The Lorentz transformationsCovariant 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’

€

Δt '= γΔt ⇒ L =L'

γ

• 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

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

Future and past• 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(2)• In the same way, all events in the region bOd are in the

absolute past relative to O; i.e. events in this region 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

r

r

s S

v1 c vcf f f

v c v1c

• 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.

'tu Δ 'tc Δ

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’ • Hence, the wavelength λ’ according to the O’ observer, is: •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

N

uct )(''

Δ

€

'=Δt'(c + u)

ν ⋅Δt0

The general relation between the wavelength and frequency of a lightwave is: wavelength = (speed of light)/(frequency). Therefore, the wavelength and the frequency the O’ observer registers are related as:

€

'=c

ν '=

Δt'(c + u)

ν ⋅Δt0

⎛

⎝ ⎜

⎞

⎠ ⎟⇒ ν '= ν

Δt0

Δt '⋅

1

1+ u /c

Now, we can use the time dilation formula:

€

Δt'

Δt0

= γ

€

ν '= ν1− u2 /c 2

1+ u /c

= ν(1− u /c)(1+ u /c)

(1+ u /c)(1+ u /c)

= ν1− u /c

1+ u /c cu

cu

/1

/1'

νν

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:

air). still to(relative source theof speed theis

air), still to(relativeobserver theof speed theis

sound; of speed theis here where'

s

o

s

o

u

u

cuc

uc

νν

If we consider an analogous situation as before, then only the source moves:

c

u

cuuc

c

u

s

ss

o

1/1

1'

then ;0put we, therefore

νννν

c

ucu

cu

cu1/1

/1

/1' 2 ννννRELATIVISTIC EXPRESSION:

4-ve

ctor

s

4-tensors

Unit tensor and metric tensor

The completely antisymmetric tensor of rank 4

Covariant derivatives

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∂∂x

μ≡∂μ;∂

∂xμ

≡∂μ

∂μ∂μ

=1

c2

∂2

∂t2−∇

2=

☐

4-dimensional velocity

Relativistic dynamics: mass and energy

€

E = hν

V

e/2 e/2

(g 1-V/c) e/2 (g 1+V/c) e/2

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

The continuity equation tells that ; thus jµ is a 4-vector.

Also Aµ is then a 4-vector.

€

∇• r

j +∂ρ

∂t= 0⇒ ∂μ j μ = 0

Electromagnetism and relativity

The electromagnetic tensor

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E i = −∂ 0A i +∂ iA0 ⇒ F 0i = E i

B i = −

ˆ 1 ˆ 2 ˆ 3

∂1 ∂ 2 ∂ 3

A1 A2 A3

Covariant expression of Maxwell’s equations

• Non ce ne sarebbe bisogno, ma…

€

∂μF μν = ∂μ ∂ μ Aν −∂ν Aμ( ) = ∂μ∂

μ( )Aν −∂ν ∂μ Aμ

( )

Ma ∂μ Aμ =∂φ

∂t+

r ∇⋅

r A = 0

⇒ ∂μ F μν = jν

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

€

qeEu = dp0

ds⇒dp0

dt = ddt

m

1− v 2

⎛

⎝ ⎜

⎞

⎠ ⎟= Fv

Decay of an unstable particle

(note: natural units - c=1 from now on)

Elastic collision (1 + 2 1’+2’)

Elastic collision: target at restp1

p’1

p’2

Compton scattering• If the incident particle 1 has 0 mass (photon)

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⇒ cosθ =E '1 E1 + m(E '1 −E1)

E '1 E1

=1+ m1

E1

−1

E '1

⎛

⎝ ⎜

⎞

⎠ ⎟

=1− m1

hν '−

1

hν

⎛

⎝ ⎜

⎞

⎠ ⎟=1−

mc

hλ '−λ( )