Four-frequency

The four-frequency of a massless particle, such as a photon, is a 4-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

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

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 $\mathbf{k}=\frac{2 \pi}{\lambda}\hat{\mathbf{n}}$ 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

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