Chapter 26:Relativity

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Chapter 26Relativity

• The Principle of Galilean Relativity

Homework Read Chap.26
Sample homework problems 2,9,19,38,53

• Frame of reference and relative velocity

• The measured velocity of an object depends on
  the velocity
• of the observer with respect to the object.

• Relative velocity relates velocities measured by
  two different
• observers, one moving with respect to the other.

• Measurements of velocity depend on the reference
  frame of the
• observer where the reference frame is a just
  coordinate system
• used to measure physical quantities such as
  velocity, acceleration
• etc. Most of time, we will use a stationary
  frame of reference, relative
• to earth, but occasionally we will use a moving
  frame of reference.

• Inertial frames of reference are those in which
  objects subjected to
• no forces move in straight lines at constant
  speed.

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The Principle of Galilean Relativity

• Principle of Galilean relativity

• Principle of Galilean relativity
• The law of mechanics must be the same in
  all inertial frames of
• reference.

• In Fig. an airplane is moving
• with a constant speed. An
• observer on the airplane finds
• the motion of the ball is des-
• cribed by the same equation
• that describes the motion of
• the ball thrown in a laboratory
• at rest on Earth.

• For an observer at rest on
• Earth who observes the movement of the ball in
  the airplane finds that
• the motion of the ball is described by the law
  of gravity and Newtons
• laws of motion.

There is no preferred frame of reference for
describing the laws of mechanics. No experiment
using the laws of mechanics can determine if a
frame of reference is moving at zero velocity or
at a constant velocity.
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Speed of light

• Speed of light

• Speed of light from EM theory (Maxwells
  equations)
• The speed of light in free space (in vacuum) is
  2.99792458x108 m/s.

• But doesnt that contradict what Galilean
  relativity says?

The observer S would expect to see
light propagate at cv not c!
To solve this problem 19th century
physicists invented existence of medium (Ether)
in which light propagates, rather than vacuum.
Pros
Cons

• Ether must be rigid and
• massless with no effect on
• planetary motion.
• - No experimental evidence.

• - Speed of light is different in
• different frames.
• Light becomes like other waves.
• Ether is absolute reference frame.

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Speed of light

• Ether and its detection

• As an attempt to detect the ether, physicists
• came up with the following idea, assuming
• that Earth was in motion through the ether
• and that there was an ether wind blowing
• through Earth
• (1) If light propagates downwind, the speed of
• light should be cv where v is the
  relative
• speed of the ether with respect to Earth.
• (2) If light propagates upwind, the speed of
• light should be c-v.
• (3) If light propagates in an intermediate
• direction, the speed of light should be
• (c2-v2)1/2.

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Michelson-Morley Experiment

• Michelson-Morley experiment

• Michelson and Morley tried to
• detect the motion of the ether with
• respect to Earth using the Michelson
• interferometer.

• If the ether wind blows to the left,
• the velocities of light that goes through
• Arm 2 are c-v (cv) for the path to M2
• (M0), So the times of flight to the right
• and to the left are, respectively
• The total time of flight for light going
• through Arm 2 is, therefore

c speed of light w.r.t. the ether.
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Michelson-Morley Experiment

• Michelson-Morley experiment (contd)

ether velocity w.r.t. Earth
light velocity w.r.t. ether
light velocity w.r.t. Earth
going up
coming down
c speed of light w.r.t. the ether.

• Now the velocities of light going through
• Arm 1 are
• The total time of flight for the trip along
• Arm 1, then, is

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Michelson-Morley Experiment

• Michelson-Morley experiment (contd)

• So if the ether theory is correct, the
• difference in flight times is

c speed of light w.r.t. the ether.
for v/cltlt1
However, they failed to see any time diff. by
trying to see a shift in fringe pattern due to
interference.
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Einsteins Principle of Relativity

• Einsteins postulates for special relativity

• The null result from Michelson-Morleys
  experiment meant that ether
• theory was not correct.

• To solve the mystery of the contradiction
  between the Maxwells equation
• and Galilean relativity, Albert Einstein
  proposed in 1905 his special theory
• of relativity based on the following two
  postulates
• 1. The principle of relativity
• All the laws of physics are the same in all
  inertial frames.
• 2. 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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Consequences of Special Relativity

• Surprise consequences

• The distance between two objects is not
  absolute.
• It is different in different inertial frames.

• The time interval between events is not
  absolute.
• It is different in different inertial frames.

• Velocities do not always add directly.

Many of these consequences can be demonstrated in
simple thought experiments (gedanken
experiments).
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Consequences of Special Relativity

• Simultaneity and relativity of time

• A boxcar is moving with constant velocity v
  w.r.t. observer O on the
• ground.
• Observer O rides in exact center of the boxcar.
• Two lightning bolts strike the ends of the
  boxcar, leaving marks on the
• the boxcar and the ground underneath.
• Observer O on the ground finds that she is
  halfway between the scorch
• marks.

• Observer O on the ground also observes that
  light waves from each
• lightning strike at the boxcar ends reach her
  at the exactly the same time.

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Consequences of Special Relativity

• Simultaneity and relativity of time (contd)

• Since each light wave traveled at c, and each
  traveled the same distance,
• the lightning strikes are simultaneous in the
  frame of the ground observer.
• When light from front flash reaches boxcar
  observer O, he has moved
• away from rear flash.
• Both light waves travel at c in the boxcars
  frame, and observer O is equi-
• distance from the lightning strikes. But light
  flashes arrive at different time!
• The lightning strikes at the boxcar ends are not
  simultaneous in the
• boxcar frame!

Two events are simultaneous in one reference
frame and are not in another. Both statements are
correct as there is no preferred inertial frame
of reference.
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Consequences of Special Relativity

• Time dilation

• Observer O is on the ground.
• Observer O is on the train moving at v relative
  to O.
• A pulse of light emitted from a laser, reflected
  from a mirror, arrives back
• at the laser after some time interval.
• This time interval is different for the two
  observers.
• Observer O light pulse travels distance 2d.
• Observer O light pulse travels farther.
• From relativity, light travels at velocity c in
  both frames.
• Time interval between the two events is longer
  for stationary observer.

Time dilation
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Consequences of Special Relativity

• Time dilation (contd)

• Time interval between events in frame O

• Time (proper time) interval between events in
  frame O

Proper time is the time interval between
two events as measured by an observer who
sees the events occur at the same position
time dilation factor
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Consequences of Special Relativity

• Time dilation (contd)

• All physical processes, including chemical and
  biological ones,
• slow down by a factor of g-1 relative to a
  clock when those processes
• occur in a frame moving with respect to the
  clock.

• Time dilation and muons (unstable elementary
  particles)

• Muons have similar properties as electrons
  except that they carry
• a non-zero muon number and their mass is 207
  times the mass of
• electrons.
• Muons have a lifetime (tp) of 2.2ms when
  measured in a reference frame
• at rest with respect to them.
• Muons, at speed close to c, can travel only 600
  m before they decay
• without time dilation effect.
• Muons produced by cosmic rays in upper
  atmosphere (5 km above
• the sea level), however, can easily be observed
  at the sea level.
• The reason for this is that the lifetime of
  muons is prolonged by a
• factor of g from the point of view of an
  observer on Earth.
• The average muon lifetime of muons at v0.99c
  (g7.1) is gtp16ms,
• and they can travel a distance of gvtp4.8 km
  before they decay.

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Consequences of Special Relativity

• Twin paradox

• Imagine twin sisters, one (Susy) of whom goes to
  the closet star a
• distance of 4.3 light-year (lyr) away. The
  other (Jane) stays on Earth.

What happens according to Jane
Event 1 Susy leaves Earth.
Event 2 Susy arrives at the star.
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Consequences of Special Relativity

• Twin paradox (contd)

• Susy, on the other hand, measures proper time
  the departure and
• the arrival events occur at the same spatial
  location.

What happens according to Susy?

• Both Susy and Jane agree on the speed (0.95c)
• But if the time intervals are different, and the
  speed is the same,
• how can distances be the same????
• The distances are NOT the same! Length
  contraction!

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Consequences of Special Relativity

• Length contraction

• The measured distance between two points depends
  on the frame
• of reference of the observer.
• Define the proper length Lp of an object as the
  length of the object
• as measured by an observer 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.
• Consider a spaceship traveling with a speed of v
  from one star to
• another, as seen by two observers one on Earth
  and the other in
• the spaceship.

length contraction
Observer on Earth (at rest w.r.t. two stars)
distance between the stars
time it takes the spaceship to complete the
voyage
Observer on the spaceship
time it takes the spaceship to complete the
voyage
distance between the stars
Length contraction takes place only along the
direction of motion
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Relativistic Momentum

• Relativistic momentum

• For the momentum conservation to be valid in
  special relativity,
• the definition of momentum needs to be
  generalized.

v is the speed of the particle and m is its mass
measured by an observer at rest with respect to
the particle.
Relativistic Addition of Velocities

• Relativistic addition of velocities

- The subscript b and d label two reference
frames. - The frame d is moving at velocity vdb
in the position x-direction relative to frame
b. - If the velocity of an object a as measured
in frame d is called vad, then the velocity of
a as measured in frame b is vab.
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Relativistic Energy and Equivalence of Mass and
energy

• Relativistic kinetic energy and rest energy

• The relativistic kinetic energy of an object is
  defined as

• The 2nd term that does not depend on the speed
  of the object is called
• the rest energy

The mass of a particle may be completely convertib
le to energy.

• Total energy

• The total energy of an object is defined as

• The total energy is the sum of the kinetic
  energy and the rest energy.

• A stationary particle with zero kinetic energy
  has an energy
• proportional to its mass.

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Relativistic Energy and Equivalence of Mass and
energy

• Energy and relativistic momentum

• From the definition of the total energy and the
  relativistic momentum,

For a particle at rest, p0
For a massless particle such as photon

• Units of energy and mass

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Relativistic Energy and Equivalence of Mass and
energy

• Example 26.8 Conversion of mass to kinetic
  energy

• The fission, or splitting, of uranium was
  discovered in 1938 by Lisa
• Meitner. The fission of begins with the
  absorption of a slow-
• moving neutron that produces an unstable
  nucleus of . The
• nucleus then quickly decays into two heavy
  fragments moving at high
• speed, as well as several neutrons. Most of the
  kinetic energy released
• in such a fission is carried off by the two
  large fragments.

• For the typical fission process
  calculate
• the kinetic energy in MeV carried off by the
  fission fragments.

(b) What percentage of the initial energy is
converted into kinetic energy?
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Pair Production and Annihilation

• Pair production

• A photon with sufficient energy can create a
  pair of particle and its
• antiparticle such as an electron and a positron
  in a electromagnetic
• field which is needed to conserve both the
  total energy and momentum.
• Two photons are needed to create a
  particle-antiparticle pair. One of
• photon can come from the electromagnetic field
  by protons in a nucleus.
• In case of an ee- pair production, the minimum
  energy of the initial
• photon is

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Pair Production and Annihilation

• Pair annihilation

• The inverse process of the pair production is
  pair annihilation in which
• a particle-antiparticle pair produce a pair of
  photons.
• An example is ee- annihilation where all the
  kinetic energies as well
• as rest energies of the electron and positron
  are 100 converted into
• kinetic energies of two photons which do not
  have rest energies.

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General Relativity

• Inertial mass and gravitational mass

• A priori there are two kinds of masses inertial
  and gravitational mass

• It appears that these two kinds of masses are
  equal mimg

• Equivalence principle

• An observer in the elevator have no way of
  knowing whether he is
• under influence of gravity or the elevator is
  moving upward by a force
• which gives acceleration.

• If the statement above is true,
• trajectory of light ray is bent
• by gravity.

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General Relativity

• Postulates of Einsteins general relativity

• Based on two ideas described in the previous
  slide, Einstein proposed
• his theory of general relativity with the
  following two postulates

• All the laws of nature have the same form for
  observers in any frame
• of reference, accelerated or not.
• In the vicinity of any given point, a
  gravitational field is equivalent to
• an accelerated frame of reference without a
  gravitational field.

Principle of equivalence

• The 2nd postulate implies that gravitational
  mass and inertial mass are
• equivalent.

• The 2nd postulate also suggests that a
  gravitational field may be trans-
• formed away at any point if we choose an
  appropriate accelerated
• frame of reference a free-falling frame.

The gravitational effect is described by the
curvature of spacetime at a given point. The
presence of mass causes a curvature of
spacetime in vicinity of the mass spacetime is
deformed due to the mass.

• One interesting effect predicted by general
  relativity is that time scales
• are altered by gravity. The presence of mass
  slows down a clock.

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General Relativity

• Mass and curvature

• Some predictions and
• observations

• Bending of light near large
• mass

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General Relativity

• Some predictions and observations (contd)

• Gravitational lens