The Doppler effect
y�
y
α�
α
x, x�
In S � (source stationary) consider wave
�
�
�
�
�
�
eiω (t +(x /c) cos α +(y /c) sin α ) ,
ω
where �k � = − (cos α� , sin α� , 0) .
c
�
Then in S (moving source), ω = γ(ω � + vkx� ) = γω � (1 − (v/c) cos α� ) giving frequency
in S Doppler shifted from ω � by factor
�
�
v
ω = γ 1 − cos α� ω � .
c
Longitudinal Doppler effect
So for example for a wave in the x-direction, α� = 0, then
�
1 − vc � �
v� �
ω
≈
1
−
ω=
ω + ... .
1 + vc
c
For a source moving away from the observer (v > 0) then ω < ω � , ie frequency
red-shifted, and frequency blue-shifted if source moving towards the observer.
Transverse Doppler effect
We now also have a transverse Doppler effect, setting α� = π/2
ω = γω �
a typical relativistic effect.
Aberration
This is the difference between α and α� ,
ky�
ky
tan α = � = �
kx
γ kx −
�
vω
c2
89
�=
sin α
�,
γ cos α + vc
�
as in S we have
eiω(t+(x/c) cos α+(y/c) sin α) ,
ω
where �k = − (cos α, sin α, 0) .
c
So
tan α − tan α�
1 + tan α tan α�
γ(cos α + v/c) − cos α
= sin α
γ cos α(cos α + v/c) + sin2 α
v
1 + v/2c cos α + O(v 2 /c2 )
=
sin α
c
1 + v/c cos α + O(v 2 /c2 )
tan(α − α� ) =
or
�
� 2 ��
v
v
v
tan(α − α ) = sin α 1 −
.
cos α + O 2
c
2c
c
�
Non-relativistic (Bradley):
vd/c
α�
d
α − α�
as tan(α − α� ) ≈ sin(α − α� ) and time of travel t = d/c, so S � moves distance vd/c,
then from sine rule sin(α − α� )/(vd/c) = sin α� /d or
v
v
sin(α − α� ) = sin α� ,
or α − α� = sin α .
c
c
7.4
Minkowski (Space–Time) diagrams
Portray x, y, z, t as a point in a four-dimensional space-time: (ct, �r) ≡ (ct, x, y, z).
• Every point P represents an event in space-time
• Under a Lorentz transformation for S → S �
s2 = c2 t2 − �r2 ,
�r2 = x2 + y 2 + z 2 ,
is a Lorentz invariant, ie s2 = s�2 (after a few lines of algebra). So

 > 0 time-like
= 0 light-like
s2 =

< 0 space-like
This decomposition remains the same in all inertial frames (ie after any Lorentz
transformation)
90
ct
P
Time like
FUTURE
Space like
PRESENT
Space like
PRESENT
O
x
Time like
PAST
light cone
Worldline
Minkowski or Space–time picture (wlog two dimensional)
• Light signal define a cone (at 45◦ as choose ct for y-axis)
• All time-like points lie within light-cone
• All space-like points lie outside light-cone
• World-lines are motion of a particle (starting at t = 0, x = 0). World-lines of
a photon lie on the light cone
Consider two space-time points P1 (ct1 , �r1 ) and P2 (ct2 , �r2 ). Then
s212 = c2 (t1 − t2 ) − |�r1 − �r2 |2 ,
also a Lorentz invariant.
Space-like separation
From s212 < 0, then x1 − x2 > c(t1 − t2 ), ie events not connected by a light signal
no causal connection
Can find a Lorentz transformation to S � where both events P1 and P2 are at the
same time, as
�
�
v
c(t�1 − t�2 ) = γ c(t1 − t2 ) − (x1 − x2 ) .
c
91
Choose
v=c
c(t1 − t2 )
< c,
x 1 − x2
so S � possible
Hence can always find Lorentz transformations so that the order of space-like events
is interchanged.
Time-like separation
From s212 > 0, then c(t1 − t2 ) > x1 − x2 , ie events connected by a light signal
causal connection possible
Cannot find a Lorentz transformation to S � where both events P1 and P2 are at the
same time, as would need
v=c
c(t1 − t2 )
> c,
x 1 − x2
so S � not possible
So cannot interchange cause and effect. However as
x�1 − x�2 = γ ((x1 − x2 ) − v(t1 − t2 )) .
can find a frame S � where x�1 = x�2 , events happen at the same place
Define proper time by
c 2 τ 2 = s2 ,
or τ =
�
t2 −
�r2
.
c2
So here we have
τ P1 − τ P2 =
�
(t1 − t2 )2 −
|�r1 − �r2 |2
.
c2
ie time measured on a clock moving with uniform speed |�r1 − �r2 |/(t1 − t2 ) between
the two events, P1 and P2 .
7.4.1
Diagramatic relation between S and S �
We now briefly consider the relationship between S and S � on a Minkowski diagram
Co-ordinate systems same at t = 0 = t�
• S � time-axis defined by x� = 0 = γ(x − vt) or
ct =
1
x,
v/c
ie straight line, gradient > 1, ie between S time-axis and light signal
92