four-frequency

The four-frequency of a massless particle, such as a photon, is a four-vector defined by

:$N^a\; =\; \backslash left(\; \backslash nu,\; \backslash nu\; \backslash hat\{\backslash mathbf\{n\}\}\; \backslash right)$

where $\backslash nu$ is the photon's frequency and $\backslash hat\{\backslash mathbf\{n\}\}$ 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

:$\backslash frac\{1\}\{c\}\backslash eta\backslash left(N^a,\; V^b\backslash right)\; =\; \backslash frac\{1\}\{c\}\backslash eta\_\{ab\}N^aV^b$

Where $\backslash eta$ is the Minkowski inner-product (+−−−) with covariant components $\backslash eta\_\{ab\}$.

Closely related to the four-frequency is the four-wavevector defined by

:$K^a\; =\; \backslash left(\backslash frac\{\backslash omega\}\{c\},\; \backslash mathbf\{k\}\backslash right)$

where $\backslash omega\; =\; 2\; \backslash pi\; \backslash nu$, $c$ is the speed of light and $\backslash mathbf\{k\}\; =\; \backslash frac\{2\; \backslash pi\}\{\backslash lambda\}\backslash hat\{\backslash mathbf\{n\}\}$ and $\backslash lambda$ 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 $c\; =\; \backslash nu\; \backslash lambda$) by

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