Relativity

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Chapter 26

• Relativity

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• Relative Motion
• (Galilean Relativity)
• Chapter 3 Section 5

http//www.physics.mun.ca/jjerrett/relative/relat
ive.html
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• Michelson Interferometer
• Chapter 25 Section 7

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Michelson Interferometer

• The Michelson Interferometer is an optical
  instrument that has great scientific importance
• It splits a beam of light into two parts and then
  recombines them to form an interference pattern
• It is used to make accurate length measurements

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Michelson Interferometer, schematic

• A beam of light provided by a monochromatic
  source is split into two rays by a partially
  silvered mirror M
• One ray is reflected to M1 and the other
  transmitted to M2
• After reflecting, the rays combine to form an
  interference pattern
• The glass plate ensures both rays travel the same
  distance through glass

Active Figure The Michelson Interferometer
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Measurements with a Michelson Interferometer

• The interference pattern for the two rays is
  determined by the difference in their path
  lengths
• When M1 is moved a distance of ?/4, successive
  light and dark fringes are formed
• This change in a fringe from light to dark is
  called fringe shift
• The wavelength can be measured by counting the
  number of fringe shifts for a measured
  displacement of M
• If the wavelength is accurately known, the mirror
  displacement can be determined to within a
  fraction of the wavelength

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Luminiferous Ether

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

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Verifying the Luminiferous Ether

• Associated with the ether was an absolute frame
  of reference in which light travels with speed c
• The Earth moves through the ether, so there
  should be an ether wind blowing
• If v is the speed of the ether wind relative to
  the Earth, the observed speed of light should
  have a maximum (a), minimum (b), or in-between
  (c) value 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 initially aligned along the direction of
  the earths motion through space

• An interference pattern was observed
• The interferometer was rotated through 90

• Should observe small, but measurable, shifts in
  the fringe pattern as orientation with the ether
  wind changes

Active Figure The Michelson-Morley Experiment
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Michelson-Morley Results

• Measurements failed to show any change in the
  fringe pattern
• No fringe shift of the magnitude required was
  ever observed
• The addition laws for velocities were incorrect
• The speed of light is a constant in all inertial
  frames of reference
• Light is now understood to be an electromagnetic
  wave, which requires no medium for its
  propagation
• The idea of an ether was discarded

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• Relativity I
• Sections 14

13
Basic Problems

• The speed of every particle of matter in the
  universe always remains less than the speed of
  light
• Newtonian Mechanics is a limited theory
• It places no upper limit on speed
• It breaks down at speeds greater than about 10
  of the speed of light (v gt .1c)
• Newtonian Mechanics becomes a specialized case of
  Einsteins Theory of Special Relativity
• When speeds are much less than the speed of light
  vltltc

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Galilean Relativity

• Choose a frame of reference
• Necessary to describe a physical event
• According to Galilean Relativity, the laws of
  mechanics are the same in all inertial frames of
  reference
• An inertial frame of reference is one in which
  Newtons Laws are valid
• Objects subjected to no forces will move in
  straight lines

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Galilean Relativity, cont.

• A passenger in an airplane throws a ball straight
  up
• It appears to move in a vertical path
• This is the same motion as when the ball is
  thrown while standing at rest on the Earth
• The law of gravity and equations of motion under
  uniform acceleration are obeyed

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Galilean Relativity, cont

• There is a stationary observer on the ground
• Views the path of the ball thrown to be a
  parabola
• The ball has a velocity to the right equal to the
  velocity of the plane
• The law of gravity and equations of motion under
  uniform acceleration are still obeyed

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Galilean Relativity, final

• The two observers disagree on the shape of the
  balls path
• Both agree that the motion obeys the law of
  gravity and Newtons laws of motion
• Both agree on how long the ball was in the air
• Conclusion There is no preferred frame of
  reference for describing the laws of mechanics

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Galilean Relativity Limitations

• Galilean Relativity does not apply to experiments
  in electricity, magnetism, optics, and other
  areas
• Results do not agree with experiments
• According to Galilean relativity, the observer S
  should measure the speed of the light pulse as
  vc
• Actually observer S measures the speed as c
• What is the problem?

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Albert Einstein

• 1879 1955
• 1905 published four papers
• Brownian motion
• Photoelectric effect
• 2 on Special Relativity
• 1916 published theory of General Relativity
• Searched for a unified theory
• Never found one

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Einsteins Principle of Relativity

• Resolves the contradiction between Galilean
  relativity and the fact that the speed of light
  is the same for all observers
• Postulates
• The Principle of Relativity All the laws of
  physics are the same in all inertial frames
• The constancy of the speed of light The speed of
  light in a vacuum has the same value in all
  inertial reference frames, regardless of the
  velocity of the observer or the velocity of the
  source emitting the light

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The Principle of Relativity

• The results of any kind of experiment performed
  in one laboratory at rest must be the same as
  when performed in another laboratory moving at a
  constant velocity relative to the first one
• No preferred inertial reference frame exists
• It is impossible to detect absolute motion with
  respect to an absolute frame of reference

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The Constancy of the Speed of Light

• Been confirmed experimentally in many ways
• A direct demonstration involves measuring the
  speed of photons emitted by particles traveling
  near the speed of light
• Confirms the speed of light to five significant
  figures
• Explains the null result of the Michelson-Morley
  experiment relative motion is unimportant when
  measuring the speed of light
• We must alter our common-sense notions of space
  and time

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Consequences of Special Relativity

• In relativistic mechanics
• There is no such thing as absolute length
• There is no such thing as absolute time
• Events at different locations that are observed
  to occur simultaneously in one frame are not
  observed to be simultaneous in another frame
  moving uniformly past the first
• In Special Relativity, Einstein abandoned the
  assumption of simultaneity

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Simultaneity Thought Experiment

• Thought experiment
• A boxcar moves with uniform velocity v
• Two lightning bolts strike the ends
• Flashes leave points A and B on the car and
  points A and B on the ground at speed c

• Observer O is midway between the points of
  lightning strikes on the ground, A and B
• Observer O is midway between the points of
  lightning strikes on the boxcar, A and B

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Simultaneity Results

• The light signals reach observer O at the same
  time
• He concludes the light has traveled at the same
  speed over equal distances
• Observer O concludes the lightning bolts occurred
  simultaneously

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Simultaneity Results, cont

• By the time the light has reached observer O,
  observer O on the car has moved
• The light from B has already moved by observer
  O, but the light from A has not yet reached him
• The two observers must find that light travels at
  the same speed
• Observer O concludes the lightning struck the
  front of the boxcar before it struck the back
  (they were not simultaneous events)

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Simultaneity Summary

• Two events that are simultaneous in one reference
  frame are in general not simultaneous in a second
  reference frame moving relative to the first
• That is, simultaneity is not an absolute concept,
  but rather one that depends on the state of
  motion of the observer
• In the thought experiment, both observers are
  correct, because there is no preferred inertial
  reference frame

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Time Dilation, Moving Observer

• The vehicle is moving to the right with speed v
• A mirror is fixed to the ceiling of the vehicle
• An observer, O, at rest in this system holds a
  laser a distance d below the mirror
• The laser emits a pulse of light directed at the
  mirror (event 1) and the pulse arrives back after
  being reflected (event 2)

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Time Dilation, Moving Observer

• Observer O carries a clock
• She uses it to measure the time between the
  events (?tp)
• The p stands for proper
• She observes events 1 and 2 to occur at the same
  place
• Light travels distance 2d c?tp
• The time interval ?tp is called the proper time
• The proper time is the time interval between
  events as measured by an observer who sees the
  events occur at the same position
• You must be able to correctly identify the
  observer who measures the proper time interval

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Time Dilation, Stationary Observer

• Observer O is a stationary observer on the Earth
• He observes the mirror and O to move with
  velocity v
• By the time the light from the laser reaches the
  mirror, the mirror has moved to the right

• The light must travel farther with respect to O
  than with respect to O

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Time Dilation, Stationary Observer

• Observer O carries a clock
• He uses it to measure the time between the events
  (?t)
• He observes events 1 and 2 to occur at different
  places
• Events separated by distance v?t
• Light travels distance c?t

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Time Dilation, Observations

• O and O must measure the same speed of light
• The light travels farther for O
• The time interval, ?t, for O is longer than the
  time interval for O, ?tp
• Observer O measures a longer time interval than
  observer O by the factor gamma

Active Figure Time Dilation
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Time Dilation, Example
v

• The time interval ?t between two events measured
  by an observer moving with respect to a clock is
  longer than the time interval ?tp between the
  same two events measured by an observer at rest
  with respect to the clock
• For example, when observer O, moving at v
  0.5c, claims that 1.00 s has passed on the clock,
  observer O claims that ?t ? ?tp (1.15)(1.00s)
  1.15 s has passed Observer O considers the
  clock of O to be reading too low a value
  running to slow
• A clock in motion runs more slowly than an
  identical stationary clock

O
O
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Time Dilation Equivalent Views

• Initial View Observer O views O moving with
  speed v to the right and the clock of O is
  running more slowly
• Equivalent View Observer O views O as the one
  who is really moving with speed v to the left and
  the clock of O is running more slowly
• The principle of relativity requires that the
  views of the two observers in uniform relative
  motion must be equally valid and capable of being
  checked experimentally

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Time Dilation Generalization

• All physical processes slow down relative to a
  clock when those processes occur in a frame
  moving with respect to the clock
• These processes can be chemical and biological as
  well as physical
• Time dilation is a very real phenomena that has
  been verified by various experiments

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Time Dilation Verification

• Muons are unstable particles that have the same
  charge as an electron, but a mass 207 times more
  than an electron
• Muons have a half-life of ?tp 2.2 µs when
  measured in a reference frame at rest with
  respect to them (a) unlikely to reach the
  Earths surface.
• Relative to an observer on earth, muons should
  have a longer lifetime of ?tp ? ?tp (b)
  likely to reach surface
• A CERN experiment measured lifetimes in agreement
  with the predictions of relativity

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Length Contraction

• The measured distance between two points depends
  on the frame of reference of the observer
• The proper length, Lp, of an object is the length
  of the object measured by someone at rest
  relative to the object
• The length of an object measured in a reference
  frame that is moving with respect to the object
  is always less than the proper length
• This effect is known as length contraction

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Length Contraction Equation

• Length contraction takes place only along the
  direction of motion

Active Figure Length Contraction
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Length Contraction, Example
v

• The length between two points L measured by an
  observer moving with respect to a ruler is
  shorter than the length Lp between the same two
  points measured by an observer at rest with
  respect to the ruler
• For example, when observer O, moving at v
  0.5c, claims that a length of 1.00 m is measured
  by a ruler, observer O claims that L Lp /?
  (1.00 m)/(1.15) 0.87 m is the measured length
  between the two points Observer O considers the
  length of O to be contracted
• A ruler in motion is contracted compared to an
  identical stationary ruler

O
O