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Black Holes

Newtonian

- Universal Mutual Gravitation Isaac Newton, in his

Principia, formulated the Law of Universal Mutual

Gravitation - Gravity is an Attractive force
- Works to bring massive objects closer together.
- Gravity is a Universal force
- Works everywhere in the Universe.
- Gravity is a Mutual force
- Works between pairs of massive objects

- Gravitational Force Force of gravity between any

two objects depends only upon - The masses of the two objects
- More massive objects feel a stronger force.
- The distance between them
- Objects closer together feel a stronger force.
- It does not depend at all on the shapes, colors,

or compositions of the two objects.

- The Law of Universal Gravitation The force of

gravitational attraction between any two massive

bodies is proportional to their masses and

inversely proportional to the square of the

distance between their centers. - The Force of Gravity is an example of an "Inverse

Square Law Force"

- Stated Mathematically
- Where
- F force due to gravity.
- M1 mass of the first object
- M2 mass of the second object
- d distance between their centers.
- G "Gravitational Force Constant"

The Flaw

- Doesnt work when you are talking about intense

gravitational force of black holes and neutron

stars - Thankfully Einstein created the principles of

relativity

- In Newtonian gravitation, an orbit is always an

ellipse. - As the gravitating body becomes more massive and

the test particle orbits it more closely, the

speed of the particle in its orbit increases

without bound, always balancing the gravitational

force. For a black hole, Newton's theory predicts

orbital velocities greater than the speed of

light,

1st Flaw

- It gave the wrong prediction for the precession

of the perihelion of Mercury's orbit. - Mercury's orbit is elliptical, as predicted by

Newton's theory of gravity, but the ellipse

doesn't stay in precisely the same place all the

time. - It precesses, which is to say that as Mercury

orbits the sun, the entire ellipse rotates about

the focal point (i.e. the sun) as shown in the in

the picture

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2nd Flaw

- It did not explain why the gravitational force on

an object was proportional to its inertial mass. - In other words it did not explain why

gravitational acceleration is independent of the

mass or composition of an object.

3rd Flaw

- It was inconsistent with the Special Theory of

Relativity. That is, if an instantaneous force of

gravitational attraction exists between two

objects then information about the location of

one object would be transmitted to another object

instantaneously by changes in the gravitational

force. Thus it would be possible to send

information faster than the speed of light.

The special theory of relativity changes our

conceptions of space and time

- This theory, published by Einstein in 1905, is

based on the notion that there is no such thing

as absolute space or time - Space and time are not wholly independent of each

other, but are aspects of a single entity called

spacetime

Special Relativity

- General Relativity developed from Special

Relativity - universal speed limit speed of light c

300,000 km/sec - Example A Nolan Ryan on a train
- train moves East at vtrain 30 m/sec (70 mph)
- Nolan, who is on the train, throws his fastball

at vball 40 m/sec (90 mph) - Nolan sees the ball move at vball 40 m/sec (90

mph) - We see the ball move at vtrainvball 70 m/sec

(160 mph) from the ground

- Example B Nolan switches on a flashlight
- Nolan turns on his flashlight pointing East
- Nolan sees the light move at c 300,000 km/sec
- Do we see the light move at vtrainc 300,000.03

km/sec? NO!! - We also see the light move at exactly c 300,000

km/sec! - Even if the train moved at vtrain 200,000

km/sec, we'd still see the light move at velocity

c!

- Time reversed case now lets throw a baseball up

from the Earth (ignoring air friction) - I throw it at 20 m/sec - it goes up 20 m and

falls back to Earth - Nolan Ryan throws it at 40 m/sec - it goes up to

80m at falls back to Earth - shoot it out of a cannon at 10 km/sec - it goes

out beyond the communication satellites and then

falls back to Earth - shoot it out of a cannon at 11 km/sec - and it

goes up and slows down, but never comes back - This is the escape velocity

The speed of light is the same to all observers,

no matter how fast they are moving

An observer will note a slowing of clocks and a

shortening of rulers that are moving with respect

to the observer

- This effect becomes significant only if the clock

or ruler is moving at a substantial fraction of

the speed of light

The general theory of relativity is our most

accurate description of gravitation

- Published by Einstein in 1915, this is a theory

of gravity - A massive object causes space to curve and time

to slow down - These effects manifest themselves as a

gravitational force - These distortions of space and time are most

noticeable in the vicinity of large masses or

compact objects

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- The theory of relativity predicts a number of

phenomena, including the bending of light by

gravity and the gravitational redshift, whose

existence has been confirmed by observation and

experiment

Escape Speed

- Escape velocity is the speed an object would need

to escape from a celestial body. - Gravity is low on an asteroid. You could throw a

ball off it, or jump off it. - Thus, low escape velocity
- The escape velocity depends on mass.

The general theory of relativity predicts black

holes

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- If a stellar corpse has a mass greater than about

2 to 3 M?, gravitational compression will

overwhelm any and all forms of internal pressure - The stellar corpse will collapse to such a high

density that its escape speed exceeds the speed

of light

Certain binary star systems probably

containblack holes

- Black holes have been detected using indirect

methods - Some binary star systems contain a black hole
- In such a system, gases captured from the

companion star by the black hole emit detectable

X rays

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Supermassive black holes exist at the centers of

most galaxies

- These are detected by observing the motions of

material around the black hole

A nonrotating black hole has only a center and

a surface

- The entire mass of a black hole is concentrated

in an infinitely dense singularity - The singularity is surrounded by a surface called

the event horizon, where the escape speed equals

the speed of light - Nothingnot even lightcan escape from inside the

event horizon

Just 3 numbers completely describe the structure

of a black hole

- A black hole has only three physical properties

mass, electric charge, and angular momentum - A rotating black hole (one with angular momentum)

has an ergoregion around the outside of the event

horizon - In the ergoregion, space and time themselves are

dragged along with the rotation of the black hole

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Falling into a black hole is an infinite voyage

- Could a black hole somehow be connected to

another part of spacetime, or even some other

universe? - General relativity predicts that such

connections, called wormholes, can exist for

rotating black holes

- Mass tells space how to curve
- Space tells mass how to move

Gravitational redshift

- light rays (i.e. photons) lose energy as they

climb out of a gravitational field - So, they shift to larger wavelength, lower energy

Gravitational Energy

- Energy is conserved - i.e. the total energy does

not change but it can be transferred into a

different form - consider a baseball in outer space - very far

from the Earth - we'll say infinitely far. - let it go from rest
- it will reach a high velocity - and gain lots of

energy of motion as it falls - energy is conserved - so where did the energy

come from?

- Gravity - we assign a negative potential energy

to an object in a gravitational field - so, the total energy is still the same as before
- lots of energy of motion and
- a negative gravitational energy that compensates

for this to allow energy conservation

- Suppose the Earth was squeezed down to half its

size, but kept the same mass - The escape velocity would be larger - 15 km/sec

in this case - the baseball would slow down from 15 km/sec to 11

km/sec by the time it reached the current radius

of the Earth

- Suppose the Earth was squeezed down to 1 cm
- the escape velocity would be c
- any smaller and its a black hole - nothing can

escape!

Sample Escapes Velocities

- Earth 11.2 km/sec (25,000 mph)
- Moon 2.4 km/sec
- 1 km asteroid 1.3 m/sec
- Sun 618 km/sec
- White Dwarf 6000 km/sec !!

Key Words

- black hole
- black hole evaporation
- equivalence principle
- ergoregion
- event horizon
- general theory of relativity
- gravitational radiation
- gravitational waves
- gravitational redshift
- Heisenberg uncertainty principle
- law of cosmic censorship
- length contraction
- Lorentz transformations

- mid-mass black hole
- no-hair theorem
- primordial black hole
- proper length (proper distance)
- proper time
- Schwarzschild radius (RSch)
- singularity
- spacetime
- special theory of relativity
- stellar-mass black hole
- supermassive black hole
- time dilation
- virtual pairs
- wormhole