The Lorentz transformations
Covariant representation of
electromagnetism
Consequences of the invariance of the speed of light:
a reminder
• From the principle of relativity it follows that the velocity
of propagation of electromagnetic interactions is the same in
all inertial systems of reference
• We assume that the velocity of propagation of interactions
is a universal constant, and equal to c
– Relativity of simultaneity
– Time dilation
– Length contraction
(time and space intervals are not absolute)
More on length contraction
•
The observers on the car, whose frame of reference we
have chosen to call S’, measure the length of their car
to be L’. They verify this by watching the front end of
the car and then the rear end of the car pass the center
of the platform, and measuring the amount of time Dt’
that passes between those two events. According to the
observers on the car, the car is at rest and the subway
platform is moving at a velocity -V relative to the car.
L’ = V Dt’
•
To an observer standing at the center of the platform when the subway car goes by, the car
has a length L that can be calculated by measuring the time that passes between the time
when the front end of the car reaches the center of the platform and the time when the rear
end of the car reaches the center of the platform: L=VDt
The interval Dt is a proper time interval for the observer standing on the platform, because the
two events whose times are being measured both happen at the center of the platform.
Therefore the time intervals are related by
•
Dt' Dt  L 
L'

Examples of time dilation
and length contraction:
The muon experiment
Think of the muon’s point
of view…
The propagation of signal
• Let’s look at two inertial reference systems K and K' with coordinate axes XYZ and
X’ Y’ Z’, where the system K' moves relative to K along the X(X') axis.
• Suppose signals start out from some point A on the X' axis in two opposite
directions. Since the velocity of propagation of a signal in the K' system, as in all
inertial systems, is equal (for both directions) to c, signals will reach points B and C,
equidistant from A, at the same time (in the K' system)
• The same two events (arrival of the signal at B and C) can by no means be
simultaneous for an observer in the K system. In fact, the velocity of a signal
relative to the K system has, according to the principle of relativity, the same value
c, and since the point B moves (relative to the K system) toward the source of its
signal, while the point C moves in the direction away from the signal (sent from A to
C), in the K system the signal will reach point B earlier than point C.
• We shall frequently use the concept of event. An event is described by the place
where it occurred and the time when it occurred. Thus an event occurring in a
certain material particle is defined by the three coordinates of that particle
• We designate the time in the systems K and K' by t and t'
Intervals
Transformation of intervals
As already shown, if ds = 0 in one inertial system, then ds' = 0 in any other system.
On the other hand, ds and ds' are infinitesimals of the same order. From these two
conditions it follows that ds2 and ds'2 must be proportional to each other:
Timelike events
Timelike events (2)
Spacelike events
•
•
•
•
Let us take some event O as our origin of time and
space coordinates. Let us now consider what relation
other events bear to the given event O. For
visualization, we shall consider only one space
dimension and the time, marking them on two axes.
Uniform rectilinear motion of a particle, passing
through x = 0 at t = 0, is represented by a straight line
going through O and inclined to the t axis at an angle
whose tangent is the velocity of the particle. Since the
maximum possible velocity is c, there is a maximum
angle which this line can subtend with the t axis.
The two lines represent the propagation of signals (with
the velocity of light) in opposite directions passing
through O (i.e. going through x=0 at t=0). All lines
representing the motion of particles can lie only in the
regions aOc and dOb. On the lines ab and cd, x = ± ct.
First consider events whose world points lie within the
region aOc. For all the points of this region c2t2-x2 >0.
The interval between any event in this region and the
event O is timelike. In this region t>0, i.e. all the
events in this region occur "after" the event O. But two
events which are separated by a timelike interval cannot
occur simultaneously in any reference system.
Consequently it is impossible to find a reference system
in which any of the events in region aOe occurred
"before" the event O, i.e. at time t<0. Thus all the
events in region aOc are future events relative to O in
all reference systems. This region can be called the
absolute future relative to O.
Future and past
•
•
•
•
Future
and
past
In the same way, all events in the region bOd are in the
absolute past relative to O; i.e. events in this region
(2)
occur before the event O in all systems of reference.
Next consider regions dOa and eOb. The interval
between any event in this region and the event O is
spacelike. These events occur at different points in
space in every reference system. Therefore these
regions can be said to be absolutely remote relative to
O.
Note that if we consider all three space coordinates
instead of just one, then instead of the two intersecting
lines of Fig. 2 we would have a “cone" x2 + y2 + z2 –
c2t2 = 0 in the 4-dimensional coordinate system x, y, z.
t, the axis of the cone coinciding with the t axis. (This
cone is called the light cone.) The regions of absolute
future and absolute past are then represented by the two
interior portions of this cone.
Two events can be related causally to each other only if
the interval between them is timelike; this follows
immediately from the fact that no interaction can
propagate with a velocity greater than the velocity of
light. As we have just seen, it is precisely for these
events that the concepts "earlier" and "later" have an
absolute significance, which is a necessary condition
for the concepts of cause and effect to have meaning.
The Lorentz transformations
The Lorentz transformations (2)
The Lorentz transformations (3)
The Lorentz transformations (4)
Length contraction, again
Transformation of velocities
Exercises at home
• Are Lorentz transformations commutative?
• Derive time dilation from the Lorentz
transformations
Doppler effect in classical physics:
General solution
vr

 1 c
f  f 
v
 1 s
c



 c  vr 
f

 c vS 


• Numerator – Receiver (observer)
– Toward +
– Away –
• Denominator – Source
– Toward –
– Away +
Doppler effect from Lorentz transformations
• Now, the signal transmitted by one observer, and received by another, is a light
wave. It makes a difference compared to the situation in Doppler effect with sound
waves: there is no medium, and the velocity of the wave (i.e., of light) is the same
for both observers. And the the “transmitter” and “receiver” move relative to each
other with such a speed that relativistic effects have to be taken into account.
• We will consider the following situation: the “transmitter” is in the frame O that
moves away with speed -u (meaning: to the left) from the frame O’ in which the
“receiver” located. At a moment the “transmitter” starts to broadcast a light wave.
u  Dt '
c  Dt '
A total of N waves sent out from O
• Δt’ – the time that elapsed between the beginning of transmission and the moment
the wave-front reached the observer O’ , measured in the O’ frame.
• For the O’ observer, the N waves sent out from the O source are stretched over the
distance uΔt’ + cΔt’ = (u+c) Δt’
Dt ' ( c  u )
• Hence, the wavelength λ’ according to the O’ observer, is:  ' 
N
•Denote the time registered in the O frame between the beginning of transmission and
the moment the wave-front reached O’ as Δt0 , and the frequency of the signal for the
O observer as ν (it is the number of waves sent out in a time unit). Therefore, the
total number of waves emitted is N = ν Δt0
Dt'(c  u)
'
  Dt0
The general relation between the wavelength and frequency of a light
wave is: wavelength = (speed of light)/(frequency). Therefore, the
wavelength and the frequency the O’ observer registers are related as:
c  Dt'(c  u) 
Dt0
1
' 

  ' 
 '    Dt0 
Dt' 1 u /c
Dt'
Now, we can use the time dilation formula:

Dt 0
1 u 2 /c 2
 ' 
1 u /c
(1 u /c)(1 u /c)

(1 u /c)(1 u /c)
1 u /c

1 u /c

1 u / c
 ' 
1 u / c
Let’s compare the relativistic DE with the classical DE for sound waves:
The general formula for the Doppler frequency shift of sound waves is:
c  uo
 ' 
c  us
where here c is the speed of sound;
uo is the speed of the observer (relative to still air),
us is the speed of the source (relative to still air).
If we consider an analogous situation as before, then only the source moves:
therefore , we put uo  0; then
c
1
 us 
 ' 

  1  
c  us
1  us / c
c

RELATIVISTIC EXPRESSION:
1 u / c
 ' 

1 u / c
1  u / c 
2
 u
  1  
 c
4-vectors
4-tensors
Unit tensor and metric tensor
The completely antisymmetric tensor of rank 4
Covariant derivatives

c 2 t 2
  
 2 
1 2

x 

x
  ;
 
☐


4-dimensional velocity
Relativistic dynamics: mass and energy
E  h
e/2
e/2
 1-V/c) e/2
 1+V/c) e/2
V
E=mc2
Relativistic momentum
The pi are NOT the components of the
space part of a 4-vector
Relativistic momentum
In addition, it is proportional to the space part of the 4-velocity
Energy-momentum vector (4-momentum)
pµ = m uµ = mcγ (1 , v/c) is instead a 4-vector
pµpµ = m2c2 = E2/c2 – p2
The photon
Relativistic force
Vector potential and scalar potential
By defining
Aµ = (Φ/c , A)
and
jµ = (cρ , j)
the above equations become
☐Aµ = μ0 jµ
r 
The continuity equation tells that   j 
 0   j   0 ; thus jµ is a 4-vector.
t
Also Aµ is then a 4-vector.
Electromagnetism and relativity
The electromagnetic tensor
E i   0 A i   i A 0  F 0i  E i
1ˆ
2ˆ
3ˆ
B i   1
A1

2
3
A2
A3
Covariant expression of Maxwell’s equations
 F

    A   A




    A



 r r
Ma   A 
  A  0
t


  F  j

• Non ce ne sarebbe bisogno, ma…


   A


Invariants constructed from the electromagnetic tensor
(segue il calcolo esplicito)
Lorentz force: covariant expression
Lorentz force: covariant expression
Motion of a particle under constant force
0
dp
qeEu 

ds
0
dp

dt
d
 m 
 Fv
dt 
2
 1  v 
Decay of an unstable particle
(note: natural units - c=1 from now on)
Elastic collision (1 + 2 1’+2’)
p’2
p1
p’1
Elastic collision: target at rest
Compton scattering
•
If the incident particle 1 has 0 mass (photon)
 1
E'1 E1  m(E'1 E1 )
1 
 cos 
 1 m 

E'1 E1
E1 E'1 
 1
1 
mc
 1 m   1
'
h ' h 
h