Environmental Engineering Reference
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the adjacent minima should equal
π/ 2 in all inertial frames. Now we notice that
the phase can be written as a Minkowski scalar product:
±
φ
=−
K
·
X ,
(12.18)
where X is a position four-vector and K
). Now since φ is a four-scalar
and X a four-vector it follows that K must also be a four-vector.
We are now in a position to re-derive the Doppler effect for light. Consider the
situation illustrated in Figure 6.4, i.e. a light source is at rest in S
=
(ω/c,
k
such that it
radiates plane waves in the direction of an observer in S .Ifin S
the light wave
has a wave four-vector given by
K =
(k ,
k , 0 , 0 )
(12.19)
then it will describe plane light waves travelling in the negative x direction and
we have used the fact that the speed of the wave is c hence ω =
ck .Asan
aside we ought to comment on our notation, which is rather standard but also
potentially rather confusing. We have used a prime on the four-vector itself ( K )
but four-vectors are frame independent objects so strictly speaking K =
K . What
the prime really indicates is that the explicit representation of K on the right hand
side is understood to be in S . Perhaps a better notation would be to write something
like K
k , 0 , 0 ) S but one rarely sees this in the literature and so we stick to
the slightly imprecise notation of Eq. (12.19). Returning to the task in hand, given
Eq. (12.19) we can immediately determine the corresponding wave four-vector in
S .Itis
(k ,
=
k
ω/c
k x
k y
k z
cosh θ sinh θ 00
sinh θ cosh θ 00
0
=
k
0
0
.
(12.20)
0
1 0
0
0
0 1
Three of these equations imply that k y =
k z =
0and k x =−
ω/c
=−
k, as it indeed
should be for a light wave, whilst the fourth implies that
k ( cosh θ
k
=
sinh θ)
γ(v)k ( 1
=
v/c).
(12.21)
Putting f
=
ω/ 2 π and k
=
ω/c :
γ(v)f ( 1
f
=
v/c)
1
v/c
f
=
(12.22)
1
+
v/c
which is Eq. (6.9).
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