Classical Relativity
Galilean Transformations
y’
y
v
u’
P
x’
x
x = x’ + vt
x’ = x - vt
Divide equations by t
u = u’ + v
u’ = u - v
Example:
A train travels through a station at a constant speed of 8.0ms-1 . One observer
sits on the train and another sits on the platform. as they pass each other, they
start their stopwatches and take measurements of a dog on the train who is
running in the same direction the train is moving.
(a) The train observer measures the velocity of the dog to be 2.0ms-1 . What is
the velocity relative to the platform observer?
(b) After 5s how far has the dog moved according to the observer on the train?
(c) After 5s how far has the dog moved according to the observer on the
platform?
8.0ms-1
Calculate the time for each boat to make a round trip
50 meters out and back along the path shown.
50m
5.0ms-1
50m
5.0ms-1
3.0 ms-1
Calculate the time for each boat to make a round trip
50 meters out and back along the path shown.
50m
5.0ms-1
50m
5.0ms-1
Michelson-Morley Experiment
An experiment using an
interferometer to detect the
motion of the Earth through the
ether.
Ether Wind
When the entire apparatus is rotated a shift in the
interference pattern should occur.
Michelson-Morley Experiment
There was no shift in the interference pattern
showing the ether did not exist, that light
could travel through a vacuum, and its speed
is independent of the source motion.
Special Relativity
Inertial Reference Frame
• Reference frames in which Newton’s law are
valid.
• Reference frames with constant velocity.
• Special Relativity only deals with events in
inertial reference frames.
• General relativity deals with non-inertial
reference frames.
Maxwell’s electromagnetic equations
• Unified electricity and magnetism.
• Predicted the existence of electromagnetic waves.
• Gave the speed of electromagnetic waves as a
constant regardless of reference frame.
c
1
m
 3.0 10
s
 o o
8
μo = Vacuum permeability
εo = Vacuum permittivity
Einstein Postulates of Special Relativity
• The laws of Physics have the same form in all
inertial reference frames.
• Light propagates through empty space with a
definite speed c independent of the speed of
the source or the observer.
Classical Relativity
1,000,000 ms-1
■ How fast is Spaceship A approaching Spaceship B?
■ Both Spaceships see the other approaching at 2,000,000 ms-1.
■ This is Classical Relativity.
1,000,000 ms-1
Einstein’s Special Relativity
0 ms-1
300,000,000 ms-1
1,000,000 ms-1
n
n
Both spacemen measure the speed of the approaching ray of light.
How fast do they measure the speed of light to be?
Special Relativity
• Stationary man
– 300,000,000 ms-1
• Man travelling at 1,000,000 ms-1
– 301,000,000 ms-1?
– Wrong!
• The Speed of Light is
the same for all observers
Time Travel!
• Time between ‘ticks’ = distance / speed of light
• Light in the moving clock covers more distance…
– …but the speed of light is constant…
– …so the clock ticks slower!
V
• Moving clocks run more slowly!
The Lorentz factor
𝛾=
1
𝑣2
1− 2
𝑐
Time dilation
Δ𝑡 = 𝛾Δ𝑡𝑜
Δto = Proper time – The time measured by a
clock at rest relative to the event.
An observer sets up an experiment to measure
the time of oscillation of a mass suspended from
a vertical spring. He measures the time period
as 2.0s. To another observer this time period is
measured as 2.66s. Calculate the relative
velocity between the two observers.
Length Contraction
𝐿𝑜
L=
𝛾
The length measured by an observer who is at
rest relative to the object.
Mary is traveling in a space ship, which is not
accelerating. To her the space ship has a length
of 100m. To Paul who is traveling in another
space ship, which is also not accelerating.
Mary’s space ship has a length of 98m.
Calculate the relative velocity of Paul and Mary.
A spaceship is traveling away from the Earth with a speed
of 0.6c as measured by an observer on the Earth. The
spaceship sends a light pulse back to Earth every 10
minutes as measured by a clock on the space ship.
(a) Calculate the distance that the spaceship travels
between light pulses as measured by
i. the observer on Earth.
ii. somebody on the space ship.
(b) If the Earth observer measures the length of the
spaceship as 60m, determine the proper length of the
spaceship.
Muon Decay
Half life of 3.1E-6s as measured in a reference frame at
which they are at rest.
Muons are created in the upper atmosphere (10km) of the
Earth from cosmic ray bombardment. These muons have
very high velocities (0.98c)
Special Relativity
■ How fast is Spaceship A approaching Spaceship B?
𝑢𝑥′ = 𝑢𝑥 − 𝑣
𝑢′𝑥 = −.08𝑐 − 0.7𝑐
𝑢′𝑥 = −1.5𝑐
impossible
v = 0.7c
𝑢𝑥 − 𝑣
=
𝑢𝑥 𝑣
1− 2
𝑐
−.8𝑐 − .7𝑐
′
𝑢𝑥 =
−.8𝑐 .7𝑐
1−
𝑐2
𝑢𝑥′
𝑢𝑥′ = 0.96𝑐
ux = -0.8c
Relativistic Mass, Energy,
and Momentum
Relativistic Mass
The mass of a moving object is greater than
than the rest mass.
m  mo
m = relativistic mass - the mass measured moving relative
to the object.
mo = rest mass – the mass measured at rest relative to the
object
the Lorentz factor
 =
Classical Energy
In classical physics if a constant force is
applied to an object it experiences a
constant acceleration. The work done by the
force is transferred to kinetic energy.
v
c
1
2
E K  m ov
2
t

Relativistic Energy
According to special relativity no object can
exceed the speed of light (c). So the
acceleration of the object must decrease,
but where does the work done by the force
transfer to…….MASS
EK = total energy – rest energy
E K  mc  m oc
2
v
2
E K  m  m o c 2
c
E K  m o  m o c 2
E K   1m oc
t
2
Let’s look at the relativistic kinetic energy
equation at low velocities.
E K   1moc
EK 
moc
2
v2
1 2
c
2
 moc 2
 v 2  2
E K  1 2 moc  moc 2
 2c 
2 2
m
c
2
2
o v
E K  moc 

m
c
o
2c 2
1
2
E K  m ov
2
Binomial expansion when x is small
1 x   1 nx
n
1
2 2
 v 
v2
1 2   1 2
2c
 c 
Example: An electron is accelerated through a potential
difference of 1.0MV. Calculate its velocity
Classical Calculation:
Eo  W  E f
UE  E K
1 2
qV  mv
2

J 
21.6E 19C1.0E6 
2eV
m

C 
v

 5.9E8
m
9.11E  31
s
IMPOSSIBLE
Example: An electron is accelerated through a potential
difference of 1.0MV. Calculate its velocity
Relativistic Calculation
Eo  W  E f
UE  E K
qV   1m oc 2
J
m 2
(1.6E 19C)(1.0E6 )  ( 1)(9.11E  31kg)(3.0E8 )
C
s
1.6E 13J  ( 1)(8.2E 14J)
1.95  ( 1)
  2.95

1
2
v
1 2
c
1
2.95 
2
v
2  .88
c
2
2
v  .88c
v  .94c
v2
1 2
c
2
v
.34  1  2
c
v2
.12  1  2
c

Lets work the same example using different units and
working in terms of c
UE  E K
qV   1moc
2
e(1.0MV )   1.511MeVc
1.0MeV   1(.511MeV )
( 1)  1.96
  2.96
2
c
2

1
v2
1 2
c
1
v2
2 1 2

c
2
v
1
2 1 2
c

1
1
v  c 1 2  c 1
2  .94c

2.96
Example: Calculate the pd necessary to accelerate an
electron to a velocity of 0.8c
The classical relation between energy and momentum
1 2
E K  mv
2
p  mv
2
p
EK 
2m

mv
2

2 2
mv
1 2
EK 

 mv
2m
2m
2
The relativistic relation between energy and momentum:
m  m o
1
m
p  mv
mo
v2
1 2
c
2
m
m2  o 2
v
1 2
c
2
mv
m 2  2  m o 2
c
m 2c 2  mv 2  m oc 2
p m v
2
2 2
E  mc 2
E 2  m 2c 4
m 2c 2  m 2v 2  moc 2
m 2c 2  m 2v 2  moc 2
c 2 m 2c 2  m 2v 2  moc 2 
m 2c 4  (m 2v 2 )c 2  moc 4
E 2  p 2c 2  moc 4

Example: What is the momentum of an electron with a
kinetic energy of 1.0MeV?
Total energy = rest energy + kinetic energy
E  .511MeV 1.0MeV  1.5MeV
E 2  p 2c 2  m o 2c 4
1.5MeV   p c  .511MeVc
2
2 2
2.25MeV 2  p 2c 2  .26MeV 2
1.99MeV 2  p 2c 2
2
1.99MeV
p2 
c2
p  1.41MeVc 1
c
2 2
4
Example: What is the speed of an electron with a
momentum of 2.0MeVc-1?