Four-frequency

The four-frequency of a massless particle, such as a photon, is a four-vector defined by 
where formula_2 is the photon's frequency and formula_3 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 formula_4 will observe a frequency
Where formula_6 is the Minkowski inner-product (+−−−)
Closely related to the four-frequency is the four-wavevector defined by
where formula_8, formula_9 is the speed of light and formula_10 and formula_11 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 formula_12) by