Taking the Lorentz transformation one step further
By studying the pole-and-barn assignment last week, you now
know that x and t are relative concepts in Special Relativity.
Thereby keeping the speed of light c constant in vacuum in both
inertial frames.
S′
S
y′ V
y
0
ct
1 β
ct
= γ(V )
x0
x
β 1
x
x′
β = V /c
s
γ(V ) =
Lambert van Eijck
1
1 − β2
TN2612 - intro special relativity - lecture 3
1
What is ux for an object observed to have ux 0 in S 0 ?
S′
S
u′
y′ V
y
x
x′
Figure: An object is observed as moving with ux 0 based on (x 0 , t 0 ). How
is the movement perceived in S?
Lambert van Eijck
TN2612 - intro special relativity - lecture 3
2
What is ux for an object observed to have ux 0 in S 0 ?
Galilei transformation (GT):
Lorentz transformation (LT):
x = x 0 + Vt 0
x = γ(V )(x 0 + β ct 0 )
y = y0
y = y0
z = z0
z = z0
t = t0
ct = γ(V )(ct 0 + β x 0 )
∆(x 0 + Vt 0 )
∆(x)
=
∆(t)
∆t
0
0
∆x + V ∆t
=
= (ux 0 + V )
∆t 0
ux =
Dx/Dt=ux
VDt/Dt=V
ux =????
uy =????
uz =????
uy = uy 0
uz = uz 0
Lambert van Eijck
TN2612 - intro special relativity - lecture 3
3
What is ux for an object observed to have ux 0 in S 0 ?
Lorentz transformation (LT):
check yourself:
x = γ(V )(x 0 + β ct 0 )
∆x = γ(V )(∆x 0 + β c∆t 0 )
y = y0
∆y = ∆y 0
z = z0
∆z = ∆z 0
ct = γ(V )(ct 0 + β x 0 )
∆ct = γ(V )(c∆t 0 + β ∆x 0 )
beta=V/c -> beta*c = V
0
∆x
∆x
∆x
γ(V )(∆x 0 + βc∆t 0 )
∆t 0 + V
ux =
=
c=
c
=
0 c
0
0
∆t
∆ct
γ(V )(c∆t + β∆x )
c + β ∆x
∆t 0
ux 0 + V
ux =
0
1 + Vcu2x
Lambert van Eijck
TN2612 - intro special relativity - lecture 3
4
What is uy , uz for an object observed to have uy 0 , uz 0 in S 0 ?
0
So we obtain for the velocity
components:
0
∆x = γ(V )(∆x + β c∆t )
∆y = ∆y 0
∆z = ∆z
0
∆ct = γ(V )(c∆t 0 + β ∆x 0 )
∆y
∆y
=
c
uy =
∆t
∆ct
∆y 0
c
=
γ(V )(c∆t 0 + β∆x 0 )
uy 0
γ(V )(1 +
V ux 0
)
c2
uz =
uz 0
γ(V )(1 +
V ux 0
)
c2
different from:
ux =
dividing top and bottom by Dt
=
uy =
∆y 0
∆t 0
0
γ(V )(c + β ∆x
∆t 0 )
Lambert van Eijck
ux 0 + V
0
1 + Vcu2x
c
TN2612 - intro special relativity - lecture 3
5
Particle collisions at relativistic speeds
Put yourself on the left
particle, so the lab-frame
(x, y ) travels with β = −3/4
(to the left). The speed of the
right particle is then:
ux
=
ux 0 +V
0
1+ V u2x
c
ux
S
3
4c
− 43 c
ux
=
− 34 c+− 34 c
−3
1+( −3
)
4 4
96
= − 100
c
y
x
Lambert van Eijck
TN2612 - intro special relativity - lecture 3
6
addition of velocities in the x direction
at the sane ct, light (dark yellow) has
travelled two x, while the object at u=0.5c
has travelled 1. so it goes at half the speed
of c
Figure: frame S 0 moves with β = 0.5 wrt S, so it travels half the distance
light would do, at any time ct. But also in the frame of S 0 an object at
speed β 0 = u 0 /c = 0.5 travels half the distance of what light would do.
How to recognize that in this plot?
Lambert van Eijck
TN2612 - intro special relativity - lecture 3
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addition of velocities in the x direction
Figure: frame S 0 moves with β = 0.5 wrt S and in the frame of S 0 an
object traveling with β 0 = u 0 /c = 0.5 travels half the distance of what
light would do (cf. red arrows). But for S the velocity of the object is
clearly not (0.5 + 0.5)c.
Lambert van Eijck
TN2612 - intro special relativity - lecture 3
8
Example: particle collisions at relativistic speeds
Put yourself on the left
particle, so the lab-frame
(x, y ) travels with β = −3/4
(to the left). The speed of the
right particle is then:
ux
=
ux 0 +V
0
1+ V u2x
c
ux
S
3
4c
− 43 c
ux
=
− 34 c+− 34 c
−3
1+( −3
)
4 4
96
= − 100
c
y
x
Lambert van Eijck
TN2612 - intro special relativity - lecture 3
9
Doppler effect: a wave observed by a moving object
L
v
u
The thin grey circles depict the maxima of the wave emitted from
the source. They leave the source in a period T after each other.
How is the wave and its frequency perceived by the moving object
using a non-relativistic (Galileian) transformation?
I
1st maximum leaves source: t1
I
2nd maximum leaves source: t2 = t1 + T with f = 1/T
Lambert van Eijck
TN2612 - intro special relativity - lecture 3
10
Doppler effect: a wave observed by a moving object
Galilei with observer moving:
t/T = number of periods
blue line: moving observer
Galilei with source moving
(ambulance):
14
12
solid red line: source moving
dotted red line: wave from source
10
8
t/T
t2'
9
t1'
8
6
7
4
0
t/T
t2
t1
6
2
0
1
u= speed of the wave
v = speed of observer
2
x/L
3
4
∆t 0 = t2 0 − t1 0
Dt' (blue) is the
uT
0
perceived period from
the observer moving at u ∆t =
u−v
v
⇒ f 0 = (1 − )f
u
Lambert van Eijck
5
5
4
3
2
1
0
−6
−4
−2
0
x/L
2
4
6
will yield: f 0 = f /(1 − vu )
TN2612 - intro special relativity - lecture 3
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Relativistic Doppler effect of EM radiation
We will assume the ’ambulance’ case again: a light source is static
at some position r 0 in S 0 and we want to study the Doppler effect
in S, while S 0 is moving at speed u.
ur= radial component
u = speed of frame
ut= tangent component
In S 0 the source has a proper frequency f0 and proper period T0 .
Lambert van Eijck
TN2612 - intro special relativity - lecture 3
12
Relativistic Doppler effect of EM radiation
I
The wave sent out by the light source has its first crest at t0 0
and the second crest at t1 0 = t0 0 + 1/f0 .
I
If the time t0 0 corresponds to some time t0 in S, it will take
an extra r /c to arrive at the origin of S: t1 = t0 + r /c, with r
as measured in S.
I
In S, the second crest comes
t2 = t0 +
source.
γ(u)
f0
γ(u)
f0
later and
+ r2 /c, with r2 the position of the moved
r and r2 are components of the radiation, so
like the radius of a sphere originating in the
source of the radiation
Lambert van Eijck
TN2612 - intro special relativity - lecture 3
13
Relativistic Doppler effect of EM radiation
1 instead of 2
If |r | >> u∆t then the length of r2 can be approximated as
r2 ≈ r + ur ∆t.
γ(u)
+ r2 /c
f0
γ(u)
= t0 +
+ (r + ur ∆t)/c
f0
γ(u)
γ(u)
= t0 +
+ (r + ur
)/c
f0
f0
t2 = t0 +
Lambert van Eijck
TN2612 - intro special relativity - lecture 3
14
Relativistic Doppler effect of EM radiation
T = t2 − t1 =
and
f =
γ(u)
(1 + ur /c)
f0
f0
γ(u)(1 +
f=1/T
ur
c )
I
Only the radial component ur appears in the equations
I
If ur = 0, there is still a Doppler effect. There is no classical
analogue of this effect.
Lambert van Eijck
TN2612 - intro special relativity - lecture 3
15
Examples of the Doppler effect
Lambert van Eijck
TN2612 - intro special relativity - lecture 3
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Examples of the Doppler effect
A police officer gives a person a ticket for driving through a red
traffic light. The driver says that the police man is mistaken.
According to the driver the light was green.
Question: What was the velocity of the car, approaching the traffic
light? Given: the wavelengths of red and green light are 690 and
530 nm, respectively.
Lambert van Eijck
TN2612 - intro special relativity - lecture 3
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