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General Relativity II
Gravitational time dilation
Curved space-time & Einstein’s theory
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The General Theory of Relativity
Einstein’s equations
Some general motivations
Consequences of GR
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Orbit of Mercury
Gravitational lensing
Gravitational waves
I : GRAVITATIONAL TIME DILATION
Recap of waves:
Waves characterized by
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Wavelength () = distance between crests
Frequency (f or ) = number of crests passing a given point
per second
Speed of a crest; c=
Energy of a wave is proportional to frequency .
The electromagnetic spectrum
Small wavelength
High Frequency
High energy
Large wavelength
Low frequency
Low energy
Remember the tower…
Laser light must lose energy
as it climbs up
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So…frequency must decrease
i.e., light is redshifted.
Gravitational redshifting
Imagine a clock based on
frequency of laser light…
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1 “tick” = time taken for fixed
number of crests to pass
Gravitational redshifting slows
down the clock.
Clocks in gravitational
fields run slowly
 GM 
t grav  1  2 tspace
cr 
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if gravitatio nal field is " weak"
How to live for a 1000 years!
Observer on Earth would see astronauts clock
running very slowly when close to black hole
– astronaut would age very slowly.
Gravitational time dilation has practical importance!
Global Positioning System (GPS)
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System of satellites that emit timing signals
Detector on Earth receives signals
Can figure out position on Earth’s surface by measuring time
delay between signals from different satellite.
Need to measuring timing signal from satellite very well!
If GR effects were not included, GPS positions would
drift from true position by kilometers per day!
MACH PRINCIPLE
Frictionless experiment
II: CURVED SPACE-TIME
Einstein pondered several things…
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Success of Special Relativity showed that space &
time were closely linked
The “tower thought experiment” suggested that
free-fall observers are (locally) free of effects of
gravity
He wanted to say that gravity was an illusion
caused by the fact that we live in an accelerating
frame…
… but there is no single accelerating frame that
works! Somehow, you need to stick together
frames of reference that are accelerating in
different directions
Einstein’s suggestion
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4-dimensional space-time is curved
Free-falling objects move on “geodesics”
(generalizations of straight lines) through curved
space-time.
Matter and energy causes space-time to bend.
What is a geodesic?
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Shortest path between two points on a surface
E.g. path flown by aircraft
Geodesics that start parallel can converge or
diverge (or even cross).
Another example – a “saddle”
Geodesics diverge
Curved space around the Earth looks
something like this…
From web site of UCSD
III: THE GENERAL THEORY
OF RELATIVITY
Within a free-falling frame, the Special Theory
of Relativity applies.
Free-falling particles/observers move on
geodesics through curved space-time
The distribution of matter and energy
determines how space-time is curved.
“Space-time curvature tells matter/energy how to move.
Matter/energy tells space-time how to curve.”
Notes:
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8G
G 4 T
c
The Einstein curvature tensor “G” is mathematical object
describing curvature of 4-D space-time.
The Stress-Energy tensor “T” is mathematical object
describing distribution of mass/energy.
Newton’s constant of gravitation “G” and the speed of light
“c” appear as fundamental constants in this equation.
This is actually a horrendous set of 10 coupled non-linear
partial differential equations!!
For weak gravitational fields, this gives Newton’s law
of gravitation.
IV: GR EFFECTS IN THE
SOLAR SYSTEM
Have already heard about bending of star
light by the Sun (detected by Eddington).
Orbit of Mercury:
Mercury does not move
in perfect ellipse but
precesses-> Vulcan?
sun
Effect called “precession of perihelion”.
Effect small - orbit twists by 5600 arc-seconds (1.56
degrees) per century
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With Newtonian gravity, can explain 5557 arcseconds/century as due to
 Gravitational effect of other planets,
 deformation of the Sun,
 non-inertial nature of Earth’s frame
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But still leaves 43 arc-seconds per century unexplained…
Using GR, Einstein predicted (with no fiddling!) that
Mercury should precess 43 arcseconds per century!
V : THE BENDING OF LIGHT
(GRAVITATIONAL LENSING)
“The Einstein Cross”
Gravitational micro-lensing
Individual stars can also make a gravitational
lens… microlensing.
Suppose we…
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Look at a distant star in our galaxy
Another massive (but dark) star passes in front…
From web site of
Ned Wright (UCLA)
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Causes apparent increases in brightness of stellar
image
VI: GRAVITATIONAL WAVES
Particular kind of phenomena (e.g. orbiting
stars) produce ripples in the space-time
curvature…
Ripples travel at speed of light through space
These are called gravitational waves.
Features of gravitational waves…
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Usually extremely weak!
Only become strong when massive objects are
orbiting close to each other.
Gravitational waves carry energy away from
orbiting objects… lets objects spiral together.
The grand challenge – to compute the spiralling
together of two black holes.
How do we know that these waves exist?
The binary pulsar (PSR1913+16)
Russell Hulse & Joseph Taylor (1974)
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Discovered remarkable double star system
Nobel prize in 1993
From Nobel Prize website
Two neutron stars orbiting each other
One neutron star is a pulsar –
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Neutron star is spinning on its axis (period of
59ms)
Emits pulse of radio towards Earth with each
revolution
Acts as a very accurate clock!
Interesting place to study GR
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Orbit precesses by 4 degree per year!
Orbit is shrinking due to gravitational waves
Very precise test of certain aspects of GR
Direct detection of
gravitational waves…
How do you search for gravitational waves?
Look for tidal forces as gravitational wave
passes
Pioneered by Joseph Weber (UMd Professor)
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Estimated wave frequency (10000Hz)
Looked for “ringing” in a metal bar caused by
passage of gravitational wave.
Weber claimed detection of waves in early 1970s
Never verified – but Weber held out to the end…
Modern experiments : LIGO
Laser Interferometer Gravitational Wave Observatory
Two L-shaped 4km components
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Hanford, Washington
Livington, Louisiana
Will become operational very soon!
Can detect gravitational waves with
frequencies of about 10-1000Hz.
VERY sensitive… need to account for
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Earthquakes and Geological movement
Traffic and people!
What will it see?
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Stellar mass black holes spiraling together
Neutron stars spiraling together
LISA
Laser Interferometer Space Antenna
Space-based version of LIGO
Sensitive to lower-frequency waves (0.0001 –
0.1Hz)
Can see
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Normal binary stars in the Galaxy
Stars spiraling into large black holes in the nearby
Universe.
Massive black holes spiraling together anywhere in
the universe!