By:%20Jennifer%20Doran

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Relativity

• By Jennifer Doran

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What was Known in 1900

• Newtons laws of motion
• Maxwells laws of electromagnetism

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Contradiction Between Laws

• Newtons Laws
• Predicted that the speed of light should depend
  on the motion of the observer and the light source

• Maxwells Laws
• Predicted that light in a vacuum should travel
  at a constant speed regardless of the motion of
  the observer or source

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Two Postulates of Special Relativity

• The laws of physics are the same for all
  non-accelerating observers
• The speed of light in a vacuum is constant for
  all observers, regardless of motion

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Inconsistencies with Classical Mechanics

• Newtons Laws state that vuV
• Einsteins postulate says that the speed of light
  is independent of the motion of all observers and
  sources.

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Classical Lorentz Transformation

• x x' u t' y y' z z' t t' OR
• x' x u t y y' z z' t t'
• Then v v' u
• And a a'
• BUT this is only good for ultltv.
• To make the transforms relativistic, assume
• x G (x' u t')
• x' G (x u t)

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Finding G

• A light pulse starts at S at t 0 S' at t' 0
• So in S, x ct, and in S, x c t', by
  Einsteins second postulate
• Then ct G (c t' u t') G (c u) t'
• And ct' G (c - u) t
• Substitute for t', then ct G(c u)G(c -
  u)t/c
• Then G2 c2 / (c2-u2) 1/(1- u2/c2)
• G 1/v(1- u2/c2)

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Time Dilation

• ?t G?tp
• ?tp is the proper time, the time between events
  which happen at the same place.
• Since G is always greater than 1, all clocks run
  more slowly according to an observer in relative
  motion.

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Time Dilation Example

• An astronaut in a spaceship traveling away from
  the earth at u 0.6c decides to take a nap. He
  tells NASA that he will call them back in 1 hour.
  How long does his nap last as measured on earth?

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Time Dilation Example

• ?tp 1 hour
• 1 (u/v)2 1 (0.6)2 0.64
• Therefore, G v(1/0.64) 1.25
• ?t 1.25 hours

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Length Contraction

• L (1/G)Lp
• Lp is the length of the object in the reference
  frame in which the object is at rest.
• All observers in motion relative to the object
  measure a shorter length, but only in the
  direction of motion.

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Length Contraction
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Length Contraction Example

• In the reference frame of a muon traveling at u
  0.999978c, what is the apparent thickness of
  the atmosphere? (To an observer on earth, the
  height of the atmosphere is 100 km.)

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Length Contraction Example

• L (1/G)Lp
• Lp 100 km
• L 100 kmv(1-(0.999978)2)
• L 0.66 km 660 m

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Relativistic Velocity Addition

• When objects are moving at relativistic speeds,
  classical mechanics cannot be used.
• v (v' u) / (1 v'u/c2)
• v velocity addition
• v' velocity of object moving in the reference
  frame of u.
• u motion of object

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Relativistic Velocity Addition Example

• A spaceship moving away from Earth at a speed of
  0.80c fires a missile parallel to its direction
  of motion. The missile moves at a speed of 0.60c
  relative to the ship. What is the speed of the
  missile as measured by an observer on Earth?

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Relativistic Velocity Addition Example

• Cannot use classical mechanics.
• v is the velocity we are looking for.
• u 0.80c the velocity of the spaceship
• v' 0.60c the velocity of the missile in the
  reference frame of the spaceship
• v (v' u) / (1 v'u/c2)
• v (0.6c 0.8c) / (1 (0.6c)(0.8c)/c2)
• v 1.40c/1.48 0.95c

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Relativistic Velocity Addition Example

• A star cruiser is moving away from the planet
  Mars with a speed of 0.58c and fires a rocket
  back towards the planet at a speed of 0.69c as
  seen from the star cruiser. What is the speed of
  the rocket as measured by an observer on the
  planet?

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Relativistic Velocity Addition Example

• v is the velocity we are looking for.
• u 0.58c the velocity of the spaceship
• v' -0.69c the velocity of the rocket in the
  reference frame of the star cruiser
• v (v' u) / (1 v'u/c2)
• v (0.58c - 0.69c) / (1 (0.58c)(-0.69c)/c2)
• v -0.11c/0.5998 -0.18c
• Compared to 0.11c

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Twin Paradox

• A clock in motion runs slower than one at rest,
  including biological clocks.
• Apparent paradox with one twin staying on earth
  and other going on a relativistic journey.
• Solution

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Twin Paradox Example

• Twin sisters Betty and Ann decide to test the
  relativity theory. Ann stays on earth. On
  January 1, 2000, Betty goes off to a nearby dwarf
  star that is 8 light years from Earth. Betty
  travels there and back at 0.8c.

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Twin Paradox Example

• What is Bettys travel time to the star according
  to Ann?
• t d/v 8 light years / 0.8c 10 years
• So Bettys total travel time according to Ann is
  20 years.

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Twin Paradox Example

• As soon as Betty reaches the star, she sends her
  sister an e-mail message saying, Wish you were
  here! via radio. When, from Anns perspective,
  does Ann receive the e-mail from Betty?
• 2010 8 yrs 2018

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Twin Paradox Example

• When Ann receives Bettys e-mail message, what is
  the date on it?
• Time dilation G 1/v(1- (0.8c)2/c2)
• So G 0.6c
• 10 years 0.6c 6 years have passed
• So the e-mail will have the date of January 2006.

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Twin Paradox Example

• How much younger is Betty than Ann when Betty
  returns from the star?
• For Betty, her return date is 2012, but for Ann,
  she returns in 2020.
• Ann is now 8 years older than Betty.

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Twin Paradox Example
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Newtonian Gravity

• Quote from Newton
• Two masses in Newtons theory inertial mass and
  gravitational mass
• Newton saw no reason why these masses should be
  equal
• BUT they are!!!

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Einstein on Gravity

• I was sitting in a chair in the patent office at
  Bern when all of a sudden a thought occurred to
  me if a person falls freely he will not feel
  his own weight. This simple thought made a deep
  impression on me. It impelled me toward a theory
  of gravitation.
• -Albert Einstein

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Principle of Equivalence

• There is no local experiment that can be done to
  distinguish between the effects of a uniform
  gravitational field in a non-accelerating
  inertial frame and the effects of a uniformly
  accelerating reference frame.

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Principle of Equivalence

• Free space view is the same as the free fall view

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General Relativity

• The presence of matter causes spacetime to warp
  or curve.
• Gµv 8?G/c4Tµv

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Schwarzschilds Solution

• Found the first exact solution of one of
  Einsteins equations
• The metric (distance relation) is
• ds2 -c2(1-2MG/c2r)dt2 dr2/(1-2MG/c2r)
  r2(dq2sin2qdf2).
• Describes space-time geometry around a spherical
  object of mass M
• Basis of tests of general relativity

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Testing the General Theory of Relativity

• Deflection of Starlight
• First tested during total solar eclipse in 1919

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Slowing Down of Clocks by Gravity

• Clock closer to the mass measures a shorter
  elapsed proper time than a clock that is further
  out.
• Rc2MG/c2, where Rc is the Schwarzschild radius.