Chapter 38A -- Relativity

Chapter 38A - Relativity
A PowerPoint Presentation by
Paul E. Tippens, Professor of Physics
Southern Polytechnic State University
© 2007

Objectives: After completing this
module, you should be able to:
• State and discuss Einstein’s s two
postulates regarding special relativity.
• Demonstrate your understanding of time
dilation and apply it to physical
problems.
• Demonstrate and apply equations for
relativistic length, momentum, mass,
and energy.

Special Relativity
Einstein’s Special Theory of Relativity, , published
in 1905, was based on two postulates:
I. I. The laws of of physics are the same for all
frames of of reference moving at at a constant
velocity with respect to to each other.
II. The free space velocity of of light c is
is
constant for all observers, independent of
of
their state of of motion. ( (c (c = 3 x 10 8 m/s)

Rest and Motion
What do we mean when we say that an object
is at rest . . . or in motion? ? Is anything at rest?
We sometimes say that
man, computer, phone,
and desk are at rest.
We forget that the Earth
is also in motion.
What we really mean is that all are moving with
the same velocity. . We can only detect motion in
reference to something else.

No Preferred Frame of Reference
What is the velocity of
the bicyclist?
We cannot say without
a frame of reference.
West
Earth
25 m/s
East
10 m/s
Assume bike moves at 25 m/s,W relative to Earth
and that platform moves 10 m/s, E relative to Earth.
What is the velocity of the bike relative to platform?
Assume that the platform is the reference, then
look at relative motion of Earth and bike.

Reference for Motion (Cont.)
To find the velocity of the bike relative to
platform, , we must imagine that we are sitting
on the platform at rest (0(
0 m/s) ) relative to it.
We would see the Earth moving westward at
at
10 m/s and the bike moving west at at 35 m/s.
Earth as Reference
West
25 m/s
East
10 m/s
Platform as Reference
West
35 m/s
East
0 m/s
Earth
0 m/s
10 m/s

Platform as Reference
West
Frame of
Reference
Consider the velocities
for three different
frames of reference.
35 m/s
East
0 m/s
Earth as Reference
West
Earth
25 m/s
0
East
10 m/s
Bicycle as Reference
West
0 m/s
East
35 m/s
10 m/s
25 m/s

Constant Velocity of Light
Platform v = 30 m/s to right relative to ground.
c
c c
c
10 m/s
10 m/s
20 m/s
40 m/s
Velocities observed
inside car
Velocities observed
from outside car
The light from two flashlights and the two
balls travel in opposite directions. . The
observed velocities of the ball differ, but the
speed of light is independent of direction.

Velocity of Light (Cont.)
Platform moves 30 m/s to right relative to boy.
c
c
30
Each observer sees
m/s c = 3 x 10 8 m/s
10 m/s
10 m/s
m/s
Outside observer sees very
different velocities for balls.
The velocity of light is unaffected by relative
motion and is exactly equal to:
c = 2.99792458 x 10 8 m/s

Simultaneous Events
The judgment of simultaneous events is also a
matter of relativity. Consider observer O T sitting on
top of moving train while observer O E is on ground.
A
A T
A E
Not simultaneous
O T
O E
Simultaneous
B E
B
B T
At t = 0, lightning
strikes train and
ground at A and B.
Observer O E sees
lightning events A E &
B E as simultaneous.
Observer O T says event B T occurs before event A T
due to motion of train. Each observer is correct!

Time Measurements
Since our measurement
of time involves judg-
ments about simul-
taneous events, we can
see that time may also
be affected by relative
motion of observers.
In fact, Einstein's theory shows that observers
in in relative motion will judge times differently -
furthermore, each is is correct.

Relative Time
Consider cart moving with velocity v under a
mirrored ceiling. A light pulse travels to ceiling and
back in time t o for rider and in time t for watcher.
d
2d
c t
t o
Light path
for rider
0
2R
c t R R
d
x
t
Light path ct
vt
R
; x
for watcher 2 2

Relative Time (Cont.)
d
c
2d
t
0
t
c
2t
v
2t
d
R
t o
Light path
for rider
2 2
c v
2t
2t
d
2
Substitution of:
d
ct
2
0
t
2 2
c v 0
t
ct0
22
2t
21
tv c 2
2

Time Dilation Equation
Einstein’s Time
dilation Equation:
t
t
1
v
0
c
2 2
t = Relative time (Time measured in frame
moving relative to actual event).
t o = Proper time (Time measured in the
same frame as the event itself).
v = Relative velocity of two frames.
c = Free space velocity of light ( (c = 3 x 10 8 m/s).

Proper Time
The key to applying the time dilation equation is
to distinguish clearly between proper time t o
and relative time t. . Look at our example:
Event
d
Frame
t o
Proper
Time
Relative
Frame
t
Relative
Time
t > t o

Example 1: Ship A passes ship B with a
relative velocity of 0.8c (eighty percent of the
velocity of light). A woman aboard Ship B
takes 4 s to walk the length of her ship. What
time is recorded by the man in Ship A?
Proper time t o = 4 s
Find relative time t
t
t
1
v
0
c
2 2
A
B
v = 0.8c
4.00 s 4.00 s
t
2 2
t = 6.67 s
1- (0.8 c) / c 1- 0.64

The Twin Paradox
Two twins are
on Earth. One
leaves and
travels for 10
years at 0.9c.
When traveler
returns, he is 23
years older due
to time dilation!
t
t
1
v
0
c
2 2
Traveling twin
ages more!
Paradox: Since motion is
relative, isn’t t it just as true that
the man who stayed on earth
should be 23 years older?

The Twin Paradox Explained
The traveling twin’s
motion was not
uniform. Acceleration
and forces were
needed to go to and
return from space.
Traveling twin
ages more!
The traveler
ages more and
not the one who
stayed home.
This is NOT science fiction.
Atomic clocks placed aboard
aircraft sent around Earth and
back have verified the time
dilation.

Length Contraction
Since time is affected by
relative motion, length
will also be different:
L
L v c
0
1
2 2
L o is proper length
L is relative length
L o
L
0.9c
Moving objects are foreshortened due to to relativity.

Example 2: A meter stick moves at 0.9c
relative to an observer. What is the relative
length as seen by the observer?
L
L v c
0
1
2 2
L(1 m) 1 (0.9 c) / c
2 2
L (1 m) 10.81 0.436 m
L o
L = ?
1 m
0.9c
Length recorded by observer:
L = 43.6 cm
If the ground observer held a meter stick, the
same contraction would be seen from the ship.

Foreshortening of Objects
Note that it is the length in the direction of
relative motion that contracts and not the
dimensions perpendicular to the motion.
Assume each holds a
meter stick, in example.
If meter stick is 2 cm
wide, each will say the
other is only 0.87 cm
wide, but they will
agree on the length.
W o
1 m=1 m
0.9c
W

Relativistic Momentum
The basic conservation laws for momentum and
energy can not be violated due to relativity.
Newton’s s equation for momentum (mv)) must be
changed as follows to account for relativity:
Relativistic
momentum:
p
mv
0
1
v
c
2 2
m o is the proper mass, , often called the rest
mass. . Note that for large values of v, , this
equation reduces to Newton’s s equation.

Relativistic Mass
If momentum is to be conserved, the relativistic mass
m must be consistent with the following equation:
Relativistic
mass:
m
m
0
1
v
c
2 2
Note that as an object is accelerated by a
resultant force, its mass increases, which
requires even more force. This means that:
The speed of of light is is an ultimate speed!

Example 3: The rest mass of an electron is
9.1 x 10 -31
kg. . What is the relativistic mass if
its velocity is 0.8c ?
m
m
0
1
v
c
2 2
- 0.8c
m o = 9.1 x 10 -31
kg
m
-31 -31
9.1 x 10 kg 9.1 x 10 kg
1 (0.6 c)
c
2 2
0.36
m = 15.2 x 10 -31 kg
The mass has
increased by 67% !

Mass and Energy
Prior to the theory of relativity, scientists
considered mass and energy as separate
quantities, each of which must be conserved.
Now mass and energy must
be considered as the same
quantity. We may express
the mass of a baseball in
joules or its energy in
kilograms! ! The motion adds
to the mass-energy.

Total Relativistic Energy
The general formula for the relativistic total
energy involves the rest mass m o and the
relativistic momentum p = mv.
E ( m c ) p c
Total Energy, E
2 2 2
0
For a particle with zero
momentum p = 0:
For an EM wave, m 0 = 0,
and E simplifies to:
E = m o c 2
E = pc

Mass and Energy (Cont.)
The conversion factor between
mass m and energy E is:
E o = m o c 2
The zero subscript refers to proper or rest values.
A 1-kg 1
block on a table has an energy
E o and mass m o relative to table:
1 kg
E o = (1 kg)(3 x 10 8 m/s) 2
E o = 9 x 10 16 J
If the 1-kg 1
block is in relative motion, its kinetic
energy adds to the total energy.

Total Energy
According to Einstein's theory, the total energy
E of a particle of is given by:
Total Energy: E = mc 2
(m o c 2 + K)
Total energy includes rest energy and energy
of motion. If we are interested in just the
energy of motion, we must subtract m o c 2 .
Kinetic Energy: K = mc 2 – m o c 2
Kinetic Energy: K = (m – m o )c 2

Example 4: What is the kinetic energy of a
proton (m o = 1.67 x 10 -27
kg) traveling at 0.8c?
m
m
0
1
v
c
2 2
+ 0.7c
m o = 1.67 x 10 -27
kg
m
-27 -27
1.67 x 10 kg 1.67 x 10 kg
1 (0.7 c)
c
2 2
0.51
; m = 2.34 x 10 -27
kg
K = (m – m o )c 2 = (2.34 x 10 -27
kg – 1.67 x 10 -17
kg)c 2
Relativistic Kinetic Energy K = 6.02 x 10 -11 -11 J

Summary
Einstein’s Special Theory of Relativity, , published
in 1905, was based on two postulates:
I. I. The laws of of physics are the same for all
frames of of reference moving at at a constant
velocity with respect to to each other.
II. The free space velocity of of light c is
is
constant for all observers, independent of
of
their state of of motion. ( (c (c = 3 x 10 8 m/s)

Summary (Cont.)
Relativistic
time:
t
t
1
v
0
c
2 2
Relativistic
length:
L
L v c
0
1
2 2
Relativistic
mass:
m
m
0
1
v
c
2 2

Summary(Cont.)
Relativistic
momentum:
p
mv
0
1
v
c
2 2
Total energy: E = mc 2
Kinetic energy: K = (m – m o )c 2

CONCLUSION: Chapter 38A
Relativity