Physics:Four-frequency

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

$N^a=(\nu ,\nu \widehat{\mathbf{𝐧}})$

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

$\frac{1}{c}\eta (N^a,V^b)=\frac{1}{c}{\eta }_{ab}{N}^a{V}^b$

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

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

$K^a=(\frac{\omega }{c},\mathbf{𝐤})$

where $\omega =2\pi \nu$, $c$ is the speed of light and $\mathbf{𝐤}=\frac{2\pi }{\lambda }\widehat{\mathbf{𝐧}}$ 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

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

See also

• Four-vector
• Wave vector

References

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