Four-frequency

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

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

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

{\frac {1}{c}}\eta \left(N^{a},V^{b}\right)={\frac {1}{c}}\eta _{ab}N^{a}V^{b}

Where $\eta$ is the Minkowski inner-product (+−−−) with covariant components $\eta _{ab}$ .

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

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

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