The four-frequency of a massless particle, such as a photon, is a four-vector defined by
N
a
=
(
ν
,
ν
n
^
)
where
ν
is the photon's frequency and
n
^
is a unit vector in the direction of the photon's motion. The four-frequency of a photon is always a future-pointing and null vector. An observer moving with four-velocity
V
b
will observe a frequency
1
c
η
(
N
a
,
V
b
)
Where
η
is the Minkowski inner-product (+---)
Closely related to the four-frequency is the wave four-vector defined by
K
a
=
(
ω
c
,
k
)
where
ω
=
2
π
ν
,
c
is the speed of light and
k
=
2
π
λ
n
^
and
λ
is the wavelength of the photon. The wave four-vector is more often used in practice than the four-frequency, but the two vectors are related (using
c
=
ν
λ
) by
K
a
=
2
π
c
N
a
