What is "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 ν 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 will observe a frequency

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

Where η 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 ω = 2πν, c is the speed of light and $\mathbf{k}=\frac{2 \pi}{\lambda}\hat{\mathbf{n}}$ and λ 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 = νλ) by

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