Ennio de Giorgi

The four-frequency of a photon is defined by

$N^{a}=\left(\nu ,\nu \mathbf {n} \right)$

where $\nu$ is the photon's frequency and $\mathbf {n}$ is a unit vector in the direction of the photon's motion. The four-frequency is always a future-pointing and null vector. An observer moving with four-velocity $V$ will observe a frequency

${\tfrac {1}{c}}\eta (N^{a},V)$

Where $\eta$ is the Minkowski inner-product (+---)

Closely related to the four-frequency is the wave four-vector defined by

$K^{a}=\left({\frac {\omega }{c}},\mathbf {k} \right)$

where $\omega =2\pi \nu$, $c$ is the speed of light and $\mathbf {k} ={\frac {2\pi }{\lambda }}\mathbf {n}$ and $\lambda$ 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 $c=\nu \lambda$) by

$K^{a}={\frac {2\pi }{c}}N^{a}$

References

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