Lecture 1213: General Relativity II

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Lecture 12/13 General Relativity II

• Gravitational time dilation
• Curved space-time Einsteins theory
• The General Theory of Relativity
• Einsteins equations
• Some general motivations
• Consequences of GR
• Orbit of Mercury
• Gravitational lensing
• Gravitational waves

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I GRAVITATIONAL TIME DILATION

• Recap of waves
• Waves characterized by
• Wavelength (?) distance between crests
• Frequency (f or ?) number of crests passing a
  given point per second
• Speed of a crest c?f
• Energy of a wave is proportional to frequency f.

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The electromagnetic spectrum
Small wavelength High Frequency High energy
Large wavelength Low frequency Low energy
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• Remember the tower
• Laser light must lose energy as it climbs up
• Sofrequency must decrease
• i.e., light is redshifted.
• Gravitational redshifting
• Imagine a clock based on frequency of laser
  light
• 1 tick time taken for fixed number of crests
  to pass
• Gravitational redshifting slows down the clock.
• Clocks in gravitational fields run slowly

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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.

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• Gravitational time dilation has practical
  importance!
• Global Positioning System (GPS)
• System of satellites that emit timing signals
• Detector on Earth receives signals
• Can figure out position on Earths 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!

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II CURVED SPACE-TIME

• Einstein pondered several things
• 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

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• Einsteins suggestion
• 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?
• 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).

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• Another example a saddle
• Geodesics diverge

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• Curved space around the Earth looks something
  like this

From web site of UCSD
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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.
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• Notes
• 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.
• Newtons 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
  Newtons law of gravitation.

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IV GR EFFECTS IN THE SOLAR SYSTEM

• Have already heard about bending of star light by
  the Sun (detected by Eddington).
• Orbit of Mercury

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• Effect called precession of perihelion.
• Effect small - orbit twists by 5600 arc-seconds
  (1.56 degrees) per century
• With Newtonian gravity, can explain 5557
  arc-seconds/century as due to
• Gravitational effect of other planets,
• deformation of the Sun,
• non-inertial nature of Earths frame
• But still leaves 43 arc-seconds per century
  unexplained
• Using GR, Einstein predicted (with no fiddling!)
  that Mercury should precess 43 arcseconds per
  century!

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From the web site of The University of Oregon
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V THE BENDING OF LIGHT(GRAVITATIONAL LENSING)
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The Einstein Cross
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Gravitational micro-lensing

• Individual stars can also make a gravitational
  lens microlensing.
• Suppose we
• Look at a distant star in our galaxy
• Another massive (but dark) star passes in front
• Causes apparent increases in brightness of
  stellar image

From web site of Ned Wright (UCLA)
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Really hard to do!
MACHO Project
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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.

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• Features of gravitational waves
• 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?

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The binary pulsar (PSR191316)

• Russell Hulse Joseph Taylor (1974)
• Discovered remarkable double star system
• Nobel prize in 1993

From Nobel Prize website
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• Two neutron stars orbiting each other
• One neutron star is a pulsar
• 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
• Orbit precesses by 4 degree per year!
• Orbit is shrinking due to gravitational waves

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• Very precise test of certain aspects of GR

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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)
• 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

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Modern experiments LIGO

• Laser Interferometer Gravitational Wave
  Observatory
• Two L-shaped 4km components
• Hanford, Washington
• Livington, Louisiana

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• Will become operational very soon!
• Can detect gravitational waves with frequencies
  of about 10-1000Hz.
• VERY sensitive need to account for
• Earthquakes and Geological movement
• Traffic and people!
• What will it see?
• Stellar mass black holes spiraling together
• Neutron stars spiraling together

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LISA

• Laser Interferometer Space Antenna
• Space-based version of LIGO
• Sensitive to lower-frequency waves (0.0001
  0.1Hz)
• Can see
• 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!

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