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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

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

1905 he Einstein that 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

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

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

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

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.

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

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

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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Lecture 15 A105 Black Holes