Special Relativity

1
Special Relativity
Physics of objects moving at very large
speeds Close to the speed of light, c 3 x 108
m/s This gets weird - why not?
2
Galilean Transformation

• Newtons Laws Work in all Inertial Reference
  Frames
• There is no special reference frame
• Measurements in one frame can be related to
  another, e.g., x x x0 if new reference
  frame is translated with respect to the first by
  x0 along x-axis.
• Frames can move at uniform velocity If the
  primed frame was at same location as unprimed
  frame at time t0, and is moving at uniform speed
  v0 along x-axis
• Galilean Tranformation x x v0 (t-t0), t
  t, v v v0
• Velocities of the frame and object in motion add
  (like vectors)
• Lengths and time intervals remain invariant in
  the two frames

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What about light?

• Do EM waves also respect Galilean addition of
  velocities?
• c 3 x 108 m/s is rather large
• v has to be large to make a measurable difference
  
• Accurate measurements were needed to resolve the
  issue
• Maxwells EM equations do not respect Galilean
  transformation
• Lorenz discovered a different transformation for
  EM equations
• Something was wrong
• At the turn of 19th century this was a very
  important unresolved problem

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Preflight 1
1) 0.5 c 2) c 3) 1.5 c
Galilean addition of velocities is NOT
valid.Speed of light is the same in all
inertial reference frames.
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Luminiferous Ether

• 19th Century physicists compared electromagnetic
  waves to mechanical waves
• Mechanical waves need a medium to support the
  disturbance
• The luminiferous ether was proposed as the medium
  required (and present) for light waves to
  propagate
• Present everywhere, even in space
• Massless, but rigid medium
• Could have no effect on the motion of planets or
  other objects

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Verifying theLuminiferous Ether

• Associated with an ether was an absolute frame
• Perhaps, the laws of E M take on their simplest
  form only in this frame of reference, where speed
  is c
• Since the earth moves through the ether, there
  should be an ether wind blowing
• If v is the speed of the ether relative to the
  earth, the speed of light should have minimum or
  maximum values depending on its orientation to
  the wind

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Michelson-Morley Experiment

• First performed in 1881 by Michelson
• Repeated under various conditions by Michelson
  and Morley
• Designed to detect small changes in the speed of
  light
• By determining the velocity of the earth relative
  to the ether

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Michelson-Morley Equipment

• Used the Michelson Interferometer
• Arm 2 is aligned along the direction of the
  earths motion through space
• The interference pattern was observed while the
  interferometer was rotated through 90
• The effect should have been to show small, but
  measurable, shifts in the fringe pattern

9
Michelson-Morley Results

• Measurements failed to show any change in the
  fringe pattern
• No fringe shift of the magnitude required was
  ever observed
• Light is now understood to be an electromagnetic
  wave, which requires no medium for its
  propagation
• The idea of an ether was discarded
• The laws of electricity and magnetism are the
  same in all inertial frames

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Addition of VelocitiesMichelson-Morley
Observation
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Relative Velocity (Light)

• _____ throws photon (3x108 m/s). How fast do I
  think it goes when I am
• Standing still
• Running 2x108 m/s towards
• Running 2x108 m/s away

3x108 m/s
3x108 m/s
3x108 m/s
Strange but True!
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Postulates of Relativity

• Laws of physics are the same in every inertial
  reference frame
• Perform experiment on train going east gives same
  results as on train going west.
• Speed of light in vacuum is c for everyone
• Measure c3x108 m/s if you are on train going
  east or on train going west
• Even if light source isnt on the train.
• Leads to Lorenz Transformation (discovered in the
  context of electromagnetism by H. Lorenz)

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Simultaneous?
A flash of light is emitted in the center of a
box. Does the light reach the sides at the same
time?
At Rest
YES
Moving
NO
Simultaneity depends on frame!
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Time Dilation
t0 is proper time Because it is rest frame of the
event
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Time Dilation
t0 is proper time Because it is rest frame of the
event
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Time Dilation Example
You and a friend are having a drinking contest.
Your friend is on a train traveling at speed
v0.9 c. By her watch, she finishes her drink in
5 seconds. Determine the time you measure, if you
are standing still at the train station.
Since drinking is happening on the train, that is
the proper time.
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Time Dilation 2
Now it is your turn to drink. According to your
watch you finish your drink in 5 seconds. How
long does your friend think it took you to finish
the drink?
Now drinking is happening at the station, so that
is the proper time.
Both people think they won!
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Space Travel Example
Alpha Centauri is 4.3 light-years from earth. (It
takes light 4.3 years to travel from earth to
Alpha Centauri). How long would people on earth
think it takes for a spaceship traveling v0.95c
to reach A.C.?
How long do people on the ship think it takes?
People on ship have proper time they see earth
leave, and Alpha Centauri arrive. Dt0
Dt0 1.4 years
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Twin Paradox
?tp
?t??tp
20
Preflight 2

• Younger than you
• Older than you
• The same age as you

Time differences are not the same in two inertial
reference frames moving at differing (large)
speeds with respect to each other.
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Muons from Cosmic Rays
Fast muons (v0.99c, ?7) travel only 600m before
they decay, in their reference frame, in which
their average lifetime is about 2.2 ?s. But,
they reach all the way to the surface of the
Earth (4800m) The reason for this is from the
point of view of an Earth bound observer, there
is time dilation and muons can travel much
longer (2.2 ?s ?).
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Twin Paradox
Twins decide that one will travel to Alpha
Centauri and back at 0.95c, while the other stays
on earth. Compare their ages when they meet on
earth.
Earth twin thinks it takes 2 x 4.5 9 years
Traveling twin thinks it takes 2 x 1.4 2.8
years
Traveling twin will be younger!
Note Traveling twin is NOT in inertial frame!
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Length Contraction
People on ship and on earth agree on relative
velocity v 0.95 c. But they disagree on the
time (4.5 vs 1.4 years). What about the distance
between the planets?
Earth/Alpha d0 v t
.95 (3x108 m/s) (4.5 years)
4x1016m (4.3 light years)
Ship d v t
.95 (3x108 m/s) (1.4 years)
1.25x1016m (1.3 light years)
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Preflight 3

• Longer than your ship
• Shorter than your ship
• Exactly the same as your ship

Lengths are not the same in two inertial
reference frames moving at differing (large)
speeds with respect to each other.
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Length Contraction Giffs
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Length Contraction Example
Sue is carrying a pole 10 meters long. Paul is on
a barn which is 8 meters long. If Sue runs
quickly v.8 c, can she ever have the entire pole
in the barn?
Paul Sure the barn is 8 meters long, and the
pole is only
Sue No way! This pole is 10 meters long and that
barn is only
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Relativistic Momentum
Relativistic Momentum
Note for vltltc pmv
Note for vc pinfinity
Relativistic Energy
Note for v0 E mc2
Note for vltltc E mc2 mv2/2
Note for vc E infinity (if m gt 0)
Massive objects cant go faster than c!
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Relativistic Momentum

• Relativistic momentum does not grow linearly with
  velocity

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Relativistic Kinetic Energy

• Relativistic kinetic energy does not grow
  quadratically with velocity

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Summary

• Physics predictions hold in all inertial frames
• Speed of light in vacuum, c, is same in
  allinertial reference frames
• Simultaneity depends on frame
• Time dilates
• Length contracts
• Energy/Momentum conserved
• For vltltc reduce to Newtons Laws