The Theory of Special Relativity

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The Theory of Special Relativity

• Ch 26

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Two Theories of Relativity

• Special Relativity (1905)
• Inertial Reference frames only
• Time dilation
• Length Contraction
• Momentum and mass (Emc2)
• General Relativity
• Noninertial reference frames (accelerating frames
  too)
• Explains gravity and the curvature of space time

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Classical and Modern Physics

• Classical Physics Larger, slow moving
• Newtonian Mechanics
• EM and Waves
• Thermodynamics
• Modern Physics
• Relativity Fast moving objects
• Quantum Mechanics very small

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10 c
Speed
Classical Relativistic
Atomic/molecular size
Size
Quantum Classical
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Correspondence Principle

• Below 10 c, classical mechanics holds
  (relativistic effects are minimal)
• Above 10, relativistic mechanics holds (more
  general theory)

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Inertial Reference Frames

• Reference frames in which the law of inertia
  holds
• Constant velocity situations
• Standing Still
• Moving at constant velocity (earth is mostly
  inertial, though it does rotate)

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• Basic laws of physics are the same in all
  inertial reference frames
• All inertial reference frames are equally valid

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Speed of Light Problem

• According to Maxwells Equations, c did not vary
• Light has no medium
• Some postulated ether that light moved through
• No experimental confirmation of ether
  (Michelson-Morley experiment)

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Two Postulates of Special Relativity

• Einstein (1905)
• The laws of physics are the same in all inertial
  reference frames
• Light travels through empty space at c,
  independent of speed of source or observer
• There is no absolute reference frame
• of time and space

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Simultaneity

• Time always moves forward
• Time measured between things can vary
• Lightning strikes point A and B at the same time
• O will see both at the same time and call them
  simultaneous

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Moving Observers 1

• On train O2 - train O1 moves to the right
• On train O1 train O2 moves to the left

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Moving Observers 2

• Lightning strikes A and B at same time as both
  trains are opposite one another

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• Train O2 will observe the strikes as simultaneous
• Train O1 will observe strike B first (not
  simultaneous
• Neither reference frame is correct.
• Time is NOT absolute

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Time Dilation

• Consider light beam reflected and observed on a
  moving spaceship and from the ground

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• Distance is shorter from the ship
• Distance is longer from the ground
• c D/t
• Since D is longer from the ground, so t must be
  too.

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• On Spaceship
• c 2D/Dto
• Dto 2D/c
• On Earth
• c 2 D2 L2
• Dt
• v 2L/Dt
• L vDt
• 2

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• c 2 D2 v2 (Dt)2/4
• Dt
• c2 4D2 v2
• Dt2
• Dt 2D
• c 1 v2/c2
• Dt Dto
• 1 - v2/c2

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• Dt Dto
• v 1 - v2/c2
• Dto
• Proper time
• time interval when the 2 events are at the same
  point in space
• In this example, on the spaceship

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Is this real? Experimental Proof

• Jet planes (clocks accurate to nanoseconds)
• Elementary Particles muon
• Lifetime is 2.2 ms at rest
• Much longer lifetime when travelling at high
  speeds

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Time Dilation Ex 1

• What is the lifetime of a muon travelling at 0.60
  c (1.8 X 108 m/s) if its rest lifetime is 2.2 ms?
• Dt Dto
• v 1 - v2/c2
• Dt (2.2 X 10-6 s) 2.8 X 10-6 s
• 1- (0.60c)2 1/2
• c2

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Time Dilation Ex 2

• If our apatosaurus aged 10 years, calculate how
  many years will have passed for his twin brother
  if he travels at
• ¼ light speed
• ½ light speed
• ¾ light speed

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Time Dilation Ex 2

1. 10.3 y
2. 11.5 y
3. 10.5 y

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Time Dilation Ex 3

• How long will a 100 year trip (as observed from
  earth) seem to the astronaut who is travelling at
  0.99 c?
• Dt Dto
• 1 - v2/c2
• Dto Dt 1 - v2/c2
• Dto 4.5 y

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Time Dilation Ex 3

• If our apatosaurus aged 10 years, and his brother
  aged 70 years, calculate the apatosaurus average
  speed for his trip. (Express your answer in terms
  of c).
• ANS 0.99 c

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Length Contraction

• Observers from earth would see a spaceship
  shorten in the length of travel

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• Only shortens in direction of travel
• The length of an object is measured to be shorter
  when it is moving relative to an observer than
  when it is at rest.

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• Dto Dt v 1 - v2/c2
• v L Dto L/v (L is from spacecraft)
• Dto
• Dt Lo/v
• Lo L
• v v v 1 - v2/c2
• L Lo v 1 - v2/c2

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• L Lo v 1 - v2/c2
• Lo Proper Length (at rest)
• L Length in motion (from stationary observer)

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Length Contraction Ex 1

• A painting is 1.00 m tall and 1.50 m wide. What
  are its dimensions inside a spaceship moving at
  0.90 c?

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Length Contraction Ex 2

• What are its dimensions to a stationary observer?
• Still 1.00 m tall
• L Lo v 1 - v2/c2
• L (1.50 m)(v 1 - (0.90 c)2/c2)
• L 0.65 m

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Length Contraction Ex 3

• The apatosaurus had a length of about 25 m.
  Calculate the dinosaurs length if it was running
  at
• ½ lightspeed
• ¾ lightspeed
• 95 lightspeed

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1. 21.7 m
2. 15.5 n
3. 7.8 m

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Four-Dimensional Space-Time

• Consider a meal on a train (stationary observer)
• Meal seems to take longer to observer
• Meal plate is more narrow to observer

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• Move faster Time is longer but length is
  shorter
• Move slower Time is shorter but length is
  longer
• Time is the fourth dimension

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Momentum and the Mass Increase

• p mov
• 1 - v2/c2
• Mass increases with speed
• mo proper (rest) mass
• m mo
• 1 - v2/c2

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Mass Increase Ex 1

• Calculate the mass of an electron moving at 4.00
  X 107 m/s in the CRT of a television tube.
• m mo
• 1 - v2/c2
• m 9.11 X 10-31 kg 9.19 X 10-31 kg
• 1 - (4.00 X 107 m/s)2/c2

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Mass Increase Ex 2

• Calculate the mass of an electron moving at 0.98
  c in an accelerator for cancer therapy.
• m mo
• v 1 - v2/c2
• m 9.11 X 10-31 kg 4.58 X 10-30 kg (5mo)
• v 1 - (0.98c)2/c2

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The Speed Limit

• Nothing below the speed of light can be
  accelerated to the speed of light
• Would require infinite energy
• Mass becomes infinite
• Length goes to zero
• Time becomes infinite

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Mass Increase Ex 2

• The apatosaurus had a mass of about 35,000 kg.
  Calculate the dinosaurs mass if it was running
  at
• ½ lightspeed
• ¾ lightspeed
• 95 lightspeed

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1. 40,415 kg
2. 53,915 kg
3. 112,090 kg

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Relativistic Momentum

• p mov
• 1 - v2/c2

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E mc2

• Particle at Rest
• E moc2
• E Total Energy
• mo Rest mass
• c speed of light
• Moving Particles
• E2 mo2c4 p2c2

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• E moc2 KE
• rest kinetic KE does not equal ½ mv2
  at
• energy energy relativistic speeds

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Emc2 Ex 1

• How much energy would be released if a p0 meson
  (mo2.4 X 10-28 kg) decays at rest.
• E mc2
• E moc2 (particle is at rest)
• E (2.4 X 10-28 kg)(3.0 X 108 m/s)2
• E 2.16 X 10-11 J

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Emc2 Ex 2

• A p0 meson (mo2.4 X 10-28 kg) travels at 0.80 c.
• Calculate the new mass 4 X 10-28 kg
• Calculate the relativistic momentum 9.6 X 10-20
  kg m/s
• Calculate the energy of the particle (E2 mo2c4
  p2c2 ) ANS 3.6 X 10-11 J

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Emc2 KE Ex 3

• What is the kinetic energy of the p0 meson in the
  former example.
• E moc2 KE
• KE E moc2
• KE 3.6 X 10-11J - (2.4 X 10-28 kg)(3.0X108
  m/s)2
• KE 1.4 X 10-11 J

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Emc2 KE

• An electron is moving at 0.999c in the CERN
  accelerator.
• Calculate the rest energy
• Calculate the relativistic momentum
• Calculate the relativistic energy
• Calculate the Kinetic energy

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Electron Volts

• 1 eV 1.6 X 10-19 J
• 1 MeV 106 eV 1.60 X 10-13 J
• What is the rest mass of an electron in MeV?
• E mc2
• E moc2 (particle is at rest)
• E (9.11 X 10-31 kg)(3.0 X 108 m/s)2
• E 8.20 X 10-14 J

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• 8.20 X 10-14 J 1 MeV 0.511 MeV/c2
• 1.60 X 10-13 J

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Relativistic Addition of Velocity

• Classical Addition
• Bus moves at 40 mph
• You walk to the front at 5 mph
• Overall speed 45 mph

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• Relativistic Addition
• Cannot simply add velocities above 0.10 c
• Length and time are in different reference frames
• Formula
• u v u
• 1 vu/c2
• u overall speed with respect to stationary
  observer
• v speed of moving object with respect to st.
  observer
• u speed of 2nd object with respect to moving
  observer

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Relativistic Addition Ex 1

• What is the speed of the second stage of the
  rocket shown with respect to the earth?

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• u v u
• 1 vu/c2
• u 0.60c 0.60c
• 1 (0.60c)(0.60c)/c2
• u 0.88 c
• (classical addition would give you 1.20c, over
  the speed of light)

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Relativistic Addition Ex 2

• Suppose a car travelling at 0.60c turns on its
  headlights. What is the speed of the light
  travelling out from the car?
• u v u
• 1 vu/c2
• u 0.60c c 1.60c
• 1 (0.60c)(c)/c2 1.60
• u c

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Relativistic Addition Ex 3

• Now the car is travelling at c and turns on its
  headlights.
• u v u
• 1 vu/c2
• u c c 2c
• 1 (c)(c)/c2 2
• u c