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[SQUEAKING]

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[RUSTLING]

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[CLICKING]

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SCOTT HUGHES: So we'll pick
up where we ended last time.

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00:00:12,040 --> 00:00:16,720
We're looking at the spacetime
of a compact spherical body,

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00:00:16,720 --> 00:00:19,060
working in what we call
Schwarzschild coordinates.

7
00:00:19,060 --> 00:00:22,810
We deduce that the line
element describing this body

8
00:00:22,810 --> 00:00:26,100
is of the form ds
squared equals negative e

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00:00:26,100 --> 00:00:29,350
to the 2 phi dt squared
plus tr squared divided

10
00:00:29,350 --> 00:00:34,780
by 1 minus 2g m of r
over r plus r squared d

11
00:00:34,780 --> 00:00:38,800
omega, where d omega is the
usual solid element, line

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00:00:38,800 --> 00:00:40,270
element.

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00:00:40,270 --> 00:00:41,050
And let's see.

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Yes, the body is
described, interior,

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00:00:48,910 --> 00:00:52,090
as a perfect fluid with
particular density profile

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00:00:52,090 --> 00:00:55,537
rho of r, a pressure
profile p of r.

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00:00:55,537 --> 00:00:56,620
That describes this thing.

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Everywhere inside some radius r
star, which gives its surface.

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00:01:03,660 --> 00:01:05,260
Sorry, I just distracted myself.

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00:01:05,260 --> 00:01:09,335
I don't know why I've
been calling this d omega.

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Should be d omega squared.

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Whatever.

23
00:01:16,440 --> 00:01:19,380
Fluff in that notation.

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00:01:19,380 --> 00:01:23,040
OK, so in the exterior of
this thing-- so for everywhere

25
00:01:23,040 --> 00:01:25,770
for r greater than our
star, it is vacuum.

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There is no density.

27
00:01:26,700 --> 00:01:28,860
There is no pressure.

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00:01:28,860 --> 00:01:31,440
Once we're outside this
thing, the only mass you see

29
00:01:31,440 --> 00:01:32,730
is the mass of the star.

30
00:01:32,730 --> 00:01:35,890
So another way of saying this
is that when-- you know what?

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00:01:35,890 --> 00:01:38,500
I'll get to that
in just a moment.

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00:01:38,500 --> 00:01:42,870
And in the exterior, the e to
the 2, 5 becomes 1 minus 2g,

33
00:01:42,870 --> 00:01:45,240
that same mass over r.

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00:01:45,240 --> 00:01:51,250
Everywhere in the interior,
the pressure, the function phi,

35
00:01:51,250 --> 00:01:53,660
and the mass r are governed
by these equations.

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00:01:53,660 --> 00:01:56,130
So the mass, as I
described last time,

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00:01:56,130 --> 00:01:58,913
it looks like a deceptively
simple spherical integral.

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00:01:58,913 --> 00:02:00,580
But just be aware
that when you do this,

39
00:02:00,580 --> 00:02:02,523
you are not integrating
over a proper volume.

40
00:02:02,523 --> 00:02:04,440
If you were to integrate
over a proper volume,

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00:02:04,440 --> 00:02:08,370
you would get a larger mass,
and such a mass, in fact,

42
00:02:08,370 --> 00:02:10,139
does have meaning to it.

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00:02:10,139 --> 00:02:14,025
And the difference between
that mass and this mass

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tells you something
about how gravitationally

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bound this object is.

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And these two equations
actually have a Newtonian limit

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associated with them.

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So this is a
relativistic version

49
00:02:27,280 --> 00:02:30,890
of the equation of
hydrostatic equilibrium,

50
00:02:30,890 --> 00:02:34,776
and this has a simple
Newtonian analog.

51
00:02:34,776 --> 00:02:36,450
Did they drop the minus sign?

52
00:02:36,450 --> 00:02:37,400
No.

53
00:02:37,400 --> 00:02:39,860
This has a simple
Newtonian analog,

54
00:02:39,860 --> 00:02:41,810
describing the
gravitational potential

55
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inside a fluid object.

56
00:02:43,255 --> 00:02:44,630
These whole things
taken together

57
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are called the
Tolman-Oppenheimer-Volkoff

58
00:02:47,390 --> 00:02:51,200
equations, or the TOV equations.

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00:02:51,200 --> 00:03:01,100
So the comment I was making
was that we require m of r star

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to equal m, the total
mass of this object.

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So that's another
condition on this.

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Once you integrate up,
then you switch over

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to the mass, the
total mass, that

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is used in the
exterior of the star.

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00:03:13,080 --> 00:03:20,490
So if you wish to
solve these things,

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what you basically do is just
choose a central density.

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You have to choose
an equation of state,

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which allows you to relate
the pressure to the density.

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00:03:44,690 --> 00:03:48,830
We'll talk about that a little
bit more later in this class.

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00:03:48,830 --> 00:03:52,610
And then you just
start integrating.

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00:03:52,610 --> 00:04:01,510
So you basically then
integrate until you find

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that the pressure equals 0.

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The radius at which this occurs
defines the star surface.

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00:04:25,320 --> 00:04:27,810
As you integrate along,
this allows you to build up.

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00:04:27,810 --> 00:04:30,990
You then build the mass profile
of the star, the pressure

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00:04:30,990 --> 00:04:33,090
profile of the star.

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00:04:33,090 --> 00:04:37,830
You are building up how the phi
changes as you integrate along

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00:04:37,830 --> 00:04:39,180
the star.

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00:04:39,180 --> 00:04:40,980
So note, if you
integrate this up,

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this is only defined up to
a constant of integration.

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00:04:44,320 --> 00:04:45,750
And so what you
then need to do is

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00:04:45,750 --> 00:04:48,997
once you have reach the surface
of the star, you know m of r.

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00:04:48,997 --> 00:04:50,580
And so what you're
going to need to do

84
00:04:50,580 --> 00:04:54,030
is adjust the phi that
you found in order

85
00:04:54,030 --> 00:04:58,440
to match with the exterior
solution 1 minus 2gm over r.

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00:04:58,440 --> 00:05:00,750
This is an exercise you will
do on an upcoming homework

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assignment.

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This is one of my favorite
assignments in the class.

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00:05:03,780 --> 00:05:06,600
It's a really good
chance to actually see

90
00:05:06,600 --> 00:05:10,540
the way we solve
these equations.

91
00:05:10,540 --> 00:05:13,170
And this kind of
an exercise, it's

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done all the time as
part of modern research,

93
00:05:15,900 --> 00:05:18,300
and you will do this for
a particularly simple kind

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of equation of state.

95
00:05:21,170 --> 00:05:24,250
So I want to look at what some
examples of objects like this

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00:05:24,250 --> 00:05:27,010
actually look like.

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00:05:27,010 --> 00:05:31,230
So what we're going
to do is consider

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00:05:31,230 --> 00:05:34,620
an unrealistic but
instructive idealized limit.

99
00:05:41,650 --> 00:05:49,750
So imagine a star that
has rho equal constant.

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That is something where no
matter how hard you squeeze it,

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00:05:55,880 --> 00:05:58,280
you cannot change its density.

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The only way that
that can happen

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is if you have an object
that is infinitely stiff.

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This corresponds to a speed
of sound, which is defined

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as dpd rho that is infinite.

106
00:06:20,130 --> 00:06:21,510
Now, of course,
no speed of sound

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can actually exceed
the speed of light.

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00:06:24,160 --> 00:06:28,000
So this is a somewhat
pathological object.

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Nonetheless, it's useful
for us for the simple reason

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that the mass function
that emerges for this

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is quite trivial.

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Rho is constant, so m of r
just goes as the volume inside

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radius r times rho.

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00:07:02,560 --> 00:07:10,620
The star then has a
total mass of 4/3 pi r,

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00:07:10,620 --> 00:07:13,370
to be pi rho r star cubed.

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00:07:16,680 --> 00:07:19,290
So we know how its density
behaves, it's a constant,

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we know how the mass
function behaves.

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The challenge is solving
for the pressure profile.

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So the pressure is governed
by taking this differential

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00:07:36,560 --> 00:07:41,800
equation and basically
plug in that mass,

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00:07:41,800 --> 00:07:45,130
plug in this mass function,
and see what you get.

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00:07:45,130 --> 00:08:17,220
So if I go and I throw
this guy in here,

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00:08:17,220 --> 00:08:18,990
this is what you end up getting.

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00:08:18,990 --> 00:08:26,816
So bear in mind as
you go through this--

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yeah, I see what I did here.

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So what I do as I just
pulled out my factor of 4/3,

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4/3 pi rho r cubed-- excuse
me, my 4/3 pi r cubed.

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00:08:38,043 --> 00:08:39,460
I factor that out,
and then cancel

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00:08:39,460 --> 00:08:43,070
it an overall factor of r
squared in the denominator.

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00:08:43,070 --> 00:08:45,980
So this is simply
what I have over here

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with this mass function defined.

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We want to solve this to find p.

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This is one of
those rare moments

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where nature and analysis
conspire and a miracle occurs.

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OK, it's a somewhat messy
looking kind of solution,

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but nonetheless, it turns
out that the pressure profile

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is determined by-- you can
manipulate this equation.

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00:09:48,130 --> 00:09:49,480
You can integrate it up.

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00:09:49,480 --> 00:09:52,660
And this ends up describing
what the pressure

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00:09:52,660 --> 00:09:59,950
profile looks like, p sub c
is the pressure at r equals 0.

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00:10:04,570 --> 00:10:06,790
We define the
surface of the star

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00:10:06,790 --> 00:10:11,050
as being the location at
which the pressure goes to 0.

143
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So by using that, so by
exploiting that condition,

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we can use this equation
to make a mapping

145
00:10:20,650 --> 00:10:24,700
between the radius of the
star and the central pressure.

146
00:10:24,700 --> 00:10:26,680
So it's a spherically
symmetric thing.

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00:10:26,680 --> 00:10:29,680
Once we have chosen
what this density is,

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00:10:29,680 --> 00:10:32,720
then there is essentially a one
parameter family of solutions.

149
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We-- if we choose a central
pressure, that determines

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00:10:37,180 --> 00:10:38,500
what the radius would be.

151
00:10:38,500 --> 00:10:40,737
Conversely, if we want to
have a particular radius,

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00:10:40,737 --> 00:10:42,820
that determines what the
central pressure must be.

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00:10:48,473 --> 00:10:49,890
Let's just do a
little of analysis

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00:10:49,890 --> 00:10:51,240
that follows from this.

155
00:10:51,240 --> 00:10:57,480
So p equals 0 at
r equals r star.

156
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That defines the surface.

157
00:11:00,870 --> 00:11:04,290
So putting all
that together, you

158
00:11:04,290 --> 00:11:06,210
can manipulate this
equation to find

159
00:11:06,210 --> 00:11:19,690
that the radius of
the star is determined

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00:11:19,690 --> 00:11:26,290
from the central
pressure, like so.

161
00:11:26,290 --> 00:11:27,760
If you prefer,
you can write this

162
00:11:27,760 --> 00:11:30,490
as an equation for the
central pressure in terms

163
00:11:30,490 --> 00:11:31,630
of the radius of the star.

164
00:12:08,870 --> 00:12:11,510
I've written this
in terms of m tot

165
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but m tot is, of
course, simply related

166
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to this radius-- excuse
me, to this density

167
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and the star's radius cubed.

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So what's the
importance of this?

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00:12:26,270 --> 00:12:32,510
This is, as I've emphasized,
a fairly idealized problem.

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Nature will never
give us an object

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00:12:34,280 --> 00:12:36,110
that has constant density.

172
00:12:36,110 --> 00:12:38,810
Any object, if you give it
a little bit of a squeeze,

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00:12:38,810 --> 00:12:40,850
the density will change.

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Causality requires
that the speed of sound

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be less than the speed of light.

176
00:12:46,470 --> 00:12:51,420
So clearly, this is a
somewhat fictional limit.

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00:12:51,420 --> 00:12:53,870
But we can learn something
very interesting about this.

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00:12:53,870 --> 00:12:59,660
Notice that this formula
for the central pressure,

179
00:12:59,660 --> 00:13:04,160
it diverges for a particular
compactness of the star.

180
00:13:10,310 --> 00:13:11,973
So p usb c goes to infinity.

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00:13:11,973 --> 00:13:13,640
Perhaps a little bit
more easily to see,

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the denominator goes to 0, for
a certain compactness, m over r.

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00:13:31,840 --> 00:13:35,860
So let's look at the value
at which the denominator goes

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to 0.

185
00:13:48,050 --> 00:13:49,420
So let's see.

186
00:13:49,420 --> 00:13:50,700
Move my 1 to the other side.

187
00:13:50,700 --> 00:13:51,640
Divide by 3.

188
00:13:51,640 --> 00:13:52,230
Square it.

189
00:14:02,960 --> 00:14:03,720
Rearrange terms.

190
00:14:17,480 --> 00:14:24,960
So if the ratio of this star's
total mass to its radius

191
00:14:24,960 --> 00:14:33,130
is such that gm over
r star exceeds 4/9

192
00:14:33,130 --> 00:14:36,165
than the pressure diverges.

193
00:14:40,510 --> 00:14:51,640
What this tells me is that I
cannot construct a physically

194
00:14:51,640 --> 00:14:57,490
allowable static object, even
using this stuff as we can

195
00:14:57,490 --> 00:15:02,050
imagine fluid, a fluid that
has an infinite sound speed.

196
00:15:02,050 --> 00:15:05,560
I cannot make a star
more compact than that.

197
00:15:09,010 --> 00:15:13,090
Making it that compact requires
infinite pressure at the core.

198
00:15:15,720 --> 00:15:19,080
This implies that stars
have a maximum compactness.

199
00:15:39,560 --> 00:15:59,510
We cannot have physically
realizable pressure profiles

200
00:15:59,510 --> 00:16:00,410
if--

201
00:16:00,410 --> 00:16:01,900
and let's turn that around--

202
00:16:01,900 --> 00:16:09,290
if the ratio of the
radius to g times

203
00:16:09,290 --> 00:16:14,840
the total mass is
smaller than 9/4.

204
00:16:14,840 --> 00:16:18,740
Now, this holds for the stiffest
possible fluid that we can even

205
00:16:18,740 --> 00:16:24,030
imagine, one which the laws of
physics actually do not permit.

206
00:16:24,030 --> 00:16:26,390
And so one infers
from this, actually,

207
00:16:26,390 --> 00:16:29,600
that this bound
basically tells me

208
00:16:29,600 --> 00:16:34,760
that given any physical fluid,
any physically realizable star

209
00:16:34,760 --> 00:16:37,820
that I can construct,
I must have

210
00:16:37,820 --> 00:16:39,590
a maximum allowed compactness.

211
00:16:42,420 --> 00:16:46,230
Putting this a little
bit more precisely

212
00:16:46,230 --> 00:16:51,740
brings us to a result that is
known as Buchdahl's theorem.

213
00:17:10,236 --> 00:17:11,819
Buchdahl's theorem
tells me that there

214
00:17:11,819 --> 00:17:33,600
is no stable spherical
fluid configuration

215
00:17:33,600 --> 00:17:41,390
in which the
configuration's radius is

216
00:17:41,390 --> 00:17:45,260
smaller than 9/4 of gm total.

217
00:17:48,220 --> 00:17:51,250
For those of you who like to
put factors of c in there,

218
00:17:51,250 --> 00:17:54,400
divide by c squared, and
this tells you something

219
00:17:54,400 --> 00:17:57,520
that you can convert
to SI units, which

220
00:17:57,520 --> 00:18:00,490
tells you how small you are
allowed to make an object.

221
00:18:04,390 --> 00:18:05,620
So this is clear.

222
00:18:05,620 --> 00:18:09,460
It emerges, and very nicely, in
this idealized but unphysical

223
00:18:09,460 --> 00:18:10,660
fluid limit.

224
00:18:10,660 --> 00:18:13,510
But it can be proven
more generally.

225
00:18:13,510 --> 00:18:16,700
You just have to make a few
assumptions about the way

226
00:18:16,700 --> 00:18:20,980
that the pressure profile is
not singular in any place.

227
00:18:24,840 --> 00:18:31,230
Proof of this can be found in
the beautiful, old textbook

228
00:18:31,230 --> 00:18:39,700
by Weinberg, section 11.6.

229
00:18:39,700 --> 00:18:42,340
Well, when you
hear something like

230
00:18:42,340 --> 00:18:44,000
that, you gotta
think to yourself,

231
00:18:44,000 --> 00:18:48,970
well, suppose I made a star
with some kind of a fluid

232
00:18:48,970 --> 00:18:55,150
and I gave it a radius of
10/4 gm total over c squared.

233
00:18:55,150 --> 00:18:57,280
And I just came along
and squeezed it.

234
00:18:57,280 --> 00:18:59,730
What would happen?

235
00:18:59,730 --> 00:19:04,750
Well, notice the word
"stable" in this definition.

236
00:19:04,750 --> 00:19:07,210
You're free to do
that, and should you

237
00:19:07,210 --> 00:19:11,110
do so, you would simply no
longer have a stable object.

238
00:19:11,110 --> 00:19:14,170
So remember, part of what
went into this analysis

239
00:19:14,170 --> 00:19:18,580
is we were assuming the
spacetime and the fluid that

240
00:19:18,580 --> 00:19:20,020
is the source of
the spacetime, we

241
00:19:20,020 --> 00:19:21,540
were assuming
everything is static.

242
00:19:21,540 --> 00:19:24,460
OK, we're making sure
everything just sits still.

243
00:19:24,460 --> 00:19:26,920
This is telling us you
can't do that if you

244
00:19:26,920 --> 00:19:28,840
want to have a star
as compact as that.

245
00:19:28,840 --> 00:19:31,930
So if you were to do
this, it would collapse.

246
00:19:31,930 --> 00:19:34,630
It would become dynamical
and the spacetime

247
00:19:34,630 --> 00:19:38,820
would transition
into something else.

248
00:19:38,820 --> 00:19:40,425
What that something
else might be

249
00:19:40,425 --> 00:19:44,668
will be a topic that we
get into a little bit more

250
00:19:44,668 --> 00:19:46,710
after we have developed
some additional material.

251
00:19:52,270 --> 00:19:54,010
So let's talk a little bit.

252
00:19:54,010 --> 00:19:57,270
This will help you with the
homework assignment, where

253
00:19:57,270 --> 00:20:01,860
you guys are going to construct
relativistic stellar models.

254
00:20:06,570 --> 00:20:12,600
Let's talk a little bit about
how we describe real objects.

255
00:20:12,600 --> 00:20:20,600
They are not of
constant density,

256
00:20:20,600 --> 00:20:26,910
and they instead
have some p that

257
00:20:26,910 --> 00:20:30,600
is a function of the density.

258
00:20:30,600 --> 00:20:34,890
It's worth noting that in
an even more general case--

259
00:20:34,890 --> 00:20:36,570
this is actually
worth a brief aside--

260
00:20:42,480 --> 00:20:46,950
the equation of state relates
the pressure to local density

261
00:20:46,950 --> 00:20:49,140
and s, where s is the entropy.

262
00:20:55,210 --> 00:20:58,210
For the kind of applications
where general relativity tends

263
00:20:58,210 --> 00:21:01,660
to be important, for
instance, when we're studying

264
00:21:01,660 --> 00:21:05,860
the stellar structure
of a neutron star,

265
00:21:05,860 --> 00:21:09,445
the fluid ends up being so
cold that you don't need

266
00:21:09,445 --> 00:21:10,570
to worry about the entropy.

267
00:21:10,570 --> 00:21:17,680
And where that comes
from is that when

268
00:21:17,680 --> 00:21:21,470
I revisit my first
law of thermodynamics

269
00:21:21,470 --> 00:21:26,620
and I include temperature and
entropy effects, so the term

270
00:21:26,620 --> 00:21:30,040
that I left out in my
earlier accounting, TDS.

271
00:21:30,040 --> 00:21:33,100
If t is small, I can
ignore that term,

272
00:21:33,100 --> 00:21:34,600
and it ends up being
something where

273
00:21:34,600 --> 00:21:38,920
my local energy,
or energy density,

274
00:21:38,920 --> 00:21:40,960
ends up only depending
on the pressure.

275
00:21:40,960 --> 00:21:43,180
But "cold" is a wiggle word.

276
00:21:43,180 --> 00:21:45,880
I have to define a scale
to say whether an object is

277
00:21:45,880 --> 00:21:48,970
cold or hot.

278
00:21:48,970 --> 00:21:52,360
There's a few notes laying
this out right in my notes.

279
00:21:52,360 --> 00:21:53,695
Let me just sketch the key idea.

280
00:21:57,170 --> 00:22:03,100
Cold depends on the
fluids, and it's worth

281
00:22:03,100 --> 00:22:05,350
noting that the kind of fluid
you're playing with here

282
00:22:05,350 --> 00:22:07,070
tend to be made out of fermions.

283
00:22:07,070 --> 00:22:10,260
And so it depends on the
fluids' Fermi temperature.

284
00:22:16,840 --> 00:22:20,620
So the Fermi
temperature is defined

285
00:22:20,620 --> 00:22:27,700
as the Fermi energy normalized
to a Boltzmann factor.

286
00:22:27,700 --> 00:22:29,770
You get the Fermi
energy by looking

287
00:22:29,770 --> 00:22:35,500
at how the energy levels are
filled in your fermion fluid

288
00:22:35,500 --> 00:22:36,976
here.

289
00:22:36,976 --> 00:22:39,310
I have a few additional
notes to lay out

290
00:22:39,310 --> 00:22:43,540
the way in which you can relate
this to the local density,

291
00:22:43,540 --> 00:22:47,790
the mass of each particle that
goes into this Fermi fluid.

292
00:22:47,790 --> 00:22:55,080
The punchline is that
for neutron stars, one

293
00:22:55,080 --> 00:22:57,190
of the cases where
we do, in fact,

294
00:22:57,190 --> 00:23:01,800
make dense general
relativistic fluid stars,

295
00:23:01,800 --> 00:23:05,430
the Fermi temperature tends
to be on the order of 10

296
00:23:05,430 --> 00:23:08,970
to the 13 Kelvin, so
about 10 trillion Kelvin.

297
00:23:08,970 --> 00:23:17,940
When we actually
observe these objects,

298
00:23:17,940 --> 00:23:23,320
they are on the order of 10 to
the 6 to 10 to the 9 Kelvin.

299
00:23:23,320 --> 00:23:27,630
So they're a factor of about
10 of the 7, 10 to the 4 to 10

300
00:23:27,630 --> 00:23:30,810
to the 7 times colder than
the Fermi temperature.

301
00:23:30,810 --> 00:23:34,170
Even though they may be a
billion Kelvin, they are cold.

302
00:23:34,170 --> 00:23:37,380
So we're going to use what are
called cold equations of state

303
00:23:37,380 --> 00:23:38,340
to describe these guys.

304
00:23:42,910 --> 00:23:48,560
So with that out of the
way, let's talk a little bit

305
00:23:48,560 --> 00:23:51,620
about the kinds of equations of
state that we will tend to use.

306
00:23:58,160 --> 00:24:01,700
So people who study the
physics of dense matter,

307
00:24:01,700 --> 00:24:06,050
a lot of their lives is
really down to understanding

308
00:24:06,050 --> 00:24:08,998
what the equation of state of
that cold matter looks like.

309
00:24:08,998 --> 00:24:10,790
Some of them are
concerned about hot matter

310
00:24:10,790 --> 00:24:13,040
as well, in which case they
might be actually worrying

311
00:24:13,040 --> 00:24:15,433
about things at tens
of trillions of Kelvin.

312
00:24:15,433 --> 00:24:17,600
But if you're looking at
astrophysical applications,

313
00:24:17,600 --> 00:24:20,970
you're generally interested
in the cold matter.

314
00:24:20,970 --> 00:24:24,950
And so they end up putting the
other very complicated models

315
00:24:24,950 --> 00:24:31,100
using QCD and effective
field theories to try

316
00:24:31,100 --> 00:24:36,350
to understand how it is
that a particular fluid

317
00:24:36,350 --> 00:24:38,690
of dense matter, how its
pressure and its density

318
00:24:38,690 --> 00:24:39,590
are related.

319
00:24:39,590 --> 00:24:41,420
And what I'm sort of
wheeling around here

320
00:24:41,420 --> 00:24:44,600
is that you generally do not
have a simple analytic form.

321
00:24:44,600 --> 00:24:47,480
You wind up with some kind of
a fairly complicated function

322
00:24:47,480 --> 00:24:51,230
that emerges from a
numerical calculation.

323
00:24:51,230 --> 00:24:52,610
It often ends up being--

324
00:24:52,610 --> 00:24:55,143
if you are a user of
this equation of state--

325
00:24:55,143 --> 00:24:56,810
it ends up being in
the form of a table.

326
00:24:56,810 --> 00:24:58,100
So they might
actually just give you

327
00:24:58,100 --> 00:24:59,642
a file that's got
a bunch of numbers,

328
00:24:59,642 --> 00:25:02,510
which says if the
density is this,

329
00:25:02,510 --> 00:25:03,710
then the pressure is this.

330
00:25:03,710 --> 00:25:06,770
And you can fit little
functions to that

331
00:25:06,770 --> 00:25:10,500
that allow you to look things
up and do your calculations.

332
00:25:10,500 --> 00:25:12,260
But it's not in the
form of a clean thing

333
00:25:12,260 --> 00:25:15,528
that you can write
down on the blackboard.

334
00:25:15,528 --> 00:25:17,820
I want something clean I can
write down the blackboard.

335
00:25:17,820 --> 00:25:25,090
So I'm going to introduce
an approximation, which

336
00:25:25,090 --> 00:25:33,190
is useful for testing
things out, test cases,

337
00:25:33,190 --> 00:25:34,920
and for pedagogy.

338
00:25:34,920 --> 00:25:41,560
What we do is we
take the pressure

339
00:25:41,560 --> 00:25:53,200
to be a power law
of the density.

340
00:25:53,200 --> 00:26:00,410
So what we do is we
write p equals k rho

341
00:26:00,410 --> 00:26:04,930
0 to the gamma, where k
and gamma are constants.

342
00:26:11,760 --> 00:26:14,730
A form that looks like this,
this is called a polytrope.

343
00:26:21,720 --> 00:26:24,690
Now, the thing which I
particularly want to highlight,

344
00:26:24,690 --> 00:26:27,990
and for those of you who
are going to do this highly

345
00:26:27,990 --> 00:26:30,360
recommended homework
exercise, please

346
00:26:30,360 --> 00:26:33,750
pay attention at this point.

347
00:26:33,750 --> 00:26:39,420
This rho 0 is not--
oh, I erased it.

348
00:26:39,420 --> 00:26:42,140
This rho 0 is not the rho--

349
00:26:42,140 --> 00:26:43,050
oh, there it is--

350
00:26:43,050 --> 00:26:46,770
it's not the rho that appears
in the equation of state.

351
00:26:46,770 --> 00:26:48,690
It's slightly different.

352
00:26:48,690 --> 00:26:52,200
Rho 0 is not the rho that
appears, for instance,

353
00:26:52,200 --> 00:26:54,398
in the t of e equations.

354
00:26:58,890 --> 00:27:05,130
Rho 0 is what is called
the rest mass density.

355
00:27:10,550 --> 00:27:14,750
It does not take into account
the fact that if I take a big--

356
00:27:14,750 --> 00:27:17,540
let's say I've got a big
bucket of nuclear fluid.

357
00:27:17,540 --> 00:27:21,620
So I take my bucket here
and I squeeze down on it.

358
00:27:24,140 --> 00:27:26,360
When I squeeze
down, its density is

359
00:27:26,360 --> 00:27:28,310
going to increase,
first of all, because I

360
00:27:28,310 --> 00:27:30,560
have decreased the volume.

361
00:27:30,560 --> 00:27:33,200
So the number of
particles remains fixed,

362
00:27:33,200 --> 00:27:35,730
but I decrease the
space in there.

363
00:27:35,730 --> 00:27:38,930
But I have also done work on
it, because this thing exerts

364
00:27:38,930 --> 00:27:41,540
a pressure that
opposes my squeezing.

365
00:27:41,540 --> 00:27:43,940
And I need to take
into account the fact

366
00:27:43,940 --> 00:27:48,050
that the work I do in
squeezing this fluid

367
00:27:48,050 --> 00:27:50,030
increases the density rho.

368
00:28:26,740 --> 00:28:29,320
So when you write out
your t of e equations,

369
00:28:29,320 --> 00:28:31,460
rho is energy density.

370
00:28:31,460 --> 00:28:35,812
All forms of energy gravitate.

371
00:28:35,812 --> 00:28:37,520
This is just the way
people traditionally

372
00:28:37,520 --> 00:28:38,690
write the equation of state.

373
00:28:38,690 --> 00:28:42,260
This is when one is
doing nuclear physics.

374
00:28:42,260 --> 00:28:43,760
There's good reasons
for doing this,

375
00:28:43,760 --> 00:28:45,980
but it's not the
most convenient form

376
00:28:45,980 --> 00:28:48,410
for the kind of calculations
that we want to do,

377
00:28:48,410 --> 00:28:54,280
and that you are going to want
to do, in the problem set.

378
00:29:00,470 --> 00:29:02,892
Fortunately, it's not too
difficult to convert, so let

379
00:29:02,892 --> 00:29:04,350
me describe to you
how you do that.

380
00:29:19,150 --> 00:29:23,760
So we are going to use the
first law of thermodynamics

381
00:29:23,760 --> 00:29:25,932
in a form in which I've
written now a couple times.

382
00:29:25,932 --> 00:29:28,140
Interestingly, it showed up
in our cosmology lecture.

383
00:29:30,960 --> 00:29:39,210
So my first law tells me du
equals minus pressure dv.

384
00:29:39,210 --> 00:29:42,080
So this is my total
energy in a fluid element

385
00:29:42,080 --> 00:29:45,202
and this is the work
done on a fluid element.

386
00:29:45,202 --> 00:30:01,154
So rho is equal to the amount
of energy in a fiducial volume.

387
00:30:04,060 --> 00:30:07,525
My rest energy-- excuse
me, my rest density--

388
00:30:11,230 --> 00:30:15,210
is the rest energy of
every little body that

389
00:30:15,210 --> 00:30:20,485
goes into this per unit volume.

390
00:30:25,530 --> 00:30:41,940
This means that I can write
du as d rho over rho 0,

391
00:30:41,940 --> 00:30:48,133
provided I throw in an
extra factor of m rest

392
00:30:48,133 --> 00:30:49,300
to get the dimensions right.

393
00:30:51,900 --> 00:31:01,920
And I can write d
volume as d1 over rho 0,

394
00:31:01,920 --> 00:31:09,190
provided I throw in
that factor of m rest

395
00:31:09,190 --> 00:31:11,400
to get the dimensions right.

396
00:31:11,400 --> 00:31:15,900
I know this looks weird
but, it's perfectly valid.

397
00:31:15,900 --> 00:31:21,930
So I'm going to rewrite my
first law of thermodynamics

398
00:31:21,930 --> 00:31:33,550
as d rho over rho 0 equals
minus pd 1 over r 0.

399
00:31:47,750 --> 00:31:49,510
Let's manipulate
that right-hand side.

400
00:31:52,660 --> 00:31:55,370
So I'm going to assume
this polytropic form.

401
00:31:55,370 --> 00:31:59,400
I'm going to use p equals
k rho 0 to the gamma.

402
00:32:03,870 --> 00:32:05,700
But I'm going to
switch that around.

403
00:32:05,700 --> 00:32:13,220
I'm going to write this
as rho 0 equals p over k

404
00:32:13,220 --> 00:32:14,580
to the power of 1 over gamma.

405
00:32:17,110 --> 00:32:25,320
So when I do that, I
get d rho over rho 0

406
00:32:25,320 --> 00:32:39,780
equals kappa 1 over gamma
over kappa pdp over p to the 1

407
00:32:39,780 --> 00:32:42,330
plus 1 over gamma.

408
00:32:56,887 --> 00:32:57,970
Pardon me just one moment.

409
00:32:57,970 --> 00:32:59,710
I did something clever
in my notes here

410
00:32:59,710 --> 00:33:02,620
and I'm just trying to
make sure I understand

411
00:33:02,620 --> 00:33:04,470
what the hell I actually did.

412
00:33:09,670 --> 00:33:11,768
So I'm going to level with you.

413
00:33:11,768 --> 00:33:13,310
I've gone through
this several times.

414
00:33:13,310 --> 00:33:16,070
There's a step in the
calculation that at this point,

415
00:33:16,070 --> 00:33:18,475
I, for some stupid
reason, didn't write down.

416
00:33:18,475 --> 00:33:20,600
I'm going to trust I knew
what I was doing, though,

417
00:33:20,600 --> 00:33:22,730
because I know the
final result was right.

418
00:33:22,730 --> 00:33:26,790
You can integrate
up both sides here.

419
00:33:26,790 --> 00:33:28,910
Oh, I think I see what I did.

420
00:33:28,910 --> 00:33:29,410
OK.

421
00:33:29,410 --> 00:33:33,440
So you integrate
up both sides here.

422
00:33:33,440 --> 00:33:35,440
And what you find
is this becomes rho

423
00:33:35,440 --> 00:33:42,480
equals p over gamma
minus 1 plus a constant.

424
00:33:46,740 --> 00:33:48,510
Yeah, not 100% sure
how I actually did

425
00:33:48,510 --> 00:33:50,010
that, so my apologies on that.

426
00:33:52,957 --> 00:33:54,790
I'm going to assume I
knew what I was doing.

427
00:33:54,790 --> 00:33:58,078
I will try to fix this and
I may post an addendum here.

428
00:33:58,078 --> 00:34:00,120
The next step, actually,
is you want to determine

429
00:34:00,120 --> 00:34:01,120
what that constant is.

430
00:34:01,120 --> 00:34:03,090
So the way you
determine the constant

431
00:34:03,090 --> 00:34:08,380
is you take advantage of the
fact that rho goes to rho 0.

432
00:34:08,380 --> 00:34:11,320
The energy density becomes
the rest energy density

433
00:34:11,320 --> 00:34:12,719
if there is no pressure exerted.

434
00:34:16,429 --> 00:34:19,070
And so this gives us our
final relationship here,

435
00:34:19,070 --> 00:34:26,210
which is that rho equals rho
0 plus p over gamma minus 1.

436
00:34:36,800 --> 00:34:41,239
OK, I will double check how I
went from line 2 line 3 there,

437
00:34:41,239 --> 00:34:46,170
but the final thing that
I have boxed online for

438
00:34:46,170 --> 00:34:49,560
is, indeed, exactly what
you need to do in order

439
00:34:49,560 --> 00:34:52,147
to build a stellar model.

440
00:34:52,147 --> 00:34:54,480
I guess I've been emphasizing
this is something you will

441
00:34:54,480 --> 00:34:57,480
do on an upcoming problem set.

442
00:34:57,480 --> 00:34:58,980
Let me just sketch the recipe.

443
00:34:58,980 --> 00:35:01,025
I've said this verbally, but let
me just write it out explicitly

444
00:35:01,025 --> 00:35:01,525
here.

445
00:35:04,920 --> 00:35:09,990
So I will give you
an equation of state.

446
00:35:09,990 --> 00:35:18,740
You then need to pick
rho 0 at r equals 0.

447
00:35:21,650 --> 00:35:24,500
Using your equation of state
and using that relationship

448
00:35:24,500 --> 00:35:28,370
between rho and rho
0, this will give you

449
00:35:28,370 --> 00:35:30,650
rho at the center,
pressure at the center.

450
00:35:33,550 --> 00:35:44,760
Set m of r at the center to 0.

451
00:35:44,760 --> 00:35:47,470
I emphasize, again, that
this may seem obvious,

452
00:35:47,470 --> 00:35:50,418
but it is somewhat important
that you get it right.

453
00:35:50,418 --> 00:35:51,960
When you do this
homework assignment,

454
00:35:51,960 --> 00:35:53,502
I'll give you a
little hint as to how

455
00:35:53,502 --> 00:35:55,130
to build that in smoothly.

456
00:35:55,130 --> 00:35:56,130
It can be a little bit--

457
00:35:56,130 --> 00:35:59,370
I don't want to say
tricky, but it's worth

458
00:35:59,370 --> 00:36:00,900
thinking about a little bit.

459
00:36:00,900 --> 00:36:10,880
Then what you do is integrate
your equations for the pressure

460
00:36:10,880 --> 00:36:15,930
in the mass from r equals 0.

461
00:36:15,930 --> 00:36:18,660
And this cannot be
done analytically.

462
00:36:18,660 --> 00:36:20,650
You have to use a
numerical integrator.

463
00:36:34,380 --> 00:36:36,660
If you have never used
one of these before,

464
00:36:36,660 --> 00:36:39,090
I will give you a
Mathematica notebook that

465
00:36:39,090 --> 00:36:40,740
demonstrates how to use it.

466
00:36:40,740 --> 00:36:42,690
This is a skill that
is worth knowing.

467
00:36:42,690 --> 00:36:46,800
The plain truth of the matter
is that the class of problems

468
00:36:46,800 --> 00:36:51,473
that are amenable to purely
analytic solutions, those

469
00:36:51,473 --> 00:36:52,140
are interesting.

470
00:36:52,140 --> 00:36:53,015
They're illustrative.

471
00:36:53,015 --> 00:36:54,720
They're good to work with.

472
00:36:54,720 --> 00:36:57,780
But they tend to be
unphysical and they're just

473
00:36:57,780 --> 00:37:00,660
not the ones that are of
interest for many things

474
00:37:00,660 --> 00:37:03,915
that we study in science.

475
00:37:03,915 --> 00:37:07,770
So show while
you're doing this--

476
00:37:07,770 --> 00:37:10,170
this is not necessary
to make your model,

477
00:37:10,170 --> 00:37:11,630
but it's very useful to do this.

478
00:37:16,210 --> 00:37:25,960
You can also integrate,
whoops, d5 er from the center.

479
00:37:31,010 --> 00:37:47,690
So a caution is that you do
not know phi at r equals 0.

480
00:37:47,690 --> 00:37:50,480
So what you should do is just
temporarily set it equal to 0.

481
00:37:50,480 --> 00:37:52,855
And what you're going to be
doing then when you integrate

482
00:37:52,855 --> 00:37:55,640
this up is you will
calculate the delta phi that

483
00:37:55,640 --> 00:38:01,680
describes your model from the
center to the surface, which

484
00:38:01,680 --> 00:38:02,850
brings me to step 4.

485
00:38:18,860 --> 00:38:27,240
When you find p equals 0,
you've hit the surface.

486
00:38:38,190 --> 00:38:43,340
So what we do is we use
the fact that p of r

487
00:38:43,340 --> 00:38:51,950
equals 0 defines the
star's radius r star.

488
00:38:51,950 --> 00:39:03,080
Once you've done that, you
now know the total mass

489
00:39:03,080 --> 00:39:05,190
and the radius.

490
00:39:05,190 --> 00:39:07,400
So you will find, when
you're doing this,

491
00:39:07,400 --> 00:39:11,750
that your numerical integrator
is not super well-behaved

492
00:39:11,750 --> 00:39:13,310
as you approach the surface.

493
00:39:13,310 --> 00:39:16,610
This is a feature, not a bug.

494
00:39:16,610 --> 00:39:20,420
What's going on is that as you
begin to approach the surface,

495
00:39:20,420 --> 00:39:23,270
the gradient and the
pressure gets quite steep.

496
00:39:23,270 --> 00:39:25,370
And so the way one
numerically integrates

497
00:39:25,370 --> 00:39:29,441
a set of couple equations
like this is by, essentially,

498
00:39:29,441 --> 00:39:34,490
if you take advantage of
the fact that an integral,

499
00:39:34,490 --> 00:39:36,560
it's what you get by sort
of dividing things up

500
00:39:36,560 --> 00:39:41,052
into tiny little pieces and
add up like little rectangles.

501
00:39:41,052 --> 00:39:43,010
And when you're solving
a differential equation

502
00:39:43,010 --> 00:39:45,470
like this, you're essentially
taking the continuum solution--

503
00:39:45,470 --> 00:39:47,637
that you guys have learned
how to do in many cases--

504
00:39:47,637 --> 00:39:49,940
and you're approximating
it by a series of smaller

505
00:39:49,940 --> 00:39:53,300
and smaller finite steps.

506
00:39:53,300 --> 00:39:55,550
Because the gradient
in the pressure

507
00:39:55,550 --> 00:39:58,640
gets large as you
approach the surface,

508
00:39:58,640 --> 00:40:00,410
numerical integrators
typically try

509
00:40:00,410 --> 00:40:04,340
taking an infinite number of
infinitesimal steps, which

510
00:40:04,340 --> 00:40:07,550
makes the CPU sad, and so it's
likely to exit with an error

511
00:40:07,550 --> 00:40:08,780
condition.

512
00:40:08,780 --> 00:40:11,570
Generally, when
that has happened,

513
00:40:11,570 --> 00:40:13,520
you've gotten an
answer that's probably

514
00:40:13,520 --> 00:40:15,800
good to within a
part in a million,

515
00:40:15,800 --> 00:40:17,240
or something like that.

516
00:40:17,240 --> 00:40:19,400
Fine for our purposes.

517
00:40:19,400 --> 00:40:21,650
If you need to do something
a little bit more careful,

518
00:40:21,650 --> 00:40:25,760
that's a subject for a
numerical analysis class.

519
00:40:25,760 --> 00:40:27,710
For us, I will
give you some hints

520
00:40:27,710 --> 00:40:32,100
on this when you begin
exploring these solutions.

521
00:40:32,100 --> 00:40:38,890
So you have an additional
boundary condition.

522
00:40:38,890 --> 00:40:48,100
You know by Birkhoff's theorem
that the Schwarzschild metric

523
00:40:48,100 --> 00:40:49,200
describes the exterior.

524
00:40:55,620 --> 00:41:07,500
That means gtt is minus 1 minus,
given by this for everywhere

525
00:41:07,500 --> 00:41:09,600
greater than r star.

526
00:41:09,600 --> 00:41:18,860
This gives us a boundary
condition that phi of r star

527
00:41:18,860 --> 00:41:30,300
must be 1/2 log 1 minus
2m total over r star.

528
00:41:30,300 --> 00:41:32,290
By enforcing this
boundary condition,

529
00:41:32,290 --> 00:41:34,620
you can go back to
your solution for phi

530
00:41:34,620 --> 00:41:36,660
and you can figure out
what the value at r

531
00:41:36,660 --> 00:41:41,040
equals 0 should have been to
give you a continuous function

532
00:41:41,040 --> 00:41:42,975
that matches at the surface.

533
00:41:46,000 --> 00:41:51,960
So that's it for
spherical stars.

534
00:41:51,960 --> 00:41:55,488
I look forward to you
doing these exercises.

535
00:41:55,488 --> 00:41:57,030
My own biases are
perhaps coming out,

536
00:41:57,030 --> 00:41:58,155
but these are a lot of fun.

537
00:42:00,600 --> 00:42:03,970
The one thing which
I will do for you,

538
00:42:03,970 --> 00:42:07,020
and I regret that my notes
didn't really have this,

539
00:42:07,020 --> 00:42:10,855
is I will try to figure out
how on earth I went from line 2

540
00:42:10,855 --> 00:42:15,720
to line 3 in this
calculation over here,

541
00:42:15,720 --> 00:42:18,900
going from the rest mass
density, the rest energy

542
00:42:18,900 --> 00:42:21,330
density to the energy density.

543
00:42:21,330 --> 00:42:23,460
My apologies that
that's not there.

544
00:42:23,460 --> 00:42:26,370
All I can say is that there are
many distractions these days

545
00:42:26,370 --> 00:42:28,512
and I overlooked that when
I was reviewing my notes

546
00:42:28,512 --> 00:42:29,970
and preparing for
today's lectures.

547
00:42:43,440 --> 00:42:48,780
I'd like to take this moment to
take a little bit of a detour.

548
00:42:48,780 --> 00:42:53,250
Let's imagine that we
have a spacetime that

549
00:42:53,250 --> 00:42:56,140
is Schwarzschild everywhere.

550
00:43:07,770 --> 00:43:25,780
In other words, it has this
form for all r, not simply

551
00:43:25,780 --> 00:43:27,580
the exterior of some object.

552
00:43:30,370 --> 00:43:34,415
We already know that this
spacetime is a vacuum solution.

553
00:43:43,590 --> 00:43:49,440
I know that t mu nu equals 0.

554
00:43:49,440 --> 00:43:50,410
Back up for a second.

555
00:43:50,410 --> 00:43:54,040
If I generate the
Einstein tensor for this,

556
00:43:54,040 --> 00:43:58,240
I will get identically
0, which implies

557
00:43:58,240 --> 00:44:01,120
that this corresponds to
a solution, which has t

558
00:44:01,120 --> 00:44:01,750
mu equals 0.

559
00:44:04,480 --> 00:44:07,690
I also know, though,
that if I examine

560
00:44:07,690 --> 00:44:11,020
the behavior of radial
geodesics in the weak field

561
00:44:11,020 --> 00:44:27,050
of this spacetime,
I find that they

562
00:44:27,050 --> 00:44:29,000
fall towards this
like an object that

563
00:44:29,000 --> 00:44:30,950
is falling towards a mass m.

564
00:44:46,090 --> 00:44:48,430
So this spacetime
appears to be something

565
00:44:48,430 --> 00:44:51,160
that is everywhere vacuum.

566
00:44:51,160 --> 00:44:53,890
There is nothing
in this spacetime,

567
00:44:53,890 --> 00:44:56,050
and that nothing
has a mass of m.

568
00:45:19,570 --> 00:45:21,490
I hope that bothers you.

569
00:45:21,490 --> 00:45:24,670
That is among the sillier
things that has been

570
00:45:24,670 --> 00:45:26,320
said in the name of physics.

571
00:45:26,320 --> 00:45:29,020
That sure sounds silly.

572
00:45:29,020 --> 00:45:32,230
But let me remind you that we,
in fact, have seen something

573
00:45:32,230 --> 00:45:37,180
very similar in a much less
complicated theory of physics.

574
00:45:50,590 --> 00:45:57,290
So if I look at the electric
field of a point charge

575
00:45:57,290 --> 00:45:58,070
at the origin--

576
00:46:03,902 --> 00:46:08,210
so that's the three vector
e is just q displacement

577
00:46:08,210 --> 00:46:09,560
factor over r cubed.

578
00:46:25,390 --> 00:46:28,240
If I compute the divergence
of this electric field--

579
00:46:33,470 --> 00:46:36,430
the divergence, of course, tells
me about the charge density--

580
00:46:36,430 --> 00:46:37,250
and I get 0.

581
00:46:41,060 --> 00:46:45,110
So this is an electric field
that has no charge density

582
00:46:45,110 --> 00:46:47,750
anywhere, but that
lack of charge density

583
00:46:47,750 --> 00:46:48,890
has a total charge of q.

584
00:47:02,840 --> 00:47:05,750
This was something that we
easily learn how to resolve.

585
00:47:05,750 --> 00:47:08,360
Usually at the MIT
curriculum, this often

586
00:47:08,360 --> 00:47:11,540
shows up when you take
a course like 8.07.

587
00:47:11,540 --> 00:47:14,150
What we do is we say, oh,
all that's going on here

588
00:47:14,150 --> 00:47:25,080
is that I have a singular
point charge at r equals 0.

589
00:47:25,080 --> 00:47:27,460
So yeah, I've got
no charge density,

590
00:47:27,460 --> 00:47:29,060
but I have a total charge.

591
00:47:29,060 --> 00:47:29,560
Fine.

592
00:47:29,560 --> 00:47:31,450
We were happy with that.

593
00:47:31,450 --> 00:47:33,760
I want you to think of
the Schwarzschild metric

594
00:47:33,760 --> 00:47:37,900
as doing something
similar for gravity.

595
00:47:37,900 --> 00:47:42,010
There is no source
anywhere, but there is mass.

596
00:47:42,010 --> 00:47:44,680
Maybe there's just something
singular and a little

597
00:47:44,680 --> 00:47:46,360
funny going on at r equals 0.

598
00:48:03,928 --> 00:48:05,470
You might be
concerned about what's

599
00:48:05,470 --> 00:48:06,800
happening there at r equal 0.

600
00:48:09,817 --> 00:48:11,275
When I say it plays
a similar role,

601
00:48:11,275 --> 00:48:14,260
it plays a similar role to
the pull on point charge.

602
00:48:28,478 --> 00:48:30,520
So there'll be nothing
there, but perhaps there's

603
00:48:30,520 --> 00:48:32,710
something funny going
on at r equals 0.

604
00:48:32,710 --> 00:48:34,348
And by the way,
the field equations

605
00:48:34,348 --> 00:48:36,640
that govern gravity, my
relativistic theory of gravity,

606
00:48:36,640 --> 00:48:37,520
they're non-linear.

607
00:48:37,520 --> 00:48:40,100
So when I say there's something
funny going on at r equals 0,

608
00:48:40,100 --> 00:48:42,550
it could be really funny.

609
00:48:42,550 --> 00:48:46,050
So we're not going to get
too worked up about that,

610
00:48:46,050 --> 00:48:49,210
but we're just going to bear in
mind this is odd. t mu equals 0

611
00:48:49,210 --> 00:48:51,350
but it has mass.

612
00:48:51,350 --> 00:48:55,460
So let's look at the
spacetime itself.

613
00:48:55,460 --> 00:49:02,480
Just staring at this, we can
see two radii where it appears

614
00:49:02,480 --> 00:49:04,690
something odd is going on.

615
00:49:18,380 --> 00:49:21,430
So you can see right away,
lots of stuff kind of blows up

616
00:49:21,430 --> 00:49:26,200
and behaves badly at r equals 0.

617
00:49:26,200 --> 00:49:30,400
And you can also see that
your gtt and your grr,

618
00:49:30,400 --> 00:49:37,270
they are behaving in a way that
is potentially problematic when

619
00:49:37,270 --> 00:49:39,397
the radius is 2gm.

620
00:49:39,397 --> 00:49:41,230
So you look at that and
think, yeah, there's

621
00:49:41,230 --> 00:49:43,280
two radii there that look sick.

622
00:49:43,280 --> 00:49:46,710
I am worried about
this spacetime.

623
00:49:46,710 --> 00:49:50,130
Well, we should be cautious.

624
00:49:50,130 --> 00:49:53,010
One of the parables that we
learned about when we studied

625
00:49:53,010 --> 00:49:57,210
linearized gravity is
that we can sometimes

626
00:49:57,210 --> 00:50:01,565
put ourselves into a coordinate
system that confuses us.

627
00:50:01,565 --> 00:50:02,940
When we study
linearized gravity,

628
00:50:02,940 --> 00:50:05,250
we found a solution
that looked everywhere.

629
00:50:05,250 --> 00:50:07,860
It looked like the entire
spacetime metric was radiated.

630
00:50:07,860 --> 00:50:10,140
And it turned out only
two of those 10 components

631
00:50:10,140 --> 00:50:10,975
were radiative.

632
00:50:10,975 --> 00:50:12,600
That turned out to
be something that we

633
00:50:12,600 --> 00:50:16,103
were able to cure by introducing
a gauge transformation.

634
00:50:16,103 --> 00:50:17,770
Doing that here's a
little bit trickier,

635
00:50:17,770 --> 00:50:20,490
but we're going to
need to think about,

636
00:50:20,490 --> 00:50:23,220
how can I more clearly call
out the physical content

637
00:50:23,220 --> 00:50:25,060
of this spacetime?

638
00:50:25,060 --> 00:50:29,670
So one of the
lessons that I hope

639
00:50:29,670 --> 00:50:31,740
has been imparted
in this class so far

640
00:50:31,740 --> 00:50:33,900
is that if you really
want understand

641
00:50:33,900 --> 00:50:37,320
the nature of gravity, you
want to go from the metric

642
00:50:37,320 --> 00:50:38,650
to the curvature.

643
00:50:38,650 --> 00:50:45,900
So what I'm going to do is
assemble an invariant scalar

644
00:50:45,900 --> 00:50:47,262
from my curvature.

645
00:50:59,570 --> 00:51:01,290
And I'm going to use
the Riemann tensor

646
00:51:01,290 --> 00:51:03,750
because I know Ricci
vanishes in this spacetime,

647
00:51:03,750 --> 00:51:07,800
so that wouldn't give
me anything interesting.

648
00:51:07,800 --> 00:51:10,320
So what I'm going to do
is assemble an object.

649
00:51:10,320 --> 00:51:16,640
I'm going to call it capital
I. And that's just Riemann

650
00:51:16,640 --> 00:51:19,700
contracted into Riemann.

651
00:51:19,700 --> 00:51:20,860
This actually has a name.

652
00:51:20,860 --> 00:51:24,200
It is known as the
Kretschmann scalar.

653
00:51:24,200 --> 00:51:25,580
And you can go in.

654
00:51:25,580 --> 00:51:27,590
You can work out all
these components.

655
00:51:27,590 --> 00:51:32,690
The gr tool that is posted
to the 8.962 website

656
00:51:32,690 --> 00:51:35,930
is something you
can explore with us.

657
00:51:35,930 --> 00:51:38,600
And this is just a number.

658
00:51:38,600 --> 00:51:47,510
Turns out to be 48 g squared
m squared over r to the sixth.

659
00:51:47,510 --> 00:51:48,590
What does this guy mean?

660
00:51:48,590 --> 00:51:55,570
Well, in an invariant way,
it's kind of Riemann squared.

661
00:51:55,570 --> 00:51:58,430
Riemann tells me to go
back and think about things

662
00:51:58,430 --> 00:51:59,670
like geodesic deviation.

663
00:51:59,670 --> 00:52:01,640
It tells me about the
strength of tides.

664
00:52:01,640 --> 00:52:12,080
So roughly speaking, square
root I is an invariant way

665
00:52:12,080 --> 00:52:14,105
of characterizing tidal forces.

666
00:52:29,380 --> 00:52:35,020
So if you're sitting around
in the Schwarzschild spacetime

667
00:52:35,020 --> 00:52:38,020
and you want to give
yourself an estimate of what

668
00:52:38,020 --> 00:52:42,040
kind of tidal forces are
likely to act on you,

669
00:52:42,040 --> 00:52:45,520
compute the Kretschmann
scalar, take its square root,

670
00:52:45,520 --> 00:52:48,370
and that'll give you an idea
of how strong they typically

671
00:52:48,370 --> 00:52:48,870
tend to be.

672
00:52:53,860 --> 00:52:58,680
So notice, when we look at this,
this tells us r equals 2gm.

673
00:52:58,680 --> 00:53:02,308
If you plug r
equals 2gm in there,

674
00:53:02,308 --> 00:53:03,350
nothing special about it.

675
00:53:14,390 --> 00:53:15,990
It's a radius just
like any other.

676
00:53:15,990 --> 00:53:24,280
As you go from 2.001gm
gm to 1.99999gm,

677
00:53:24,280 --> 00:53:25,420
it increases a little bit.

678
00:53:25,420 --> 00:53:27,210
Of course, it's got the 1
over r to the sixth behavior,

679
00:53:27,210 --> 00:53:29,190
but it's not like there's
a sudden transition,

680
00:53:29,190 --> 00:53:34,070
or anything particularly special
happens right at that radius.

681
00:53:34,070 --> 00:53:40,010
But it is hella
singular at r equals 0.

682
00:53:47,070 --> 00:53:51,270
So sure enough, r equals
0 is a place where

683
00:53:51,270 --> 00:53:53,700
tidal forces blow up.

684
00:54:01,380 --> 00:54:02,340
OK, fine.

685
00:54:02,340 --> 00:54:04,590
We're going to need to do a
little bit more work then,

686
00:54:04,590 --> 00:54:07,560
because I still want understand,
yeah, OK, r equals 2gm.

687
00:54:07,560 --> 00:54:10,710
There's no diverging
tidal forces there,

688
00:54:10,710 --> 00:54:13,230
but that metric still
looks wacky at that point.

689
00:54:13,230 --> 00:54:14,970
So what is going on there?

690
00:54:19,167 --> 00:54:21,250
So let's think about the
geometry of the spacetime

691
00:54:21,250 --> 00:54:23,190
in the vicinity of 2gm.

692
00:54:30,920 --> 00:54:33,080
So let's imagine.

693
00:54:33,080 --> 00:54:36,020
Let's do the following exercise.

694
00:54:36,020 --> 00:54:51,150
Suppose I draw a circle
at some radius r that's

695
00:54:51,150 --> 00:54:56,250
in the theta equals
pi over 2 plane.

696
00:54:56,250 --> 00:55:00,220
So I'm just sweeping
around in phi.

697
00:55:00,220 --> 00:55:01,320
I'm making this like so.

698
00:55:05,140 --> 00:55:11,220
So here is my r cosine phi axis.

699
00:55:11,220 --> 00:55:15,310
Here is my r sine phi axis.

700
00:55:15,310 --> 00:55:20,830
Here is my circle of radius r.

701
00:55:20,830 --> 00:55:24,490
And let's ask,
what is the surface

702
00:55:24,490 --> 00:55:29,090
area that this guy sweeps out
as an advance forward in time?

703
00:55:29,090 --> 00:55:32,970
So as this thing
goes forward in time,

704
00:55:32,970 --> 00:55:39,730
it sort of sweeps out a
cylinder in a spacetime diagram.

705
00:55:39,730 --> 00:55:45,070
Let's compute the proper area
associated with this cylinder

706
00:55:45,070 --> 00:55:47,800
that this circle is sweeping
out as it moves forward in time.

707
00:55:51,180 --> 00:55:54,900
So the surface
area of my tube, I

708
00:55:54,900 --> 00:55:59,580
integrate from some start
time to some end time.

709
00:56:05,810 --> 00:56:09,800
I'm going to integrate
around in phi,

710
00:56:09,800 --> 00:56:12,020
and then the proper
area element that I

711
00:56:12,020 --> 00:56:18,258
need to do this is going
to be gtt g5 phi, the 1/2.

712
00:56:18,258 --> 00:56:20,800
There's actually a minus sign
in there to get the sign right.

713
00:56:20,800 --> 00:56:21,980
Let's write it like this.

714
00:56:25,190 --> 00:56:28,540
To remind you how I do this,
think of this area element

715
00:56:28,540 --> 00:56:29,680
as a 2 volume.

716
00:56:29,680 --> 00:56:32,080
Go back to some of our
earlier discussion of defining

717
00:56:32,080 --> 00:56:37,180
integrals in spacetime, and this
is the proper area associated

718
00:56:37,180 --> 00:56:39,880
with a figure that has some
extent in time and extent

719
00:56:39,880 --> 00:56:40,420
in angle.

720
00:56:50,207 --> 00:56:51,290
So let's compute that guy.

721
00:57:08,650 --> 00:57:11,450
So I take my
Schwarzschild metric.

722
00:57:11,450 --> 00:57:17,040
g5 phi in the theta equals
pi over 2 plane is just r.

723
00:57:17,040 --> 00:57:25,110
gtt is the square root
of 1 minus 2gm over r.

724
00:57:25,110 --> 00:57:27,840
So the area of my
tube is going to be

725
00:57:27,840 --> 00:57:37,840
r, integrate from my start
time to my end time, dt.

726
00:57:54,080 --> 00:57:54,920
So this is easy.

727
00:57:54,920 --> 00:58:05,790
So I get a 2 pi 2 pi r square
root 1 minus 2g m over r,

728
00:58:05,790 --> 00:58:08,660
and let's just say my
interval is delta t.

729
00:58:12,550 --> 00:58:15,550
Notice what happens
as I take the radius

730
00:58:15,550 --> 00:58:17,590
of this thing down to 2gm.

731
00:58:22,140 --> 00:58:25,176
This goes to 0.

732
00:58:25,176 --> 00:58:28,230
This r goes to 2gm.

733
00:58:28,230 --> 00:58:32,910
If I go inside 2gm, I don't
even want to compute that.

734
00:58:32,910 --> 00:58:36,950
Something has gone awry.

735
00:58:36,950 --> 00:58:40,590
But look, I can draw
this thing just fine.

736
00:58:40,590 --> 00:58:42,450
Clearly, there's
a surface there.

737
00:58:42,450 --> 00:58:44,447
It's got to have an
area associated with it.

738
00:58:44,447 --> 00:58:46,530
Why are you telling me
that the area of this thing

739
00:58:46,530 --> 00:58:49,860
is 0 in that limit and is
a nonsense integral if I

740
00:58:49,860 --> 00:58:50,940
go inside this thing?

741
00:58:53,664 --> 00:58:57,460
Well, what's happening
is we have uncovered

742
00:58:57,460 --> 00:59:00,070
a coordinate singularity.

743
00:59:00,070 --> 00:59:12,870
The time coordinate is
badly behaved as we--

744
00:59:12,870 --> 00:59:22,215
not well-- as we approach
this radius, r equals 2gm.

745
00:59:28,950 --> 00:59:34,180
Let me give you an analogy
that describes, essentially--

746
00:59:34,180 --> 00:59:41,180
it's something that is
very similar to that tube,

747
00:59:41,180 --> 00:59:43,220
that world tube
that I just drew.

748
00:59:43,220 --> 00:59:45,785
But let me do it in a,
perhaps, more familiar context.

749
00:59:48,370 --> 00:59:54,430
Suppose I want to
draw a sphere, and all

750
00:59:54,430 --> 00:59:59,620
that I know about a sphere
is that it has got two

751
00:59:59,620 --> 01:00:04,830
coordinates to cover it, an
angle phi and an angle theta.

752
01:00:04,830 --> 01:00:14,420
And so I could say,
OK, here is my sphere.

753
01:00:20,280 --> 01:00:23,550
Here is theta equals
0, phi equals 0.

754
01:00:23,550 --> 01:00:25,170
Here is phi equals pi over 2.

755
01:00:25,170 --> 01:00:27,180
Here's phi equals pi.

756
01:00:27,180 --> 01:00:29,025
Pi equals 3 pi over 2.

757
01:00:29,025 --> 01:00:32,214
Phi equals 2 pi.

758
01:00:32,214 --> 01:00:33,990
Theta equals pi over 2.

759
01:00:33,990 --> 01:00:34,980
Theta equals pi.

760
01:00:38,130 --> 01:00:38,880
There's my sphere.

761
01:00:42,020 --> 01:00:47,650
So this chart that I've
just drawn here, it's true.

762
01:00:47,650 --> 01:00:53,090
This does represent
the coordinate system,

763
01:00:53,090 --> 01:01:01,695
but it's a horrible rendering
of a sphere's geometry.

764
01:01:06,980 --> 01:01:09,440
What I didn't realize when
I wrote this down here is

765
01:01:09,440 --> 01:01:16,070
that, in fact, at theta
equals 0 and theta equals pi,

766
01:01:16,070 --> 01:01:19,850
every phi value should be
collapsed to a single point.

767
01:01:36,770 --> 01:01:40,450
This is reflecting the fact
that if you look at a globe,

768
01:01:40,450 --> 01:01:45,160
all lines of longitude cross the
north pole and the south pole.

769
01:01:45,160 --> 01:01:49,780
Every value of the
azimuthal angle

770
01:01:49,780 --> 01:01:51,370
on the surface of
the earth, they

771
01:01:51,370 --> 01:01:53,980
become singular at the north
pole and the south pole.

772
01:01:53,980 --> 01:01:57,400
This drawing, well,
it's like one of those,

773
01:01:57,400 --> 01:01:59,650
I forget the names of them,
but the various renderings

774
01:01:59,650 --> 01:02:01,240
of a map that try
to take the earth

775
01:02:01,240 --> 01:02:03,460
and write it on a flat
space, and you wind up

776
01:02:03,460 --> 01:02:06,970
with Greenland being three
times the size of Africa,

777
01:02:06,970 --> 01:02:08,090
or something like that.

778
01:02:08,090 --> 01:02:12,225
And it's because there should
be zero area at the top here.

779
01:02:12,225 --> 01:02:13,600
As you approach
the top the area,

780
01:02:13,600 --> 01:02:15,058
it should be getting
much stronger.

781
01:02:15,058 --> 01:02:17,800
And when you do representation
of your map like this,

782
01:02:17,800 --> 01:02:19,950
you're spreading
everything way, way out.

783
01:02:22,530 --> 01:02:23,700
What is going on?

784
01:02:23,700 --> 01:02:27,050
And why this idea of
drawing this world tube that

785
01:02:27,050 --> 01:02:29,670
is swept out by my
circle of radius

786
01:02:29,670 --> 01:02:32,250
r as it advances
forward in time?

787
01:02:36,580 --> 01:02:40,690
That drawing does not
account for the fact

788
01:02:40,690 --> 01:02:46,660
that the Schwarzschild
time coordinate is singular

789
01:02:46,660 --> 01:02:50,700
as you approach r equals 2gm.

790
01:02:50,700 --> 01:03:04,880
It's going to turn out all
times t map to a single sphere,

791
01:03:04,880 --> 01:03:07,530
and r equals 2gm.

792
01:03:10,890 --> 01:03:14,580
To get some insight into
what's going on here,

793
01:03:14,580 --> 01:03:16,194
let's do a little
thought experiment.

794
01:03:27,810 --> 01:03:30,390
What I'm going to do
is imagine I'm at rest

795
01:03:30,390 --> 01:03:32,610
in the Schwarzschild spacetime.

796
01:03:32,610 --> 01:03:35,310
So let's say that I'm
at some finite radius r.

797
01:03:35,310 --> 01:03:36,570
I am not in a weak field.

798
01:03:36,570 --> 01:03:39,870
OK, maybe I am at
something like r equals

799
01:03:39,870 --> 01:03:42,760
4gm, or something like that.

800
01:03:42,760 --> 01:03:48,765
And what I'm going to do is drop
a little rock, drop a particle.

801
01:03:58,820 --> 01:04:03,810
So I'm going to drop a
particle from r equals r0.

802
01:04:03,810 --> 01:04:06,560
I'm going to integrate
the geodesic equation,

803
01:04:06,560 --> 01:04:09,500
and I'm going to parameterize
what its radial motion looks

804
01:04:09,500 --> 01:04:10,820
like as a function of "time."

805
01:04:35,890 --> 01:04:38,578
I put "time" in quotes
here because you

806
01:04:38,578 --> 01:04:40,120
should be saying at
this point, well,

807
01:04:40,120 --> 01:04:42,910
you just told me that time is
doing something kind of funny

808
01:04:42,910 --> 01:04:43,410
here.

809
01:04:43,410 --> 01:04:45,420
What do you mean by that?

810
01:04:45,420 --> 01:04:47,880
I'm actually going to do this
for two different notions

811
01:04:47,880 --> 01:04:48,380
of time.

812
01:04:51,810 --> 01:05:00,330
I'm going to do this for
the coordinate time t,

813
01:05:00,330 --> 01:05:06,510
and I'm also going to
do this for proper time,

814
01:05:06,510 --> 01:05:14,980
tau, as measured along
that world, that infall.

815
01:05:21,240 --> 01:05:22,990
So I'm not going to
go through the details

816
01:05:22,990 --> 01:05:23,830
of this calculation.

817
01:05:23,830 --> 01:05:27,100
It's a straightforward,
moderately tedious exercise.

818
01:05:30,308 --> 01:05:32,350
I would just quote to you
what the result ends up

819
01:05:32,350 --> 01:05:33,106
looking like.

820
01:05:52,470 --> 01:05:54,950
So let's first write down
what the solution looks like,

821
01:05:54,950 --> 01:05:58,530
parameterized by
the proper time.

822
01:05:58,530 --> 01:06:03,593
So this is most easily written
as tau, proper time, and 2gm.

823
01:06:03,593 --> 01:06:05,010
Essentially, I'm
just going to use

824
01:06:05,010 --> 01:06:06,900
it to set a system of units.

825
01:06:06,900 --> 01:06:10,710
I write this thing
as a function of r.

826
01:06:10,710 --> 01:06:12,150
My solution turns out to be--

827
01:06:27,734 --> 01:06:29,940
it looks like this.

828
01:06:29,940 --> 01:06:33,230
So if I were to make a plot of
what this thing's motion looks

829
01:06:33,230 --> 01:06:46,020
like as a function of
time, so here's our 0.

830
01:06:46,020 --> 01:06:50,030
Here is r of tau.

831
01:06:52,930 --> 01:06:56,520
And let's just put in, for
fun, let's say this is 2gm.

832
01:07:02,090 --> 01:07:04,550
Zoom, fallen.

833
01:07:04,550 --> 01:07:10,630
You reach r equals 0
in finite proper time.

834
01:07:10,630 --> 01:07:14,350
The parable of the Kretschmann
scalar is that as you do so,

835
01:07:14,350 --> 01:07:17,470
the tidal forces acting
on you are diverging.

836
01:07:17,470 --> 01:07:21,340
So if you have any last
wishes, send them out

837
01:07:21,340 --> 01:07:23,290
because you're not going
to have a lot of time

838
01:07:23,290 --> 01:07:26,340
to tell people about them.

839
01:07:30,710 --> 01:07:34,480
Let's now write it as a
function of coordinate time t.

840
01:07:38,620 --> 01:07:42,457
This ends up being--

841
01:07:42,457 --> 01:07:45,040
bear with me while I write this
out, this is slightly lengthy.

842
01:08:23,399 --> 01:08:25,210
OK, so what I mean
on this last line

843
01:08:25,210 --> 01:08:27,069
is if you want to get
the complete solution,

844
01:08:27,069 --> 01:08:28,736
just write both of
these functions down.

845
01:08:28,736 --> 01:08:33,279
Again, subtract them off
and place the r's with r0.

846
01:08:33,279 --> 01:08:41,870
When you look at this,
here's what you see.

847
01:08:45,870 --> 01:08:51,412
The motion expressed in times
of the coordinate time t

848
01:08:51,412 --> 01:08:55,609
asymptotically approaches
the radius 2gm,

849
01:08:55,609 --> 01:09:00,069
but it never quite reaches it.

850
01:09:00,069 --> 01:09:03,700
As t goes to infinity,
it eventually reaches--

851
01:09:03,700 --> 01:09:06,359
so r, you can see it
appearing in the behavior

852
01:09:06,359 --> 01:09:07,979
of this natural log.

853
01:09:07,979 --> 01:09:11,258
r gets to 2gm as t
goes to infinity.

854
01:09:15,569 --> 01:09:22,399
So as measured by clocks
on the infalling body,

855
01:09:22,399 --> 01:09:24,920
it rapidly reaches r equals 0.

856
01:09:24,920 --> 01:09:26,630
According to this
coordinate time,

857
01:09:26,630 --> 01:09:28,550
it never even
crosses r equals 2gm.

858
01:09:31,180 --> 01:09:34,540
What the hell is
going on with that?

859
01:09:34,540 --> 01:09:44,649
Well, to give a little
bit of insight into this,

860
01:09:44,649 --> 01:09:47,450
it's useful to stop for a
second and ask ourselves,

861
01:09:47,450 --> 01:09:52,370
what is that coordinate
time t actually measuring?

862
01:09:52,370 --> 01:09:58,375
So let me write down
the Schwarzschild metric

863
01:09:58,375 --> 01:09:59,500
and let's think about this.

864
01:10:20,480 --> 01:10:24,990
So kind of hard to see
what t means in this,

865
01:10:24,990 --> 01:10:26,480
but let's consider a limit.

866
01:10:26,480 --> 01:10:33,260
Suppose I consider observers
who are very far away.

867
01:10:33,260 --> 01:10:37,220
If I look at people who are at
r, much, much larger than 2gm.

868
01:10:44,110 --> 01:10:55,810
For such observers,
spacetime looks like this,

869
01:10:55,810 --> 01:11:00,850
and this is nothing more than
flat spacetime in spherical

870
01:11:00,850 --> 01:11:02,980
coordinates.

871
01:11:02,980 --> 01:11:05,740
This is what we call an
asymptotically flat spacetime.

872
01:11:05,740 --> 01:11:09,040
As you get sufficiently
far away from the source,

873
01:11:09,040 --> 01:11:11,770
it looks just like
flat spacetime.

874
01:11:11,770 --> 01:11:14,153
essentially, special
relativity rules apply.

875
01:11:14,153 --> 01:11:15,820
And that gives us
some insight into what

876
01:11:15,820 --> 01:11:18,370
this coordinate t means.

877
01:11:18,370 --> 01:11:21,220
The t that we are using in
the Schwarzschild coordinate

878
01:11:21,220 --> 01:11:32,200
system, this is time as
measured by distant observers.

879
01:11:38,760 --> 01:11:42,750
Tau is time, as measured
by this infalling observer.

880
01:11:42,750 --> 01:11:46,200
So what we are seeing here
is the infalling observer

881
01:11:46,200 --> 01:11:52,170
crosses 2gm, reaches r equals
0, and has a very short life.

882
01:11:52,170 --> 01:11:55,900
But those who are using
clocks, adapted to things very,

883
01:11:55,900 --> 01:11:59,850
very far away, never
even see it cross 2gm.

884
01:11:59,850 --> 01:12:01,890
Why is that?

885
01:12:01,890 --> 01:12:04,710
Well, we will pick this
up in the next lecture,

886
01:12:04,710 --> 01:12:07,620
but let me remind you
that when we initially

887
01:12:07,620 --> 01:12:11,550
began working on this subject,
one of the very first lectures,

888
01:12:11,550 --> 01:12:14,280
we talked about something called
the Einstein synchronization

889
01:12:14,280 --> 01:12:17,400
procedure, where what we did
was we imagined spacetime

890
01:12:17,400 --> 01:12:20,310
was filled with a conceptual
lattice of measuring

891
01:12:20,310 --> 01:12:21,960
rods and clocks.

892
01:12:21,960 --> 01:12:25,020
And we synchronized
all of those clocks

893
01:12:25,020 --> 01:12:31,530
by requiring that the time
delay between different clocks

894
01:12:31,530 --> 01:12:33,390
is synchronized
according to the time it

895
01:12:33,390 --> 01:12:37,090
takes for light to travel
from one to the other.

896
01:12:37,090 --> 01:12:40,110
This is telling us we
are actually working--

897
01:12:40,110 --> 01:12:41,700
when we use
Schwarzschild time, we

898
01:12:41,700 --> 01:12:45,150
are working in a system
that reflects an underlying

899
01:12:45,150 --> 01:12:47,340
inheritance from
special relativity.

900
01:12:47,340 --> 01:12:50,880
These are clocks that have been
synchronized by the Einstein

901
01:12:50,880 --> 01:12:53,290
synchronization procedure.

902
01:12:53,290 --> 01:12:55,680
And so the pathological
behavior that we

903
01:12:55,680 --> 01:13:00,360
see here, it must ultimately owe
to the behavior of these clocks

904
01:13:00,360 --> 01:13:02,490
that we use to define
our coordinate system,

905
01:13:02,490 --> 01:13:04,560
and the behavior of
those clocks is linked

906
01:13:04,560 --> 01:13:07,410
to the behavior of light.

907
01:13:07,410 --> 01:13:11,370
So in order to get insight as to
what is going on with this, why

908
01:13:11,370 --> 01:13:14,430
is it that if I use a clock
adapted to the infalling body,

909
01:13:14,430 --> 01:13:17,370
I see painful death, but
if I use a clock adopted

910
01:13:17,370 --> 01:13:19,410
to someone very far
away, I don't even

911
01:13:19,410 --> 01:13:23,195
see it approach that dangerous
r equals to a radius.

912
01:13:23,195 --> 01:13:24,570
In order to resolve
that mystery,

913
01:13:24,570 --> 01:13:29,100
I'm going to need to examine
what the motion of light looks

914
01:13:29,100 --> 01:13:31,170
like in this spacetime.

915
01:13:31,170 --> 01:13:34,340
We'll pick that up
in the next lecture.