The four-frequency of a massless particle, such as a photon, is a four-vector defined by
![{\displaystyle N^{a}=\left(\nu ,\nu {\hat {\mathbf {n} ))\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2636e73f6680da5d417c798267866de895f1b91)
where
is the photon's frequency and
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
will observe a frequency
![{\displaystyle {\frac {1}{c))\eta \left(N^{a},V^{b}\right)={\frac {1}{c))\eta _{ab}N^{a}V^{b))](https://wikimedia.org/api/rest_v1/media/math/render/svg/90c33b30e8490f1de74ac3f364e2966c6cd90e01)
Where
is the Minkowski inner-product (+−−−) with covariant components
.
Closely related to the four-frequency is the four-wavevector defined by
![{\displaystyle K^{a}=\left({\frac {\omega }{c)),\mathbf {k} \right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e294b3d5bf16527c89ca1b82ffa7a432e05692cd)
where
,
is the speed of light and
and
is the wavelength of the photon. The four-wavevector is more often used in practice than the four-frequency, but the two vectors are related (using
) by
![{\displaystyle K^{a}={\frac {2\pi }{c))N^{a))](https://wikimedia.org/api/rest_v1/media/math/render/svg/ccd26bc2bca1ef37bd74a350ab0a79769a88b8de)