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Lecture 16 A105 Stephen Hawking | Brian Woodahl | YouTubeToText
YouTube Transcript: Lecture 16 A105 Stephen Hawking
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This content explores the nature of black holes, from their potential formation in the early universe and the mathematical description of their size, to the theoretical implications of quantum physics on their existence and the speculative concept of wormholes.
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Guys, let's continue um our discussion
of the black holes. And so for today,
So, a hawking um
that during the um early stages of the
big bang, that's the theory that
describes the birth of our universe and
its evolution.
stages of the Big [Music]
[Music]
formed. And this is because you
have extreme fluctuations in the
spaceime. And it's possible that a
significant amount of mass could have
concentrated in a very small
volume and such that the gravity would
overwhelm the other three forces and
this black hole would form. So
primordial black holes could have
formed. Uh if you do the calculation on
the range of the masses, they're
actually very small. So not only are
these uh black holes early in the in in
our universe, I mean that they formed
early in our universe, but they would
have had very small masses. So, uh range of
grams up to
Speaking of masses on black holes, uh
there's no theory on the upper limit of
any black hole. Uh we expect the masses
could be limitless. um just the amount
of matter that falls in there the black
hole is going to increase in mass. So
there's so let me let me just say here
for any black holes not even these but
limit. All right, next
topic is
was the first to
use Einstein's GR so
GR to determine the size of a black hole.
So that GR is short for general
relativity was first to use general
relativity to determine
the size of a. Now it's a special type
of black hole. It's one that is not
hole. And the formula the Schwarz shout
formula looks like this. The radius of
the black hole which is called the
Schwarz child
radius is 2 big G big
M over
C^2. We'll talk about the terms, but
labeled SW child
formula. Okay. So, let's talk about the
That's the short shell
terms. That will be incidentally in
meters. Two is the number two and it the
universe 6.67
* 10 -
11 m cubed kilogram over seconds
squared. And then uh m is the mass of
And then the last there in the numerator
light another
universe approximately 3 * 10
uh the Schwarz child formula basically gives
gives
us for a given mass if that mass is
constrained within this radius then that
object is obviously a black
hole. So with that we can kind of ask
the following question. If we we think
about the earth, if we were to take the
earth and compress it down to a very
small size, how small would we have to
make the earth to turn it into a black
hole? And we can use the Schwarz child
uh formula which will give us the
Schwarz child radius to answer that
were
radius need to be? And of course that's
radius. Okay. So the only input to our
shortch formula is simply the mass of
the earth.
So m in our formula it's going to be the
mass of the
earth and we need that in kilograms and
in kilograms that's approximately 6 * 10 24
24
kg and then everything else is just the
constants. So we can just plug them in
show the shorts child
radius. So I've got a two and I've got the
the
here two big g big m / c^ squ. So the
big m is the mass of the earth. Okay. So
g is the gravitational constant
6 67 * 10
-1 mass of the earth 6 * 10
24 speed of light approximately 3 * 10 8
I'm going to square
that and this works out to be
about 9 mm which is close to
10 mm which is approximately 1
cm either way. You know a millimeter is
10us 3 c m 10 the minus2. So basically
if you take the
earth and shrink it down so its radius
the entire earth fits inside a 1 cm
radius turns into a black
hole. All
right. So let's go back to a picture
just so we understand what it means when
we talk about the short radius. So the
picture that I always like to do is we
circle and then in the center of that
circle this is a black hole by the way
singularity and that
is where all the mass is concentrated.
And then this surface here is the event
horizon. And that's the point where the
escape velocity is equal to the speed of
light. That's the point of no return,
the event
horizon. Outside of the event horizon,
with sufficient energy in a rocket ship
or whatever, you can escape. Inside the
event horizon, no amount of energy is
sufficient for you to escape the black
hole. So this is like a one-way gate.
You know, it's like the door to Hotel
California. You can check in, but you
can never check out once you pass through
through
there. You know, Hotel California,
you're not getting back
out. And no matter how much energy you
have in your uh spaceship, okay? So then
the distance from the singularity out to
the event
horizon is the short
So event horizon I mean it's another
important word
light. So anything that has mass not
So one of the early physicists in
the study of black holes was John
Archbald Wheeler and he came up with
this no hair theorem and basically if
you just go with Einstein's theory of
general relativity it's a classical
theory has no quantum physics in it. The
no hair theorem states the following
that there's only three physical
quantities that define all black holes.
And that's what we mean by the no hair
theorem. So there's only
only three
Wheeler. Incidentally, later on we're
going to talk about um some other
important physicists. One of them will
be Richard Fineman. So, John Wheeler was
Fineman's uh uh adviser when when
Fineman was working on his PhD.
So only three physical quantities that
describe all black
holes and uh this is let's talk about
the three quantities and then uh this is
you know again classical general
relativity. So no quantum once we have a
quantum theory of gravity quantum theory
of general
relativity we can't say for certain that
no hair theorem will be a good theory at
that point. it it probably may need to
be modified. So I'll just put in here
uh this comes out of classical general
relativity. So classical GR
okay and here are the three quantities the
the
mass. So what's the mass of the black
hole and then how quick the spin how
quickly is it rotating? The technical
word for spin of any macroscopic body in
physics is angular momentum. So I'll
write out angular momentum and then when
spin spin you know the faster it spin.
Okay. And then the third is if there
happens to be a charge imbalance you
would see that. So electrical
charge generally though we expect that
when matter falls into a black hole an
equal amount of positive electrical
charge and and negative electrical
charge fall in so that the net
electrical charge is zero but it is
possible you could have an imbalance and
you would actually be able to measure
that. So electrical
charge those are the three quantities.
So all black holes have various numbers
Okay. And the last thing I want to say
is there's these and it comes out of the
two. So let's
let's
let's do a new bullet here. Uh
Roy Kerr.
So Kerr was a very important uh
physicist that took the work that was uh
done by Schwarzchild and extended it.
Now Schwarz's work work was basically on
a static black hole. Black hole that's
not rotating. That's an unrealistic
black hole because we always have to
satisfy conservation of angular
momentum. And so whenever matter falls
into a black hole, we expect there to be
a net angular momentum. And so the next
person that took the work of
Schwarzchild and extended that and
allowed for the fact that black holes
are likely to be rotating was Roy Kerr.
So before I dro Roy Kerr, let me just
make the following
statement. If you have a black hole and
you treat it as a static black hole
where it's non-rotating, that's called a
Schwarz child black hole. So a nonrotating
Okay, non-rotating. So, it's just
static. And of course, we recognize
that's it's the easiest case. If you
take the field equations, the general
relativistic field equations, and you
hold the black hole, the easiest way is
to just say the the hole is not
rotating, it's static and equations are
easier. Uh then if we have a more
realistic black hole, one that is
rotating, then that's where Roy Kerr
came in. So Roy [Music]
And in honor of Roy Kerr, we call that a
Um, so Schwarz child and the cur
and I should have added this was done in
Next topic is just want to discuss some
of the interesting
features if we have a person an
astronaut who falls into the black hole.
So a new bullet here
So imagine we have an astronaut and you
know we are in a spaceship orbiting
above the event horizon of a black hole
and tie a rope around the astronaut's
waist and we open up the cargo bay and
lower the astronaut
in. So imagine we're back on the
spaceship in the cargo bay looking down.
So from we're going to look at it from
our viewpoint. So
from our
viewpoint, we watch the astronaut as we
lower the astronaut towards the event
horizon. So from our
viewpoint and in no particular order
here. So and you know he's initially
kicking, screaming, I don't want to go.
Don't lower me down. So as we get lower
him closer and closer, what we start to
see is his motion slow
down. So we we notice that his motions slow
down. Let's say we gave him a flashlight
because we knew, you know, he'd be
scared. So, if you have a flashlight,
you know, shine it and you're not afraid
of the dark. And we look at the light
coming from his flashlight. And what we
notice is it'll get shifted to the reds
due to the gravitational red shift. So,
redshifted. Let's say we gave him a
watch to wear. We watch the watch and we
actually watch the second hand. And what
we'll see is it'll tick more
slowly. So his
And then finally what we start to see is
the gravitational force is so great it's
we're lowering him in feet first. Okay.
So we start to see that the force of
gravity will really start to pull on him
and stretch him
out. So he gets stretched like spaghetti.
Okay, that's how we we see it. So from
our viewpoint. Now let's do it from his
viewpoint. You know what? What? Okay.
So, from the astronaut's
viewpoint, the one that's getting lowered
in. Okay. So, I'll try to contrast it
with the ones we did up here. So, up
here, we saw that his motion slow down,
but to him, no, his motions are normal.
Everything runs normal.
He's kicking and screaming. His motions
motions are
normal.
Um, and in his vicinity, the flashlight
is normal. There's no red shifting. So the
light is
normal from his flashlight. So he, you
know, shines it on other pieces of
matter that are drifting by him and the
light to him is normal. So the light is
If he looks at his watch, he sees his
watch tick normal. Tick tick tick tick.
But when one of the things that he'll
start to see due to the curvature you
know it's like a funnel if we go back
and we think you know that spacetime is
bent you know extreme. So what can
happen is uh he can see all the stars
due to the extreme curvature.
So he can and he doesn't have to tip
turn his head. Just in any direction he
And actually in some cases the light
that leaves the back of his head can
see the back of his head or helmet. Give
him a helmet. He's an astronaut in a
the back of his head, but you know helmet.
helmet.
And it's also depending on the size of
the black hole and then if the rotation
of it and then where he is. It's
possible that he can see light that
possibly
see what he
he
was doing say 10 minutes ago, possibly
even up to an hour ago. You know, he
sees Oh, I sees what the cargo bay
opened up and he was lowered down. So,
he could possibly see what he was doing 10
10
minutes. I'm just using that. It's It
could be any about 10 minutes ago, you
know, to an hour, say, or hours
right. And then of course he looks down
and he sees that
his, you know, ankles,
ankles,
toes, calves are getting stretched like spaghetti.
spaghetti.
sees his
his [Music]
[Music] ankles,
toes,
Spaghetti. All right, lights from his
viewpoint. Okay, next
topic jumps back
to the work done by uh Stephven Hawking.
In fact, this is his big contribution to
physics and why he's so
radiation. So remember I've been telling
you how we really need to have a quantum
theory of gravity. And actually what
Hawking did here is he attempted to kind
of if you will paste on the effects of
quantum physics on the classical black
hole and see if there was an interesting
effect and there is and we this
interesting effect is that black holes
are not forever objects. They actually
slowly evaporate over time and they
will eventually
time. Let me step back here though and
explain something. As matter falls into
a black hole, the mass of the black hole
is increasing. And as the mass
increases, the short shell radius
increases. If you go back and look at
the short shell, it's proportional to
the mass. The more mass in the black
hole, the larger the black hole. You
know, the greater the short radius.
There's there's a point though that
there's only a finite amount of uh
matter in the universe. And so
eventually the black
hole in any region of space. There's no
more matter in that in that region of
space. So the matter is not falling into
the black hole. So then that's the
maximum size of the black hole. And then
you get the effect of the Hawking
radiation. And slowly over time the
Hawking radiation leaks out of the black
hole and it carries away mass from the
black hole. from a black hole gets a
little smaller. And that's what we mean
by the evaporation. So, we're way off in
the future where all the matter has
fallen into the black hole. There's no
more matter falling into the black hole.
And now the black hole is not static.
It's slowly going to shrink. It's going
to lose the matter. Just to give you a
time scale on this, I said it takes a
long time. So, let's let's look at a
very small black hole. Say one that's
five times the mass of the sun. So
hole
takes about 10 62
years to fully
evaporate. So this is 10 raised to the
So it's a very long time. You know the
universe is only 14 billion years. No
only 14
billion years
old. Now a billion is 10 the 9 and then
another factor is 10. So that's 10 the
10. So the universe is 10 to the 10
years. So this is 52 orders of magnitude
greater than the actual age of the
universe. So it's a very very long time
for the black hole to
evaporate. So why do they evaporate?
It's a quantum uh physics effect. Uh
basically there's virtual particles
right above the event horizon and if one
of the particle so they come out of the
vacuum and they exist for a very brief
moment of time and in general they go
back into the vacuum but if one of those
virtual particles falls into the event
horizon below the event horizon then the
other one becomes a real particle and if
that one does not fall in that
represents mass lost to the interior of
the black hole and so that's how the
particles are
And again, it's a very slow
process. Um, I don't know. I can draw
this. I'll try to draw it. So, all
right. So, there's our event horizon.
And in general, we'll have a virtual
pair of quantum particles come out of the
the
vacuum and they'll go back into the
vacuum. So, there's where they come out.
They live for a little bit of time, very
short per bit of time, and then they
together go back in. And this is
happening all around us, right in front
of your face, everywhere. All of our
fundamental theories of the
electromagnetic, the strong and the weak
depend upon the this existence of these
virtual pair that come out of the vacuum
exists and go back in. But sometimes
what can happen, they'll come
out and then one of them will fall below
the event horizon and go in here. And
then this one does not have a partner
then to go back into the vacuum and if
it does not fall back in and it travels
away that represents a loss of mass to
the black hole. So this little particle
travels away and that the particle
traveling away that's what Hawking calls
that's hawking hawking
wormholes. So, classical general
relativity. Again, we don't have a
quantum theory, so all bets are off once
the quantum theory of general relativity
of gravity shows up. This wormholes, who
knows? But so
with Einstein's normal general
relativity with no quantum effects built
into it, you know, his original theory
that
connect to
locations. And though that tunnel then
wormhole. Let's draw a picture, show you
an example. There's a couple of examples
draw. So,
um, show you how this this can be
utilized to our
Let's say we have the earth here and
star. So this is the
earth and this is the star and the star
may be a nearby star that we want to get
in a spaceship and want to travel to that
that
star. Um, so we get in a spaceship and
so this surface here is our four-dimensional
spaceime. So we get in the our spaceship
here and we travel to the star and let's
So the distance here traveling through our
our
space 25 lys 25 light years
away. If people back on earth if if our
spaceship travels you know 98 99% of the
speed of light people back on earth
would see that it would take us about 25
years a little longer 25 years to reach
the star. Okay so that's that's normal
space flight anything. So however though
if we look at classical general
relativity there's solutions that allow
for the following situation. You can
open up a little tunnel
here and that little tunnel then leaves
our four-dimensional space-time sheet
and travels through
And this passage, if you
will, through
hyperspace is the wormhole. So this
wormhole. Acid, you know, think of it as
kind of a
And you know this may be half a
lightyear in length whereas if you go
here it'd take 25 light years. So
instead assuming you know everything
works out the spaceship goes down through
through
here and they take 6 months instead of
25 years you know it's measured by
people back on
earth. Okay. So that's one way. This is
where you're connecting to spatial
points. So just to describe
shortcut. If you think about it,
actually you could even beat light
because somebody here on Earth could
turn on a flashlight and point it out to
that star and that light would take 25
years to reach the star and then the
spaceship in here, you know, it's only
going to take them, you know, six,
seven, eight months. So they'd be but
again they're not they're not beating
the light on the manifold of our
fourdimensional spacetime. They're
actually exploiting higher dimensional
space hyper space. Okay. So that's so
this is a wormhole where we
connect to uh
space locations but you can also arrange
the wormhole along the time
axis. So there's another way. So let me
say so you could act. So, just to back
back up here, you could uh
actually
light. But again, you're not you can't
violate going faster than light on the
same space time. But here, you're taking
a shortcut. You know, it's like it is
almost like you run a marathon race, but
part of the race goes through a force
and the part of
the the race, you know, gets kind of,
you know, you turns and comes back and
then you you know, you're sneaking, you
take a little shortcut path through that
forest, get back on the track. It's, you
know, that's the sort of thing we're
talking about. Okay. So, you could
actually beat a beam of light. All
right. Now let me there's another way
you can
um position the wormhole and that's
along the time
axis. So page 10. So uh here we have the
time axis. So here's time
uh I don't know. Here's you
you
now and then here will be you in the
future and then back here is you in the
past. Okay. So what we can do is we
could the same classical general
theory. We'll put the wormhole
opening right there. But then what we'll
do is to we'll swing it around.
and put the opening right there. I got
to draw all these lines like that
that
gosh. Okay. So, you could jump in
here, tra travel along, and
then you're going to go end up out here.
time. Back in
time. So when you position the
wormhole along the um time axis, another
machine. And you could put it you could
put the opening up here and go into the
future. You could go forward in time.
Here I've got going back in time up
here. So this is the
past and up here's the future of course
because time time is marching forward
you know along that direction. So you
could go back in time and that's another
way of thinking is we call it a time
machine. Then you get into some issues.
Well, you go back in
time. Yeah. What happens if you prevent
your parents or your grandparents from
gosh. We have answers to that question,
but that's that's later on. Um, but the
question
have a quantum theory of
gravity. We don't have that
yet. So we take the wormholes with a
grain of salt. In fact, people have
shown in similar just like how
Hawking kind of if you will pasted some
quantum physics on the surface of a
classical black hole and recognize that
the black holes will evaporate via
Hawking radiation. Physicists have done
the same thing with the wormholes is
when you start to paste some quantum
physics inside the tunnel of the
wormhole you the stability of the
wormhole comes into question. In fact,
what they've even shown is if you just
shoot even a single photon, a particle
of light down a wormhole where you've
kind of paste it on the quantum physics
that the wormhole collapses.
So it
again classical GR allows for it but
until we have a full quantum theory I
wouldn't put a lot of hope on the
wormhole till we have that quantum
theory and we can say for certain that
they're stable over significantly long
time intervals
uh then we know they exist and and
that's kind of where we're at. So we
take them with a grain of salt with the
classical theory.
All right, guys. We'll end there and
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