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

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

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

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

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

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

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

See also

• Four-vector
• Wave vector

References

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