3/31/2016
Chapter 25 Lecture
Special
Relativity
WHITEBOARD VECTOR ANALYSIS
An analogy: A boat race
 Consider a process involving two identical
boats in a race on a wide river. Which
boat returns to the starting dock first?
Ether or no ether?
 The work of Maxwell and Hertz led to the
conclusion that light propagation could be
explained by changing electric and
magnetic fields that do not require any
medium to travel.
 Before this work, physicists were
searching for ether.
 This search produced an unexpected
outcome that eventually changed the
way we think about space and time.
Testing the existence of ether
Albert Michelson and Edward Morley
experiment - 1887
 Imagine that ether fills the solar system and
is stationary with respect to the Sun.
 Because Earth moves around the Sun at a
speed of about 3.0 x 104 m/s, ether should
be moving past Earth at this speed.
 Shining
light
waves
parallel
and
perpendicular to the ether's motion relative
to Earth is similar to sending boats parallel
and perpendicular to a flowing river.
The Michelson-Morley Experiment
(1887)
Albert Michelson and Edward Morley
experiment - 1887
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Testing the existence of ether
Albert Michelson and Edward Morley
experiment - 1887
 Outcome: no matter how the interferometer
orientation was changed, the interference
pattern did not change.
 Possible conclusions:
 There is not ether through which light
travels.
 There is ether, but it is stuck to Earth’s
surface and does not move relative to the
interferometer.
Testing the existence of ether
 Physicists were reluctant to accept this
result.
 Perhaps the ether is at rest relative to
Earth.
 If the ether is attached to Earth, then
as Earth rotates around its axis and
orbits the Sun, the ether should become
twisted.
 This would cause light coming from
stars to be slightly deflected on its way
to Earth
 No one observed such an effect.
Postulates of special relativity
INVARIANCE
 Using Newton's laws yields consistent
results, regardless of the inertial reference
frames (not accelerating) used—a feature
known as invariance.
Einstein’s Thought Experiments
Source emits light
 Maxwell's equations have to be written
differently for different observers, with a
different speed of light in each case.
Observer measures speed of light
Einstein embarked on an intellectual journey to
determine what the speed of light would appear to be
from the reference frame of various observers.
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Einstein’s Thought Experiment
Would the detector observe light traveling at 0.5c?
Would the detector observe light traveling at 1.5c?
Would the detector observe light traveling at 2c??
The light would stay in place???
This doesn’t seem to make any sense!!!
Einstein's two postulates
Postulate:
A postulate is a statement that is assumed to
be true. It is not derived from anything.
Einstein's two postulates
1. The laws of physics are the same in all
inertial reference frames.
 Newton's second law remains the same
regardless of the inertial reference frame in
which you choose to apply it.
•
There is no experiment (mechanics, electricity,
magnetism, thermodynamics) that is affected by the
motion of an inertial reference frame.
•
If you are in a reference frame that is moving at a
constant velocity, there is no way to find out
whether or not you are moving (because you are not
moving, in an absolute sense! All motion is
relative!)
It is the starting point for a logical argument
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There is no absolute rest frame in the
universe.
All motion is relative. One could never
say that a reference frame is “at rest” in
an absolute sense.
Now, on to this speed
of light thing…
Einstein's two postulates
2. The speed of light in a vacuum is measured
to be the same in all inertial reference frames.
 The speed of light in a vacuum measured
by observers in different inertial reference
frames is the same regardless of the
relative motion of those observers.
The source emits light at speed c.
The detector will observe light traveling at speed c!
Light always travels at speed c, regardless of
the observer’s reference frame!
What would the detector measure?
Light traveling at speed c!
The detector would still see light traveling at 3
x 108 m/s!
This leads to some very bizarre
results about the nature of space
and time.
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Experimental evidence for the
constancy of light speed
Simultaneity
 So far, we have assumed that the time
interval for an object moving from one
point to another is independent of the
reference frame.
 The speed of gamma rays is measured in
the lab to be precisely the speed of light,
despite being produced by the decaying
pion, which was already moving near light
speed relative to the lab.
 The second postulate of the special theory
of relativity made physicists completely
rethink their ideas about time.
 This result supports postulate 2.
Simultaneity of events in two
different inertial reference frames
Simultaneity of events in two
different inertial reference frames
 You are in the middle of the train car and the
detectors are at rest with respect to you; you
note that the light reaches the front and back
detectors at the same time.
Simultaneity of events in two
different inertial reference frames
Simultaneity of events in two
different inertial reference frames
 Your friend is moving with respect to the
detectors and notes that the light reaches
the back detector before the front
detector.
 Phenomena that become significant
only in high-speed circumstances are
called relativistic effects.
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Implications of the difference in
observations
 Events that happen at the same time in
one reference frame do not necessarily
occur at the same time in another
reference frame.
c
WHITEBOARD:
What would each of the following observers
measure for the speed of light emitted by the
bulb?
v = 0.9c
v = 0.9c
 This difference gets larger as the speeds
involved become higher, but the effect
becomes significant only if the speeds are
a substantial fraction of light speed.
c
The speed of light is constant for all
observers!
They will ALL measure the speed of light as
c.
v=0
Then what about this case?
v = 0.9c
v = 0.9c
This scenario is could never happen, because no
source or observer could travel at the speed of light!
v=0
It is fundamentally forbidden by the laws of physics!
No matter what the motion of the source or the observer,
the observed speed of light will be 3 x 108 m/s.
If this were not true, then it would be possible to
determine an “absolute” frame of reference, which is
quite simply not the case in the universe.
0.999c
According to physics as we currently know it,
no object that has mass could ever travel at
the speed of light or faster than the speed of
light!
0.999c
The detector will still measure the speed of light to be
3 x 108 m/s.
Craziness!
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Whiteboard: Einstein’s Thought Experiment
A moving train cart is rigged so that a pulse of light starts at its floor
and is detected when it reaches the ceiling.
First, let’s consider an observer that is on the train, at rest relative to
the source of light
detector
By the time the
light reaches the
ceiling, the clock
that is at rest
relative to the
source will have
elapsed by an
amount t0.
detector
c = d / t0
Write an expression for d in terms of c and t0.
t0 is the time that it takes the light to reach the ceiling from the
reference frame of a person that is at rest (v = 0) relative to the source.
Now you’re going to need to stretch your conception of reality.
d = ct0
t0 is the time that it takes the light to reach the ceiling from the
reference frame of a person that is at rest relative to the source.
A little bit stranger now!
Now, imagine that the train is moving to the right with constant speed v
relative to an observer on the side of the tracks.
We generally think of time as moving forward at a constant rate,
the same for all reference frames.
The light pulse will still shine on the same part of the ceiling, but...
However, by accepting Einstein’s second postulate, we reach some
very surprising conclusions regarding the passage of time in a
reference frame that is in motion relative to another.
All that is required to achieve the result it Einstein’s
second postulate and some algebra!
v
Let’s see!
According to this observer, the light traveled a greater
distance to get there!
What will the observer that is outside the train
(moving relative to the light source) determine?
v
t
v
ct
Write an expression for
d in terms of c, t and v.
Remember! All observers
see light traveling at speed c.
ct
d
vt (where v is the speed of the train)
d
vt
t is the time that it takes the light to reach the ceiling from the reference
frame of a person that is in motion relative to the source (blue Ruggles).
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Stationary Observer
ct
d
Observer Moving Relative to Source
t0
vt
t
c t =v t +d
2 2
2 2
2
d = ct0
d = c2t 2 - v2t 2
d =t c -v
2
v
d = t c2 - v2
Since the observers will certainly agree on the height of the train (but not
the amount of time that it took light to travel those different distances!)…
2
Where t is the amount of time that has elapsed
on the observer’s clock that is not on the train.
ct0 = t c2 - v 2
Rearranging some terms (I leave the algebra as an exercise for you to
do on your own if you are interested in this stuff – you should be!)
What does this mean??
We end up with a relationship between the measurements of time that
the each observer will make about the same event, from different
reference frames.
Since the speed of light is constant in all
reference frames, but different observers will see
light travel different distances based on their own
relative motion, time itself must elapse at
different rates for different observers.
Time elapsed according to
an observer inside the train
v2
t0 = t 1- 2
c
Speed of the train
Speed of light
Time elapsed according to an
observer outside the train
This is known as the equation for time dilation.
This means that the person on the tracks will
measure a larger time interval than the person
on the train for the same event!
Give this some thought!
WHITEBOARD
 A spaceship moves past Earth at a speed of
2.6 x 108 m/s. The ship's captain carries a light
that flashes each time his heart beats.
According to the captain, a flash occurs every
1.0 s. Which time interval elapses between
flashes according to an observer on Earth?
t0 = 2 seconds
Length Contraction, Simultaneity and Velocity
Addition
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Meet the Muon
Muons are formed when high-energy protons from the solar
wind hit Carbon nuclei in our atmosphere.
After being formed, they fly outward at speeds of up to 99.5% of
the speed of light!
However, muons are very unstable particles. A muon at rest
exists for only about 2.2 x 10-6 seconds before it decays.
Muon Whiteboard: Part 1
A muon is formed by a nuclear reaction in the high
atmosphere 5 km above Earth’s surface. The muon
travels straight downward at a speed of 0.995c.
From the reference frame of the muon, how far will
it travel in the 2.2 x 10-6 seconds before it decays?
d = 656.7 m
According to classical physics, the muon will decay
long before it reaches Earth’s surface!
d = 656.7 m
Muon Whiteboard: Part 2
However, muons from
the high atmosphere can
be regularly found
striking the surface of
the Earth!
A muon travels straight downward at a speed of
0.995c. From the reference frame of an observer on
Earth, how far will it travel in the 2.2 x 10-6 seconds
before it decays?
How is this possible?
Note: The time for the muon to decay applies within the
reference frame of the muon itself – not to the
observer!
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Muons are Evidence of Special
Relativity!
Time elapsed according
to the rest frame
(muon’s ref frame)
Time elapsed according to
an outside observer
t0 = t 10.9952 c 2
2.2 ´10 s = t 1c2
-6
v2
c2
t = 22.0 x 10-6 s
But hey, wait a second…
How can it be possible that the muon travels only
650 m in its own reference frame, but travels a
whole 6,500 m in our reference frame?
It either hits the ground or it doesn’t!
…right?
d = 6,575 m!
From the reference frame of the Earth,
the muon has plenty of time to reach the ground!
As it turns out, the 650 m that the muon travels in its own
reference frame is equivalent to the 6,500 m that it travels in
ours.
Time is not the only
quantity that is relative to
the observer.
Lengths are also relative
Another Thought Experiment
In the year 2500, an astronaut takes a trip
to Vega - a distant star. The trip is a
distance of 25.3 light-years, as measured
by an observer on Earth. The astronaut
travels at a speed of 0.99c
In formulating special
relativity, Einstein showed
that space and time are
linked in the most
fundamental way.
How will the astronaut see this?
Whiteboard: Length Contraction!
From the astronaut’s reference frame, Earth and
Vega are moving at 0.99c, and their ship is at
rest.
a) How much time will the trip take, according to
each of the observers?
b) What is the distance between Earth and Vega,
according to each of the observers?
The astronaut and the Earth observer will agree on
their relative velocity, but that’s about all they will
agree upon!
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Two Different Stories – Both Correct!
Length contraction
 Consider an arrow flying across a lab that moves
past a clock at rest with respect to the lab.
Time:
25.56
years
Time:
3.61
years
The only thing they will agree upon is their relative velocity
Proper length
 The arrow's proper length is the length measured
in a reference frame in which the arrow is
stationary.
 In this case, it is the reference frame defined
by the arrow itself.
 The two events occurred at the same place in the
lab reference frame, so that is the proper
reference frame.
Length Contraction
Lengths are shorter to observers who are moving
relative to the object being measured.
Length measurement of an
observer moving relative to
the object being measured
v2
L = L0 1- 2
c
 The proper length is always the longest length
measured for a given object.
Lengths only contract
along the direction of motion
Rel. speed of
object/observer
Speed of light
Length measurement of an
observer at rest relative to the
object being measured
Both observers will measure the other’s lengths
to be contracted (and both will be correct!)
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Whiteboard: Length Contraction
L = L0 1How fast would a meter stick have to move for it
to become a half-meter stick from your
reference frame?
Whiteboard: Laying Down
v2
c2
0.5 = 1 1-
v2
c2
v = 0.866c
Solution
An astronaut is resting on a bed inclined at an angle
theta above the floor of a spaceship.
Lengths contract along the direction of relative motion. This
will cause x to contract, while y is constant.
From the reference frame of an observer who is moving
to the right with a speed close to c, is the angle that the
bed makes with the floor (a) greater than, (b) less than,
or (c) equal to the angle as observed by the astronaut?
Therefore, the moving observer will measure a larger theta
than the astronaut!
Whiteboard
“It takes the muon
22.0 μs to hit the
ground, which is
5000 m away.”
By combining the concepts of time dilation and length
contraction, describe the journey of a muon traveling
downward at 0.995c from the reference frame of…
a) A Earth observer
b) The muon
“It takes the ground
2.2 μs to hit me,
starting from just
499.37 m away.”
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WHITEBOARD
THE TWIN PARADOX
 An arrow flies past a person standing on
Earth. When at rest with respect to Earth,
the arrow's length was measured to be
1.00 m. Determine the arrow's length L as
measured by the person on Earth when
the arrow moves:
A. At speed 0.90c.
B. At speed 300 m/s.
L = 0.436 m
L=1m
Less time elapses in a moving reference frame than a
stationary one!
Whiteboard: The Twin Trip Paradox
Two twins are born simultaneously on Earth
in the year 2500. One of them lives a happy
life on Earth, and the other is put on a
spaceship and travels the Universe at
extremely high speeds, approaching the
speed of light. When the twin returns to
Earth from his journey, he finds that his
brother is 100 years old, while he is only 25
years old!
“When you’re
talking to a pretty
girl, an hour feels
like a second. When
you put your hand
on a red-hot ember,
a second feels like
an hour. That’s
relativity.”
How fast did the spaceship need to travel in
order for this to happen?
v = 0.968c
RELATIVISTIC MOMENTUM
 When we use the classical definition of
momentum to analyze collisions at high
speed, we find that even for an isolated
system, the momentum of the system is
constant in some reference frames but not
in others.
RELATIVISTIC MOMENTUM
p = mv
p  m
p = mv
 To get an improved relativistic expression
for momentum, we use the proper time
interval.
p  m
 No speed restriction !
x
t0
p
1
x
t  1 
m
2
v
c2
p
2

v
c2
x
t
mv
1
v2
c2
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Relativistic momentum
WHITEBOARD
The relativistic momentum of an object of
mass m in a reference frame where the
object is moving at velocity v is:
mv
p
1
 Speed restriction !
 An electron is moving at a speed of 0.9c.
Compare its momentum as calculated using
a nonrelativistic equation and using a
relativistic equation.
p = 2.46x10-22 kgm/s
2
v
c2
p = 5.64x10-22 kgm/s
RELATIVISTIC ENERGY
SUMMARY OF MATH MODELS
 An electron is accelerated through
potential difference of 300,000 V.
U E  q  V
2  q  V
mE
v
KE F  KEi  q  V

a
v2
c2
v2
L = L0 1- 2
c
1

v
mE  v 2
 q  V
2
v = 3.25x108 m/s
19
Relativistic energy
L
v2
c2
L0

v2
c2
p   mv
 Any object with mass has rest energy:
Rest Energy
Rest Energy
E0 = mc2
Total Energy
E = mc2
Kinetic Energy
KE = mc2 - mc2
KE =
1
Rest energy of particles
A point like object of mass m has socalled rest energy because of its mass
mc2(
t0  t  
mv
p
1

2  1.6 x10 C  300,000V 
9.1131 kg
KE F  q  V
(difference between total
energy and rest energy)
t0 = t 1-
E0 = mc2
 Rest energy can be converted into other
forms of energy.
– 1)
 The rest energy of the Sun is being
slowly converted via nuclear fusion
reactions into internal energy.
 Speed restriction !
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Electron volt
WHITEBOARD
An electron volt (1 eV) is the increase in
kinetic energy of an electron when it moves
across a 1.0 V potential difference:
 The mass of an electron is 9.11 x 10−31 kg.
The mass of a proton is 1.67 x 10−27 kg.
Determine the electron and proton rest
energies in joules and in electron volts.
1 eV = 1.6x10-19 J
ELECTRON
WHITEBOARD
PROTON
E0 = 8.199x10-14 J
E0 = 1.503x10-10 J
E0 = 512437.5 eV
E0 = 9.394x108 eV
Energy for an electron crossing a
large potential difference:
Accelerating a particle
 On average, each year about 2 x 1010 J of
electric and chemical potential energy is
converted to cool and warm your home. If
rest energy could be converted for this
purpose, how much mass equivalent of
rest energy would be needed?
m = 2.22x10-7 kg
1/10 the mass of one of the hairs on
your head
WHITEBOARD
 An electron in a particle accelerator
accelerates through a potential difference
of 1x106 V. What is its final speed?
E0i  E0 f  KE f  Uq f
0  KE f  Uq f
KE f  Uq f

m  c2
1
1
v2
c2
v  c  1
mc 2  1  e  V f
e V f

1
 = 2.9515
1
2
DOPPLER
EFFECT
v = 2.82x108 m/s
v = 0.94c
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Youtube Links
THE DOPPLER EFFECT ALSO APPLIES TO
LIGHT!
If a source of light is moving toward an observer, the
light that the observer receives will have a higher
frequency and shorter wavelength than would normally
be received!
This is called blueshift. (Light is shifted toward the
blue end of
the spectrum – higher frequency)
If a source of light is moving away from an observer, the
received light will have a lower frequency and longer
wavelength than normal!
 http://www.youtube.com/watch?v=Y5KaeCZ_
AaY
This is called redshift. (Light is shifted toward the
red end of
the spectrum – lower frequency)
DOPPLER EFFECT
It is the apparent change in the frequency
of a wave caused by relative motion
between the source of the wave and the
observer
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Doppler effect for EM waves
WHITEBOARD
 Physicist Dr. R. Wood ran a red light while
driving his car and was pulled over by a police
officer. Dr. Wood explained that because he
was driving toward the red light, he actually
observed it as green due to the Doppler effect.
Dr. Wood was then given a very expensive
speeding ticket. Should Dr. Wood go to court to
argue the speeding ticket?
fo =5.4x1014 Hz
fs =4.5x1014 Hz
WHITEBOARD
(solution)
 v 
f o  f s  1  rel 
c 

Multiply by c
 v 
c  f o  c  f s  1  rel 
c 

vrel  f s  c  f o  c  f s
vrel  f s  c   f o  f s 
vrel 
c   fo  f s 
fs
c  f o  f s  c  vrel 
f

vrel  c   o  1
 fs 
c  f o  c  f s  vrel  f s
Vrel = 6x107 m/s
Hubble's law
Hubble's discovery of the expansion
of the universe
 At the beginning of
the 20th century, it
was believed that
the universe was
static.
 Hubble concluded
that the universe
is expanding
because light
emitted by distant
galaxies is shifted
toward longer
wavelengths.
General relativity
 The special theory of relativity allows us to
compare
measurements
made
by
observers in two reference frames that
move at constant velocity relative to each
other.
 Einstein was able to generalize this
invariance to all reference frames,
including
noninertial
(accelerated)
reference frames.
 The result is the general theory of
relativity.
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Principle of equivalence
An early testing experiment
 An object of large mass, such as the Sun,
causes space to curve more than does an
object of smaller mass, such as Earth.
 This curvature deflects light by a measurable
angle.
Another testing experiment
 The theory of general relativity solved another
problem that had plagued astronomers since the
early 1800s.
 The elliptical path of Mercury around the Sun
was known to slowly rotate, a phenomenon
called precession.
Gravitational waves and black holes
Gravitational time dilation and red
shift
 Objects do not just curve space around
them; they also alter the rates at which time
passes around them.
 The closer a point is to the massive object,
the more slowly time passes there.
 One consequence of gravitational time
dilation is gravitational redshift.
 If EM waves are emitted from a region
closer to a massive object and observed in
a region farther away, the observed
frequency will be lower than the emitted
frequency.
Gravitational waves
 Einstein showed that spacetime is curved by the
presence of mass and changes shape when that
mass moves.
 These changes can propagate as gravitational
waves that ripple through the vacuum at the
speed of light.
 General relativity also predicts the existence of
black holes.
https://www.youtube.com/watch?v=4GbWfNHtHRg
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