Chapter 39 Relativity

1
Chapter 39 Relativity January 21 Galilean
relativity Introduction 1. Syllabus 2. Modern
physics Quantum mechanics and relativity 3.
Methods of learning Modern Physics 1)
Conceptions. 2) Principles. 3) Practices. 4)
Lets go slowly. 39.1 The principle of
Galilean relativity

• Basic problems of Newtonian mechanics
• The motion of objects whose speeds approach that
  of light. e.g., accelerating an electron beam.
• Speed of light. It is constant according to
  Maxwells equations. It is varying according to
  Galilean relativity.
• Inertial frame of reference A reference frame in
  which objects subjected to no forces will
  experience no acceleration.
• Any system moving at constant velocity with
  respect to an inertial reference frame must also
  be in an inertial frame.
• There is no absolute inertial frame of
  reference.
• Compare to Coordinate systems.

2
Principle of Galilean relativity The laws of
mechanics are the same in all inertial frames of
reference.

• Galilean relativity Example
• A child in a truck throws a ball straight up.
• For the child The path of the ball is in a
  vertical line.
• For the roadside observer The path of the ball
  is a parabola.
• However, both observers agree that the motion
  obeys the law of gravity and Newtons laws of
  motion.
• There is no preferred frame of reference for
  describing the laws of mechanics.
• Quiz 39.1
• Event A physical occurrence. Can be described
  using the coordinates (x, y, z, t). e.g.,
  sparking, meeting, colliding,...

3
Galilean space-time transformation
equations Suppose inertial frame S' moves
relative to S with a constant velocity v along x
and x' axes. The origins of S and S' coincide at
t 0. If the observers in S and S' describe an
event with (x, y, z, t) and (x', y', z', t'), then
Galilean velocity transformation equation
u is for particle velocity and v is for relative
velocity
Quiz 39.2 What if the ball is thrown in the
opposite direction?
4
Read Ch39 1 Homework Ch39 1,2 Due January 30
5
January 23 Michelson-Morley experiment
Speed of light Galilean relativity does not
apply to electricity, magnetism, or optics.
Maxwells equations imply that the speed of light
has a fixed value in all inertial frames c
3108 m/s.
Light in an ether wind ? Physicists in the late
1800s thought light moved through a medium called
the ether. The speed of light would be c only in
an absolute frame at rest with respect to the
ether.
The orbital motion of the earth produces an ether
wind. Assume v is the velocity of the ether wind
relative to the earth. Various resultant light
velocities can be observed.
6
39.2 The Michelson-Morley experiment
The experiment was designed to determine the
velocity of the earth relative to the
hypothetical ether by detecting small changes in
the speed of light.

• In the Michelson interferometer, Arm 2 is aligned
  along the direction of the earths motion through
  space.
• The speed of light in the earth frame should be
  c - v as the light approaches M2, and c v as
  the light is reflected from M2.
• The interference fringes should shift while the
  interferometer was rotated through 90. The shift
  is calculated to be measurable ( 0.44 fringe).

• Measurements failed to show any change in the
  fringe pattern.
• Ether hypothesis is wrong.
• Light is an electromagnetic wave requiring no
  medium for its propagation

7
Read Ch39 2 No homework
8
January 26 Einsteins principle of relativity
Albert Einstein (1879-1955)
A German-born theoretical physicist. One of the
greatest physicists of all time. Best known for
the theory of relativity. Contributed to quantum
theory and unified field theory. 1921 Nobel Prize
in Physics for the photoelectric effect. Now
"Einstein" Great intelligence and genius.
9
39.3 Einsteins principle of relativity
Resolves the contradiction between Galilean
relativity and the fact that the speed of light
is the same for all observers.
Two postulates 1. Principle of relativity The
laws of physics must be the same in all inertial
reference frames. 2. Constancy of the speed of
light The speed of light in a vacuum has the
same value c 3.00108 m/s in all inertial
reference frames, regardless of the velocity of
the observer or the velocity of the source
emitting the light.

• Einsteins principle of relativity is a
  generalization of the principle of Galilean
  relativity. The results of any kind of experiment
  (mechanics, electricity, magnetism, optics,
  thermodynamics, ) performed in a laboratory at
  rest must be the same as when performed in a
  laboratory moving at a constant speed.
• The constancy of the speed of light is required
  by the first postulate. It also 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, and be ready to see some surprising
  results from Einsteins principle of relativity.
  

10
39.4 Consequences of the special theory of
relativity
11
Read Ch39 3,4 No homework
12
January 28 Simultaneity
Simultaneity and the relativity of time
A thought experiment A boxcar moves with uniform
velocity. Events Two lightning bolts strike the
car and leave marks A' and B' on the car, and A
and B on the ground. Light signals were sent out
at the strikes (not necessary).
Observer O is on the ground, midway between A and
B. The light reaches him at the same time. He
concludes that the lightning bolts struck A and B
simultaneously.
Observer O' is in the boxcar, midway between A'
and B'. By the time the light reached O, O' has
moved. The signal from B' has already passed O',
but the signal from A' has not yet reached him.
Observer O' concludes that the lightning struck
B' before it struck A'. ? Two events that are
simultaneous in one reference frame are in
general not simultaneous in another reference
frame. Simultaneity is not an absolute concept.
It depends on the state of motion of the
observer.
13
The thought experiment revisited Events Obser
ver O Observer O' Event 1 A and A'
sparks. (x1, t1) (x1', t1') Event 2 B and B'
sparks. (x2, t2) (x2', t2') (Event 3) The two
spark pulses meet. (x3, t3) (x3 ', t3 ')
Facts Observer O Sees Observer O'
Sees Conclusions Observer O They
sparked simultaneously. Observer O'
B' and B sparked first.
14
Read Ch39 4 No homework
15
January 30 Time dilation
A thought experiment A mirror is fixed to the
ceiling of a vehicle moving with speed v.
Observer O' is at rest in the vehicle. She holds
a flashlight a distance d below the
mirror. Event 1 The flashlight emits a pulse of
light directed at the mirror. Event 2 The pulse
arrives back after being reflected.
Observer O' carries a clock and she measures the
time interval between the events to be ?tp 2d/c.
Observer O is stationary on the earth. He
observes that the light travels farther than O'
sees. He measures the time interval between the
events as
Time dilation 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. ? A moving clock runs slower.
16
The g factor
Think why.
Proper time interval tp The time interval
between events as measured by an observer who
sees the events occur at the same point in space.
(Eigenzeit ? own-time)
Generalization of time dilation If a clock is
moving with respect to you, the time interval
between ticks of the moving clock is observed to
be longer than that of an identical clock in your
reference frame. ?All physical processes are
measured to slow down when these processes occur
in a frame moving with respect to the observer.
Quiz 39.3, 39.4
17
Read Ch39 4 Homework Ch39 5,6,8 Due February
6
18
February 2 Time dilation Applications
Time dilation is a real phenomenon that has been
verified by various experiments.
Airplane flights In 1972 time intervals measured
with four macroscopic cesium clocks in jet flight
were compared to time intervals measured by
Earth-based reference clocks. Flying clocks were
found to lose time ( tens of ns), and the
results were in good agreement with the
predictions of the special theory of relativity.
?You save life time while flying!
Decay of muons Muon particles with q e, m
207me, half-life time ?tp 2.2 µs measured in a
reference frame at rest with respect to
them. Relative to an observer on the Earth,
flying muons should have a lifetime of ??tp. This
explains why large number of muons reach the
surface of the earth.
19
Example 39.1 What is the period of the
pendulum? Example 39.2 How long was the trip?
The twin paradox Situation Speedo, one of the
twins, travels at v 0.95c to Planet X 20 light
years from the earth. After he reaches there he
immediately returns to the earth at the same
speed. When Speedo returns, he has aged 13 years,
but his brother Goslo has aged 42
years. Paradox Speedo thinks that he was at
rest, while Goslo and the Earth raced away from
him and then headed back toward him. Therefore,
Goslo should have aged less. Question Whose
hair has turned white?

• Answer
• Theory of special relativity only applies to
  reference frames moving at uniform speeds.
• Speedo must experience a series of accelerations
  during the journey. Therefore he is not in an
  inertial frame and cannot apply the theories of
  special relativity.
• Goslo can apply the time dilation formula. He
  finds that Speedo have aged 13 years.

Quiz 39.5
20
Read Ch39 4 Homework Ch39 12,14 Due February
13
21
February 4 Length contraction
Proper length Lp The proper length of an object
is the length of the object measured by someone
at rest relative to the object.
A thought experiment Events A spacecraft
traveling with a speed v leaving one star and
reaching another. Observer on the earth
Measures the distance between the stars as the
proper length Lp. The time interval for finishing
the voyage is Dt Lp/v. Observer in the
spacecraft Measures the proper time interval for
the voyage because the events occur at the same
position for him. Dtp Dt /g. He concludes that
the distance between the stars is L v Dtpv Dt
/g Lp /g .
Length contraction
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 LLp
/g. ?A moving ruler shrinks. Length contraction
takes place only along the direction of motion.
22
Proper length and the proper time The proper
length is measured by an observer for whom the
end points of the length remain fixed in space.
The proper time interval is measured by someone
for whom the two events take place at the same
position in space. Moving muons revisited The
observer on the muon Measures the proper
lifetime, but the travel distance is shorter
because of length contraction. The observer on
the earth Measures the proper travel distance,
but the lifetime is longer because of time
dilation. The outcome of the experiment is the
same for both observers.
Quiz 39.6, 39.7
Example 39.3 A voyage to Sirius
23
Read Ch39 4 Homework Ch39 9,10 Due February
13
24
February 11 Length contraction Applications
Space-time graphs

• For a certain reference frame, ct is the ordinate
  and position x is the abscissa.
• A path (trajectory) of an object through
  space-time is called a world-line.
• World-lines for light are diagonal lines.
• Example the space-time graph of the twin paradox.

25
Example 39.4 The pole-in-the-barn paradox The
relativity of simultaneity.
26
Read Ch39 4 Homework Ch39 13 Due February 20
27
February 13 Relativistic Doppler effect
Relativistic Doppler effect
Relativistic Doppler effect The frequency shifts
for light emitted by atoms observed by an
observer who has a relative speed with respect to
the light source. A consequence of time dilation.
Scenario A person is running with velocity v
toward a light source fixed on the
ground. Events The two wavefronts meet the
person.
Observer standing on the ground
Observer who is running
28
If a light source and an observer approach each
other with a relative speed v, the frequency
measured by the observer is Things moving
toward us appear more blue, while things moving
away from us appear more red.
Example Red shift (from XUV) of galaxies ? Most
galaxies are moving away from us.
Problem 39.18 Fines on speeding and on running
through a red signal (650 nm ?520nm).
29
Read Ch39 4 Homework Ch39 16,17,18 Due
February 20
30
February 16, 18 Lorentz transformation
39.5 The Lorentz transformation equations
Suppose O meets O' at (xx' 0, y y' 0,z z'
0,t t'0). Event A spark occurs at point
P. Observer in S describes the event with (x, y,
z, t). Observer in S ' describes the event with
(x', y', z', t').
Lorentz transformation equations
Equations that converts between two different
observers measurements of space and time
S' ?S
Matrix form
31

• In relativity, space and time are not separate
  concepts but rather closely interwoven with each
  other.
• When v ltltc, Lorentz transformation reduces to
  Galilean transformation.

A pair of events Lorentz transformation
equations in difference form S (x1, y1, z1,
t1) and (x2, y2, z2, t2) S' (x'1, y'1, z'1,
t'1) and (x'2, y'2, z'2, t'2).
32
Example 39.5 Simultaneity, time dilation and
length contraction revisited
33
Space-time interval
In general,
?The space-time interval is invariant in inertial
reference frames.
34
Relativistic Doppler effect revisited
35
Read Ch39 5 Homework Ch39 19,20,22 Due
February 27
36
February 20 Lorentz velocity transformation
39.6 The Lorentz velocity transformation equations
S' is moving at v relative to S. Event the
motion of the object.

• If v ltlt c, then ux'? ux-v, Lorentz transformation
  reduces to Galilean transformation.
• If uxc, then ux'c. The speed of light does not
  depend on the motion of the frame.

37
S' ?S

• The two observers do not agree on (Things related
  to space and time)
• The time interval between events that take place
  at the same position in one of the reference
  frames.
• The distance between two points that remain fixed
  in one of the frames.
• The velocity components of a moving object.
• Whether or not two events occurring at different
  places are simultaneous.
• The two observers agree on
• Their relative speed v with respect to each
  other.
• The speed c of any ray of light
• The simultaneity of two events take place at the
  same position and same time.

38
Quiz 39.8 Example 39.6 Two space crafts 1)
Identify the observers and the event. 2) What is
the velocity of craft A as observed by the crew
on craft B? 3) What is the relative velocity
between A and B as observed by us? Example 39.7
Leaders of the pack
39
Read Ch39 6 Homework Ch39 23,24 Due February
27
40
February 23,25 Relativistic momentum
39.7 Relativistic linear momentum
Suppose we observe a collision in the reference
frame S and the momentum and kinetic energy of
the system are conserved, as in the figure
? Now lets move to the S' frame, which moves
at constant v -u relative to S. Question Will
the momentum be conserved in the S' frame?
First let us use classical momentum
definition p mu, and the Lorentz velocity
transformation
Final
Initial
? Classical momentum is not conserved after
Lorentz transformation.
41
To achieve linear momentum conservation in all
inertial frames, the definition of momentum must
be modified. New definition must satisfy two
conditions 1) The linear momentum of an isolated
system must be conserved in all collisions. 2)
The relativistic linear momentum p of a particle
must approach the classical value mu when u
approaches zero.
Solution The relativistic linear momentum of
any particle is defined as
42
Now let us test our example again in the S' frame
using the new definition of momentum
Initial
Final
? Relativistic momentum is conserved after
Lorentz transformation.
43
The relativistic force acting on a particle whose
linear momentum is p
It is impossible to accelerate a particle from
rest to a speed of u c
? The speed of light is the speed limit of the
universe. It is the maximum possible speed for
energy and information transfer.
Example 39.8 Linear momentum of an electron.
44
Read Ch39 7 Homework Ch39 26,27,29 Due March
6
45
February 27, March 2 Relativistic energy
39.8 Relativistic energy
Einsteins postulates ? Redefinition of linear
momentum ? Redefinition of energy. Work done by
force F on a particle
?Relativistic kinetic energy
46
When u ltlt c,

• Mass is a form of energy.
• A particle has energy by virtue of its mass
  alone.
• A small mass corresponds to an enormous amount of
  energy.

47
The relativistic energy momentum relation
Example For photon,
Quiz 39.9
Example 39.9
48
39.9 Mass and energy
The conversion from mass to energy in nuclear
reactions (fission and fusion)
Example 39.10
49
Read Ch39 8-9 Homework Ch39
31,35,37,43,52,58 Due March 6