Physics:Four-frequency
The four-frequency of a massless particle, such as a photon, is a four-vector defined by
- [math]\displaystyle{ N^a = \left( \nu, \nu \hat{\mathbf{n}} \right) }[/math]
where [math]\displaystyle{ \nu }[/math] is the photon's frequency and [math]\displaystyle{ \hat{\mathbf{n}} }[/math] 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 [math]\displaystyle{ V^b }[/math] will observe a frequency
- [math]\displaystyle{ \frac{1}{c}\eta\left(N^a, V^b\right) = \frac{1}{c}\eta_{ab}N^aV^b }[/math]
Where [math]\displaystyle{ \eta }[/math] is the Minkowski inner-product (+−−−) with covariant components [math]\displaystyle{ \eta_{ab} }[/math].
Closely related to the four-frequency is the four-wavevector defined by
- [math]\displaystyle{ K^a = \left(\frac{\omega}{c}, \mathbf{k}\right) }[/math]
where [math]\displaystyle{ \omega = 2 \pi \nu }[/math], [math]\displaystyle{ c }[/math] is the speed of light and [math]\displaystyle{ \mathbf{k} = \frac{2 \pi}{\lambda}\hat{\mathbf{n}} }[/math] and [math]\displaystyle{ \lambda }[/math] 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 [math]\displaystyle{ c = \nu \lambda }[/math]) by
- [math]\displaystyle{ K^a = \frac{2 \pi}{c} N^a }[/math]
See also
References
- Woodhouse, N.M.J. (2003). Special Relativity. London: Springer-Verlag. ISBN 1-85233-426-6.
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