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


 * $$\frac{1}{c}\eta\left(N^a, V^b\right) = \frac{1}{c}\eta_{ab}N^aV^b$$

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


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