The four-frequency of a massless particle, such as a photon, is a four-vector defined by
:<math>N^a = \left( \nu, \nu \hat{\mathbf{n \right)</math>
where <math>\nu</math> is the photon's frequency and <math>\hat{\mathbf{n</math> 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 <math>V^b</math> will observe a frequency
:<math>\frac{1}{c}\eta\left(N^a, V^b\right) = \frac{1}{c}\eta_{ab}N^aV^b</math>
Where <math>\eta</math> is the Minkowski inner-product (+−−−) with covariant components <math>\eta_{ab}</math>.
Closely related to the four-frequency is the four-wavevector defined by
:<math>K^a = \left(\frac{\omega}{c}, \mathbf{k}\right)</math>
where <math>\omega = 2 \pi \nu</math>, <math>c</math> is the speed of light and <math display="inline">\mathbf{k} = \frac{2 \pi}{\lambda}\hat{\mathbf{n</math> and <math>\lambda</math> 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 <math>c = \nu \lambda</math>) by
:<math>K^a = \frac{2 \pi}{c} N^a</math>
See also
- Four-vector
- Wave vector