Physics 334 Modern Physics

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Physics 334Modern Physics
Credits Material for this PowerPoint was adopted
from Rick Trebinos lectures from Georgia Tech
which were based on the textbook Modern Physics
by Thornton and Rex. Many of the images have been
used also from Modern Physics by Tipler and
Llewellyn, others from a variety of sources
(PowerPoint clip art, Wikipedia encyclopedia
etc), and contributions are noted wherever
possible in the PowerPoint file. The PDF handouts
are intended for my Modern Physics class, as a
study aid only.
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CHAPTER 4Special Theory of Relativity 1

• 4-1 Foundations of Special Relativity
• The Experimental Basis of Relativity
• Einsteins Postulates
• 4-2 Relationship between Space and Time
• The Lorentz Transformation
• Time Dilation and Length Contraction
• The Doppler Effect
• The Twin Paradox and Other Surprises

Albert Michelson(1852-1931)
It was found that there was no displacement of
the interference fringes, so that the result of
the experiment was negative and would, therefore,
show that there is still a difficulty in the
theory itself - Albert Michelson, 1907
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Newtonian (Classical) Relativity

• Newtons laws of motion must be implemented with
  respect to (relative to) some reference frame.

A reference frame is called an inertial frame if
Newtons laws are valid in that frame. Such a
frame is established when a body, not subjected
to net external forces, moves in rectilinear
motion at constant velocity.
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Difference Between Inertial and Non-Inertial
Reference Frame
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Newtonian Principle of Relativity

• If Newtons laws are valid in one reference
  frame, then they are also valid in another
  reference frame moving at a uniform velocity
  relative to the first system.
• This is referred to as the Newtonian principle of
  relativity or Galilean invariance.

If the axes are also parallel, these frames are
said to be Inertial Coordinate Systems
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The Galilean Transformation

• For a point P
• In one frame S P (x, y, z, t)
• In another frame S P (x, y, z, t)

The Inverse Relations
1. Parallel axes 2. S has a constant relative
velocity (here in the x-direction) with respect
to S. 3. Time (t) for all observers is a
Fundamental invariant, i.e., its the same for
all inertial observers.
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A need for ether

• In Maxwells theory, the speed of light, in terms
  of the permeability and permittivity of free
  space, was given by
• Thus the velocity of light is constant
• Aether was proposed as an absolute reference
  system in which the speed of light was this
  constant and from which other measurements could
  be made.
• Properties of Aether
• Low density
• Elasticity
• Transverse waves
• Galilean transformation

Maxwells equations are not invariant under
Galilean transformations. The Michelson-Morley
experiment was an attempt to show the existence
of aether.
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Michelson-Morley experiment
Michelson and Morley realized that the earth
could not always be stationary with respect to
the aether. And light would have a different
path length and phase shift depending on whether
it propagated parallel and anti-parallel or
perpendicular to the aether.
Perpendicular propagation
Parallel and anti-parallel propagation
Supposed velocity of earth through the aether
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Michelson-Morley Experimental Analysis
Exercise 4-1 Show that the time difference
between path differences after 90 rotation is
given by
Recall that the phase shift is w times this
relative delay
or

• The Earths orbital speed is v 3 104 m/s ,
  and the interferometer size is L 1.2 m, So
  the time difference becomes 8 10-17 s, which,
  for visible light, is a phase shift of 0.2 rad
  0.03 periods

The Michelson interferometer shouldve revealed a
fringe shift as it was rotated with respect to
the aether velocity. MM expected 0.4 of the width
of a fringe, and could only see 0.01 equal to the
uncertainty in the measurement.
Interference fringes showed no change as the
interferometer was rotated.
Thus, aether seems not to exist!
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Einsteins Postulates

• Albert Einstein was only two years old when
  Michelson and Morley reported their results.
• At age 16 Einstein began thinking about
  Maxwells equations in moving inertial systems.
• In 1905, at the age of 26, he published his
  startling proposal the Principle of
  Relativity.
• It nicely resolved the Michelson and Morley
  experiment (although this wasnt his intention
  and he maintained that in 1905 he wasnt aware of
  Michelson and Morleys work)

Albert Einstein (1879-1955)
It involved a fundamental new connection between
space and time and that Newtons laws are only an
approximation.
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Einsteins Two Postulates

• With the belief that Maxwells equations must be
  valid in all inertial frames, Einstein proposed
  the following postulates
• The principle of relativity The laws of physics
  are the same in all inertial reference frames.
• The constancy of the speed of light The speed
  of light in a vacuum is equal to the value c,
  independent of the motion of the source.

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Relativity of Simultaneity

• In Newtonian physics, we previously assumed that
  t t.
• With synchronized clocks, events in S and S can
  be considered simultaneous.
• Einstein realized that each system must have its
  own observers with their own synchronized clocks
  and meter sticks.
• Events considered simultaneous in S may not be in
  S.
• Also, time may pass more slowly in some systems
  than in others.

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The constancy of the speed of light
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Lorentz Transformation

• Exercise 4-2 The equations for a spherical
  wavefronts in S is
• x2y2z2c2t2 , Show that the equation for the
  spherical wavefronts in S cannot be
  x2y2z2c2t2 in the Galilean transformation.
• Exercise 4-3 Show that x g (x vt) so that
  x g (x vt) , yields the g factoid
• and that for small velocities

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Lorentz Transformation

• Exercise 4-4 Use x g (x vt) and x g
  (x vt) , to find t g (t v x /c2)
• Exercise 4-5 Use x g (x vt) and t g (t
  v x /c2) to show that the equations for
  spherical wave fronts in S and S are the same.

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Lorentz Transformation Equations
A more symmetrical form
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Properties of g

• Recall that b v / c lt 1 for all observers.

g equals 1 only when v 0. In general
Graph of g vs. b (note v lt c)
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The complete Lorentz Transformation
If v ltlt c, i.e., ß 0 and g 1, yielding the
familiar Galilean transformation. Space and time
are now linked, and the frame velocity cannot
exceed c.
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Relativistic Velocity Transformation
Exercise 4-6 Suppose a shuttle takes off quickly
from a space ship already traveling very fast
(both in the x direction). Imagine that the
space ships speed is v, and the shuttles speed
relative to the space ship is u. What will the
shuttles velocity (u) be in the rest frame?
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The Inverse Lorentz Velocity Transformations

• If we know the shuttles velocity in the rest
  frame, we can calculate it with respect to the
  space ship. This is the Lorentz velocity
  transformation for ux, uy , and uz. This is
  done by switching primed and unprimed and
  changing v to v

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Lorentz velocity transformation
Example As the outlaws escape in their really
fast getaway ship at 3/4c, the police follow in
their pursuit car at a mere 1/2c, firing a
bullet, whose speed relative to the gun is 1/3c.
Question does the bullet reach its target a)
according to Galileo, b) according to Einstein?
vpg 1/2c
vog 3/4c
vbp 1/3c
police
outlaws
bullet
vpg velocity of police relative to ground vbp
velocity of bullet relative to police vog
velocity of outlaws relative to ground
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Galileos addition of velocities
In order to find out whether justice is met, we
need to compute the bullet's velocity relative to
the ground and compare that with the outlaw's
velocity relative to the ground.
In the Galilean transformation, we simply add the
bullets velocity to that of the police car
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Einsteins addition of velocities
Due to the high speeds involved, we really must
relativistically add the police ships and
bullets velocities
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Gedanken (Thought) experiments
It was impossible to achieve the kinds of speeds
necessary to test his ideas (especially while
working in the patent office), so Einstein used
Gedanken experiments or Thought experiments.
Young Einstein
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The complete Lorentz Transformation
If v ltlt c, i.e., ß 0 and g 1, yielding the
familiar Galilean transformation. Space and time
are now linked, and the frame velocity cannot
exceed c.
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Time Dilation and Length Contraction
More very interesting consequences of the Lorentz
Transformation

• Time Dilation
• Clocks in S run slowly with respect to
  stationary clocks in S.
• Length Contraction
• Lengths in S contract with respect to the same
  lengths in stationary S.

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We must think about how we measure space and time.
In order to measure an objects length in space,
we must measure its leftmost and rightmost points
at the same time if its not at rest. If
its not at rest, we must ask someone else
to stop by and be there to help out.
In order to measure an events duration in time,
the start and stop measurements can occur at
different positions, as long as the clocks are
synchronized. If the positions are
different, we must ask someone else to stop
by and be there to help out.
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Proper Time

• To measure a duration, its best to use whats
  called Proper Time.
• The Proper Time, t, is the time between two
  events (here two explosions) occurring at the
  same position (i.e., at rest) in a system as
  measured by a clock at that position.

Same location
Proper time measurements are in some sense the
most fundamental measurements of a duration. But
observers in moving systems, where the
explosions positions differ, will also make such
measurements. What will they measure?
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Time Dilation and Proper Time
Franks clock is stationary in S where two
explosions occur. Mary, in moving S, is there
for the first, but not the second. Fortunately,
Melinda, also in S, is there for the second.
Mary and Melinda are doing the best measurement
that can be done. Each is at the right place at
the right time.
If Mary and Melinda are careful to time and
compare their measurements, what duration will
they observe?
S
Frank
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Time Dilation

• Mary and Melinda measure the times for the two
  explosions in system S as t1 and t2 . By the
  Lorentz transformation

This is the time interval as measured in the
frame S. This is not proper time due to the
motion of S .
Frank, on the other hand, records x2 x1 0 in
S with a (proper) time t t2 t1, so we have
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Time Dilation

• 1)  ?t gt ?t(? gt1) the time measured between
  two events at different positions is greater
  than the time between the same events at one
  position this is time dilation.
• 2) The events do not occur at the same space and
  time coordinates in the two systems.
• 3) System S requires 1 clock and S requires 2
  clocks for the measurement.
• 4) Because the Lorentz transformation is
  symmetrical, time dilation is reciprocal
  observers in S see time travel faster than for
  those in S. And vice versa!

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Time Dilation Example Reflection
S
S
Frank
Mary
Fred
Exercise 4-7 Show that the event in its rest
frame (S) occurs faster than in the frame thats
moving compared to it (S).
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Time stops for a light wave
Because
And, when v approaches c
For anything traveling at the speed of light
In other words, any finite interval at rest
appears infinitely long at the speed of light.
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Proper Length
When both endpoints of an object (at rest in a
given frame) are measured in that frame, the
resulting length is called the Proper Length.
Well find that the proper length is the largest
length observed. Observers in motion will see a
contracted object.
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Length Contraction

• Frank Sr., at rest in system S, measures the
  length of his somewhat bulging waist
• Lp xr - xl
• Now, Mary and Melinda S, measure it, too, making
  simultaneous measurements (tl tr ) of the
  left, xl , and the right xr endpoints
• Frank Sr.s measurement in terms of Marys and
  Melindas

? Proper length
Moving objects appear thinner!
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Length contraction is also reciprocal.
So Mary and Melinda see Frank Sr. as thinner than
he is in his own frame. But, since the Lorentz
transformation is symmetrical, the effect is
reciprocal Frank Sr. sees Mary and Melinda as
thinner by a factor of g also. Length
contraction is also known as Lorentz
contraction. Also, Lorentz contraction does not
occur for the transverse directions, y and z.
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Lorentz Contraction
v 10 c
A fast-moving plane at different speeds.
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Experimental Verification of Time Dilation
Cosmic Ray Muons Muons are produced in the
upper atmosphere in collisions between ultra-high
energy particles and air-molecule nuclei. But
they decay (lifetime 1.52 ms) on their way to
the earths surface
No relativistic correction
Top of the atmosphere
Now time dilation says that muons will live
longer in the earths frame, that is, t will
increase if v is large. And their average
velocity is 0.98c!
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Detecting muons to see time dilation

• At 9000 m it takes muons (9000/0.998c 30 µs)
  about 15 lifetimes to reach earth. If No 108
  and t 15t, N 31 muons should reach earth.

From relativistic approach, the distance traveled
is only 600m at that speed in 1 lifetime (2 µs)
and therefore N 3.68 x 107 Experiments have
confirmed this relativistic prediction

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Space-time Invariants
This is a quantity that is invariant under
Lorentz transformation. It is defined in the
following way
(?s)2 (c2?t2) - ?x2 ?y2 ?z2

• The quantity ?s2 between two events is invariant
  (the same) in any inertial frame.
• ?s is known as the space-time interval between
  two events.

There are three possibilities for ?s2 ?s2 0
?x2 c2 ?t2, and the two events can be connected
only by a light signal. The events are said to
have a light-like separation. ?s2 gt 0 ?x2 gt c2
?t2, and no signal can travel fast enough to
connect the two events. The events are not
causally connected and are said to have a
space-like separation. ?s2 lt 0 ?x2 lt c2 ?t2,
and the two events can be causally connected. The
interval is said to be time-like.
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Space-time

• When describing events in relativity, its
  convenient to represent events with a space-time
  diagram.
• In this diagram, one spatial coordinate x,
  specifies position, and instead of time t, ct is
  used as the other coordinate so that both
  coordinates will have dimensions of length.
• Space-time diagrams were first used by H.
  Minkowski in 1908 and are often called Minkowski
  diagrams. Paths in Minkowski space-time are
  called world-lines.

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Particular Worldlines
Stationary observers live on vertical lines. A
light wave has a 45º slope.
Worldline is the record of the particles travel
through spacetime, giving its speed (1/slope) and
acceleration (1/rate of change of slope).
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The Light Cone
The past, present, and future are easily
identified in space-time diagrams. And if we add
another spatial dimension, these regions become
cones.
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The Doppler Effect
The Doppler effect for sound yields an increased
sound frequency as a source such as a train (with
whistle blowing) approaches a receiver and a
decreased frequency as the source recedes.
Christian Andreas Doppler (1803-1853)

• A similar change in sound frequency occurs when
  the source is fixed and the receiver is moving.
• But the formula depends on whether the source or
  receiver is moving.
• The Doppler effect in sound violates the
  principle of relativity because there is in fact
  a special frame for sound waves. Sound waves
  depend on media such as air, water, or a steel
  plate in order to propagate. Of course, light
  does not!

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Waves from a source at rest
Viewers at rest everywhere see the waves with
their appropriate frequency and wavelength.
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Recall the Doppler Effect
A receding source yields a red-shifted wave, and
an approaching source yields a blue-shifted
wave. A source passing by emits blue- then
red-shifted waves.
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The Relativistic Doppler Effect

• So what happens when we throw in Relativity?
• Exercise 4-8 Consider a source of light (for
  example, a star) in system S receding from a
  receiver (an astronomer) in system S with a
  relative velocity v. Show that the frequency can
  be obtained from
• Where f0 is the proper frequency
• Exercise 4-9 What would be the frequency if the
  source was approaching?
• Exercise 4-10 Use the results from exercise 8
  and 9 to deduce the expressions for
  non-relativistic velocities.

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Using the Doppler shift to sense rotation
The Doppler shift has a zillion uses.
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Using the Doppler shift to sense rotation
Example The Sun rotates at the equator once in
about 25.4 days. The Suns radius is 7.0x108m.
Compute the Doppler effect that you would expect
to observe at the left and right limbs (edges) of
the Sun near the equator for the light of
wavelength l 550 nm 550x10-9m (yellow light).
Is this a redshift or a blueshift?
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Aether Drag
Exercise 4-12 In 1851, Fizeau measured the
degree to which light slowed down when
propagating in flowing liquids.
Fizeau found experimentally
This so-called aether drag was considered
evidence for the aether concept. Derive this
equation from velocity addition equations.
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Lorentz-FitzGerald Contraction

• Exercise 4-13 Lorentz and FitzGerald, proposed
  that the null test of Michelson Morleys
  experiment can be explained by using the concept
  of length contraction to explain equal path
  lengths and zero phase shift. Show that this
  proposition can work.

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The Twin Paradox

• The Set-up
• Mary and Frank are twins. Mary, an astronaut,
  leaves on a trip many lightyears (ly) from the
  Earth at great speed and returns Frank decides
  to remain safely on Earth.
• The Problem
• Frank knows that Marys clocks measuring her age
  must run slow, so she will return younger than
  he. However, Mary (who also knows about time
  dilation) claims that Frank is also moving
  relative to her, and so his clocks must run slow.
  
• The Paradox
• Who, in fact, is younger upon Marys return?

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The Twin-Paradox Resolution

• Franks clock is in an inertial system during the
  entire trip. But Marys clock is not. As long as
  Mary is traveling at constant speed away from
  Frank, both of them can argue that the other twin
  is aging less rapidly.
• But when Mary slows down to turn around, she
  leaves her original inertial system and
  eventually returns in a completely different
  inertial system.
• Marys claim is no longer valid, because she
  doesnt remainin the same inertial system.
  Frank does, however, and Mary ages less than
  Frank.

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Twin Paradox

• Exercise 4-14 A clock is placed in a satellite
  that orbits Earth with a period of 108 min. (a)
  By what time interval will this clock differ from
  an identical clock on Earth after 1 year? (b) How
  much time will have passed on Earth when the two
  clocks differ by 1.0 s? (Assume special
  relativity applies and neglect general
  relativity.)