Lecture 15 General Theory of Relativity

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Lecture 15General Theory of Relativity

• ASTR 340
• Fall 2006
• Dennis Papadopoulos

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Special Theory of Relativity

• Einsteins postulates for Special Relativity
• Laws of physics look the same in any inertial
  frame of reference.
• The speed of light is the same in any inertial
  frame of reference
• Strange consequences of S.R.
• Time dilation and length contraction
• Relativity of simultaneity and ordering of events
• Equivalence and conversion of mass and energy

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Motivation for General Relativity Einsteins
tower

• So far, we have ignored the effects of gravity.
  Is this really okay??
• Consider another thought experiment, to test
  whether light can be unaffected by gravity.
• Consider a tower on Earth
• Shine a light ray from bottom to top
• When light gets to top, turn its energy into
  mass.
• Then drop mass to bottom of tower, in Earths
  gravity field
• Then turn it back into energy

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• If we could do this, then we could get energy
  from nothing!
• Original energy in light beam Estart
• Thus, mass created at top is mE/c2
• Then drop mass at bottom of tower it has picked
  up speed (and energy) due to the effects of
  gravitational field.
• When we turn it back into energy, we have
  EendEstartEgrav
• But, we started off with only Estart we have
  made energy! Were rich!

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• Clearly, our assumption must be wrong
• light must be affected by gravity.
• But gravity does not appear in Maxwells
  equations, which govern light
• Thus, Maxwells equations are not exactly valid
  in the reference frame of Earths surface, where
  there is gravity.
• The Earths surface must not be an inertial frame
  of reference!

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Recap SR

• Einsteins postulates for Special Relativity
• Laws of physics look the same in any inertial
  frame of reference.
• The speed of light is the same in any inertial
  frame of reference
• Strange consequences of S.R.
• Time dilation and length contraction
• Relativity of simultaneity and ordering of events
• Equivalence and conversion of mass and energy
• Behavior of light in gravity field (tower
  experiment)
• Energy of light must vary in a gravitational
  field to ensure that massenergy is conserved

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GR Birds eye View

• In GR time is variable and ever changing it even
  has shape
• It is bound up inextricably interconnected
  with 3 space dimensions creating a new dimension
  spacetime
• Spacetime is the most challenging and
  non-intuitive concept time is part of space,
  rather than eternal, absolute and immutable so
  that nothing can affect its steady tick

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Rubber Mattress or Sheet Analogy

• Gravity is the product of bending of spacetime
  in the presence of mass (or energy)
• Every mass creates a little depression in the
  fabric of cosmos (like a mattress)
• Gravity is the is not a force but a consequence
  of spacetime warping

Gravity does not exist and what makes stars and
planets move is the distortion of space time by
the presence of masses
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Naked space
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GR POSTULATES

• PRINCIPLE OF EQUIVALENCE IN THE VICINITY OF ANY
  POINT, A GRAVITATTIONAL FIELD IS EQUIVALENT TO AN
  ACCELERATED FRAME OF REFERENCE IN THE ABSENCE OF
  GRAVITATIONAL EFFECTS
• THE LAWS OF NATURE HAVE THE SAME FORM IN ANY
  FRAME OF REFERENCE, WHETHER ACCELERATED OR NOT

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Principle of Equivalence
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PRINCIPLE OF EQUIVALENCE
No experiment in an isolated space can
distinguish between a gravitational field and an
equivalent uniform acceleration.
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No experiment would help you distinguish between
being weightless far out in space and being in
free-fall in a gravitational field.
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Elevator at rest on Earth equivalent to elevator
being pulled by accelerating rocket in deep space
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Floating Astronauts
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ARTIFICIAL GRAVITY
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WHAT ABOUT LIGHT
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EQUIVALENCE PRINCIPLES

• The weak equivalence principle
• All objects are observed to accelerate at the
  same rate in a given gravitational field.
• Therefore, the inertial and gravitational masses
  must be the same for any object.
• This has been verified experimentally, with
  fractional difference in masses ?10-11
• As a consequence, the effects of gravity and of
  inertial forces (fictitious forces associated
  with accelerated frames) cannot locally be
  distinguished

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Interior of elevator free-falling on Earth is
equivalent to interior of elevator floating
freely in deep space
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The Eddington Test

• 1919 the first accessible total Solar eclipse
  since Einstein postulated SEP
• Arthur Eddington
• Famous British Astronomer
• Lead expedition to South America to observe
  eclipse
• Was looking for effects of gravitational light
  bending by searching for shifts in positions of
  stars just next to the Sun.

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Bending of starlight by Suns gravity
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GEN RELAT PREDICTION Light bends when it passes
by massive objects. The more the mass the larger
it bends.
Observation During solar eclipse stars along the
same line of sight with the Sun are seen on a
shifted position.
GR gives accurate prediction. SR half of the
observed shift. Newton no shift
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Galaxies between the earth and a quasar can
produce multiple images. From bending one can
estimate the mass of galaxy
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The Einstein Cross
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This picture, released to commemorate Hubble's
sixth anniversary, shows several blue,
loop-shaped objects that are actually multiple
images of the same galaxy. The duplicate images
were produced by a cosmic lens in space the
massive cluster of yellow elliptical and spiral
galaxies near the photograph's center. This
cosmic lens, called a gravitational lens, is
created by the cluster's tremendous gravitational
field, which bends light from a distant object
and magnifies, brightens, and distorts it. How
distorted the image becomes and how many copies
are made depends on the alignment between the
foreground cluster and the more distant galaxy.
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THE STRONG EQUIVALENCE PRINCIPLE

• Einstein introduced the strong equivalence
  principle when gravity is present, the inertial
  frames of Special Relativity should be identified
  with free-falling frames of reference.
• More generally, all inertial and freely-falling
  reference frames are equivalent, and there is no
  (local) experiment that can distinguish them
• What does this mean???

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Angular Frequency
Period T, Frequency n1/T, Angular frequency
w2pn2p/T Units T in secs, n in
Hertz1/sec Angular frequency in rad/sec
On time T the red dot makes a complete circle -gt
2p radians or 360 degrees. It turns one radian on
a time T/2p
Notice that angular frequency is independent of
radius e.g. rpm
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Time dilation in GR
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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.