Special Relativity
Aims
 To understand that frames of reference are central to special relativity
 To understand the phenomena of length contraction and time dilation
 To undertake group work and practice presentation skills
In this experimental tutorial you will be divided into three groups. Each group will
undertake a tutorial question and will then present and explain their results to the
other groups. You will be marked as a group on your answers to the questions below,
and on your contribution to your group and the presentation to the other students.
Discretionary marks will be awarded to students that made particularly noteworthy
contributions.
You have 25 minutes to perform the tutorial question and to prepare your
presentation.
Group 1: Particle Lifetime
The B- meson is a type of particle consisting of a bottom quark and an anti-up quark.
This meson is unstable and quickly decays into other particles. Figure 1 is an event
display picture showing the production and decay of this particle as observed at the
ALEPH experiment on the Large Electron Positron (LEP) collider at the European
centre for particle physics (CERN).
The LEP collider was an accelerator that collided electrons and their anti-matter
equivalents positrons. The energy of the electron and positron beams was equal. The
event display picture is in the plane transverse to that of the incident electrons and
positron beams.
In the event shown two B-mesons were produced at the point marked IP. There were
no other products in the reaction. One of the B mesons (a B-) was cleanly
reconstructed in the detector and can be seen in Figure 1 to travel along the X axis
decaying at the point marked B-.
1.1 What is the energy of the B- meson in Figure 1 in GeV?
1.2 Using the world average value for the mass of the B meson, find its velocity.
1.3 What is the lifetime of this particle as observed in the ALEPH detector ?
1.4 What is the lifetime of the particle in its rest frame ? Compare your answer with
the world average value for this particle.
1.5 Produce a presentation (max 10 mins including discussion) to explain the relevant
concepts to your peers.
Figure 1: Production and decay of a B- meson observed at the ALEPH experiment at
LEP
Hints:
 The centre of mass energy (ECM) of the accelerator is given on the figure.
The units are GeV. The centre-of-mass energy is the total energy of the
electron-positron system before the collision.
 Remember E  mc 2
 The display shows only the X-Y plane.
 The web site of the particle data group, who produce the world average values
for the masses and lifetimes of particles, is at http://pdg.lbl.gov/ (the link is
included on moodle).
 The web site of CERN is at http://www.cern.ch/ (the link is included on
moodle).
 If you wish to include the event picture in your presentation it can be found on
the P1X moodle site.
Group 2: The Relativistic Police Car
A new police car has been developed, with a relativistic top speed, and is being tested
on Great Western road in Glasgow. The warning light flashes once every second
when the car is stationary.
The police car is being driven by P.C. Amy, when in the distance she notices her
colleague P.C. Bob driving the exact same car directly towards her at speed v relative
to her frame of reference.
v
2.1
What is the time between the flashes of light being emitted by Bob in Amy’s
frame of reference. What is this phenomena known as ? (Note this is not the
time between the light flashes being observed by Amy, we will discuss that
below.)
2.2
If the warning light flashes are emitted by Bob every 2 seconds in Amy’s
frame then what speed must the car be travelling at, relative to Amy’s frame of
reference?
Consider Figure 2 and the distance, d, travelled by Bob in Amy’s frame between
consecutive flashes from his warning light. The new distance that the light pulse will
have to travel to reach Amy is less than before, as Bob has moved closer to Amy
between the flashes.
y’
y
d
v
O’
O
Bob’s frame
Amy’s frame
x’
x
Figure 2
2.3
Show that the time T between Amy observing the flashes is:
T
cv
T0
cv
where T0 is the time between flashes in the rest frame of the car.
This is the relativistic Doppler shift. Note, that T is now shorter than To, whereas
time dilation alone would make the time longer.
Hints:
 Say, one flash is emitted at T=0 when O’ coincides with location A in Amy’s
frame.
 The relation you found in 2.1 tells you how long it will be in Amy’s frame
before the next flash is emitted.
 The time between flashes observed by Amy needs to be reduced. The light
travels a distance d shorter to Amy than before, due to Bob’s movement in
Amy’s frame.
2.4
Hence, find the frequency of the flashes (not the frequency of the light).
Hints:

1
 v 
1  2 
 c 
2

1
 v
1  
 c

1
 v
1  
 c
This same story can be told from the perspective of Bob as he drives along Great
Western road.
He notices Amy off in the distance driving directly towards him at a speed v2 relative
to his frame of reference and would observe the same time between flashes coming
from Amy’s warning light as shown in 2.3.
Amy and Bob stop before crashing into each other and park at the side of the road.
They notice that both warning lights emit the same red light but are both convinced
that they saw blue flashes.
2.5
Is it possible to determine who is correct in their observation and explain
your reasoning?
2.6
Produce a presentation (max 10 mins including discussion) to explain the
relevant concepts to your peers.
Group 3: The pole and the barn
A pole-vaulter runs into a barn. The barn doors are open on both ends, so he just runs
right through. His friend is standing by the barn, stationary with respect to the barn.
Afterwards the friend comments that the pole-vaulter’s pole at one instant just fitted
totally inside the barn.
“No it didn’t” says the pole-vaulter, “at no time was my pole entirely in the barn!”
To settle their dispute, the two friends measure the pole when it is at rest and find that
it is l0 long. The barn is similarly a length L0 when measured by a stationary observer.
If the pole-vaulter was running at a speed v (and he is a very very fast runner, so v is
not too much less than the speed of light c), what is the length of the pole and the
distance between the barn doors while the pole-vaulter is running through, as seen by
3.1.a) the pole-vaulter?
3.1.b) the friend stationary with respect to the barn?
What are the relationships between the pole length and the barn length for both
observations to be true ?
3.2.a) State the relationship for the pole-vaulter’s claim than the pole is longer than
the barn in his frame of reference
3.2.b) State the equality for the friend’s claim that the pole just fits inside the barn.
3.3. a) Draw a sketch graph in the friend’s reference frame with distance along the xaxis and time along the y-axis. Mark the position of the barn doors. Now draw the
position of the front of the pole versus time and the position of the back of the pole
versus time.
Hints:
 Remember the pole just fits inside the barn at one instant.
 Distance / time = velocity !
3.3 b) Now draw a similar graph for the pole-vaulter’s reference frame.
Hints:
 The pole is stationary in the pole vaulter’s frame.

Remember, the pole doesn’t fit inside the barn !
3.4. Think about what would happen in the following scenario. The friend enlists the
help of another spectator to stand at the barn’s exit door, while he stands at the barn’s
entrance door. The barn’s exit door is initially closed and the entrance door open.
Since he and his accomplice are in the same reference frame, they agree that the pole
just fits inside the barn. When the pole is entirely inside the barn they open/close their
doors at the same time to allow the pole through i.e. simultaneously the exit door is
opened to let the pole leave the barn and the entrance door closed. By considering the
graph you drew in part 3b) explain what the pole-vaulter sees?
3.5. Produce a presentation (max 10 mins including discussion) to explain the relevant
concepts to your peers.
Further work
The following questions are related to the topic covered by this experimental tutorial.
 Exercise book questions D83-D85
 Mastering Physics Dynamics 7: Relativity.
“The lifetime of the speeding muon” is particularly relevant to the particle lifetime
question of group 1.
“When time flies… it runs more slowly” is particularly relevant to the relativistic
Police Car question of group 2.
“The space-time interval” are particularly relevant to the pole and barn question of
group 3.
Demonstrators' Answers, Hints, Marking Scheme and Equipment List.
The student’s are likely to find these difficult – you will need to brush up on your
relativity and give them lots of help. The three problems are in order of difficulty, so
get group 1 to present before group 2, before group 3.
Group 1 – Particle Lifetime:
1.1
ECM=89.44 GeV from Figure 1.
Momentum is conserved and since initial electron and positron momenta are equal
and opposite, initial and final momenta is 0.
Hence, each B meson carries ECM/2 of total energy=44.72 GeV.
1.2
http://pdg.lbl.gov/ -> particle listings -> mesons -> Bottom mesons ->
B± mass 5.28 GeV/c2
E  mc 2
  44.72 / 5.28  8.47
1
 
v
1  
c
2
2
 1 
v

  1  
 8.47 
c
v  0.993c
2
99% of speed of light, so highly relativistic
1.3
Velocity=distance/time
Distance from picture = 3.2mm
Lifetime in lab frame = 3.2x10-3/0.993c=1.07 x 10-11 s
This is calculated using only the XY projection shown, we do not have the image of
the RZ projection. Hence this is a minimum value for the lifetime of this particle. For
example, if the angle of the particle travel was at 45 degrees to this plane the distance
in the RZ projection would be the same and the actual lifetime would be sqrt(2)
longer, 1.52 x 10-11 s .
1.4
Time dilation
Proper time (lifetime of particle in its rest frame) =  lifetime of particle in lab frame
Hence proper time = 1.07 x 10-11 s / 8.47 = 1.26 x 10-12 s
This value is a minimum because of the XY projection effect discussed above.
If the experiment is repeated many times a range of values will be expected as, like in
radioactive atom decay, a range of lifetimes is expected.
The world average value for the mean lifetime is
http://pdg.lbl.gov/ -> particle listings -> mesons -> Bottom mesons ->
B± lifetime 1.64x10-12 s
These values are in reasonable agreement, the relativistic effect is clearly needed to
explain the observed lifetime in the lab frame.
Possible questions for presentation –
1.1 how much of this energy is in the K.E. of the particle ?
This energy is mostly in the (relativistic) kinetic energy of the particle which is
(-1)mc2
and partially in its rest mass which is mc2
1.2 what is the difference between applying the non-relativistic equation for
momentum and the correct relativistic equation in this frame of reference ?
Relativistic p  mv
1.3 how were the tracks in the image reconstructed ? (not covered in lectures yet, so
get them to speculate)
Charged tracks leave hits in detectors (most important for this is silicon vertex
detector) surrounding the collision point. These hits are joined together by a join-thedots style software algorithm and the particle trajectories reconstructed from them.
The image shows these reconstructed trajectories from the particles that were
reconstructed. The B meson is inferred from the gap between the interaction point of
the electron and positron and the point from which the decay particles were
reconstructed.
1.4 Explain this situation from the point of view of an observer travelling along with
the B meson.
This observer would measure the proper lifetime of the B meson but would see the
meson travel a shorter distance by a factor of  due to length contraction.
Marks
Mark questions communally for each group. Presentation marks can be given
communally or to individuals as appropriate. Award discretionary marks to those that
have made particular notable contributions.
Section 1
1.1 group
1.2 group
1.3 group
1.4 group
1.5 group/individual
Mark
1
1
1
2
3
Discretionary marks individual
TOTAL
2
10
Group 2 – Relativistic Police Car
2.1 time dilation
so:
2.2
T  T0
T0 = 1 gives T = 
2
1
2 
1
v2
c2
v2 1

c2 2
v2 1
1  2 
4
c
2
3 v
  2
4 c
 v  0.87c
 1
2.3
Firstly:
d  vt  vT0
Next:
T t
Since:

we find:
T
therefore
T
d
v
 v
 T0  T0   1  T0
c
c
 c
1
 v2 
1  2 
 c 
1
 v
1  
 c
 v
1  
 c
 v
1  
 c

1

 v
1  
 c

1
 v
1  
 c
 v
 1    T0
 v  c
1  
 c
T0 
1
cv
cv
T0
2.4
Knowing:
f 
therefore:
f 
1
T
c  v 
c  v 
f0
2.5
They are both correct. Bob was moving in Amy’s frame of reference. Amy
was moving in Bob’s frame of reference.
Bob and Amy both saw the frequency of the flashes increase. The same would
be true of the frequency of the light itself, the frequency of the light would increase.
Hence the light could become blue, as blue light (c/400nm) has a higher frequency
than red light (c/700nm).
Marks
Mark questions communally for each group. Presentation marks can be given
communally or to individuals as appropriate. Award discretionary marks to those that
have made particular notable contributions.
Section 1
2.1
2.2
2.3
2.4
2.5
2.6 group/individual
Discretionary marks individual
TOTAL
Mark
1
1
1
1
1
3
2
10
Possible questions for presentation
2.1
What is the time between flashes for the driver of the car?
Answer: T0 = 1 second
What would the answer be in Newtonian mechanics without relativity for the
driver?
Answer: T = T0 = 1 second
What would the answer be in Newtonian mechanics without relativity for the
observer Calum?
Answer: still T = T0 = 1 second
2.3 / 2.4 What if the cars were travelling away from each other at speed v?
Answer: reverse the sign of the velocities
T
f 
cv
cv
cv
cv
T0
f0
2.5 The velocity of stars can be measured using a technique known as red-shift - can
you now explain how this works ? Stars are receding from the earth, light is shifted
towards lower frequency (higher wavelength), hence red-shift.
Group 3 – The Pole and the Barn
3.1
a) The length of the pole, as measured by the pole-vaulter is l0 since it is stationary
with respect to him. The distance between the barn doors is length contracted to
b) The barn is stationary with respect to the friend, so he sees that it is a length L0
while the pole appears length contracted to
with  as before.
3.2 The pole-vaulter’s claim that he never sees the pole entirely in the barn is true if
while the friend sees the pole just entirely in the barn if
1

l0  L0
3.3 Friend’s frame:
(pole)
Pole front
Time pole just
fits inside barn
Barn back
(barn)
Pole-vaulter’s frame:
t2 Front of
barn and back
of pole
coincide
t1 back of barn
and front of
pole coincide
Pole front
Barn back
(barn)
3.4 The paradox is resolved by realising that simultaneity is relative.
Friend’s frame - The door-closers open/shut the doors simultaneously. This is seen in
figure 3a), the doors are opened/closed at the time the pole just fits inside the barn.
The front of the pole reaches the barn back door and the door is opened to let it pass
through. The back door of the barn is closed at the same time just as the pole goes into
the barn.
Pole Vaulter’s frame – Friend 1 closes the exit door of the barn at time t1, when the
front of the pole reaches the back of the barn. At this time in the pole vaulter’s frame
the back of the pole is still outside the barn. Friend 2 closes the entrance door when
the back of the pole passes through the front door of the barn at time t2
So, the friend sees that both doors are closed simultaneously, but the pole vaulter sees
that the back door of the barn is opened before the front door is closed i.e. events that
are simultaneous in one frame are not simultaneous in other frames.
Marks
Mark questions communally for each group. Presentation marks can be given
communally or to individuals as appropriate. Award discretionary marks to those that
have made particular notable contributions.
Section 2
3.1 a/b) group
3.2 a/b) group
3.3 a) group
3.3 b) group
3.4 group
3.5 group/individual
Discretionary marks individual
TOTAL
Mark
1
1
1
1
1
3
2
10
Possible questions for presentation
3.2 Does the pole fit inside the barn when the pole and barn are both stationary ?
No, the pole only just fits when it is relativistically contracted.
3.2 Would the pole fit inside the barn if the pole and barn were in the same frame but
you flew past at a relativistic speed ?
No, both the pole and the barn would be contracted by the same factor of  depending
on the relative velocity of the space-ship and the pole-barn frame.
3.3 How does diagram a) change as the velocity of the pole increases ?
The gradient of the graphs increase – as this represents the velocity of the pole.
As the gradient increases  increases and the length of the pole decreases, so the pole
fits comfortably inside the barn.
3.3 What is the maximum gradient of the lines representing the pole on the graph ?
Maximum velocity is speed of light.
3.4 Is it possible for the pole vaulter to travel at a speed such that he does see the
events as simultaneous ?
No, the barn frame is the only frame in which the events are simultaneous.
If the pole vaulter travels faster the barn will shrink for him and the time difference
between the doors opening/closing will get longer.
If he travels slower the time differences will get shorter – but in the friend’s frame the
pole will not be contracted as much and no longer fit inside the barn.