general relativity and black holes - cornell university · lec 19: general relativity 7 historical...
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Lec 19: General Relativity 1
General Relativity and Black Holes
Lecture 19
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Lec 19: General Relativity 2
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
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Lec 19: General Relativity 3
General Relativity■ A Theory of Gravity
■ Albert Einstein■ 1916
■ Incorporates accelerated motions into Special Relativity Albert Einstein
(1879 – 1955)
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Lec 19: General Relativity 4
When you’re really famous …
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Lec 19: General Relativity 5
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.
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Lec 19: General Relativity 6
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.
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Lec 19: General Relativity 7
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)
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Lec 19: General Relativity 8
■ 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
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Lec 19: General Relativity 9
■ 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
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Lec 19: General Relativity 10
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
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Lec 19: General Relativity 11
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
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Lec 19: General Relativity 12
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
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Lec 19: General Relativity 13
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.
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Lec 19: General Relativity 14
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.
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Lec 19: General Relativity 15
Curvature of space-time
■ Empty space is “flat” space-time.■ Space with matter is “curved” space-time.
Sun
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Lec 19: General Relativity 16
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
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Lec 19: General Relativity 17
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
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Lec 19: General Relativity 18
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.
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Lec 19: General Relativity 19
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).
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Lec 19: General Relativity 20
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
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Lec 19: General Relativity 21
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?
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Lec 19: General Relativity 22
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.
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Lec 19: General Relativity 23
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
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Warped Space Time
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Lec 19: General Relativity 25
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
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Lec 19: General Relativity 26
The Event Horizon■ The event horizon
is located at Rs.■ Anything inside the
event horizon is gone from sight forever (nothing can escape).
Rs
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Event Horizon