Today: Review of SR
Exam1: Tomorrow, 7:30-9:00pm, DUANE G1B30
Bring your calculators (but no PCs or cell phones please).
What exactly did we do during the first few weeks?
Chapter 1: Spacetime
• Galileo transformation: Classical relativity
• Michelson-Morley  'c' is same in all inertial frames
• Einstein's postulates: Incompatible with Galilean relativity
(c’ ≠ c – u)!
• Consequences are 'time dilation' and 'length contraction'
 Lorentz transformation  Velocity transformation
• Spacetime interval: Invariant under Lorentz transformation
Chapter 2: Relativistic Mechanics
Re-definition of important physical quantities to preserve
conservation laws under LT:
- Momentum
- Force
- Kinetic Energy
- Rest Energy
- Total Energy
Einstein’s
Postulates of Relativity
(Abbreviated versions)
Let's start with a few
important concepts
#1: Physics (including E&M) is the same in
all inertial frames.
#2: The speed of light is the same in all
inertial frames.
Event
What does the Lorentz tr. do?
• Where something is depends on when you
check on it and on the movement of your
own reference frame.
• Time and space are not independent
quantities; they are related by Lorentz Tr.
• Definition: An event is a measurement of
where something occurs at what time.
The Lorentz (and Galileo) transformations convert
the coordinates x,y,z,t of an event (x,y,z,t) in a
frame S to the corresponding coordinates x',y',z',t'
of another frame S'. The way LT is presented here
requires the following:
The frame S' is moving along the x-axes of the frame S with
the velocity v (measured relative to S) and we assume that
the origins of both frames overlap at the time t=0.
y
y'
(x,y,z,t)
S
Events are not invariant under Lorentz transformation! In fact
the LT converts the coordinates of an event from one frame to
another; such as from S: (x,y,z,t) to S' (x',y',z',t')
Transformations
If S’ is moving with speed v in the positive x direction
relative to S, then the coordinates of the same event
in the two frames are related by:
Galilean transformation
(classical)
Lorentz transformation
(relativistic)
S'
x
z
x'
v
z'
Velocity transformation
(a consequence of the Lorentz transformation)
If S’ is moving with speed v in the positive x
direction relative to S, and the origin of S and
S' overlap at t=0, then the coordinates of the
same event in the two frames are related by:
Lorentz transformation
(relativistic)
Velocity transformation
(relativistic)
ux=Δx/Δt
…
Remark: This assumes (0,0,0,0) is the same event in both frames.
(x',y',z',t')
The Proper Time between two events is measured
by clocks at rest in a reference frame in which the
two events:
a) Occur at the same time.
b) Occur at the same place.
Proper time
Proper time: Time interval Δt = t2 – t1 between two
events (x1,y1,z1,t1) and (x2,y2,z2,t2) measured in the
frame, in which the two events occur at the same
spatial coordinates, i.e. time interval that can be
measured with one clock.
v
c) Are separated by the distance a light signal
can travel in the given time interval.
d) Are causally related
Same spatial coordinates means:
x1 = x2
y1 = y2
z1 = z2
e) None of these.
A particle is found to decay 2 µs in its own reference
frame. You observe such a particle in the laboratory,
traveling at a speed of 0.99 c. How long does it take
to decay in the lab reference frame?
a) 2 µs
b) 0.28 µs
c) 14 µs
d) 4.6 µs
e) 4.6 ms
Time dilation:
Δt’ = γ·Δtproper
The proper length of a rigid object is the length of
the object measured…
A – …in any inertial frame.
B – …by the speed of light.
C – …in the one inertial frame in which both ends
of the object have the same event coordinates.
D – …in the frame in which the object is not
rotating.
E – …in the inertial frame in which the object is at
rest.
Proper length
Proper length: Length of object measured at rest /
object measured in the frame where it is at rest (use
a ruler)
A meter stick moves at 0.99 c in the direction of its
length through a laboratory. According to
measurements in the laboratory, the meter stick
length is:
a) 0 m
b) 0.089 m
c) 0.14 m
... -3
-2
-1
0
1
2
3 ...
d) 0.28 m
e) 1 m
Proper mass (‘rest mass’)
Proper mass: Mass of an object measured in its
rest-frame.
Length in
moving frame
Length in stick’s
rest frame
(proper length)
Spacetime interval
Say we have two events: (x1,y1,z1,t1) and (x2,y2,z2,t2).
Define the spacetime interval (sort of the "distance")
between two events as:
With:
Spacetime interval
In HW3 you showed that the spacetime interval
has the same value in all reference frames! i.e.
Δs2 is “invariant” under Lorentz transformations.
Application of the Space-time interval
Two events occur in the frame S.
Event 1 (x1 = -0.5s*c, t1=1s)
Event 2: (x2 = 0, t2 = 2s)
What’s the proper time between these two events?
Simultaneity, time dilation & length contraction
As a consequence of Einstein's second postulate of
relativity ('The speed of light is the same in all inertial
frames of reference') we came to interesting
conclusions:
Peep?
- Relativity of simultaneity
- Time dilation
- Length contraction
(Δs)2 = (cΔt)2 - (Δx)2= (c*1s)2 – (0.5*c)2 = 0.75 c2
A) 0 s
Proper time: Δx’ ≡ 0
B) 0.25 s
 0.75 c2 = (c* Δtproper)2 – (0)2
C) 0.5 s
 Δtproper = 0.87 s
D) 0.75s
E) None of the above
All these effects are summarized in
a set of equations:
The Lorentz transformation
Application: Lorentz transformation
v
A(x=0)
τ0 = 0
Some applications of LT
?
B
…
A
τ1 = 1s
v
B
Two clocks (one at A and one at B) are synchronized. A third
clock flies past A at a velocity v. The moment it passes A all
three clocks show the same time τ0 = 0 (viewed by observers
in A and B. See left image.)
What time does the third clock show (as seen by an observer
at B) at the moment it passes the clock in B? The clock in B is
showing τ1 = 1s at that moment.  Use Lorentz transformation!
A) γ · (τ1-τ0)
D) (τ1-τ0) / γ
B) γ(τ1-τ0)(1 – v/c2)
E) γ(τ1-τ0)(1 + vx'/c2)
C) γ(τ1-τ0)(1 + v2/c2)
Hint: Use the following frames:
v
x'
x
B
A(x=0)
τ0 = 0
?
…
A
τ1 = 1s
v
B
?
…
A
τ1 = 1s
v
B
The clock travels from A to B with speed v. Assume A is
at position x = 0, then B is at position x = v·τ, τ=(τ1-τ0)
Use this to substitute x in the Lorentz transformation:
C) γ(τ1-τ0)(1 + v2/c2)
The moving clock shows the proper time interval!! Δtproper = Δt / γ
Lucy
x'
A(x=0)
τ0 = 0
B
What time does the third clock show (as seen by an observer
at B) at the moment it passes the clock in B? The clock in B is
showing τ1 = 1s at that moment.  Use Lorentz transformation!
B) γ(τ1-τ0)(1 – v/c2)
E) γ(τ1-τ0)(1 + vx'/c2)
v
x
Two clocks (one at A and one at B) are synchronized. A third
clock flies past A at a velocity v. The moment it passes A all
three clocks show the same time τ0 = 0 (viewed by observers
in A and B. See left image.)
A) γ · (τ1-τ0)
D) (τ1-τ0) / γ
Hint: Use the following systems:
 We get exactly the expression of the time dilation!
v
?
Lucy
v
?
... -3
-2
-1
0
George
1
2
3 ...
... -3
George has a set of synchronized clocks in reference frame
S, as shown. Lucy is moving to the right past George, and
has (naturally) her own set of synchronized clocks. Lucy
passes George at the event (0,0) in both frames. An
observer in George’s frame checks the clock marked ‘?’.
Compared to George’s clocks, this one reads
A) a slightly earlier time
B) a slightly later time
C) same time
-2
-1
0
1
George
2
3 ...
The event has coordinates (x = -3, t = 0) for George.
In Lucy’s frame, where the ? clock is, the time t’ is
, a positive quantity.
‘?’ = slightly later time
Remember this one?
Lucy
... -3
-2
-1
v
0
Lucy
1
2
x
3 ...
... -3
-2
-1
0
1
x
2
3 ...
v
?
Ethel
Ethel
Lucy has a set of synchronized clocks in her reference
frame S, as shown. Ethel is moving to the left past Lucy,
and has her own set of synchronized clocks. Ethel passes
Lucy at the events (x=x'=0,t=t'=3pm). An observer in
Lucy’s frame checks the clock marked ‘?’. Compared to
Lucy’s clocks (all showing 3pm at that moment), this one
reads…
The event has coordinates (x=3,t=3pm) in Lucy's frame.
Lucy sees Ethel travel to the left with velocity v (meaning
that v is negative as seen by Lucy). Therefore, in Lucy’s
frame the observer reads the ?-clock as:
>1
Positive
slightly after 3pm!
Why did Lucy (at x=0) need a helper at x=3 to figure that out?

? reads a time after 3pm.
Example from previous exam
Velocity transformation
A high-speed train is traveling at a velocity of v = 0.5c. The
moment it passes over a bridge it launches a cannon ball
with a velocity of 0.4c straight up (as seen by the train
conductor). What is the velocity of the ball right after it was
launched as seen by an observer standing on the bridge?
A high-speed train is traveling at a velocity of v = 0.5c. The
moment it passes over a bridge it launches a cannon ball
straight up (as seen by the train conductor) with a velocity of
0.4c. What is the velocity of the ball right after it was
launched as seen by an observer standing on the bridge?
Situation seen by the onlooker on the bridge:
Cannon ball right after
firing the cannon.
v
Attach reference frame S to the train:
Observer is in frame S' traveling from right to left (v is negative!!)
Now use the velocity
transformation:
ux = 0
uy = 0.4c
y
S
x
y'
S'
v = -0.5c
x'
Velocity transformation
How fast and in which direction relative to
you would a source of violet light (400 nm)
have to be travelling in order to have you see
it’s emitted light as red (650 nm)?
}
u'x = 0.5c
u'y = 0.346c
a) 0.65 c towards you
b) 0.65 c away from you
Velocity transf.
ux = 0
uy = 0.4c
y
c) 0.45 c towards you
y'
d) 0.45 c away form you
e) 0.2 c away from you
S
x
S'
v = -0.5c
x'
Definitions:
We redefined several physical quantities to
maintain the conservation laws for momentum
and energy in special relativity.
Now let's do some
relativistic mechanics!
Relativistic momentum:
Relativistic force:
(Chapter 2)
Relativistic Energy:
E = γmc2 = K + mc2
(K: kinetic energy)
Important relations:
(These relations result from the previous definitions)
Rest energy of an object:
Erest = mc2
Kinetic energy of an object:
K = (γ-1)mc2
Energy – momentum relation: E2 =(pc)2 + (mc2)2
Momentum of a mass-less particle: p =E/c
Velocity of a mass-less particle:
u=c
Do neutrinos have a mass?
Neutrinos are elementary particles. They come in three
flavors: electron, muon, and tau neutrino (νe,νµ, ντ). The
standard model of particle physics predicted such particles.
The prediction said that they were mass-less.
The fusion reaction that takes place in the sun (H + H  He)
produces such νe. The standard solar model predicts the
number of νe coming from the sun.
All attempts to measure this number on earth revealed only
about one third of the number predicted by the standard
solar model.
Do neutrinos have a mass? (cont.)
Bruno Pontecorvo predicted the ‘neutrino oscillation,’ a
quantum mechanical phenomenon that allows the neutriono
to change from one flavor to another while traveling from the
sun to the earth.
Why would this imply that the neutrinos have a
mass?
A 1 kg ball is heading towards you at a speed of 0.8 c. How
much momentum does it have compared to what it would
have in Galilean relativity?
a) It is the same
b) 5/3 times as much
Massless particles travel at the speed of light!
c) 3/5 times as much
i.e. γ  ∞, and therefore, the time seems to be standing
still for the neutrino:
d) None of the above
ΔtEarth = γ · Δtneutrino(“proper”)
In the HW: muon or pion experiments. The half-live time of the
muons/pions in the lab-frame is increased by the factor γ.
How does nuclear power work?
A proton (mass 1.6e-27 kg) is travelling at 0.99 c. How
much Kinetic energy does it have?
a) 6e-11 Joules
K = (γ-1)mc2
b) 2e-10 Joules
The total rest energy of the particle equals the sum
of the rest energy of all constituents plus the total
internal energy, such as binding energy (–EB)
Mc2 = Σ(mi c2) – EB
Application: Nuclear Fusion:
c) 9e-10 Joules
Isotope mass:
(1 u ≈ 1.66·10-27 kg)
d) 1e-9 Joules
Deuterium: 2.01355321270 u
Helium 4: 4.00260325415 u
 1kg of Deuterium yields ~0.994 kg of Helium 4.
e) 4e-8 Joules
Energy equivalent of 6 grams:
E0 = mc2 = (0.006 kg)·(3·108 m/s )2 = 5.4·1014 J
Enough to power ~20,000 American households for 1 year!
Length of an object
Length contraction
... -3
-2
-1
0
1
2
3 ...
(Consequence of time dilation and vice versa)
The length, measured in the
stick’s rest frame, is its
proper length.
Length of an object
S
... -3
-2
-1
S’
0
1
2
Another observer
comes whizzing by at
speed v. This
observer measures
the length of the stick,
and keeps track of
time.
v
3 ...
0
Event 1 – Origin of S’ passes left end of stick.
-2
-1
0
1
S’
2
3 ...
S
... -3
-2
-1
0
1
S’
2
v
3 ...
0
Event 1 – Origin of S’ passes left end of stick.
Event 2 – Origin of S’ passes right end of stick.
S
... -3
Length of an object
v
0
Conclusion:
Speeds are the same (both refer to the relative speed).
And so
In frame S’:
length of stick = L’ (this is what we’re looking for)
time between measurements = Δt’ (this is proper time)
speed of frame S is -v = -L’/Δt’
In frame S:
length of stick = L (this is the proper length)
time between the two events = Δt
speed of frame S’ is v = L/Δt
Length in moving frame
Length in stick’s rest frame
(proper length)
Length contraction is a consequence of time dilation
(and vice-versa).
Relationship of Energy and momentum
Recall:
= γmc2
p = γmu
Total Energy: E
Momentum:
Therefore: p2c2 = γ2m2u2c2 = γ2m2c4 · u2/c2
From the momentum-energy relation E2 = p2c2 + m2c4
we obtain for mass-less particles (i.e. m=0):
use:
 p2c2 = γ2m2c4 – m2c4
=E2
This leads us the momentum-energy relation:
or:
E2 = (pc)2 + (mc2)2
E2 = (pc)2 + E02
Application: Massless particles
E = pc , (if m=0)
p=γmu and E=γmc2  p/u = E/c2
Using E=pc leads to:
u=c
, (if m=0)
Massless particles travel at the speed of light!!
… no matter what their total energy is!!