12. Stellar Mass Black Holes

0:01  0:04Professor Charles
Bailyn: Okay, 
0:04  0:08here's the plan for today.
I want to do one last foray 
0:08  0:12into relativity theory.
And this is going to be a 
0:12  0:16tricky one, so I hope you're all
feeling mentally strong this 
0:16  0:18morning.
If not, wegosh, 
0:18  0:21we should have ordered coffee
for everyone. 
0:21  0:25And, in so doing,
I want to introduce one key 
0:25  0:30concept, and also answer at
least three of the questions 
0:30  0:36that you guys have asked before
in a morein more depth, 
0:36  0:39and also relate the whole thing
back to black holes. 
0:39  0:44And then, having done that,
we'll have some more questions. 
0:44  0:46And then, having done that,
I want to get back to 
0:46  0:48astronomy;
that is to say, 
0:48  0:52to things in the sky that
actually manifest these 
0:52  0:55relativistic effects.
So, that's where we're going 
0:55  0:57today.
And along the way, 
0:57  1:03as I said, we'll deal with some
of the questions you've been 
1:03  1:11asking in a deeper kind of way.
In particularso, questions. 
1:11  1:13Watch out for the answers to
these questions. 
1:13  1:17Somebody asked,
"What's special about special 
1:17  1:21relativity, and what's general
about general relativity?" 
1:21  1:22How do they relate?

1:22  1:26

1:26  1:29So, we'll come back to that one.
Somebody also asked, 
1:29  1:35"Why use the speed of light to
convert time into space and vice 
1:35  1:39versa, to get them in the same
coordinate system?" 
1:39  1:52So, why use c to convert
time to space and vice versa? 
1:52  1:54And then, also,
there was the question of, 
1:54  1:57you know, "What is the
mathematical formulation of 
1:57  2:02general relativity?"
So, how to express general 
2:02  2:08relativity in some kind of
equation. 
2:08  2:11

2:11  2:15And we'll get to the key
equation, which is something 
2:15  2:17called a metric,
for general relativity, 
2:17  2:22and then we're going to stop.
Because to go on from there is 
2:22  2:26fairly heavy calculus and we're
just not going to do that. 
2:26  2:29But I want to get at least that
far. 
2:29  2:35Okay, so let's go back to
special relativity for a minute. 
2:35  2:45So, special relativity.
Flat spacetime, no gravity. 
2:45  2:49

2:49  2:53And you'll recall what happens.
As you get close to the speed 
2:53  2:56of light, all sorts of things
that you thought were kind of 
2:56  2:57constant and properties of
objects, 
2:57  3:01like mass and length and
duration, and duration of time, 
3:01  3:04and things like that,
all start to get weird and 
3:04  3:07change.
So, length, time, 
3:07  3:15mass, all these things,
vary with the velocity of the 
3:15  3:21person doing the measuring.
And so, you could ask the 
3:21  3:23question, is there anything that
doesn't vary? 
3:23  3:26Is there anything that's an
invariant? 
3:26  3:30And the answer is, yes.
There are some things that 
3:30  3:35don't vary.
So, some things are invariant. 
3:35  3:40

3:40  3:44And Einstein actually said
later in his career that it's 
3:44  3:47actually the invariants that are
important, not the things that 
3:47  3:50change.
And so, he should have called 
3:50  3:54his theory invariant theory
instead of relativity theory. 
3:54  3:56Think of what that would have
done to pop philosophy. 
3:56  3:59Instead of saying,
"everything is relative," all 
3:59  4:03this stuff, you would have had
the exact same theory. 
4:03  4:05You would have called it
invariance theory. 
4:05  4:08And the pop philosophy
interpretation of this would be, 
4:08  4:12"some things never change."
And it would have been a whole 
4:12  4:16different concept in three in
the morning dorm room 
4:16  4:19conversations.
Okay, so some things are 
4:19  4:23invariant, what things?
Now, let me first give you a 
4:23  4:28little bit of a metaphor and
then come back to how this 
4:28  4:32really works in spacetime.
Supposing you're just looking 
4:32  4:35at an xycoordinate
system and you have two points 
4:35  4:40in a twodimensional space.
So, here's a point and here's a 
4:40  4:42point.
Now, if you arrange for some 
4:42  4:46kind of coordinate
systemhere's a coordinate 
4:46  4:48system.
This is x, 
4:48  4:52this is yand you ask
how far apart these points are. 
4:52  4:55Well, you can do thatlet's
see. 
4:55  5:00They're separated in x
by this amount here, 
5:00  5:05which we'll call delta [Δ]
x. And they're separated 
5:05  5:11in y by this amount here,
and that's Δ y. 
5:11  5:14But of course,
those quantities depend on the 
5:14  5:17orientation of your coordinate
system. 
5:17  5:23If I now take this coordinate
system and I shift it like this, 
5:23  5:27now it's going to be totally
different. 
5:27  5:36Now I'm going to have x
look like this and I'm going to 
5:36  5:42have y,
Δ y look like that. 
5:42  5:44So Δ y has gotten a
whole lot smaller. 
5:44  5:47Δ x has gotten a whole
lot bigger. 
5:47  5:51And all I did was twist the
coordinate system. 
5:51  5:53Yeah?
Student: [Inaudible.] 
5:53  5:54Professor Charles
Bailyn: You still get the 
5:54  5:55same distance,
thank you very much, 
5:55  5:58that's exactly right.
The distance is the invariant. 
5:58  6:01The xcoordinate and the
ycoordinate, 
6:01  6:04those vary with the coordinate
system, but the distance is the 
6:04  6:06same.
That's exactly the point. 
6:06  6:11And so, the quantitylet me
summarize this. 
6:11  6:19For points on a 2D space,
Δ x varies. 
6:19  6:25Δ y varies.
But there is a quantity that is 
6:25  6:32invariant and that iswell,
let's call it (Δ x^(2)) 
6:32  6:36+ (Δ y^(2)),
which is the distance squared, 
6:36  6:40(Δ D^(2)).
And this is invariant. 
6:40  6:44

6:44  6:48It doesn't matter which way you
twist things around, 
6:48  6:51that willthat quantity will
remain the same. 
6:51  7:05So, now, imagine that you've
got events in spacetime. 
7:05  7:12So an event in spacetime has
three spatial coordinates and 
7:12  7:17one time coordinate.
So, it's basically a point in a 
7:17  7:25fourdimensional space.
And as your velocity changes, 
7:25  7:33the distance and time also
change. 
7:33  7:36That's the equivalent of
rotating the coordinate system. 
7:36  7:40But there is something that
doesn't change, 
7:40  7:46and let me write that down.
This is usually given the Greek 
7:46  7:48letter Tau [T]
squared. 
7:48  7:55And this is equal to (Δ
x^(2)) + (Δ 
7:55  8:01y^(2)) + (Δ
Z^(2))  c^(2) 
8:01  8:07(ΔT)^(2).
And this is invariant. 
8:07  8:10

8:10  8:17This is an invariant interval,
sometimes called proper 
8:17  8:20distance.
And as you change your 
8:20  8:23velocityas the space,
as the mass, 
8:23  8:27as the time all changethis
quantity, 
8:27  8:30describing the separation of
two eventsso, 
8:30  8:33this describes the separation
of two events, 
8:33  8:37that quantity changesdoesn't
change. 
8:37  8:41That quantity is invariant.
Okay, so now, 
8:41  8:44this answers one of the
questions that was asked before. 
8:44  8:49Why does one use c^(2)
or c to transform the 
8:49  8:53space coordinate into the time
coordinate and back? 
8:53  8:56It's because you need the
c^(2) out here in order 
8:56  8:59to make this invariant.
If you're calculating the 
8:59  9:02distance, if you use
x^(2) plus 1/2 
9:02  9:06y^(2) or some other
constant times y^(2), 
9:06  9:09you're not going to get
something that's invariant. 
9:09  9:11And it's only when you use the
c, here, 
9:11  9:14that you end up with something
that's invariant. 
9:14  9:18And so, if you think about
these as representing the four 
9:18  9:21coordinates of the system,
it's clear this one coordinate 
9:21  9:24is x, one is y,
one is z, just as you 
9:24  9:27would expect.
And then, there's this other 
9:27  9:30coordinate, which is c
times T, 
9:30  9:35but it's negative so it has to
be times the square root of 1. 
9:35  9:38So, the four coordinates in
spacetime can be thought of as 
9:38  9:42x, y, Z and i cT,
if you want to think of it that 
9:42  9:44way.
And the time coordinate is 
9:44  9:47imaginary, because when you
square it, you have to end up 
9:47  9:52with a negative number.
Don't worry about the details 
9:52  9:55of that.
But the presence of the 
9:55  9:59c^(2) here is why you
have to use c, in 
9:59  10:04particular, to get from time to
space and back. 
10:04  10:08And that's necessary because
this is the thing that doesn't 
10:08  10:11vary with velocity.
All right. 
10:11  10:15So, this is actually kind of a
weird expression. 
10:15  10:19Because unlike the distance
between two pointsand you'll 
10:19  10:22notice, these three terms put
together, 
10:22  10:24that's just the distance
squared, ordinarily, 
10:24  10:27but that is not invariant
anymore, 
10:27  10:29because there's this other term
here, which can vary. 
10:29  10:33Unlike the distance,
this doesn't have to be 
10:33  10:36positive.
You've got three different 
10:36  10:41cases here.
This interval can be zero for 
10:41  10:45different points.
In ordinary distance it can 
10:45  10:48only be zero if the two points
are the same, 
10:48  10:50but this can be zero for
different points, 
10:50  10:53for different events.
It can be zero, 
10:53  10:55it can be negative,
and it can be positive. 
10:55  10:59So, what does that mean?
What happens when it's zero? 
10:59  11:06So, if the interval is zero,
that means that the distance 
11:06  11:11between the events in
lightyears, 
11:11  11:17for example,
is equal to the time separation 
11:17  11:24in years, because
that'sbecause this term has to 
11:24  11:30be exactly equal to that term
there, 
11:30  11:33in order for them to subtract
out and get zero. 
11:33  11:36And the c^(2) converts
from lightyears to years and 
11:36  11:38back again.
And so, what does that mean? 
11:38  11:47That means if you emit a photon
at one event, 
11:47  11:56that same photon can,
if it's going in the right 
11:56  12:05direction, be present at the
second event. 
12:05  12:10So, if you ride along with
light you'll see bothyou'll 
12:10  12:13participate in both these
events. 
12:13  12:16So, you sit at event one.
You flash a light. 
12:16  12:21You ride along with the
expanding light waves from that 
12:21  12:25event and you get to something
one lightyear away, 
12:25  12:30exactly a year later.
And so, if the second event is 
12:30  12:34one lightyear away in distance
and a year later in time, 
12:34  12:38that same photon will be
present at the second event, 
12:38  12:41as at the first event.
So, things that have one of 
12:41  12:46these intervals of zero are
separated by an appropriate 
12:46  12:51amount so that the same ray of
light can participate in both of 
12:51  12:52them.

12:52  12:57

12:57  13:03So, if the intervallet's keep
that up there for a minute. 
13:03  13:14If the interval is negative,
what does that mean? 
13:14  13:19The distance is less than the
light travel time. 
13:19  13:22

13:22  13:28So, the photon is already past
the second event. 
13:28  13:34

13:34  13:39So, if you were to emit a ray
of light at event number one, 
13:39  13:46it would have passed the second
event by the time it occurred. 
13:46  13:51The photon has already gone by.
And similarly, 
13:51  14:00if the interval is positive,
then the light photon hasn't 
14:00  14:09reached event twohasn't yet
reached event two. 
14:09  14:14Now, this is important,
because this means that you 
14:14  14:19can't communicate from event one
to event two. 
14:19  14:23

14:23  14:28So, if you're at event two,
you don't know what happened at 
14:28  14:31event one.
Because even if you'd sent out 
14:31  14:34a signal, a radio signal or
whatever, it would not have 
14:34  14:37reached you by the time event
two takes place. 
14:37  14:41So you can't communicate from
event one to event two. 
14:41  14:46And similarly,
you can't travel from event one 
14:46  14:53and reach event two,
because you'd have to go faster 
14:53  14:57than the speed of light to do
it. 
14:57  15:00These kinds of intervals,
these negative intervals, 
15:00  15:04these are called timelike,
because the time term is larger 
15:04  15:09than the distance turn.
And these kinds of intervals 
15:09  15:15are called spacelike intervals.
And you can only travel or 
15:15  15:19communicate over timelike
intervals. 
15:19  15:22Yes?
Student: What are these 
15:22  15:24socalled events?
Professor Charles 
15:24  15:27Bailyn: So,
they're eventsyou can think 
15:27  15:30of them as points in spacetime.
So, they have a particular 
15:30  15:33position in space and a
particular point in time. 
15:33  15:35So, they can be described by
four numbers, 
15:35  15:37three spatial coordinates and a
time coordinate. 
15:37  15:41They can be anything.
You know, turning on a light, 
15:41  15:45doing anything you want to do.
Receiving a photon, 
15:45  15:48whatever it is.
But they are points in a 
15:48  15:52fourdimensional spacetime and
therefore require four numbers 
15:52  15:53to describe them.

15:53  15:56

15:56  16:00And you can only get from one
to another if they're separated 
16:00  16:03from a timelike,
that is to say, 
16:03  16:05a negative, interval.

16:05  16:08

16:08  16:21Okay, so this expression,
which I'll write down again, 
16:21  16:26this is called a metric.

16:26  16:32

16:32  16:36And the particular metric that
I've written down here is the 
16:36  16:38metric for flat space.
Because remember, 
16:38  16:40this is special relativity.
There's no masses, 
16:40  16:42no curvature of space,
none of that stuff, 
16:42  16:47yet.
This is the metric for flat 
16:47  16:53space with no mass present.
And there are many other 
16:53  16:55metrics possible.
Any time you add mass or do 
16:55  16:58other things,
you get different kinds of 
16:58  17:01metrics more complicated than
this. 
17:01  17:04So, what's special about
special relativity is that you 
17:04  17:09use the metric appropriate for
flat space as opposed to the 
17:09  17:13many other different kinds of
metrics that you can use in 
17:13  17:16general relativity,
which has a much more general 
17:16  17:19form for the metric.
I should say, 
17:19  17:24you can also write this down.
You can write down the spatial 
17:24  17:29terms here in polar coordinates.
Remember polar coordinates? 
17:29  17:31Polar coordinates,
you describe the position in 
17:31  17:34space, instead of with x,
y, Z, you describe it with 
17:34  17:37a radius, a distance from zero,
and some angles. 
17:37  17:39And it turns out,
that's convenient to do so. 
17:39  17:43Let me write this down in polar
coordinates, or in polar. 
17:43  17:50

17:50  17:52That's an r.
Let me write that explicitly. 
17:52  17:57

17:57  18:01This is an Omega,
that's some angle. 
18:01  18:03And then the T thing
remains the same. 
18:03  18:06

18:06  18:10And I've pulled a little bit of
a notational fast one on you 
18:10  18:12here.
I've gone away from the deltas 
18:12  18:14and I've written these down as
d. 
18:14  18:17This is the differential
d. 
18:17  18:19Those of you who have taken
some calculus will remember 
18:19  18:22this.
If this were a calculusbased 
18:22  18:26course I would explain why I did
that, but I'm not going to. 
18:26  18:30So, just allow me this slight
of hand, here. 
18:30  18:32For technical reasons,
these have to be differential. 
18:32  18:33Yes?
Student: But you do need 
18:33  18:34a second angle term for the
[inaudible] 
18:34  18:35Professor Charles
Bailyn: I do need a second 
18:35  18:37angle term.
I should saygood point. 
18:37  18:41You need two angles and a
distance in three space. 
18:41  18:47This capital Omega here is
actuallyOmega squared is 
18:47  18:50actually sin θ,
d θ, 
18:50  18:55d φ,
which is the correct form. 
18:55  18:57And so you could write out both
terms here, but in fact, 
18:57  19:00this one isn't going to change.
But you're absolutely right. 
19:00  19:04In principle,
you need two angles. 
19:04  19:09Okay, why have I done this?
Excellent question. 
19:09  19:12I ask myself a question at this
point. 
19:12  19:17Where am I going?
What I want to do now is write 
19:17  19:22down a different metric.
A metric that actually involves 
19:22  19:27curved space and the presence of
a mass. 
19:27  19:29And this is something called
the Schwarzschild metric. 
19:29  19:33Remember Schwarzschild?
He had a radius. 
19:33  19:36

19:36  19:44And this is the appropriate
metric for the presence of a 
19:44  19:52single point mass at the center
of the coordinate system, 
19:52  19:57at R = 0.
That's why I put it into polar 
19:57  20:01coordinates, because the
presence of the mass is going to 
20:01  20:05change the spacetime as a
function of distancefrom 
20:05  20:08radial distance from where the
mass is. 
20:08  20:11And so, it's much more
convenient for the Schwarzschild 
20:11  20:14metric to use this in polar
coordinates. 
20:14  20:18So, here's the Schwarzschild
metric, (d T^(2)), 
20:18  20:21that's thethis is the
interval, 
20:21  20:28is equal to (dR) / (1 
R_s / 
20:28  20:31R).
So, that's just like the flat 
20:31  20:34term, except with something in
the denominator there. 
20:34  20:38Plus R^(2),
d Omega squared, 
20:38  20:42that's just like the flat
metric. 
20:42  20:48And then thewhoops this had
better be  c^(2) (1  
20:48  20:54R_S / R)
(cT) ^(2). 
20:54  20:57Where R_S is
the Schwarzschild radius, 
20:57  21:01which we've had before,
which is 2GM / 
21:01  21:04c^(2).
Okay, so this is just like the 
21:04  21:08flat metric with two exceptions.
It's got a term in the radial 
21:08  21:12part of this 1 
R_S / R. 
21:12  21:17And it's got that same term,
but this time in the numerator, 
21:17  21:23in the time term here.
Now, what do you do with 
21:23  21:28thiswith such an equation?
Well, we've donein special 
21:28  21:31relativity, we've dealt with
these kinds of things. 
21:31  21:33What you do is you start taking
the limiting cases. 
21:33  21:37You say, okay,
what happens when it's getting 
21:37  21:39really close to flat on the one
hand, 
21:39  21:42and what's happening when it's
getting very seriously 
21:42  21:48relativistic on the other hand?
So let's do that. 
21:48  21:54If R_S /
R goes to zero, 
21:54  21:58then the metric turns into the
flat metric. 
21:58  22:01Because if R_S
/ R = 0, 
22:01  22:04this term disappears because
it's 1  0, and it just cancels. 
22:04  22:08This term disappears and you
recover the flat metric. 
22:08  22:13This happens in two cases.
If the mass goes to zero, 
22:13  22:18then R_S goes
to zero, and you recover the 
22:18  22:22flat metric.
Or if R gets really big, 
22:22  22:27then R_S /
R goes to zero, 
22:27  22:31and again, you recover the flat
metric. 
22:31  22:36So inthere are two situations
where Schwarzschild metric 
22:36  22:40blends smoothly into the
ordinary flat space. 
22:40  22:44One is if the mass is zero,
that's not surprising. 
22:44  22:46If the mass is zero then
spacetime isn't curved. 
22:46  22:48Or alternatively,
if you're really, 
22:48  22:51really far away from the mass.
If R is much, 
22:51  22:53much bigger than the
Schwarzschild radius, 
22:53  22:56you're way out there.
There's no gravitational effect. 
22:56  23:01Spacetime remains flat.
So, these thethis is the 
23:01  23:05limiting case where you recover
special relativity. 
23:05  23:09Now, the other case is when
R gets close to the 
23:09  23:12Schwarzschild radius and
approaches it. 
23:12  23:16So then 1 
R_S / R 
23:16  23:20approaches zero,
because these two are going to 
23:20  23:25get closer and closer together.
1  1 = 0. 
23:25  23:28What happens then?
So, this is now, 
23:28  23:30first of allin physical terms
you're getting really close to 
23:30  23:33the Schwarzschild radius.
So now, what happens to the 
23:33  23:39metric if you do that?
The dR term gets very 
23:39  23:46big, because it's got a zero in
the denominator. 
23:46  23:54The dT term gets really
small, because it's got that 
23:54  24:00thing that's going to zero in
the numerator. 
24:00  24:02Fine.
What does that mean? 
24:02  24:05Well, remember,
this is the negative term. 
24:05  24:08This is the positive term.
So the positive term is getting 
24:08  24:10really, really big.
The negative term is getting 
24:10  24:14really, really small.
And that means that all 
24:14  24:21intervals are gradually becoming
spacelike. 
24:21  24:27

24:27  24:30What do I mean by that?
Well, the negative term is 
24:30  24:32getting small.
One of the positive terms is 
24:32  24:35getting big.
So the sum of those tends to be 
24:35  24:37positive.
It's becoming more and more 
24:37  24:39positive.
Positive intervals are these 
24:39  24:42spacelike intervals,
and you can't communicate or 
24:42  24:45travel across spacelike
intervals. 
24:45  24:48When you get all the way to the
Schwarzschild radius, 
24:48  24:51this blows up completely,
becomes infinite. 
24:51  24:54This becomes zero,
and there are no timelike 
24:54  24:57intervals.
There are no timelike 
24:57  25:00intervals that cross the event
horizon. 
25:00  25:05That's why you can't get out.
This takes us back to the basic 
25:05  25:13principle of black holes.
So, cannot communicate or 
25:13  25:20travel over spacelike
intervals. 
25:20  25:25

25:25  25:31And so, you can't cross
R equals the 
25:31  25:36Schwarzschild radius.
All right. 
25:36  25:39Let's see.
Let me write the thing down 
25:39  25:40again, here, for you.

25:40  25:57

25:57  26:01Okay, so that's the metric
we're worrying about here. 
26:01  26:07And now, let's think about what
happens inside the Schwarzschild 
26:07  26:09radius.
R less than 
26:09  26:12R_Sch
warzschild. 
26:12  26:18That means the dR term
becomes negative, 
26:18  26:24because 1 
R_S / R. 
26:24  26:27If R_S is
bigger than R then this 
26:27  26:29term isthis term is greater
than 1, 
26:29  26:32and this whole thing is less
than zero, and the signs change. 
26:32  26:38And the dT term becomes
positive. 
26:38  26:42So, that means this is the
timelike term, 
26:42  26:46where this one is the
spacelike term, 
26:46  26:48because it's positive.

26:48  26:53

26:53  26:56That's what I meant three,
four, five lectures ago, 
26:56  26:59when I said that inside the
Schwarzschild radius, 
26:59  27:02when you're inside the
Schwarzschild radius, 
27:02  27:04space and time reverse.
It's a sign change in the 
27:04  27:06metric.
That's what it means. 
27:06  27:16And you can only travel along
negative intervals. 
27:16  27:19

27:19  27:23That means you have to move in
R the same way outside 
27:23  27:27the Schwarzschild radius you
have to move in T. 
27:27  27:30But notice it's only the radial
term. 
27:30  27:33This term hasn't changed.
You could go around in circles, 
27:33  27:36but whatever you do,
you still have to move, 
27:36  27:40as it turns out,
toward the center of the thing 
27:40  27:43in radius, in order to have a
timelike interval. 
27:43  27:51And so, motion in R is
required for inside the 
27:51  27:59Schwarzschild radius,
whereas motion in T is 
27:59  28:02required outside.

28:02  28:09

28:09  28:11So space and time reverse.

28:11  28:16

28:16  28:19All of which is very nice,
but I've left out 
28:19  28:22somethingI've left something
out, 
28:22  28:24which is the factinside the
event horizon, 
28:24  28:27how do you know that this is
still the metric? 
28:27  28:30One could invent some function
that looks just like this 
28:30  28:32outside the Schwarzschild
radius, 
28:32  28:34but then looks like something
else inside the Schwarzschild 
28:34  28:36radius.
And because no communication 
28:36  28:40across the Schwarzschild radius
is possible, you'd never be able 
28:40  28:43to test it.
And so, this is how one gets 
28:43  28:46away with doing nontestable
physics. 
28:46  28:48You say, well,
we're just going to assume that 
28:48  28:51the metric hasn't changed.
Why should it change? 
28:51  28:53After all, it's the same
equation. 
28:53  28:56But inside the Schwarzschild
radius you can't actually test 
28:56  28:58this.
Outside the Schwarzschild 
28:58  29:01radius you can test it,
because you see whether objects 
29:01  29:05behave as they ought to behave
in a space that's curved in this 
29:05  29:08particular wayin a spacetime
that's curved in this particular 
29:08  29:12way.
And so, this is what I meant 
29:12  29:18by, five classes ago,
by saying space and time 
29:18  29:23reverse.
These two quantities reverse 
29:23  29:28their signs.
All right, that's as far as we 
29:28  29:32can go, because the next thing
that one would want to do is, 
29:32  29:36you find out what the equation
is for finding out how things 
29:36  29:39move in these curved
spacetimes. 
29:39  29:42Basically, you remember,
things go from one event to 
29:42  29:45another in the shortest possible
path, that's the equivalent of a 
29:45  29:48straight line.
That means if you integrate 
29:48  29:50over dT,
it has to be minimized. 
29:50  29:53So, you minimize this integral.
That tells you how things move. 
29:53  29:57We're not going to do that.
Sighs of relief? 
29:57  30:03And becausefor obvious
reasons. 
30:03  30:06So, this is as far as we can
go, just to write down the 
30:06  30:09metric here.
So, let me know pause for 
30:09  30:13questions, and then we're going
to go back and talk about 
30:13  30:17astronomyabout things in the
sky that actually exhibit these 
30:17  30:20relativistic behaviors.
Yes? 
30:20  30:22Student: You were
talking before about intervals, 
30:22  30:23and how all the intervals are
negative. 
30:23  30:27What exactly is one interval?
[Inaudible] 
30:27  30:28Professor Charles
Bailyn: Sorry? 
30:28  30:30Student: What exactly is
one interval? 
30:30  30:31[Inaudible]
Professor Charles 
30:31  30:32Bailyn: Oh,
an interval. 
30:32  30:35So, what I'm doing is I'm
taking two events, 
30:35  30:39each of which is one of these
points in spacetime, 
30:39  30:42and I'm asking,
"What is the interval between 
30:42  30:44them?"
What isyou could measure the 
30:44  30:47distance between them,
you could measure the time from 
30:47  30:49one event to another.
But as it turns out, 
30:49  30:52those aren't invariants.
And so, there's this other 
30:52  30:55thing, the metric,
which is invariant. 
30:55  31:00And so, that's a measurean
invariant measure of how 
31:00  31:05separated these two events are.
So, you take two events and you 
31:05  31:08ask yourself,
"Are they separated by zero, 
31:08  31:11a positive quantity or a
negative quantity?" 
31:11  31:13Where by separation,
I mean, this curious 
31:13  31:16combination of space and time.
Student: So the 
31:16  31:19intervals are before the metric,
before the interval? 
31:19  31:21Professor Charles
Bailyn: Yeah, 
31:21  31:23it's an intervalthink oflet
me go back to the analogy I 
31:23  31:26started with.
Here'sin two spatial 
31:26  31:31dimensions, x and
x, here are two points. 
31:31  31:33And depending on how I set the
coordinate system up, 
31:33  31:36the xdistancethe
x difference between them 
31:36  31:38and the y difference
between them can change, 
31:38  31:41but the distance is always the
same. 
31:41  31:47So now, I've got two points
witheach with and x, 
31:47  31:53y, Z and a T. And
depending on how I change my 
31:53  31:59velocity or my coordinates the
particular values of x, 
31:59  32:04y, Z,
and T can change, 
32:04  32:10but this (Δ T^(2)) defined
byI'm in flat space now, 
32:10  32:12right?
This separation, 
32:12  32:16this interval between those two
points, this is the 
32:16  32:20invariantin the same way that
the distance between two points 
32:20  32:23doesn't change if you change the
coordinate system, 
32:23  32:27even though the x and
y separations do. 
32:27  32:29Student: [Inaudible]
Professor Charles 
32:29  32:32Bailyn: It gives youno.
Well, it combines these four 
32:32  32:35things into one thing that
doesn't change. 
32:35  32:40That's the point.
Yes? 
32:40  32:44Student: Is there a way
that you canlike as an 
32:44  32:49interval from zero basically
describes two events that appear 
32:49  32:51simultaneous?
Professor Charles 
32:51  32:53Bailyn: As theyeah
exactly. 
32:53  32:55Student: So,
is that then collapsed into a 
32:55  32:57Newtonian theory,
things with twothings appear 
32:57  32:59simultaneously if they happen at
the same time  Professor 
32:59  33:03Charles Bailyn: Well okay.
So, there's two different ways 
33:03  33:06things can appear,
quote, simultaneously. 
33:06  33:11One is if they are two
different events in spacetime 
33:11  33:16and light travels from one,
and they're exactly separated 
33:16  33:20bythe amount of time between
them is the same as the distance 
33:20  33:23between them if you multiply by
c^(2). 
33:23  33:26They can also appear
simultaneously if they are the 
33:26  33:31same point as each other.
And then everything goes to 
33:31  33:34zero.
And it's only in that second 
33:34  33:40case that itthat the Newtonian
concept of simultaneous kicks 
33:40  33:43in.
Simultaneous is usually taken 
33:43  33:47to mean that the time separation
is zero. 
33:47  33:49Two things happen
simultaneously when they happen 
33:49  33:52at the same time.
Student: [Inaudible] 
33:52  33:56Professor Charles
Bailyn: On Earth thethe 
33:56  34:01distances and the velocities and
the gravitational fields are 
34:01  34:05never so strong that you have
any troublethat the Δ T 
34:05  34:11changes significantly depending
on what your point of view is. 
34:11  34:14So, in our everyday life,
we have a strong concept of 
34:14  34:16simultaneity.
It's two things that happen at 
34:16  34:17the same time.
Turns out, though, 
34:17  34:20that if you move at close to
the speedif you observe two 
34:20  34:23events to be at the same time
and I'm moving at close to the 
34:23  34:26speed of light,
I don't observe those two 
34:26  34:28events at the same time,
even though you do. 
34:28  34:32And so, at that point,
you have to abandon the 
34:32  34:37Newtonian concept that Δ T = 0
tells you that two events are 
34:37  34:40simultaneous.
And the whole concept of 
34:40  34:42simultaneity takes on a
different task. 
34:42  34:47Other questions, yes?
Student: [Inaudible] 
34:47  34:48Professor Charles
Bailyn: Okay, 
34:48  34:51so these units can be any units
of length you like provided 
34:51  34:53thatany units of length you
like, 
34:53  35:02provided the time units are
related to it by c. 
35:02  35:04That is to say,
if your distance units are 
35:04  35:07lightyears, your time units
have to be years. 
35:07  35:11If your distance units are
meters, then your time units are 
35:11  35:14some kind of meter,
lightmetersecond thing. 
35:14  35:18And so, the only restriction on
the units you use is that the 
35:18  35:22time and the space units have to
be convertible into each other 
35:22  35:24through c^(2).
Or, alternatively, 
35:24  35:28another way of saying it is,
you use any units you like, 
35:28  35:32and as long as you express the
speed of light in those units. 
35:32  35:35If you have a time unit and a
space unit, if you're inif 
35:35  35:38you're measuring your space in
furlongs and your time in 
35:38  35:41fortnights,
as long as your c is in 
35:41  35:45furlongs per fortnight,
it's going to come out okay. 
35:45  35:47So, as long as it's convertible.

35:47  35:50

35:50  35:56Other questions?
Okay, if you don't get all the 
35:56  36:00details and nuance of what I've
said this period, 
36:00  36:04don't worry too much about it.
I just wanted to get the 
36:04  36:08concept of the metric out there
and show you how, 
36:08  36:13if you look at that equation
these concepts of space and time 
36:13  36:16reversing,
and so forth, 
36:16  36:19have a kind of mathematical
consequence, as well as just 
36:19  36:21spouting words.
And if you get, 
36:21  36:23sort of, the basic outline of
the argument, 
36:23  36:24that's fine.

36:24  36:28

36:28  36:39Okay, back to actual things in
actualthat actually exist. 
36:39  36:47So, what I want to talk about
now is evidence for general 
36:47  36:54relativity from astronomical
objectsreal black holes, 
36:54  36:58stuff like that.
Now, one of the curious things 
36:58  37:01about this is that when Einstein
thought all this stuff up, 
37:01  37:05he thought it up from basically
these philosophical concerns 
37:05  37:08about mass – that the inertial
mass turned out to always be 
37:08  37:13equal to the gravitational mass.
Why would that be? 
37:13  37:16And there wasn't,
when he thought it up, 
37:16  37:22a great body of evidence for
his theory in the real world. 
37:22  37:25This is in contrast to special
relativity. 
37:25  37:27Special relativity,
there were all these 
37:27  37:29experiments that needed to be
explained. 
37:29  37:32General relativityvery,
very little. 
37:32  37:35In fact, when Einstein first
put forward the theory in 1917, 
37:35  37:39there was only one thing that
had ever been observed that 
37:39  37:42actually showed an effect of
general relativity, 
37:42  37:46and that was the orbit of
Mercury, which you're reading 
37:46  37:49about for this week's problem
set. 
37:49  37:56So, just going back a little
bit, in the nineteenth century, 
37:56  38:03people had observed the orbits
of planets in great detail. 
38:03  38:06And they found out that two of
the planets were moving in ways 
38:06  38:10they couldn't quite explain.
There were very small 
38:10  38:13deviations from the predictions
orbit. 
38:13  38:16In particular,
the orbit of Uranus was a 
38:16  38:20little weird.
And that was quickly explained 
38:20  38:24by the presence of an
unknownhitherto unknown 
38:24  38:27planet,
which was also exerting a 
38:27  38:31gravitational force on Uranus
and pulling it out of the orbit 
38:31  38:35that it should have been,
by a very small amount, 
38:35  38:38because the gravitational force
of another planet is very small 
38:38  38:41compared to that of the Sun.
But by the middle of the 
38:41  38:43nineteenth century they could
measure such things. 
38:43  38:48And they therefore predicted
the presence of this other 
38:48  38:53planet, of Neptune,
and they calculated where it 
38:53  38:58should be.
And some guy went off and 
38:58  39:04observed in that spot and found
itpredicted presence of 
39:04  39:10Neptune and discovered it in the
predicted place. 
39:10  39:13Big triumph!
Everybodyif they had had 
39:13  39:16Nobel Prizes back then,
they would have won it for 
39:16  39:19this, for sure.
And then, there was a whole big 
39:19  39:21kerfuffle because they couldn't
decide whether the French guy 
39:21  39:24had done it before the English
guy, or vice versa. 
39:24  39:28And they argued with each other
for decades about who gets the 
39:28  39:30credit.
But in scientific terms, 
39:30  39:32there was a prediction,
and the prediction was 
39:32  39:34verified.
Excellent news. 
39:34  39:42Now, there was also a problem
with the orbit of Mercuryalso 
39:42  39:48perturbed, from what you would
expect. 
39:48  39:50And having had this big triumph
in the Outer Solar System, 
39:50  39:52they figured,
well, we know how to deal with 
39:52  39:54this.
There's got to be another 
39:54  39:58planet in there.
So, they predict the presence 
39:58  40:04of a planet called Vulcan,
which then disappears from the 
40:04  40:10scientific literature until it's
resurrected by Star Trek. 
40:10  40:14But Vulcanthe concept of
Vulcan was, this was going to be 
40:14  40:17a planet that's closer to the
Sun than Mercury. 
40:17  40:20That's why they haven't been
able to find it, 
40:20  40:22because it's too near the Sun
to be easily observed. 
40:22  40:27And it's going to pull on
Mercury in such a way that it's 
40:27  40:31going to explain the problems
with the orbit of Mercury. 
40:31  40:37And so, they then look for
Vulcan in the predicted place, 
40:37  40:39and they find it.

40:39  40:42

40:42  40:47And then somebody else finds it.
And they find it many times and 
40:47  40:51each time it's differentall
different. 
40:51  40:54And it gradually becomes clear
that everybody's fooling 
40:54  40:56themselves.
That there's nothis is a 
40:56  40:59really hard observation to make,
right? 
40:59  41:01Because the thing is right near
to the Sun. 
41:01  41:07And so, it turns out that all
of this is wrong. 
41:07  41:12None of these observations are
really any good. 
41:12  41:18It's not repeatableso,
not really. 
41:18  41:21And so, after some attempts to
find Vulcanand then, 
41:21  41:24they rule out the presence of
Vulcan in various places. 
41:24  41:27So, then the people calculating
the orbits have to go back and 
41:27  41:28say, well, if Vulcan isn't
there, 
41:28  41:32maybe there are two or three
planets combining together to do 
41:32  41:35the thing that we originally
wanted Vulcan to do. 
41:35  41:37This gets sort out of control
after a while. 
41:37  41:39And at a certain point,
people just kind of give up, 
41:39  41:41and they say,
well, it's a great big mystery 
41:41  41:44about Mercury.
And after a while, 
41:44  41:48after that, people kind of even
stopped caring. 
41:48  41:51Because, you know,
we know Newton's laws worked. 
41:51  41:54This is just some weirdness
about Mercury that we don't 
41:54  42:00understand.
And then, when Einstein creates 
42:00  42:10his new theory of gravity,
he then computes in the new 
42:10  42:16theory of Mercury's orbit.

42:16  42:19

42:19  42:26And he now gets something that
agrees with the observations, 
42:26  42:30without the need for a new
planet. 
42:30  42:36

42:36  42:40And so, what happened was,
Mercury's orbit is a little 
42:40  42:42different from the Newtonian
prediction. 
42:42  42:45The general relativity
prediction is a little bit 
42:45  42:48different in just the same way
to explain this problem that 
42:48  42:52people had been trying to solve
for fifty years unsuccessfully. 
42:52  42:57And so, this was the first
verification, 
42:57  43:04empirical verification,
of general relativity. 
43:04  43:07And if you think about it,
you would expect that Mercury 
43:07  43:10would be the place you would
find this out. 
43:10  43:13For Mercury,
R_S / 
43:13  43:17R, this is the
Schwarzschild radius of the Sun 
43:17  43:21because that's what's doing the
gravitating, 
43:21  43:24is the biggest in the Solar
System. 
43:24  43:28

43:28  43:31Because the R,
the distance from Mercury to 
43:31  43:33the Sun, is the smallest of any
of the planets in the Solar 
43:33  43:38System.
And so, the relativistic 
43:38  43:48effects, the general relativity
effects, are relatively large. 
43:48  43:51But, you know,
this is still a really small 
43:51  43:53number.
This is 3 kilometers, 
43:53  43:55the Schwarzschild radius of the
Sun. 
43:55  43:56Mercury is way out there
somewhere. 
43:56  44:00So, even though this is the
mostthis is the biggest 
44:00  44:03relativistic effect in the Solar
System, it still isn't that 
44:03  44:06huge.
Let me just remind you what 
44:06  44:09this effect is.
Here's the Sun. 
44:09  44:14Mercury's going around the Sun.
And it's going around in a 
44:14  44:18slightly elliptical orbit.
I'm going to draw a very 
44:18  44:20elliptical orbit,
here, but it's really not that 
44:20  44:22big.
And there's a point in the 
44:22  44:25orbit where it is closest to the
Sun. 
44:25  44:28That point is called the
perihelion. 
44:28  44:34"Peri" for close,
"helios" for Sunof Mercury. 
44:34  44:37And in the Newtonian theory,
you should have exactly the 
44:37  44:40same orbit every time.
You should come back and the 
44:40  44:44perihelion should be in the same
place in each successive orbit. 
44:44  44:47The orbit doesn't move or
doesn't change. 
44:47  44:51But, in general relativity,
the perihelion moves. 
44:51  44:55So, after a while the
perihelion will be here. 
44:55  44:59The whole orbit will kind of
tip this way, 
44:59  45:03and it'll look like this.
So, this is the perihelion 
45:03  45:03later.

45:03  45:09

45:09  45:13And it looks like that.
And the angle which the 
45:13  45:17perihelion makes with the Sun
has changed. 
45:17  45:21This angle is called the angle
of the perihelion. 
45:21  45:25And this precesses.
So this is called the 
45:25  45:28precession of the perihelion.

45:28  45:31

45:31  45:34And it's measured in some angle
per time. 
45:34  45:37Because the question is,
"How long does it take for the 
45:37  45:40perihelion to precess across
some angle?" 
45:40  45:45And the key number for Mercury
is 43 arc seconds. 
45:45  45:47Remember arc seconds?
Those are small angles. 
45:47  45:55Per century – a really small
movement, but something that can 
45:55  46:01be measured, and had been
measured. 
46:01  46:03And it's not surprising that
this is small, 
46:03  46:06because the relativistic
effects are going to be small, 
46:06  46:08because the Schwarzschild
radius of the Sun is really 
46:08  46:11small compared to the size of
the orbit of Mercury. 
46:11  46:18But this was observed before
Einstein made his theory. 
46:18  46:22Nobody understood it.
Einstein came up with his 
46:22  46:24theory.
It turned out it predicted a 
46:24  46:28precession of the perihelion in
a way that Newton didn't, 
46:28  46:30and it turned out to work out
precisely. 
46:30  46:34So, that was good.
And at the time Einstein 
46:34  46:38published the theory,
this was the only piece of 
46:38  46:44evidence that it was correct.
Pretty small empirical 
46:44  46:52verification.
And so, let's just write down 
46:52  46:59the fable, here.
This is Einstein and the 
46:59  47:06precession of the perihelion.
And there are two versions of 
47:06  47:08the moral.
Sometimes in textbooks, 
47:08  47:11you know, they make a big deal
out of this. 
47:11  47:13They say, oh,
there was this terrible problem 
47:13  47:15with Mercury,
and then Einstein came along 
47:15  47:18with this great new theory,
solved that problem. 
47:18  47:21In the same way that they say,
there was this terrible problem 
47:21  47:23with the speed of light being
constant from all frames, 
47:23  47:26and Einstein came along with
special relativity and solved 
47:26  47:30that problem.
That's a misreading of history. 
47:30  47:32This was a byproduct of
Einstein. 
47:32  47:35It wasn't that there was a
problem with the data and he 
47:35  47:40went out to try and fix the
theory to conform with the data. 
47:40  47:46There was very little data.
So, the moral here is aesthetic 
47:46  47:53considerations,
aestheticperhaps you want to 
47:53  48:01call this philosophical,
considerations can lead to a 
48:01  48:08good new theory because he
didn't really do it to explain 
48:08  48:11the data.
This is, however, 
48:11  48:14the only time I can think of
where this actually happened 
48:14  48:17this way.
Every other major advance in 
48:17  48:23science came about because the
observers or the experimenters 
48:23  48:30had a problembut not G.R.
Only for general relativity. 
48:30  48:33Now, subsequent to that,
between 1917, 
48:33  48:37when this theory was
promulgated, and now, 
48:37  48:44there have been a variety of
tests of general relativity 
48:44  48:50using astronomical objects.
You always have to use 
48:50  48:53astronomicalor almost always
have to use astronomical objects 
48:53  48:55to test this,
because you need really strong 
48:55  48:58gravitational fields,
and it's hard to produce a 
48:58  49:00really strong gravitational
field in the laboratory. 
49:00  49:03You're kind of limited to what
the Earth provides you with, 
49:03  49:06and that isn't such a strong
gravitational field. 
49:06  49:07We computed,
at some point, 
49:07  49:10the Schwarzschild radius of the
Earth, R / 
49:10  49:13R_Sch
warzschild is the relevant 
49:13  49:15quantity,
and it's really small from the 
49:15  49:17Earth's gravitational field.
So, it's hard to see these 
49:17  49:20things.
So, all these tests tend to be 
49:20  49:23astronomical in nature.
Since then, a variety of tests 
49:23  49:28of G.R., and the punch line is
that it passes all of them. 
49:28  49:32G.R. is still a good theory.
There's no contradictory data. 
49:32  49:36But the tests still aren't as
strong as you might have hoped 
49:36  49:40they would be.
And we'll talk about some of 
49:40 those on Thursday.
 Titre:
 12. Stellar Mass Black Holes
 Description:

Frontiers/Controversies in Astrophysics (ASTR 160)
One last key concept in Special Relativity is introduced before discussion turns again to black celestial bodies (black holes in particular) that manifest the relativistic effects students have learned about in the previous lectures. The new concept deals with describing events in a coordinate system of space and time. A mathematical explanation is given for how space and time reverse inside the Schwarzschild radius through sign changes in the metric. Evidence for General Relativity is offered from astronomical objects. The predicted presence and subsequent discovery of Neptune as proof of General Relativity are discussed, and stellar mass black holes are introduced.
00:00  Chapter 1. Invariance in Special Relativity
10:10  Chapter 2. Invariant Intervals and the Schwarzschild Metric
21:01  Chapter 3. Schwarzschild Sign Changes and SpaceTime Reversals
36:27  Chapter 4. Evidence for General Relativity in AstronomyComplete course materials are available at the Open Yale Courses website: http://open.yale.edu/courses
This course was recorded in Spring 2007.
 Équipe:
 Veduca
 Projet :
 Fronteiras/Controvérsias na Astrofísica  Yale
 Durée:
 49:42
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