Lesson plan
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^aV^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 \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
Article summary
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^aV^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 \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
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Discussion questions
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