General Relativity and Black Holes - Cornell University · Lec 19: General Relativity 7 Historical Background ... Clock A runs faster than clock B

Lec 19: General Relativity 1

General Relativity and Black Holes

Lecture 19


Lecture Topics■ General Relativity■ The Principal of Equivalence■ Consequences of General Relativity

■ slowing of clocks■ curvature of space-time

■ Tests of GR■ Escape Velocity■ Black Holes


General Relativity■ A Theory of Gravity

■ Albert Einstein■ 1916

■ Incorporates accelerated motions into Special Relativity Albert Einstein

(1879 – 1955)


When you’re really famous …


Einstein’s Insight■ Newton’s law of gravity worked

very well in predicting planetary motions.

■ But Einstein wondered how gravity could be made consistent with Special Relativity.

■ Einstein’s insight was the “Principal of Equivalence”

■ He realized that a gravitational field would bend light rays.

■ He also realized that Euclidean geometry would not apply.


Principle of Equivalence■ Gravity and acceleration due to a force

are indistinguishable.■ In a small local environment (must be a

small enough “box”)

■ This is the foundation of General Relativity.


Historical Background■ Proved a relationship between symmetries in

physics and conservation principles (1915 or so).

■ This information was used by Einstein and is used in many areas of physics.

■ Noether was going to be a language teacher but became interested in mathematics.

■ David Hilbert and Felix Klein fought to get her on the faculty at University of Göttingen. The battle took four years, but she was finally appointed in 1919.

■ She remained there until 1932 when the Nazis caused her dismissal because she was Jewish.

■ She accepted a visiting professorship at Bryn Mawr College and also lectured at the Institute for Advanced Study at Princeton.

Emmy Noether(1882 – 1935)


■ Imagine yourself in a closed room.■ By the principle of equivalence you could not

tell if you were on earth or in space in an accelerating rocket.

Gravity

AcceleratingRocket ship


■ Now imagine yourself falling in a closed room.■ By the principle of equivalence you could not

tell if you were falling towards earth or floating in space.

Fallingundergravity

Floating in space


Gravity and Time■ Imagine two clocks in

an accelerating rocket.■ Clock A is in the front.■ Clock B is in the back.

■ Clock A emits pulses 1 second apart.

■ How far apart are they at Clock B?

AcceleratingRocket ship

A

B


Clocks in GR

NO ACCELERATION■ Flash emitted from A■ To reach B it travels a

distance d1.■ Since the rocket is not

accelerating, we have for the next flash

d2 = d1

■ Flashes arrive one second apart.

A

B

A

B

“A” emitsflash

“B” receives flash

d1

A

B

A

Bd2


Clocks in GRACCELERATING

■ Flash emitted from A■ To reach B it travels a

distance d1.■ Accelerating rocket ⇒

rocket is traveling faster for next flash

⇒ d2 < d1

■ Flashes arrive less than 1 second apart.

A

B

A

B

“A” emitsflash

“B” receives flash

d1

A

B

A

Bd2


Clocks in GR■ Clock A runs faster than clock B.■ The equivalence principle states

■ Gravity and Acceleration are the same.■ Therefore, the same thing happens in a

gravitational field!■ A clock on a mountain top will run faster

than a clock at sea level.


Consequences of GR■ GR changes our concepts of space and

time (gravity and geometry are linked).■ Einstein no longer thought of gravity as

a force but a curvature of space-time.■ Space is “curved” by massive objects

causing objects to fall toward them.


Curvature of space-time

■ Empty space is “flat” space-time.■ Space with matter is “curved” space-time.

Sun


Bending of light■ Near a massive object,

GR predicts that light will be deflected.

■ GR predicts 1.75” for light grazing the Sun.

■ Measurements of stars during a solar eclipse verified this to within 1%. (Eddington - 1919).

Sun


Time delay of light

■ Near a massive object, GR predicts that light will travel a longer path due to curved space-time.

■ Verified by timing signals from Viking spacecraft passing by the Sun.

Sun

At Earth

Viking


Binary Pulsar■ Two neutron stars orbiting

one another■ Work done at Arecibo Observatory■ Orbit period = 8 hr,

Orbit speed = 0.1c !!■ Serves as a test of General

Relativity■ Precession (movement of orbit) on

sky.■ Decay of orbit due to Gravitational

Radiation. (New type of radiation!!)■ General relativity has been

proven over and over to be correct.


Gravitational Redshift■ Light from the surface of a massive

object will be redshifted. ■ The more massive and/or more compact an

object, the greater the redshift.■ ~ 0.01 A for the Sun.■ ~ 1 A for a white dwarf.■ Gravitational redshift verified to 0.01% by

hydrogen masers (one in space, the other on the ground).


Escape velocity■ 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.■ The escape velocity depends on mass &

radius


Escape velocity■ Escape velocity is the speed an object would

need to escape from a celestial body.■ The escape velocity depends on mass.■ Examples:

■ Earth: 11.2 km/sec (25,000 mph)■ Moon: 2.4 km/sec■ 1 km asteroid: 1.3 m/sec (you could jump off it!)■ Sun: 618 km/sec■ White Dwarf: 6000 km/sec !!

■ How high can the escape velocity get?


Dark Stars■ Rev. John Mitchell - 1783■ An object more massive than the Sun

could have an escape velocity greater than the speed of light!

■ Today we call this object a black hole.■ An object from which no light can escape.


Making a “Dark Star”■ Suppose the escape velocity of an object was

equal to the speed of light.

Rs = Schwarzchild radius

Putting in numbers:MR 3s = Rs in km

M in solar masses

Warped Space Time


How big are black holes?Object Mass (Msun) Rs

Star 10 30 kmStar 3 9 kmSun 1 3 kmEarth 3 x 10-6 9 mm


The Event Horizon■ The event horizon

is located at Rs.■ Anything inside the

event horizon is gone from sight forever (nothing can escape).

Rs

Event Horizon

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General Relativity and Black Holes - Cornell University · Lec 19: General Relativity 7 Historical Background ... Clock A runs faster than clock B