symmetric.
00:40:35.630 --> 00:40:36.970
And the same thing here.
00:40:36.970 --> 00:40:40.080
As long as an object is
spherically symmetric,
00:40:40.080 --> 00:40:42.870
the gravitational field
outside of the object
00:40:42.870 --> 00:40:46.820
will always look
like that formula.
00:40:46.820 --> 00:40:50.660
Now there are still
two cases-- outside
00:40:50.660 --> 00:40:53.440
of the object could be
larger than or smaller
00:40:53.440 --> 00:40:57.060
than this Schwarzschild radius.
00:40:57.060 --> 00:41:00.340
So for an object like the
sun, the Schwarzschild radius,
00:41:00.340 --> 00:41:02.940
we could calculate it--
and it's calculated
00:41:02.940 --> 00:41:05.390
in the notes-- it's about
two or three kilometers.
00:41:08.300 --> 00:41:10.710
Hold on and I'll tell
you more accurately.
00:41:10.710 --> 00:41:13.380
It's 2.95 kilometers,
the Schwarzschild radius
00:41:13.380 --> 00:41:15.790
of the sun.
00:41:15.790 --> 00:41:18.090
But the sun, of course,
is much bigger than that.
00:41:18.090 --> 00:41:21.020
And that means that the sun
doesn't have a Schwarzschild
00:41:21.020 --> 00:41:22.670
horizon.
00:41:22.670 --> 00:41:28.350
That is, at 2.95 kilometers
from the center of the sun
00:41:28.350 --> 00:41:30.030
there's still sun.
00:41:30.030 --> 00:41:31.110
It's not outside the sun.
00:41:31.110 --> 00:41:33.443
This metric only holds outside
the spherically symmetric
00:41:33.443 --> 00:41:34.990
object.
00:41:34.990 --> 00:41:36.610
So it does not hold
inside the sun.
00:41:36.610 --> 00:41:39.830
The place where this has
the apparent singularity
00:41:39.830 --> 00:41:41.852
the metric is not valid at all.
00:41:41.852 --> 00:41:43.310
So there is nothing
that even comes
00:41:43.310 --> 00:41:46.150
close to anything
worth talking about,
00:41:46.150 --> 00:41:47.800
as far as the
Schwarzschild singularity
00:41:47.800 --> 00:41:49.770
for an object like the sun.
00:41:49.770 --> 00:41:53.820
But if the sun were compressed
to a size smaller than 2.95
00:41:53.820 --> 00:42:00.040
kilometers with the same
mass, then these factors
00:42:00.040 --> 00:42:03.950
would be relevant at the
places where they vanish.
00:42:03.950 --> 00:42:09.290
And whatever consequences they
have, we would be dealing with.
00:42:09.290 --> 00:42:12.930
Even though r equals r
Schwarzschild is not a singular
00:42:12.930 --> 00:42:15.670
point, it is still
a special point.
00:42:15.670 --> 00:42:19.655
What you can show-- we won't--
but what we can show is that
00:42:19.655 --> 00:42:21.840
that is the horizon.
00:42:21.840 --> 00:42:24.600
Meaning that if an object
falls inside this Schwarzschild
00:42:24.600 --> 00:42:30.191
radius, there is no trajectory
that will ever get it out.
00:42:30.191 --> 00:42:30.690
Yes?
00:42:30.690 --> 00:42:34.866
AUDIANCE: Say a star is just
incredibly dense at its core.
00:42:34.866 --> 00:42:36.850
Is it possible to
have suppression
00:42:36.850 --> 00:42:41.082
of some fractional life of
a star that's from that mass
00:42:41.082 --> 00:42:42.059
that it's contained?
00:42:42.059 --> 00:42:43.808
Or like a fusion
reaction that is going on
00:42:43.808 --> 00:42:46.220
with the net radius?
00:42:46.220 --> 00:42:49.572
PROFESSOR: OK, could there be
a horizon inside of a star?
00:42:49.572 --> 00:42:51.280
I think is what you're
asking, basically.
00:42:51.280 --> 00:42:52.870
AUDIANCE: One that
actually affects the--
00:42:52.870 --> 00:42:53.885
PROFESSOR: One that
really is a horizon.
00:42:53.885 --> 00:42:54.926
AUDIANCE: That's outside.
00:42:54.926 --> 00:42:57.050
PROFESSOR: Right.
00:42:57.050 --> 00:42:58.790
If this were the sun
you were describing,
00:42:58.790 --> 00:43:00.623
this formula would just
not be valid inside.
00:43:00.623 --> 00:43:02.010
There would be no
horizon inside.
00:43:02.010 --> 00:43:05.070
But you're asking a
real valid question.
00:43:05.070 --> 00:43:09.930
If a star had, for some
reason, a very dense spot
00:43:09.930 --> 00:43:12.350
in the middle, could
it actually form
00:43:12.350 --> 00:43:14.870
a horizon inside the material?
00:43:14.870 --> 00:43:16.412
And the answer
is, yes, it could.
00:43:16.412 --> 00:43:17.370
It would not be stable.
00:43:17.370 --> 00:43:27.960
The material would ultimately
fall in, but it could happen.
00:43:27.960 --> 00:43:29.003
Yes?
00:43:29.003 --> 00:43:31.461
AUDIANCE: So like our galaxy
has a super massive black hole
00:43:31.461 --> 00:43:32.180
in the center.
00:43:32.180 --> 00:43:32.790
PROFESSOR: That's right.
00:43:32.790 --> 00:43:34.748
Our galaxy does have a
super massive black hole
00:43:34.748 --> 00:43:35.760
in the center.
00:43:35.760 --> 00:43:35.896
AUDIANCE: Yeah.
00:43:35.896 --> 00:43:37.604
So you can consider
that as like a larger
00:43:37.604 --> 00:43:40.570
mass that has black hole, area?
00:43:40.570 --> 00:43:42.180
PROFESSOR: Right!
00:43:42.180 --> 00:43:43.860
Right!
00:43:43.860 --> 00:43:44.650
That's right.
00:43:44.650 --> 00:43:47.114
The comment is that
if we go from a star
00:43:47.114 --> 00:43:48.530
to something bigger
than a star we
00:43:48.530 --> 00:43:52.030
have perfectly good
example in our own galaxy,
00:43:52.030 --> 00:43:53.780
where there is a black
hole in the center,
00:43:53.780 --> 00:43:56.090
but there is still mass that
continues outside of that.
00:43:56.090 --> 00:43:58.300
And the black hole is
accreting, more matter
00:43:58.300 --> 00:44:00.940
does keep falling in,
it's not really stable.
00:44:00.940 --> 00:44:05.510
But it certainly does
exist, and can exist.
00:44:09.082 --> 00:44:09.915
Any other questions?
00:44:17.840 --> 00:44:21.135
Well, our goal now is
to calculate a geodesic.
00:44:28.050 --> 00:44:32.380
And I will just
calculate one geodesic.
00:44:32.380 --> 00:44:36.480
I will calculate what happens if
an object starts at some fixed
00:44:36.480 --> 00:44:41.270
radius at rest and is released
and falls into this black hole.
00:44:47.370 --> 00:44:50.890
I first want to just rewrite
the geodesic equation in terms
00:44:50.890 --> 00:44:54.720
of variables that are more
appropriate for this case.
00:44:54.720 --> 00:44:57.970
When I wrote that, I had a mind
just calculating the geodesics
00:44:57.970 --> 00:45:00.010
in space, looking
for the shortest
00:45:00.010 --> 00:45:02.731
path between two points.
00:45:02.731 --> 00:45:04.480
The geodesic that we're
talking about when
00:45:04.480 --> 00:45:07.500
we're talking about an
object in general relativity
00:45:07.500 --> 00:45:09.870
moving along the
geodesic is a geodesic
00:45:09.870 --> 00:45:12.570
that's a time-like geodesic.
00:45:12.570 --> 00:45:14.510
That is, any increment
along the geodesic
00:45:14.510 --> 00:45:17.300
is a time-like interval,
or following a particle.
00:45:17.300 --> 00:45:22.030
Particles travel on time-like
trajectories in relativity.
00:45:22.030 --> 00:45:26.700
So the usual notation for
time is something like tau
00:45:26.700 --> 00:45:30.260
rather than s, which is
why I wrote it this way.
00:45:30.260 --> 00:45:33.140
ds squared is just defined to be
minus c squared d tau squared.
00:45:33.140 --> 00:45:36.680
So d tau squared has no
more or less information
00:45:36.680 --> 00:45:40.050
than ds squared, but it has the
opposite sign and a difference
00:45:40.050 --> 00:45:42.850
by a factor of c
squared, as well.
00:45:42.850 --> 00:45:48.230
And another change in notation
which is a rather universal
00:45:48.230 --> 00:45:53.200
convention is that, when we talk
about space alone we use Latin
00:45:53.200 --> 00:45:55.810
indices, ijk..
00:45:55.810 --> 00:45:58.260
When we talk about spacetime,
where one of the indices
00:45:58.260 --> 00:46:01.500
might be 0 referring
to the time direction,
00:46:01.500 --> 00:46:05.840
then we usually use Greek
indices, mu, nu, lambda.
00:46:05.840 --> 00:46:11.280
So I'm going to rewrite the
geodesic equation using tau
00:46:11.280 --> 00:46:13.790
as my parameter instead
of s, since we're
00:46:13.790 --> 00:46:16.430
talking about proper time
along the trajectory instead
00:46:16.430 --> 00:46:17.890
of distances.
00:46:17.890 --> 00:46:21.250
And using Greek letters instead
of Latin letters, because we're
00:46:21.250 --> 00:46:24.131
talking about spacetime
rather than just space.
00:46:24.131 --> 00:46:25.630
So otherwise what
I'm going to write
00:46:25.630 --> 00:46:26.713
is just identical to that.
00:46:26.713 --> 00:46:30.020
So really is nothing more
than a change in notation.
00:46:30.020 --> 00:46:43.090
d d tau of g mu nu,
dx super nu d tau.
00:46:43.090 --> 00:46:49.520
And it is equal to 1/2 times
the partial of g lambda
00:46:49.520 --> 00:47:04.135
sigma with respect to x nu dx
lambda d tau dx sigma d tau.
00:47:08.221 --> 00:47:10.220
Now you might want to go
through the calculation
00:47:10.220 --> 00:47:12.540
and make sure of the
fact that now we're
00:47:12.540 --> 00:47:14.960
dealing with a metric
which is not positive,
00:47:14.960 --> 00:47:17.210
definite, doesn't
make any difference.
00:47:17.210 --> 00:47:18.444
But it doesn't.
00:47:18.444 --> 00:47:19.860
It does mean that
now we certainly
00:47:19.860 --> 00:47:24.040
have possibilities of getting
maxima and stationary points as
00:47:24.040 --> 00:47:28.260
well as minima, because
of the variety of signs
00:47:28.260 --> 00:47:30.220
that appear in the metric.
00:47:30.220 --> 00:47:32.590
But otherwise, the calculations
of the geodesic equation
00:47:32.590 --> 00:47:35.040
goes through exactly
as we calculated it.
00:47:35.040 --> 00:47:38.240
And the only thing
I'm doing here,
00:47:38.240 --> 00:47:40.320
relative to what we
have there, is just
00:47:40.320 --> 00:47:43.125
changing the notation a bit
to conform to the notaion that
00:47:43.125 --> 00:47:45.680
is usually used for talking
about spacetime trajectories.
00:48:04.060 --> 00:48:07.300
Since we're talking
about radio trajectories,
00:48:07.300 --> 00:48:09.640
we're just going to
release a particle at rest
00:48:09.640 --> 00:48:11.690
and then it will fall
straight towards the center
00:48:11.690 --> 00:48:15.860
of our spherical object,
we know by symmetry
00:48:15.860 --> 00:48:17.580
that it's not going
to be deflected
00:48:17.580 --> 00:48:19.750
in the positive theta
or the negative theta,
00:48:19.750 --> 00:48:22.720
or the positive phi or
negative phi directions,
00:48:22.720 --> 00:48:24.735
because that would
violate isotropy.
00:48:24.735 --> 00:48:26.455
It would violate the
rotational symmetry
00:48:26.455 --> 00:48:28.260
that we know as
part of this metric.
00:48:28.260 --> 00:48:31.800
This Is just the metric of
the surface of the sphere.
00:48:31.800 --> 00:48:34.220
So theta and phi will just
stay whatever values they have
00:48:34.220 --> 00:48:35.920
when you drop this object.
00:48:35.920 --> 00:48:38.600
So we will not even talk
about theta and phi.
00:48:38.600 --> 00:48:41.680
We will only talk
about r and t, how
00:48:41.680 --> 00:48:44.880
particle falls in as
a function of time.
00:48:44.880 --> 00:48:49.770
And then it turns out to be
useful to just first write down
00:48:49.770 --> 00:48:51.600
what the metric itself tells us.
00:48:51.600 --> 00:48:53.015
And we'll divide by d tau.
00:48:53.015 --> 00:48:55.280
So we could talk about
derivatives with respect
00:48:55.280 --> 00:48:57.130
to tau.
00:48:57.130 --> 00:48:59.200
So changing an overall
sign, since everything's
00:48:59.200 --> 00:49:00.700
going to be negative
and we'd rather
00:49:00.700 --> 00:49:03.780
have everything be
positive, we can just
00:49:03.780 --> 00:49:08.750
rewrite the metric
equation as saying,
00:49:08.750 --> 00:49:16.750
that c squared is equal
to 1 minus 2 GM over rc
00:49:16.750 --> 00:49:22.240
squared, times c
squared times dt
00:49:22.240 --> 00:49:37.580
d tau squared minus 1 minus 2
GM over rc squared inverse times
00:49:37.580 --> 00:49:41.180
dr d tau squared.
00:49:45.070 --> 00:49:47.290
So this is nothing more
than rewriting this equation
00:49:47.290 --> 00:49:49.915
saying d theta is equal
to 0 and d phi will be 0.
00:49:52.490 --> 00:49:54.100
Written this way,
though, it tells us
00:49:54.100 --> 00:49:59.285
that we can find dt d tau, for
example, if we know dr d tau.
00:49:59.285 --> 00:50:01.720
And we also know where we
are, you know, little r.
00:50:01.720 --> 00:50:03.950
And we'll be using
that, shortly.
00:50:06.590 --> 00:50:08.060
To continue a little
further, we're
00:50:08.060 --> 00:50:10.185
going to introduce some
abbreviations just so we're
00:50:10.185 --> 00:50:12.100
don't have to write so much.
00:50:12.100 --> 00:50:16.760
I'm going to define
little h of r
00:50:16.760 --> 00:50:22.680
as just one minus r
Schwarzschild over r.
00:50:26.500 --> 00:50:33.580
And this is also 1 minus
2 GM over rc squared.
00:50:33.580 --> 00:50:36.030
That's a factor
that keeps recurring
00:50:36.030 --> 00:50:38.281
in our expression
for the metric.
00:50:38.281 --> 00:50:38.780
Yes?
00:50:38.780 --> 00:50:40.321
AUDIANCE: The second
to last equation
00:50:40.321 --> 00:50:43.837
is supposed to be a c squared
in between the two parenthesis?
00:50:43.837 --> 00:50:44.670
PROFESSOR: Probably.
00:50:47.780 --> 00:50:48.600
Yes, thank you.
00:50:52.610 --> 00:50:55.359
G squared, right?
00:50:55.359 --> 00:50:55.900
Thanks a lot.
00:51:09.660 --> 00:51:11.510
In terms of h of
r, we can rewrite
00:51:11.510 --> 00:51:15.370
that equation
slightly more simply.
00:51:15.370 --> 00:51:17.720
I'm going to bring
things to the other side
00:51:17.720 --> 00:51:20.970
and write it as c
squared times dt
00:51:20.970 --> 00:51:32.680
d tau squared is equal to c
squared h inverse of r plus h
00:51:32.680 --> 00:51:41.720
to the minus 2 of r
times dr d tau squared.
00:51:45.656 --> 00:51:48.710
This is just a rewriting
of the above equation,
00:51:48.710 --> 00:51:52.057
making use of the new notation
that we've introduced.
00:51:52.057 --> 00:51:53.640
And this is the form
we will be using.
00:51:53.640 --> 00:51:55.056
It explicitly tells
us how to find
00:51:55.056 --> 00:51:57.455
dt d tau in terms
of other things.
00:51:57.455 --> 00:52:00.450
So dt d tau is not independent.
00:52:04.750 --> 00:52:07.870
Since we know dt d tau
in terms of dr d tau.
00:52:07.870 --> 00:52:11.400
If We get an expression for dr
d tau we're sort of finished.
00:52:11.400 --> 00:52:14.420
We could find everything
we want to know about t
00:52:14.420 --> 00:52:17.250
from the equation we just wrote.
00:52:17.250 --> 00:52:20.440
So it turns out that
all we need to do
00:52:20.440 --> 00:52:23.340
to calculate this
radial trajectory
00:52:23.340 --> 00:52:26.710
is to look at the
component of the metric
00:52:26.710 --> 00:52:32.140
where that free index, mu, mu
is the index that's not summed,
00:52:32.140 --> 00:52:34.880
we're going to
set mu equal to r.
00:52:34.880 --> 00:52:37.740
Remember mu is a number that
corresponds to a coordinate.
00:52:37.740 --> 00:52:39.573
And we're going to set
mu equal to the value
00:52:39.573 --> 00:52:41.170
that corresponds to
the r coordinate.
00:52:41.170 --> 00:52:46.040
And that will be sufficient
to get us our answer.
00:52:46.040 --> 00:52:56.300
When we do that, the equation
becomes d d tau of g sub r.
00:52:56.300 --> 00:53:00.540
Now the second index, nu
in the original expression,
00:53:00.540 --> 00:53:04.965
is summed from 0 to 3
for the gr case, where
00:53:04.965 --> 00:53:07.180
we have four coordinates,
one time and three
00:53:07.180 --> 00:53:11.030
spatial coordinates,
but we only need
00:53:11.030 --> 00:53:16.210
to write the terms where
gr nu variable is non-zero.
00:53:16.210 --> 00:53:19.410
And the metric
itself is diagonal.
00:53:19.410 --> 00:53:22.700
So if one index is a
little r, the other index
00:53:22.700 --> 00:53:25.330
has to also be r,
or else it vanishes.
00:53:25.330 --> 00:53:27.960
So the only value of nu
that contributes to the sum
00:53:27.960 --> 00:53:30.890
is when nu is also equal
to the r coordinate.
00:53:30.890 --> 00:53:37.900
So we get g sub rr d
xr-- which, in fact I'll
00:53:37.900 --> 00:53:43.540
write it as just dr. x super r
is just the r coordinate, which
00:53:43.540 --> 00:53:54.974
we also just call r times
d tau is equal to 1/2 dr.
00:53:54.974 --> 00:53:57.270
And now, on the
right-hand side, we're
00:53:57.270 --> 00:54:00.780
summing over lambda and sigma.
00:54:00.780 --> 00:54:04.940
And lambda and sigma have to
have the property that g sub
00:54:04.940 --> 00:54:07.866
lambda sigma depends on r,
or else the first factor
00:54:07.866 --> 00:54:08.365
will vanish.
00:54:11.030 --> 00:54:14.466
And furthermore,
g sub lambda sigma
00:54:14.466 --> 00:54:16.590
has be non-zero, for the
values of lambda and sigma
00:54:16.590 --> 00:54:19.190
that you want, which means that
lambda and sigma for this case
00:54:19.190 --> 00:54:20.898
has to be equal to
each other, because we
00:54:20.898 --> 00:54:23.980
have no off-diagonal
terms to our metric.
00:54:23.980 --> 00:54:29.410
So the only contributions we get
are from g sub rr and g sub tt.
00:54:29.410 --> 00:54:37.280
So you get the derivative with
respect to r of g sub rr times
00:54:37.280 --> 00:54:42.740
dr d tau squared.
00:54:42.740 --> 00:54:46.980
This become squared, because
lambda is equal to sigma.
00:54:46.980 --> 00:55:04.070
And then plus 1/2 drg sub
tt times dt d tau squared.
00:55:15.860 --> 00:55:20.970
And note that buried in
here is, if we expand this,
00:55:20.970 --> 00:55:22.810
the second derivative
of r with respect
00:55:22.810 --> 00:55:26.330
to time-- respect to tau.
00:55:26.330 --> 00:55:30.760
So we can extract
that and solve for it.
00:55:30.760 --> 00:55:35.870
And things like dt d tau
will appear in our answer,
00:55:35.870 --> 00:55:38.856
initially, because
it's here already.
00:55:38.856 --> 00:55:43.220
But we could replace dt d
tau by this top equation
00:55:43.220 --> 00:55:46.090
and eliminate it
from our results.
00:55:46.090 --> 00:55:50.400
And I'm going to skip
the algebra, which
00:55:50.400 --> 00:55:52.600
is straightforward,
although tedious.
00:55:52.600 --> 00:55:55.950
I urge you to go
through it in the notes.
00:55:55.950 --> 00:55:58.580
But the end result ends up
being remarkably simple,
00:55:58.580 --> 00:56:02.530
after a number of cancellations
that look like surprises.
00:56:02.530 --> 00:56:05.730
And what you find in
the end-- and it's just
00:56:05.730 --> 00:56:09.200
the simplification of this
formula, nothing more-- you
00:56:09.200 --> 00:56:15.490
find that d squared
r d tau squared is
00:56:15.490 --> 00:56:22.810
just equal to minus Newton's
constant times the mass
00:56:22.810 --> 00:56:23.910
divided by r squared.
00:56:31.950 --> 00:56:34.100
Now this is rather shocking,
and even looks exactly
00:56:34.100 --> 00:56:35.285
like Newtonian mechanics.
00:56:38.129 --> 00:56:40.420
However, even though it looks
like Newtonian mechanics,
00:56:40.420 --> 00:56:43.670
it's not really the same
as Newtonian mechanics,
00:56:43.670 --> 00:56:47.330
because the variables don't
mean quite the same thing.
00:56:47.330 --> 00:56:49.850
First of all, even
r does not really
00:56:49.850 --> 00:56:55.390
mean radius in the same sense
as radius is defined by Newton.
00:56:55.390 --> 00:56:57.030
In Newtonian
mechanics, radius is
00:56:57.030 --> 00:56:58.860
the distance from the origin.
00:56:58.860 --> 00:57:01.560
If we wanted to know the
distance from the origin,
00:57:01.560 --> 00:57:04.690
we would have to
integrate this metric.
00:57:04.690 --> 00:57:07.214
And in fact, there isn't
even an actual origin here,
00:57:07.214 --> 00:57:09.380
because you would have to
go through the singularity
00:57:09.380 --> 00:57:10.254
before you get there.
00:57:10.254 --> 00:57:12.122
And you really can't.
00:57:12.122 --> 00:57:16.581
That integral is not
really even defined.
00:57:16.581 --> 00:57:18.830
Although, of course, if we
had something like the sun,
00:57:18.830 --> 00:57:20.871
where the metric was
different from this small r,
00:57:20.871 --> 00:57:22.850
then we could integrate
from r equals 0,
00:57:22.850 --> 00:57:24.910
and that would define
the true radius,
00:57:24.910 --> 00:57:26.450
distance from the center.
00:57:26.450 --> 00:57:27.480
But it would not be r.
00:57:27.480 --> 00:57:31.992
It would be what you got by
integrating with the metric.
00:57:31.992 --> 00:57:33.450
So r has a different
interpretation
00:57:33.450 --> 00:57:37.100
than it does for
Newtonian physics.
00:57:37.100 --> 00:57:40.310
I might add, it still has
a simple interpretation.
00:57:40.310 --> 00:57:46.260
If you look at this metric, the
tangential part, the angular
00:57:46.260 --> 00:57:50.970
part, is exactly what you
have for Euclidean geometry.
00:57:50.970 --> 00:57:54.050
It's just r squared times the
same combination of d theta
00:57:54.050 --> 00:57:57.820
and d phi as appears on
the surface of a sphere.
00:57:57.820 --> 00:58:01.870
So little r is sometimes called
the circumferential radius,
00:58:01.870 --> 00:58:06.790
because it really does give you
the circumference of circles
00:58:06.790 --> 00:58:09.160
at that radial coordinate.
00:58:09.160 --> 00:58:11.650
If we went around in
a circle at a fixed r,
00:58:11.650 --> 00:58:14.450
the circle would involve
varying phi, for example,
00:58:14.450 --> 00:58:16.440
over a range of 2
pi, we really would
00:58:16.440 --> 00:58:21.460
see a total circumference
of 2 pi little r.
00:58:21.460 --> 00:58:22.990
So r is related
to circumferences
00:58:22.990 --> 00:58:26.420
in exactly the way as it
is in Euclidean geometry.
00:58:26.420 --> 00:58:27.920
But it's not related
to the distance
00:58:27.920 --> 00:58:31.330
from the origin in the same way
as it is in Euclidean geometry.
00:58:34.350 --> 00:58:38.700
In addition, tau, here,
is not the universal time
00:58:38.700 --> 00:58:41.910
that Newton imagined.
00:58:41.910 --> 00:58:48.780
But rather, tau is measured
along the geodesic.
00:58:51.690 --> 00:58:53.190
It is just ds
squared, but remember,
00:58:53.190 --> 00:58:56.490
ds squared is being measured
along the geodesic, which
00:58:56.490 --> 00:58:58.450
means that it is, in
fact, the proper time
00:58:58.450 --> 00:59:01.550
as it would be measured
by the person falling
00:59:01.550 --> 00:59:05.310
with the object
towards the black hole.
00:59:05.310 --> 00:59:09.514
So tau is proper time as
measured by the falling object.
00:59:09.514 --> 00:59:10.930
And that follows
from what we know
00:59:10.930 --> 00:59:12.800
about the meaning of
the metric itself.
00:59:17.660 --> 00:59:21.767
OK, that said we would
now like to just study
00:59:21.767 --> 00:59:22.975
this equation more carefully.
00:59:50.417 --> 00:59:51.875
And since the
equation itself still
00:59:51.875 --> 00:59:55.150
has the same form as what
you get from Newton, if you
00:59:55.150 --> 00:59:57.700
remember what you would
have done if this was 801,
00:59:57.700 --> 01:00:00.480
you can, in fact, do
exactly the same thing here.
01:00:00.480 --> 01:00:03.230
And what you probably would
have done, if this was 801,
01:00:03.230 --> 01:00:08.270
would be to recognize that this
equation can be integrated.
01:00:08.270 --> 01:00:17.200
We can write the equation
as d d tau of 1/2
01:00:17.200 --> 01:00:29.720
dr d tau squared
minus GM/r equals 0.
01:00:29.720 --> 01:00:31.180
I you carry out
these derivatives
01:00:31.180 --> 01:00:32.796
you would get that equation.
01:00:32.796 --> 01:00:34.170
And this is just
the conservation
01:00:34.170 --> 01:00:38.780
of energy version of
the force equation.
01:00:55.410 --> 01:00:59.710
And that tells us that this
quantity is a constant.
01:00:59.710 --> 01:01:14.731
If we drop the object from
some initial position, r sub 0,
01:01:14.731 --> 01:01:18.040
and we drop it with
no initial velocity,
01:01:18.040 --> 01:01:21.800
we just let go of it at
r sub 0, that tells us
01:01:21.800 --> 01:01:24.620
what this quantity
is when we drop it.
01:01:24.620 --> 01:01:27.490
It's minus GM over r sub 0.
01:01:27.490 --> 01:01:30.120
This piece vanishes if there
is no initial velocity.
01:01:30.120 --> 01:01:32.036
And that means it will
always have that value.
01:01:35.390 --> 01:01:41.750
And knowing that, we
can write dr d tau
01:01:41.750 --> 01:01:46.000
is equal to-- just
solving for that--
01:01:46.000 --> 01:02:03.980
minus the square root of 2GM
times r0 minus r over r r0
01:02:03.980 --> 01:02:08.100
I've collected two terms and put
them over a common denominator
01:02:08.100 --> 01:02:09.480
and added them.
01:02:09.480 --> 01:02:12.440
So this is not quite as
obvious as it might be.
01:02:12.440 --> 01:02:15.004
But this is just the
statement that that quantity
01:02:15.004 --> 01:02:16.920
has the same value as
it did when you started.
01:02:23.700 --> 01:02:27.550
Now this can be
further integrated.
01:02:27.550 --> 01:02:35.840
We can write it as
dr over-- bringing
01:02:35.840 --> 01:02:39.420
all this to the other
side-- is equal to d tau.
01:02:39.420 --> 01:02:40.950
And then integrate both sides.
01:02:45.480 --> 01:02:47.850
Notice when I bring
this to the other side
01:02:47.850 --> 01:02:50.150
and bring the d tau to
the right., everything
01:02:50.150 --> 01:02:52.190
on the left-hand side
now only depends on r.
01:02:52.190 --> 01:02:56.080
So this is just an explicit
integral over r that we can do.
01:03:16.160 --> 01:03:20.120
And I will just tell you that
when the integral is done
01:03:20.120 --> 01:03:24.550
we get a formula for
tau as a function of r.
01:03:32.529 --> 01:03:35.570
And it's equal to
the square root
01:03:35.570 --> 01:03:46.920
of r sub 0 over
2GM times r0 times
01:03:46.920 --> 01:03:55.900
the inverse tangent of
the square root of r0
01:03:55.900 --> 01:04:08.070
minus r over r plus the square
root of r times r0 minus r.
01:04:14.500 --> 01:04:18.780
So when r equals r0, this gives
us 0, and that's what we want.
01:04:18.780 --> 01:04:24.470
When we start we're at r0,
or time 0, or proper time 0.
01:04:24.470 --> 01:04:29.280
And then as r gets smaller,
as it falls in, time grows.
01:04:29.280 --> 01:04:31.920
And this gives us the
time as a function of r.
01:04:31.920 --> 01:04:33.920
We might prefer to have
r as a function of time,
01:04:33.920 --> 01:04:37.440
but that formula can't really
be inverted analytically.
01:04:37.440 --> 01:04:38.710
So that's the best we can do.
01:04:43.680 --> 01:04:46.000
Now one thing that
you notice from this
01:04:46.000 --> 01:04:49.100
is that nothing
special happens as r
01:04:49.100 --> 01:04:51.907
decreases all the way to 0.
01:04:51.907 --> 01:04:53.490
Even when you plug
in r equals 0 here,
01:04:53.490 --> 01:04:55.270
you just get some finite number.
01:04:55.270 --> 01:04:58.270
So in a finite amount
of time, the observer
01:04:58.270 --> 01:05:00.790
would find himself falling
through the Schwarzchild
01:05:00.790 --> 01:05:04.830
horizon and all the
way to r equals 0.
01:05:04.830 --> 01:05:07.880
I didn't mention it but r
equals 0 is a true singularity.
01:05:07.880 --> 01:05:11.340
Our metric is also
singular when r equals 0.
01:05:11.340 --> 01:05:14.820
These quantities
all become infinite.
01:05:14.820 --> 01:05:19.610
And physically what
would happen is
01:05:19.610 --> 01:05:24.530
that, as the object falling
in approaches r equals 0,
01:05:24.530 --> 01:05:26.732
the tidal forces,
that is the difference
01:05:26.732 --> 01:05:29.190
in the gravitational force on
one part of the object verses
01:05:29.190 --> 01:05:32.810
another, will get
stronger and stronger.
01:05:32.810 --> 01:05:35.310
And objects will
just be ripped apart.
01:05:35.310 --> 01:05:40.050
And the ripping apart occurs as
being spaghetti-ized, that is,
01:05:40.050 --> 01:05:42.054
the force on the front
gets to be very strong
01:05:42.054 --> 01:05:43.470
compared to the
force on the back.
01:05:43.470 --> 01:05:46.011
So I'll just get stretched out
along the direction of motion.
01:05:53.500 --> 01:05:56.920
Now the curious
thing is what this
01:05:56.920 --> 01:05:59.240
looks like if we think
of it not as a function
01:05:59.240 --> 01:06:01.140
of the proper time
measured by the wrist
01:06:01.140 --> 01:06:05.950
watch of the object
falling in but rather,
01:06:05.950 --> 01:06:10.070
we could try to describe it
in terms of our external time
01:06:10.070 --> 01:06:11.010
variable.
01:06:11.010 --> 01:06:15.800
The variable t that appears
in the Schwarzchild metric.
01:06:15.800 --> 01:06:18.470
And to do that, to
make the conversion,
01:06:18.470 --> 01:06:23.630
we want to calculate what the
dr dt is, instead of dr d tau.
01:06:23.630 --> 01:06:26.450
Like maybe an analogous
formula, in terms of t.
01:06:29.690 --> 01:06:33.540
And to get that, we use
simply chain rule here.
01:06:33.540 --> 01:06:38.080
dr dt is equal to dr d
tau-- which we've already
01:06:38.080 --> 01:06:45.540
calculated-- times d tau dt.
01:06:45.540 --> 01:06:49.870
And d tau dt is 1 over dt d tau.
01:06:49.870 --> 01:06:52.330
If you just have two variables
that depend on each other.
01:06:52.330 --> 01:06:56.490
The derivatives are just
the inverse of each other.
01:06:56.490 --> 01:07:00.220
So this could be written
as dr d tau-- which
01:07:00.220 --> 01:07:04.430
we've calculated--
divided by dt d tau.
01:07:07.410 --> 01:07:10.900
And dt d tau we've really
already calculated as well,
01:07:10.900 --> 01:07:13.740
because it's just given
by this formula here.
01:07:26.530 --> 01:07:29.470
So we could write out what
that is and figure out
01:07:29.470 --> 01:07:33.550
how it's going to behave
as the object approaches
01:07:33.550 --> 01:07:34.645
the Schwarzchild radius.
01:07:54.430 --> 01:08:00.080
So it becomes dr dt
is equal to, I'll
01:08:00.080 --> 01:08:07.520
just write the numerator as dr
d tau given by that expression.
01:08:07.520 --> 01:08:10.180
But what's behaving
in a more peculiar way
01:08:10.180 --> 01:08:17.890
is the denominator, which
is h inverse of r plus c
01:08:17.890 --> 01:08:29.820
to the minus 2, h to the minus
2 of r times dr d tau squared.
01:08:35.149 --> 01:08:39.160
So now we want to look at
this function h inverse of r.
01:08:39.160 --> 01:08:41.145
And this just means 1/h of r.
01:08:41.145 --> 01:08:44.460
It doesn't mean
functional inverse.
01:08:44.460 --> 01:08:51.773
That is just equal to r
over r minus r Schwarzchild.
01:08:51.773 --> 01:08:53.439
And we're going to
be interested in what
01:08:53.439 --> 01:08:56.255
happens when r gets to be
very near r Schwarzchild,
01:08:56.255 --> 01:08:58.950
because that's where the
interesting things happen,
01:08:58.950 --> 01:09:02.148
as you're approaching
the Schwarzchild horizon.
01:09:02.148 --> 01:09:04.189
And that means that the
behavior of the numerator
01:09:04.189 --> 01:09:04.890
won't be important.
01:09:04.890 --> 01:09:06.264
The denominator
will be going up,
01:09:06.264 --> 01:09:08.399
and that's what will
control everything.
01:09:08.399 --> 01:09:11.899
So we can approximate
this as just
01:09:11.899 --> 01:09:16.510
r Schwarzchild over r
minus r Schwarzchild.
01:09:16.510 --> 01:09:18.540
And this is for r
near r Schwarzchild.
01:09:22.653 --> 01:09:24.444
We've replaced the
numerator by a constant.
01:09:28.520 --> 01:09:30.069
And then if we look
at this formula,
01:09:30.069 --> 01:09:34.319
this is going to blow up
as we approach the horizon.
01:09:34.319 --> 01:09:35.970
This is the square
of that quantity.
01:09:35.970 --> 01:09:40.040
It will blow up faster than the
first power of that quantity.
01:09:40.040 --> 01:09:41.939
And therefore,
this will dominate,
01:09:41.939 --> 01:09:44.950
the denominator
of the expression.
01:09:44.950 --> 01:09:47.130
We can ignore this.
01:09:47.130 --> 01:09:50.840
When this dominates, the
dr d tau pieces cancel.
01:09:50.840 --> 01:09:51.520
So that's nice.
01:09:51.520 --> 01:09:54.770
We don't even need to think
about what the dr d tau is.
01:09:54.770 --> 01:10:06.890
And what we get near
the horizon is simply
01:10:06.890 --> 01:10:13.564
a factor of c times r
minus r Schwarzchild over r
01:10:13.564 --> 01:10:14.105
Schwarzchild.
01:10:18.290 --> 01:10:20.770
It's basically just h.
01:10:20.770 --> 01:10:22.810
This becomes upstairs
with a plus sign.
01:10:22.810 --> 01:10:25.470
And the square root turns it
into h instead of h squared.
01:10:29.350 --> 01:10:30.700
So this is the inverse of that.
01:10:37.220 --> 01:10:40.290
OK, now if we try to
play the same game here
01:10:40.290 --> 01:10:44.780
as we did here, to determine
what our time variable behaves
01:10:44.780 --> 01:10:48.180
as a function of r, instead
of the proper time variable
01:10:48.180 --> 01:10:59.340
tau, what we find is that t
of r-- this is for r near r
01:10:59.340 --> 01:11:07.830
Schwarzchild-- is
about equal to minus
01:11:07.830 --> 01:11:18.895
r sub s over c times
the integral up to r
01:11:18.895 --> 01:11:25.665
of dr prime over
r prime minus rs.
01:11:37.080 --> 01:11:37.915
This is dr dt.
01:11:42.126 --> 01:11:43.990
Yeah, this was dr dt
from the beginning.
01:11:43.990 --> 01:11:46.200
I forgot to write the r somehow.
01:11:46.200 --> 01:11:47.659
AUDIANCE: Doesn't
that [INAUDIBLE]?
01:11:47.659 --> 01:11:50.283
PROFESSOR: Yeah, I didn't write
the lower limit of integration.
01:11:50.283 --> 01:11:51.800
I was about to comment on that.
01:11:51.800 --> 01:11:54.700
The integrand that we're writing
is only a good approximation
01:11:54.700 --> 01:11:57.910
whenever we're near r.
01:11:57.910 --> 01:12:01.030
So whatever happens near the
lower limit of integration,
01:12:01.030 --> 01:12:02.830
we just haven't done accurately.
01:12:02.830 --> 01:12:06.010
So I'm going to just not write
a lower limit of integration
01:12:06.010 --> 01:12:08.020
here, meaning that
we're interested only
01:12:08.020 --> 01:12:12.586
in what happens as the
upper limit of integration r
01:12:12.586 --> 01:12:13.960
becomes very near
r Schwarzchild.
01:12:13.960 --> 01:12:15.626
And everything will
be dominated by what
01:12:15.626 --> 01:12:18.067
happens near the upper
limit of integration.
01:12:18.067 --> 01:12:19.817
AUDIANCE: So would you
just integrate over
01:12:19.817 --> 01:12:21.380
on [INAUDIBLE] for that?
01:12:21.380 --> 01:12:22.963
PROFESSOR: That's
right, that's right.
01:12:22.963 --> 01:12:26.640
We just integrated over a small
region near, r Schwarzchild.
01:12:53.670 --> 01:12:56.050
Nu r, which is also about
equal to r Schwarzchild.
01:12:59.530 --> 01:13:02.280
And the point is, that this
diverges logarithmically
01:13:02.280 --> 01:13:05.680
as r approaches r Schwarzchild.
01:13:05.680 --> 01:13:14.920
So it behaves approximately
as minus r Schwarzchild
01:13:14.920 --> 01:13:20.640
over c times the logarithm
of r minus r Schwarzchild.
01:13:25.960 --> 01:13:28.220
So as r approaches
r Schwarzchild,
01:13:28.220 --> 01:13:30.820
this quantity that's the
argument of the logarithm
01:13:30.820 --> 01:13:32.470
gets closer and closer to 0.
01:13:32.470 --> 01:13:36.294
It gets smaller and
smaller approaching 0.
01:13:36.294 --> 01:13:37.960
But the logarithm of
a very small number
01:13:37.960 --> 01:13:40.672
is a negative number, a
large negative number.
01:13:40.672 --> 01:13:42.130
And then there's
a minus sign here.
01:13:42.130 --> 01:13:44.680
You get a large positive
number and it diverges.
01:13:44.680 --> 01:13:48.020
As r approaches r
Schwarzchild the time variable
01:13:48.020 --> 01:13:50.520
approaches infinity.
01:13:50.520 --> 01:13:53.590
And that means that
at no finite time
01:13:53.590 --> 01:13:57.665
does the object ever reach
the Schwarzchild horizon.
01:13:57.665 --> 01:14:00.510
But as seen from the outside,
it takes an infinite amount
01:14:00.510 --> 01:14:04.100
of time for the object to
reach the Schwarzchild horizon.
01:14:04.100 --> 01:14:07.600
As time gets larger and larger,
the object gets and closer
01:14:07.600 --> 01:14:11.200
to the Schwarzchild horizon,
asymptotically approaching it
01:14:11.200 --> 01:14:12.809
but never reaching it.
01:14:12.809 --> 01:14:14.350
So this, of course,
is very peculiar,
01:14:14.350 --> 01:14:17.620
because from the point
of view of the person
01:14:17.620 --> 01:14:19.500
falling into the
black hole, all this
01:14:19.500 --> 01:14:21.984
just happens in a finite amount
of time and is over with.
01:14:21.984 --> 01:14:23.400
From the outside,
it looks like it
01:14:23.400 --> 01:14:25.480
takes an infinite
amount of time.
01:14:25.480 --> 01:14:28.920
And weird things like this can
happen because of the fact that
01:14:28.920 --> 01:14:33.250
in general relativity time is
a locally measured variable.
01:14:33.250 --> 01:14:35.490
You measure your time,
I measure my time.
01:14:35.490 --> 01:14:36.810
They don't have to agree.
01:14:36.810 --> 01:14:40.510
And in this case, they can
disagree by an infinite amount,
01:14:40.510 --> 01:14:44.460
which is rather bizarre,
but that's what happens.
01:14:44.460 --> 01:14:47.610
So according to classical
general relativity,
01:14:47.610 --> 01:14:50.200
when an object falls
into a black hole,
01:14:50.200 --> 01:14:53.020
from the point of view
of the object nothing
01:14:53.020 --> 01:14:56.900
special would happen as that
object crossed the Schwarzchild
01:14:56.900 --> 01:14:59.020
horizon.
01:14:59.020 --> 01:15:01.020
Everybody believed that
that was really the case
01:15:01.020 --> 01:15:02.570
until maybe a couple years ago.
01:15:02.570 --> 01:15:04.696
Now it's controversial,
actually.
01:15:04.696 --> 01:15:06.070
At the classical
level, everybody
01:15:06.070 --> 01:15:07.195
believes that's still true.
01:15:07.195 --> 01:15:08.880
I mean, classical
general relativity
01:15:08.880 --> 01:15:10.470
says that an object can fall
through the Schwarzchild
01:15:10.470 --> 01:15:12.180
horizon and then
nothing happens.
01:15:12.180 --> 01:15:14.820
It's not really a singularity.
01:15:14.820 --> 01:15:18.140
But the issue is that when
one incorporates, or attempts
01:15:18.140 --> 01:15:20.700
to incorporate, the effects of
quantum theory, which nobody
01:15:20.700 --> 01:15:23.660
really knows how to do in
a totally reliable way,
01:15:23.660 --> 01:15:27.330
then there are
indications that there's
01:15:27.330 --> 01:15:30.600
something dramatic happening
at the Schwarzchild horizon.
01:15:30.600 --> 01:15:33.100
The phrase that's often
used for what people think
01:15:33.100 --> 01:15:37.360
might be happening at the
horizon is the word firewall.
01:15:37.360 --> 01:15:40.240
So whether or not there is
a firewall at the horizon,
01:15:40.240 --> 01:15:42.932
is not settled at this point.
01:15:42.932 --> 01:15:44.890
Certainly, though,
classical general relativity
01:15:44.890 --> 01:15:46.750
does not predict the firewall.
01:15:46.750 --> 01:15:49.410
If it exists, all the arguments
that say it might exist
01:15:49.410 --> 01:15:51.450
are based on the quantum
physics of black holes,
01:15:51.450 --> 01:15:54.655
and black hole evaporation,
and things like that.
01:15:54.655 --> 01:15:56.030
As you know quantum
mechanically,
01:15:56.030 --> 01:15:57.740
the black holes are
not stable, either,
01:15:57.740 --> 01:16:01.210
if they evaporate--
as was derived
01:16:01.210 --> 01:16:05.479
by Stephen Hawking
in, I think, 1974.
01:16:05.479 --> 01:16:07.020
But that's strictly
a quantum effect.
01:16:07.020 --> 01:16:11.005
It would go to 0 as h bar
goes to 0, and, at the moment,
01:16:11.005 --> 01:16:13.580
we're only talking about
classical general relativity.
01:16:13.580 --> 01:16:16.220
So the black hole that we're
describing is perfectly stable.
01:16:16.220 --> 01:16:18.480
And nothing happens if you
fall through the horizon.
01:16:18.480 --> 01:16:20.120
Except from the
outside, it looks
01:16:20.120 --> 01:16:22.203
like it would take an
infinite amount of time just
01:16:22.203 --> 01:16:24.087
to reach the horizon.
01:16:24.087 --> 01:16:24.920
So we'll stop there.
01:16:24.920 --> 01:16:26.420
I guess I'm not
going to get to talk
01:16:26.420 --> 01:16:32.360
about the energy
associated with radiation.
01:16:32.360 --> 01:16:35.380
But we'll get to
that on Thursday.
01:16:35.380 --> 01:16:37.900
So see you folks on Thursday.