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

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

Where $\eta$ 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 $\omega=2 \pi \nu$ , $c$ is the speed of light and $\mathbf{k}=\frac{2 \pi}{\lambda}\hat{\mathbf{n}}$ and $\lambda$ is the wavelength of the photon. The wave four-vector 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

References

Woodhouse, N.M.J. (2003). Special Relativity. London: Springer-Verlag. ISBN 1-85233-426-6.

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Four-frequency

See also

References