Ennio de Giorgi

The four-frequency of a photon is defined by

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

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

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

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

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

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

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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