Four-frequency

The four-frequency of a massless particle, such as a photon, is a four-vector defined by

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

where ${\displaystyle \nu }$ is the photon's frequency and ${\displaystyle {\hat {\mathbf {n} }}}$ 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 ${\displaystyle V^{b}}$ will observe a frequency

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

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

Closely related to the four-frequency is the four-wavevector defined by

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

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

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