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Lecture 15 A105 Black Holes | Brian Woodahl | YouTubeToText
YouTube Transcript: Lecture 15 A105 Black Holes
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This content explains the formation of neutron stars and black holes from massive stars, and then delves into Einstein's theories of Special and General Relativity, which are crucial for understanding the physics of black holes.
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Guys, let's go ahead and get started.
Today, uh, we're going to talk about
And I better put a U case in here uh
uh
with the following where the mass is
between 8 and less than
sun and less than
25 times the mass of the sun.
star, you know, after the supernova. So,
Oh my gosh. We saw last time that the
super giants which arise
uh from main sequence stars having a
mass greater than eight times the mass
of the sun but less than 25 times the
mass of the sun uh form the neutron star
after the supernova. And just to refresh
star, it is the neutron degeneracy
pressure that balances further
gravitational collapse. So I'll just say
in the neutron
Now, as long as the mass of the neutron
star is less than three times the mass
of our sun, the degeneracy pressure of
neutrons is sufficient to repulse
gravitational collapse. So, important
sense here.
So, as
long as the
the
less [Music]
[Music]
repulse gravity.
Okay, you know sometimes students get
confused. This is the mass of the
neutron star and here is you know what
the main sequence mass started out with.
Okay, but now now we're focusing on the
actual mass of the neutron star. It has
to be less than three times the mass of the
the
sun. If it's greater then the gravity is
going to collapse. It's going to be
stronger than the neutron genery
pressure and it's going to collapse it
uh to form the black hole. So I'll just say
say
but if the
mass is greater I'm talking about the
mass of the neutron star itself. If if
the mass is greater
greater
than 3 * the mass of our
sun, gravity [Music]
wins
and neutron star collapses to form the
Okay, now we want to get into some of
the uh the physical attributes of a
black hole. But before we do that, we
have to look a little at uh Einstein's
theory of
relativity. So um we're going to look at
the the first theory we're going to look
relativity relativity. Oh my gosh. Here
in
1905 he Einstein that is
is
publishes it's SR SR he he 1920
Einstein publishes oh my
gosh SR here we go S SR SR SR and the
basis of SR is two postulates so there's
two postulates and with the two
postulates then all the important laws
of special relativity can be arrived at
derived at however you want to say it so
the two postulates are as follows and
I'm kind of watering down some of the
formalism here uh the laws of physics
uh technically the laws of physics are
the same uh in every non uh accelerating
uh reference frame. uh another word, but
I think just if I soften it like this
and say laws of physics hold everywhere
in the universe, we're good with that.
Maybe maybe I should spell that
correctly here. Just wipe that out.
Okay, that's one. And then the second is
that no matter who you are and no matter
what your speed is and no matter the
source of the light, that when you go to
measure the speed of light, everybody
measures the speed of light as the same value.
value.
observers and as we have discussed
that's letter C and rounding it off it's
about 3 * 10 to the 8 m/s. Another way
to think about it, 670 million miles per
million m
m
hour. Okay, so it's the same for all. Uh
so these are the two
postulates. Okay. And then out of those
then you get a plethora of important
uh constraints um and equations as well.
So these give rise to the following effects.
effects.
These two the two
postulates give
rise and to the
Oh gosh. In no particular order. No
particular order here,
should I don't want to number them
either because I just there's no So, I
did a hyphen here. So, time
dilation. Time
dilation. In a nutshell, what this means
is moving
clock. Now, if I take this clock, it
tick tick tick and I throw it and we
watch the second hand. and I throw it
really fast, we'll see that the second
hand will tick more slowly than it would
if the clock was was was
stationary. Um, and that's time
dilation. And that applies to
all matter. You know, it applies to
blocks of wood and house plants and
people. Anytime you're traveling fast or
relative to a stationary uh clock, a
stationary person, whatever, your clock,
the fact that you're traveling uh with a
great velocity is going to tick more
slowly. Now, the velocity has to be up
on the order, you know, percentage of
the speed of light. And so, in general,
that's why you never notice it. We just
don't travel fast enough really to
notice time dilation. Uh, another interesting
interesting
So uh distances parallel to the velocity
direction will decrease in
length. I got
a I had another example here I show you.
So we hit the clock. Let me get the the ruler
ruler
out. This a
ruler. 6 in
mechanics steel ruler. Now, if I take it
and throw it really fast and you were to
measure the length of it be slightly
less than than um 6 in. And the faster I
throw it, the more that it contracts
along that length. And I have to throw
it so that it's in the same direction as
the length of the ruler. Okay. You know,
if I turn it this way, damn thing, and
then I throw it this way, then of course
the width of it would shrink. If I take
it this way and throw it that way, the
length of it, that's what we mean by
distances, you know, along the velocity
direction will decrease in length. And
again, the effect is only noticeable
when those velo when you're the speed
that the ruler is traveling at is on the
order of the speed of light. And so, in
general, we don't see it unless you
observe something that's moving very
quickly. i.e. ate the speed of light
you're not going to see this effect. So
it's for ordinary common garden variety
everyday velocities which are very small
relative to 670 million miles an hour 3
* 10 8 meters per second just don't have
the velocity those are too small you're
not going to see the effect. Another
important effect that came out of
special relativity was this equivalence
between matter and energy and that's
this very famous equation. Energy is
equal to the amount of matter mass times
the speed of light squared. So E= MC^2.
So matter and energy are the same thing.
So,
um, we note if we go back to these first
two, the way that the time ticks is
relative to the speed and the space
occupied by that object is modified. I
uh again dependent upon the speed. So we
see an al uh an intimate connection
between space and time and then this was
an important realization made by
Einstein is that we must
treat the time variable on equal footing
with the spatial variables and that led
to this concept known as
time
equals and this led to the concept known
uh is four-dimensional because of this.
We have three spatial dimensions. We
have front and back that's one. Left and
right that's two. Up and down that's
three. And also time that's four. So
this concept of spacetime represents a
four-dimensional entity. Okay. And that
and so that's this idea that you put
time treated equally as the spatial
dimensions and the sum of those then you
got the three that come from space and
then another comes from time. That's a
four dimensions. Okay. But now prior to
Einstein was in the era of Newton the
idea was that time was absolute and that
there was some sort of if you want to
think about it some master grandfather
clock somewhere in the universe and then
that all the other clocks were
referenced off of that master to to to
to grandfather clock and what Einstein
showed you know mainly here because the
length contra uh length d time dilation
that time is, you know, relative to the
that person and their motion through u
the space and so that if you move more
quickly then your time is going to slow
down relative to a stationary observer.
Okay? So no longer is time absolute time
is relative. Okay? And of course with
length contraction spatial dimensions
are relative. They're dependent upon the
motion of the actual object. And then
that's where the word relativity comes
from. Okay, so that's special
relativity. And special relativity only
gets us so far to fully grasp the
important features of the black hole. We
got to talk about his next theory of
relativity, which is a general theory of
relativity. So we start a new page on
the page
four. And so we go back and Einstein publishes
publishes
SR in 1905.
And so for 10 years he after he
published that he works on the general
theory of relativity. So
relativity and abbreviate that GR. So we
have SR and we have GR. Okay. So this is
uh there's basically one postulate of GR
and then with that one postulate
postulate you get the field equation and
the field equation
then it connects how the matter and
energy then distort the the local
spaceime and then the motion through
that spaceime gives the appearance or
the illusion of the force of gravity.
I'm getting kind of ahead. We'll talk
about it later on, but okay.
So, so I want to go kind of in the
saying. So, one right here one postulate
and then if that one
postulate then the field equation can be
written. So, one postulate and the one
postulate is goes by a special name of
the equivalence principle. So okay let's
write down the special name of of it and
then I'll write in words what it is draw
a picture and then we of course have a
nice picture on the web page it'll be
much better than the one that we can
draw. So the one question is known as
the equivalence principle in a nutshell
in glossing over some of the uh the
nuances here. Uh
a rocket ship. I didn't know how to say
this, but I
1g
earth. Oh my gosh. That's the equivalence
equivalence
principle. Bad draw
picture. All right. So, here we go. Here
house on Earth. And there's no windows
in this house. There's just a one-way
door. You go in there and we lock the
door. And you're in there. There's a
chair and a lamp and a bunch of books
and a video games and food and water and
an exercise uh facility and a facility
to cook and uh entertainment loca uh a
room and there's sleeping room, you
know, gymnasium, bathroom, all that. But
there's no windows and you can't look
outside. Now, you're on the
earth. We put you in there on the earth. Okay.
But we can also put you in that same
house. It's just again, this is a
special house. And then put rocket
motors and then put you out in space far
away. Again, you've got oxygen, water,
food, and turn on accelerate the rocket
motors so that this
structure has an acceleration equal to
1g. Of course, that 1g is what you
experience on Earth due to Earth's
gravity. And what this equivalence
principle is is the following is you can
again there's no windows you can't look
out in either one of these situations.
space and this is on
earth and this was the connection
Einstein is that the gravity and
acceleration are one and the same. Okay.
So there's no test you can
do there. There is a caveat classically
there's no test you can do. Um but there
are there are some ways around it but
okay don't there's actually if if the
house is really big you can actually
measure the tidal force and that'll
that'll tell you something. So but and
then there's quantum tests. You surely
do you uh you you see
unrrew radiation in this that you don't
see here. So that's one way you could
um this is equivalence principle. Okay.
Einstein use that then to get the
general theory of relativity. And then
in a nutshell, let's see, I guess I'm on
page five. Um
Um
yeah. So what happens then is then with
that you get the field equation and the
field equation then tells you the
following. You get matter and energy in
a particular region then will warp the
space in that region. So GR using this
one postulate you get the field equation
which in a nutshell tells you this
matter and because we saw equals MC² the
equivalent matter and energy. So matter and
energy curves, bends,
bends the
spacetime and then that the motion of
you through that bent or curved spaceime
then is gives the illusion of the force
of gravity.
time that then gives this uh force of
gravity. Oh, I feel the force of gravity.
So as an example in our own solar system
think of the sun as very massive and so
the sun bends the space inward and if we
think of about a a way of drawing this
in a two-dimensional manner draw a
sphere here and label it the sun and
Whoop. Okay. So, this is the bent spacetime.
Uh one of the important consequences of
this is the
um so uh
uh
bent as it travels. Boy, what a long bullet.
spacetime and draw a picture page
six. Oh my
gosh. So here the one of the tests that
we were able to do is the following. So
And uh we're looking at uh starlight
star. And we projected the star to be
out here. So this is what we call the
And this a distant star but its actual
position is actually over
here. So this is the actual position and
the reason is is when the light leaves
the star because it travels
through the region of space near the sun
it comes in and it gets
bent. And so if we look from
overhead there, it's like that.
Okay. And so that was an early test that
we were able to do that verifies the
validity of the field equation which is
a byproduct of general relativity and of
course the equivalence principle.
Um so here's an example where we see that
that
there's the spatial part is bent and
there's also time is affected. So time is
So if I go back to this picture, I have
um or you know whatever watch whatever.
So there's a clock far away where the
space is flat. And then I have a clock
down here where we're the clock is close
to the
sun. And I carefully watch the second
hand on these two clocks. This second
hand here will tick more slowly relative
to that. So right here ticks
ticks
slower relative to this. This is a
normal tick because it's out here where
it's flat and here we have high
curvature. So any region where you have
high curvature the time in that region
is going to tick slower relative to
spacetime. So another way to think about
is you live a long time if you are in a
region of high curvature neutron star a
normal star and then of course near a
black hole. Okay. Uh next topic relating
to uh gravity and that's
So anytime an electromagnetic signal
leaves a region of high curvature, the
wavelengths will get stretched out.
They'll go to longer wavelengths, which
if you think in the visible, that's
shifting all the colors to the red. So
the gravitational red shift goes like
reds
as an electromagnetic signal wave whatever.
So, if I go back to this drawing and I have
have
a say a light source here and I point it
up, you know, say I have a flashlight
and I turn it on and the light leaves
the flashlight here and it travels away.
Excuse me. So as it travels away all the
the spectra in that light will get
shifted towards the reds and that's due
Okay. Um one of the early successful
tests of the in Einstein theory of
general relativity was the measure well
I should say was the agreement in the
motion of of Mercury uh as it orbits the sun.
So
the the name is the shift in the
perihelion of Mercury.
it and draw a picture. So when we look
and we see how um Mercury orbits the
sun, the Mercury is the closest planet
in the solar system that orbits the sun and
and
the the motion there's so Mercury's
orbit is not a perfect circle. It's got
some slidy
electricity and that elliptical path actually
actually
slowly precesses slowly
moves and we have we've observed that
you know through telescopes we've
actually mapped out the fact that that
happens and then prior to Einstein using
Newton's laws if you come up with a
calculation of how much it should
precess and you compare that to what you
measure or what you you know observe and
measure there was a
disagreement. Well, it turns out that in
Einstein's theory of general
relativity there's a
contribution which just goes by the word
tech the technical word desitter
precession and when you add in the
effect of the ditter
precession then the number you get is
exactly equal to what we measure when we
look at it through the telescope. So we
have an we have accurate agreement between
between
uh Einstein's theory and then what we
see in the motion of Mercury.
So, I don't know in a nutshell,
sun, and I'm going to exaggerate this.
And here's Mercury. And then if we watch
it, the way that it orbits, it it walks
around this thing like this.
this.
Okay. Now, here's
Mercury. And I've exaggerated it.
And we see so here the long axis and
then the next time it's here and here
and it's really microscopic but I've ex
exaggerated it. So again everything's
not we're back to our old favorite not to
scale. Now the
perihelion is the close approach. So in
this first one it's right there and then
in this one it's right there. So those
shift. There's two of them. Then in this
one it'd be over here. I somehow lost
all my lines here. I there's another one
in there. I guess. Yeah. One, two,
three. There should be one, two. I guess
there's three. They one on top of each
other. So this one somewhere right
there, I guess. And this one is for that
one. And that one is for that one. This
is the app he helium which is the far
away point. So there's the app healing
there, the app heel there. And they
could have called the shift in the app he
helium, but they called it the
perihelion, which is the close point.
Appalium is a far away point. Far away
point. There's the close point.
point.
uh the result of
And before we had GR Newton's laws, we
didn't we had a there was a discrepancy
between what you saw and what Newton's
law said it should be. All right. All
right. Let's go back. That's enough of
the intro material on Einstein's theory
of general relativity to go back and
pick up. Remember, we're talking about
black holes. So let's go back and then
talk about we kind of left off where we
have a neutron star and there's a upper
mass limit three times the mass of the
sun. And so now let's do a new bullet
forms.
So main
sequence star having a mass greater than
25. See um
um
remember eight less than eight white
dwarf between 8 and 25 neutron star
greater than 25 black hole. So
main sequence star. So again, this is a
star on the main sequence line. So main sequence
sequence
greater
than 25 times the mass of the
here.
Once the iron core remember that's the
end of the layered structure of uh
core [Music]
[Music]
Sakar and that's the
1.4 the mass of the
sun the electron degeneracy breaks down
it can't repulse gravity and collapse
uh begins. So
uh once the iron core mass exceeds genocar
genocar um
uh cannot
cannot
So, gravity then goes to work on those
electrons and squeezes electron and and
protons to produce
neutrons. Neutrons. neutrons. So then
you got you already had neutrons there.
You know, you started with iron. Gravity
takes the electrons and the protons in
the iron and squeezes them together
makes neutrons. You already had some
neutrons there because you know you got
iron in the nucleus there's iron. I mean
there's neutrons. So you got neutrons, a
bunch of neutrons. So then you have
neutron degeneracy pressure. And in the
neutron star that's the stable. The
neutron degeneracy pressure can push
back against gravity.
If the mass of the main sequence is
above the 25, no the neutron degeneracy
pressure not going to be able to hold
forms this this word black
hole. So the black hole
is is such that the escape velocity
um is greater than the speed of light.
So not even
light can escape from the black hole.
And so that's where you this word black
dense
that not Even
Even
field. Remember light is the fastest
So where another way is
the escape velocity of a black hole
Now you have to be
careful. There's a point where this the
escape velocity is equal to the speed of
light and then that's what we call the
event horizon. However, matter that's
getting sucked into the black hole can
collide with other matter that's getting
sucked in the black hole. There's
actually tidal forces that slam the
matter together and that then can
produce electromagnetic radiation. If
that happens above the event horizon,
that signal can leak away from the black
hole. But it's the any signals that are
generated below the event horizon that
are not able to escape. And it's
sometimes people get confused. Oh,
you're getting this you're getting
X-rays from black light that no signals
come off from. Well, that that signal is
produced above the event
horizon. Um, all right.
Now, let me just kind of finish out here
the end.
relativity gr
gr
the details of the structure inside the black
Now, if another way to draw a black hole
is this, and and we'll pick up on this
next time, but you draw a
horizon. That's the point of no return.
Above that the escape velocity is less
than the speed of light. At the event
horizon, the escape velocity is equal to
the speed of light. Inside the event
horizon, the escape velocity is greater
than the speed of light. And then the
mass is all concentrated as at a point
here in the center called the singularity.
And so when I say Einstein's general
relativity fails at describing the
details of the structure of spacetime
inside the black hole, what we mean is
this region in here. This region in
here, general relativity fails. And the
reason is is inside there, you're going
to need a quantum theory of
it. [Music]
[Music]
Theory of
of [Music]
gravity. We have no quantum theory of gravity.
we have been unable to quantize general
relativity. Now, there's a lot of things
we're working on to try to get there.
theory. But we're not there
because the theories of super string,
there's so many different theories and
we can't find the one that applies to
our universe. and to do tests to be able
to determine okay this is one a good
super string theory we can keep and oh
this is one we can discard those tests
require very high energies energies way
many orders of magnitude beyond what we
can do today
so we're kind of stuck on that there's
some other possible theories uh to
generate a quantum theory of gravity but
right now
the most of the people are leading
towards super strength theory and we'll
basically there is no
particles. If you get down to a very
very tiny scale of just little tiny
strings and it's and the strings could
be open-ended or they could the ends
could connect but it's those little
vibrating strings then that produce
basically everything. They produce the
particles, they produce the forces, and
they even produce the structure of space
and time itself. So that's in a nutshell
super string theory. All right, we'll
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