2.1 The Need for Aether

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CHAPTER 2Special Theory of Relativity 2

• 2.1 The Need for Aether
• 2.2 The Michelson-Morley Experiment
• 2.3 Einsteins Postulates
• 2.4 The Lorentz Transformation
• 2.5 Time Dilation and Length Contraction
• 2.6 Addition of Velocities
• 2.7 Experimental Verification
• 2.8 Twin Paradox
• 2.9 Space-time
• 2.10 Doppler Effect
• 2.11 Relativistic Momentum
• 2.12 Relativistic Energy
• 2.13 Computations in Modern Physics
• 2.14 Electromagnetism and Relativity

Albert Einstein (1879-1955)
Do not worry about your difficulties in
Mathematics. I can assure you mine are still
greater. Albert Einstein
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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 familiar Galilean transformation. Space and time
are now linked, and the frame velocity cannot
exceed c.
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Simultaneity
Timing events occurring in different places can
be tricky. Depending on how theyre measured,
different events will be perceived in different
orders by different observers.
0
L
-L
Due to the finite speed of light, the order in
which these two events will be seen will depend
on the observers position. The time intervals
will be Fred -2L/c Frank 0 Fil 2L/c
But this obvious position-related simultaneity
problem disappears if Fred and Fil have
synchronized watches.
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Synchronized clocks in a frame
Its possible to synchronize clocks throughout
space in each frame. This will prevent the
position-dependent simultaneity problem in the
previous slide. But there will still be
simultaneity problems due to velocity.
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Simultaneity
So all stationary observers in the explosions
frame measure these events as simultaneous.
What about moving ones?
Compute the interval as seen by Mary using the
Lorentz time transformation.
0
L
-L
Mary experiences the explosion in front of her
before the one behind her. And note that Dt is
independent of Marys position!
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2.5 Time Dilation and Length Contraction
More very interesting consequences of the Lorentz
Transformation

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

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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, T0, 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 K where two
explosions occur. Mary, in moving K, is there
for the first, but not the second. Fortunately,
Melinda, also in K, 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?
K
Frank
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Time Dilation

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

This is the time interval as measured in the
frame K. This is not proper time due to the
motion of K .
Frank, on the other hand, records x2 x1 0 in
K with a (proper) time T0 t2 t1, so we have
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Time Dilation

• 1)  T T0 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 K requires 1 clock and K requires 2
  clocks for the measurement.
• 4) Because the Lorentz transformation is
  symmetrical, time dilation is reciprocal
  observers in K see time travel faster than for
  those in K. And vice versa!

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Time Dilation Example Reflection
Let T be the round-trip time in K
Frank
Mary
Fred
K
K
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Reflection (continued)
The time in the rest frame, K, is
or
or
So the event in its rest frame (K) occurs faster
than in the frame thats moving compared to it
(K).
or
or
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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 K, measures the
  length of his somewhat bulging waist
• L0 xr - xl
• Now, Mary and Melinda measure it, too, making
  simultaneous measurements ( ) of the
  left, , and the right 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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2.6 Addition of Velocities
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?

• Taking differentials of the Lorentz
  transformation here between the rest frame (K)
  and the space ship frame (K), we can compute
  the shuttle velocity in the rest frame (ux
  dx/dt)

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The Lorentz Velocity Transformations

• Defining velocities as ux dx/dt, uy dy/dt,
  ux dx/dt, etc., we find

with similar relations for uy and uz
Note the gs in uy and uz.
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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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Relativistic velocity addition
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Example Lorentz velocity transformation
Capt. Kirk decides to escape from a hostile
Romulan ship at 3/4c, but the Romulans follow at
1/2c, firing a matter torpedo, whose speed
relative to the Romulan ship is 1/3c. Question
does the Enterprise survive?
vRg 1/2c
vEg 3/4c
vtR 1/3c
Romulans
Enterprise
torpedo
vRg velocity of Romulans relative to galaxy vtR
velocity of torpedo relative to Romulans vEg
velocity of Enterprise relative to galaxy
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Galileos addition of velocities
We need to compute the torpedo's velocity
relative to the galaxy and compare that with the
Enterprise's velocity relative to the galaxy.
Using the Galilean transformation, we simply add
the torpedos velocity to that of the Romulan
ship
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Einsteins addition of velocities
Due to the high speeds involved, we really must
relativistically add the Romulan ships and
torpedos velocities
The Enterprise survives to seek out new worlds
and go where no one has gone before
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Example Addition of velocities
We can use the addition formulas even when one of
the velocities involved is that of light. At
CERN, neutral pions (p0), traveling at 99.975 c,
decay, emitting g rays in opposite directions.
Since g rays are light, they travel at the speed
of light in the pion rest frame. What will the
velocities of the g rays be in our rest frame?
(Simply adding speeds yields 0 and 2c!) Parallel
velocities
Anti-parallel velocities
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Aether Drag
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.
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Aether Drag
Armand Fizeau (1819 - 1896)
Let K be the frame of the water, flowing with
velocity, v. Well treat the speed of light in
the medium ( u, u ) as a normal velocity in the
velocity-addition equations. In the frame of the
flowing water, u c / n
which was what Fizeau found.
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2.7 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

• It takes 6.8 ms for the 2000-m path at 0.98c,
  about 4.5 times the muon lifetime. So, without
  time dilation, of 1000 muons, we expect only 1000
  x 2-4.5 45 muons at sea level.

Since 0.98c yields g 5, instead of moving 600 m
on average, they travel 3000 m in the Earths
frame.
In fact, we see 542, in agreement with
relativity! And how does it look to the muon?
Lorentz contraction shortens the distance!
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2.8 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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Atomic Clock Measurement

• Two airplanes traveled east and west,
  respectively, around the Earth as it rotated.
  Atomic clocks on the airplanes were compared with
  similar clocks kept at the US Naval Observatory
  to show that the moving clocks in the airplanes
  ran slower.

vrotation vplane
Travel Predicted Observed Eastward -40 23
ns -59 10 ns Traveling twin Westward 275 21
ns 273 7 ns Stay-at-home twin
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There have been many rigorous tests of the
Lorentz transformation and Special Relativity.
Particle Accuracy Electrons 10-32 Neutrons 10-31 P
rotons 10-27
Quantum Electrodynamics also depends on Lorentz
symmetry, and it has been tested to 1 part in
1012.
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2.9 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
x
Stationary observers live on vertical lines. A
light wave has a 45º slope.
Slope of worldline is c/v.
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Worldlines and Time
Observers at x1 and x2 see whats happening at x
x3 at t 0 simultaneously. Alternatively, an
event occurring at x3 can be used to synchronize
clocks at x1 and x2.
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Moving Clocks
Observers in a frame moving at velocity, v, will
see the event happening at x x3 at t 0 at
different times.
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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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Space-time Interval and Metric

• Recall that, since all observers see the same
  speed of light, all observers, regardless of
  their velocities, must see spherical wave fronts.

s2 x2 y2 z2 c2t2 (x)2 (y)2
(z)2 c2 (t)2 (s)2
This interval can be written in terms of the
space-time metric
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Space-time Invariants

• 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 0 ?x2 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 and the two events can be causally connected. The
interval is said to be time-like.