Chapter 28 – Special Relativity
1. A physical activity that takes place at a definite point
in space and time is called:
A. a sport.
B. an event.
C. a happening.
D. a locale.
2. Which of the following was not discussed in the first
4 sections of Ch 28?
A.
B.
C.
D.
Wave-particle duality
Proper time interval
Time dilation
Length contraction
What’s special about special relativity?
• In 1905, Einstein’s first paper
on relativity dealt only with
inertial reference frames
(constant velocity).
• 10 years later, he published a
more encompassing theory of
relativity that considered
accelerated motion and it’s
connection to gravity. This
was a discussion of “general”
relativity.
• His earlier work was special in
that it was more specific
28.1 Events and Inertial Reference Frames
An event is a physical “happening” that occurs at a certain place and
time.
To record the event, each observer uses a reference frame that consists of a
coordinate system (space) and a clock (time).
An inertial reference frame is one in which Newton’s law of inertia is
valid. An inertial reference frame can be moving, but it cannot be accelerating. In
spite of its centripetal acceleration, the Earth is considered an inertial reference
frame, because the effects of its rotation and orbit are relatively minor.
Reference Frames
•Extend infinitely far in all
directions. You can be thousands
of miles away, yet still in the same
reference frame
•Observers are at rest in their
reference frames
•A reference frame is not the same
as a point of view. Therefore all
observers at rest relative to each
other share the same reference
frame and will view time and length
the same way.
3.4 Relative Velocity without relativity
One example of an inertial reference frame is
on the ground. This observer is at rest in his
reference frame.
The train traveling at a constant speed
is another frame of reference.
Anybody sitting down in the train is an
observer at rest in his reference frame.
3.4 Relative Velocity
Let’s say the train is traveling at 9m/s relative
to the ground: vTG = +9 m/s, where +
indicates to the right.
An observer at rest in the trains sees
the man on the ground traveling at
vGT = -9m/s, where – indicates to the
left.
Is one of these reference frames more
“real” than the other?
3.4 Relative Velocity (See Chapter 3)
A passenger inside the train starts to walk up to the bar car in
the front of the moving train. Observers in their seats note that
he is traveling at vPT = 2 m/s.
What is his velocity (v PG ) according to the ground-based
observer?
A. + 2m/s, B. +7 m/s C. +9 m/s D. +11m/s
3.4 Relative Velocity
Same situation, but now imagine the passenger moving to the
rear of the train (that’s left). What is his speed vPG now?
A. + 2m/s, B. +7 m/s C. +9 m/s D. -7 m/s
3.4 Relative Velocity
Same situation, but now imagine the passenger moving to the
rear of the train (that’s left). What is his speed vPG now?
Answers: when going towards the front, the ground-based
observer sees him going at 11 m/s to the right.
When going towards the rear of the train, it’s 7 m/s, still to the
right.



v PG  v PT  v TG
Ocean waves are approaching the beach at
10 m/s (vWG ). A boat heading out to sea
travels at 6 m/s (vBG ). How fast (speed only,
not direction) are the waves moving in the
boat’s reference frame (vWB )?
A. 4 m/s
B. 6 m/s
C.16 m/s
D.10 m/s
Ocean waves are approaching the beach at
10 m/s (vWG ). A boat heading out to sea
travels at 6 m/s (vBG ). How fast (speed only,
not direction) are the waves moving in the
boat’s reference frame (vWB )?
4 m/s
B. 6 m/s
C. 16 m/s
D. 10 m/s
A.
Using the same logic as with the train:
VWG = VWB + VBG
and therefore:
VWB = VWG – VBG where these must be
added/subtracted as vectors.
The event and the observation of
the event do not always occur at
the same time
Mr. A, at rest in the reference frame, stands at the origin,
looking in a positive direction. Mr. B, at rest in the same
reference frame, stands at x = 900 m, looking in a negative
direction. A firecracker explodes somewhere between
them. Mr. B sees the light flash at t = 3µs. Mr. A sees the
flash at t = 4µs. Light travels at 300 m/µs. In this reference
frame:
a. Where did the explosion happen?
b. When did the explosion happen?
a. Where did the explosion happen?
- Light travels at 300 m/µs. It took 1µs longer for the light to
get to Mr. A, implying Mr. A’s position is 300 m farther
away from the explosion that that of Mr. B. If x is the
position of the explosion:
(x – 0m) = (900m – x) + 300m, so x = 600m
b. When did the explosion happen?
- It took 2 µs for the light to go from the explosion to Mr. A at
600 m from the explosion. So t = (4-2) µs = 2 µs.
Mr. A and Mr. B observed the event at
different times, but after some calculations,
they can agree upon the time and place it
occurred.
Simultaneity
An experimenter in an inertial reference
frame stands at the origin looking in the
positive x direction. At t = 3.0 µs she sees
firecracker 1 explode at x = 600 m. At t =
5.0 µs, she sees firecracker 2 explode at x
= 1200m. Light travels at 300 m/ µs. Are
the two explosions simultaneous? If not,
which one happens first?
Simultaneity
An experimenter in an inertial reference frame
stands at the origin looking in the positive x
direction. At t = 3.0 µs she sees firecracker 1
explode at x = 600 m. At t = 5.0 µs, she sees
firecracker 2 explode at x = 1200m. Light travels
at 300 m/ µs. Are the two explosions
simultaneous? If not, which one happens first?
Ans: We see that the first explosion occurred at t
= I µs, and then took 2 µs to travel to the
experimenter. The light from the second
firecracker took 1200 m/300 µs/s or 4 µs to
travel. It too occurred at t = 1 µs. The events
therefore are simultaneous.
28.2 The Postulates of Special Relativity
THE POSTULATES OF SPECIAL RELATIVITY
1. The Relativity Postulate. The laws of physics are the same
in every inertial reference frame. All inertial reference frames are
equally valid.
2. The Speed of Light Postulate. The speed of light in a vacuum,
measured in any inertial reference frame, always has the same value
of c, no matter how fast the source of light and the observer are
moving relative to one another.
Number 2 seems to contradict our idea of inertial reference frames.
Why should moving light be different from a moving train or a boat?
Einstein’s Principle of Relativity
• Maxwell’s equations are true in all inertial
reference frames.
• Maxwell’s equations predict that electromagnetic
waves, including light, travel at speed c = 3.00
× 108 m/s.
• Therefore, light travels at speed c in all
inertial reference frames.
Every experiment has found that light travels at 3.00
× 108 m/s in every inertial reference frame,
regardless of how the reference frames are moving
with respect to each other.
28.3 The Relativity of Time: Time Dilation
TIME DILATION
•If you are in the
same reference frame
as the light clock,
• Δt0 = 2D/c where D
is the distance
between mirror and
receiver, and c is the
speed of light.
28.3 The Relativity of Time: Time Dilation
An observer on the earth sees the light pulse travel a greater distance
between ticks (2s as opposed to 2D). But she still measures the speed of
light in the spaceship as c (postulate #2).
Therefore, her clock reads a Δt > Δt0
28.3 The Relativity of Time: Time Dilation
This is the time interval the earth-based observer
would read. Since v< c, the denominator is less than
1, so Δt (Earth-based observer time interval) is be
greater than Δt0 (astronaut observer time interval). It
can be shown that:
t 
to
1 v2 c2
28.3 The Relativity of Time: Time Dilation
PROPER TIME INTERVAL
The time interval between two events that occur at the same position
is called the proper time interval
In general, the proper time interval between events is the time
interval measured by an observer who is at rest relative to the
events.
In the light clock example, the proper time interval was measured by
the astronaut, because from his reference frame, both events (light
leaving source, light hitting detector) happened at the same position.
leaving but not by the earth-bound observer.
t 
Proper time interval =
to
1 v2 c2
to
Dilated time interval =
t
From the Sun to Saturn
Who measures the proper time interval?
From the Sun to Saturn
From the Sun to Saturn
This implies it is the astronaut who measures the
proper time interval, Δt0 .
From the Sun to Saturn
From the Sun to Saturn
From the Sun to Saturn
t 
to
1 v2 c2
to find the proper time interval,Δt0
t0  t 1  v 2 c 2
Δt0 = 2310 s
From the Sun to Saturn
t0  t 1  v 2 c 2
to find the proper time interval, Δt0
Δt0 = 2310s
From the Sun to Saturn
A tree and a pole are 3000 m apart. Each is
suddenly hit by a bolt of lightning. Mark, who is
standing at rest midway between the two, sees the
two lightning bolts at the same instant of time.
Nancy is at rest under the tree. Define event 1 to be
“lightning strikes tree” and event 2 to be “lightning
strikes pole.” In Nancy’s frame of reference, does
event 1 occur before, after or at the same time as
event 2?
A. before event 2
B. after event 2
C. at the same time as event 2
A tree and a pole are 3000 m apart. Each is
suddenly hit by a bolt of lightning. Mark, who is
standing at rest midway between the two, sees the
two lightning bolts at the same instant of time.
Nancy is at rest under the tree. Define event 1 to be
“lightning strikes tree” and event 2 to be “lightning
strikes pole.” For Nancy, does event 1 occur before,
after or at the same time as event 2?
A. before event 2
B. after event 2
C. at the same time as event 2
Nancy sees event 1 before event 2, but she would agree
they were simultaneous since she and Mark are in the
same reference frame.
28.4 The Relativity of Length: Length Contraction
The shortening of the distance between two points is one
example of a phenomenon known as length contraction.
Length contraction:
L0 is the proper length,
the length between 2
points measured by an
observer at rest with
respect to them.
Note that the observer
who experiences the
proper time interval, is
not the one who
measures the proper
length
v2
L  Lo 1  2
c
28.4 The Relativity of Length: Length Contraction
Example 4 The Contraction of a Spacecraft
An astronaut, using a meter stick that is at rest relative to a cylindrical
spacecraft, measures the length and diameter to be 82 m and 21 m
respectively. The spacecraft moves with a constant speed of 0.95c
relative to the earth. What are the dimensions of the spacecraft,
as measured by an observer on earth.
28.4 The Relativity of Length: Length Contraction
For this problem, the earthbound observer would determine the
distance to Alpha Centaur to be the proper length, L0, as shown in the
picture, and the observer in the spaceship would see the contracted
length.
However, it is the astronaut that sees the proper length of the
spaceship she is traveling in, while experimenters will measure it at
the contracted length.
v2
2
L  Lo 1  2  82 m  1  0.95c c   26 m
c
The distance from the sun to
Saturn
The distance from the sun to
Saturn
v2
L  Lo 1  2
c
The distance from the sun to
Saturn
v2
L  Lo 1  2
c
L = 0.62 x 1012 m
Beth and Charles are at
rest relative to each
other. Anjay runs past at
velocity v while holding a
long pole parallel to his
motion. Anjay, Beth, and
Charles each measure
the length of the pole at
the instant Anjay passes
Beth. Rank in order,
from largest to smallest,
the three lengths LA, LB,
and LC.
A.
B.
C.
D.
E.
LA = LB = LC
LB = LC > LA
LA > LB = LC
LA > LB > LC
LB > LC > LA
Beth and Charles are at
rest relative to each
other. Anjay running at
close to the speed of light,
runs past at velocity v
while holding a long pole
parallel to his motion.
Anjay, Beth, and Charles
each measure the length
of the pole at the instant
Anjay passes Beth. Rank
in order, from largest to
smallest, the three
lengths LA, LB, and LC.
A.
B.
C.
D.
E.
LA = LB = LC
LB = LC > LA
LA > LB = LC
LA > LB > LC
LB > LC > LA
Simultaneity
An experimenter in an inertial reference
frame stands at the origin looking in the
positive x direction. At t = 3.0 µs she sees
firecracker 1 explode at x = 600 m. At t =
5.0 µs, she sees firecracker 2 explode at x
= 1200m. Light travels at 300 m/ µs. Are
the two explosions simultaneous? If not,
which one happens first?
Simultaneity
An experimenter in an inertial reference frame stands at the
origin looking in the positive x direction. At t = 3.0 µs she
sees firecracker 1 explode at x = 600 m. At t = 5.0 µs,
she sees firecracker 2 explode at x = 1200m. Light
travels at 300 m/ µs. Are the two explosions
simultaneous? If not, which one happens first?
Ans: We see that the first explosion occurred at t = I µs,
and then took 600 m/300 µs/m or 2 µs to travel to the
experimenter. The light from the second firecracker took
1200 m/300 µs/m or 4 µs to travel. It too occurred at t =
1 µs. The events therefore are simultaneous. The
experimenter would agree to that, even though she sees
them at different times.
Ryan (stationary) and Peggy (moving with
the fireworks) do relativity
Ryan and Peggy are both
half-way between the
firecrackers, which are
attached to the train.
The signal box on the train
has 2 light detectors.
• If the light flash is seen by
the right detector first, the
signal light will turn green.
• If the light flash is seen
simultaneously, it will turn
red.
Ryan’s frame of reference
• Ryan observed the
firecrackers go off
simultaneously
• He was halfway between
the firecrackers
• The light traveled at equal
speeds from the different
directions.
• He concludes that
firecrackers exploded
simultaneously.
• Note that although light waves reach Ryan
simultaneously, he would think that since Peggy and the
signal box move right, the light wave from the right
explosion is detected before the light wave from the left.
• This would mean that the signal box light turns green.
Peggy’s frame of reference
• Are the explosions
simultaneous in Peggy’s
frame of reference?
• If so, since explosions
occurred equidistant from
her, the light would travel at
equal speeds from the
different directions, so both
flashes would reach her at
the same time.
• Signal box light would turn
red.
• But we already saw the light
go green!
Sequence of events in Peggy’s reference frame
• The signal light is seen as
green in both reference
frames and so must show the
same color.
• For the light to be green in
Peggy’s reference frame, the
right firecracker has to go off
before the left one.
• This is the only way that light
coming from equally distant
points would arrive at
different times.
• Signal box light would turn
green, which agrees with
Ryan.
A tree and a pole are 3000 m apart. Each is
suddenly hit by a bolt of lightning. Mark, who is
standing at rest midway between the two, sees the
two lightning bolts at the same instant of time.
Nancy is flying her rocket at v = 0.5c in the
direction from the tree toward the pole. The
lightning hits the tree just as she passes by it.
Define event 1 to be “lightning strikes tree” and
event 2 to be “lightning strikes pole.” For Nancy,
does event 1 occur before, after or at the same
time as event 2?
A. before event 2
B. after event 2
C. at the same time as event 2
A tree and a pole are 3000 m apart. Each is suddenly hit by a
bolt of lightning. Mark, who is standing at rest midway
between the two, sees the two lightning bolts at the same
instant of time. Nancy is flying her rocket at v = 0.5c in the
direction from the tree toward the pole. The lightning hits the
tree just as she passes by it. Define event 1 to be “lightning
strikes tree” and event 2 to be “lightning strikes pole.” For
Nancy, does event 1 occur before, after or at the same time
as event 2?
A. before event 2
B. after event 2
C. at the same time as event 2
If Nancy had been midway between the tree and pole, this would
have been equivalent to Peggy and Ryan. But events occur in
the same order for all observers in the reference frame. Nancy’s
time frame for the two events is shorter
Conservation of momentum
• The Newtonian
momentum of an object
is defined as the product
of its mass and velocity
(mv).
• Conservation of
momentum of a system
of objects before and
after they interact, is a
law of physics that is
valid in all inertial
reference frames.
28.5 Relativistic Momentum
When the speed of an
object is close to c, the
effects of relativity must be
taken into account.
The calculation for
relativistic momentum is:
mv
p
1 v2 c2
The graph shows the effects
of relativity are not
significant until the objects
moves at some fraction of c
The relativistic momentum
is always larger than its
non-relativistic counterpart.
28.5 Relativistic Momentum
Often it is convenient to ask
for the ratio of relativistic to
non-relativistic momenta for
an object. This number can
be thought of as the
“relativistic factor” and can
be calculated as:
p
1

mv
1 v2 c2
This number is always
larger than one!
A collision of an electron with a
target in a particle
accelerator produces a
muon that moves forward
with a speed of 0.95c
relative to the laboratory.
The muon’s mass is 1.90 x
10-28 kg.
Determine the “relativity
factor”, the factor by which
the relativistic momentum is
greater than the classical
momentum.
A collision of an electron with a
target in a particle
accelerator produces a
muon that moves forward
with a speed of 0.95c
relative to the laboratory.
The muon’s mass is 1.90 x
10-28 kg. The relativistic
factor is:
p
1

mv
1 v2 c2
= 3.20
The relativistic momentum is
3.20 times that of the nonrelativistic momentum.
28.6 The Equivalence of Mass and Energy
In a short addendum to his original
paper, Einstein showed that the total
energy, and not only the kinetic
energy, of an object was dependent
on its speed and mass:
If the object is at rest relative to its
reference frame, the denominator
becomes 1 and we get the the most
famous equation in the world:
The ratio of E/E0 is the relativistic
factor:
E
mc2
1 v2 c2
Eo  mc 2
Relativistic Kinetic Energy
• It can be shown,
by use of a
binomial
expansion of the
square root term
that when v<<c,
KE = ½ mv2.
KE  E  Eo


1
KE  mc 
 1
 1 v2 c2



2
Kinetic energy and total
energy
Kinetic energy and total
energy
Kinetic energy and total
energy
Note the difference between the values for resting energy
and kinetic energy for an object moving at non-relativistic
speeds.
Kinetic energy and total
energy
For the electron, m = 9.11 x 10-31 kg, start by calculating the
relativistic factor E/E0:
E
1

E0
1 v2 c2
EXAMPLE 37.12 Kinetic
energy and total energy
28.6 The Equivalence of Mass and Energy
Example 8 The Sun is Losing Mass
The sun radiates electromagnetic energy at a rate of 3.92x1026W.
What is the change in the sun’s mass during each second that it
is radiating energy? What fraction of the sun’s mass is lost during
a human lifetime of 75 years.
28.6 The Equivalence of Mass and Energy


Eo 3.92 1026 J s 1.0 s 
m  2 
 4.36 109 kg
2
c
3.00 108 m s





m
4.36 109 kg s 3.16 107 s
12


5
.
0

10
msun
1.99 1030 kg