**Required math:vectors, algebra**

**Required physics: basics of relativity**

The world line of a photon is a null line, so any vector tangent to a photon’s world line is a null vector. This means in particular that a four-velocity cannot be defined for a photon, since all four-velocities have to satsify the condition

Another consequence of this is that, if we require a photon’s momentum to be parallel to its world line, then the momentum must also be a null vector. Since the four-momentum is defined as the vector with equal to the energy and the other three components equal to the three-momentum, this means that for photons

In particular, if the photon is moving in the direction, then and .

From quantum mechanics, we know (well, it’s one of the postulates of quantum mechanics anyway) that

where is the frequency of the photon and is Planck’s constant. We can combine this with the Lorentz transformation of the photon’s four-momentum to get a formula for the Doppler shift.

The Doppler effect occurs because the observer is moving relative to a light source. If light is being emitted by a source such as a star, then the light will have a particular frequency (or in general, mixture of frequencies, but we’ll concentrate on monochromatic light), which can be measured as the number of peaks in the wave that pass a fixed point in one second. If the observer moves towards the light source, then in that second, he will pass a greater number of peaks in the wave, and thus the frequency of the light appears higher, or *blue-shifted*, since for visible light, the colour appears shifted towards the blue end of the spectrum. Similarly, if the observer moves away from the light source, the frequency appears lower and the light is *red-shifted*.

Note that this effect does not violate the postulate of the constancy of the speed of light, which is fundamental to relativity. The light itself still moves at the same speed relative to the moving observers; what changes is the frequency, and hence the energy, of the light that is observed.

If the photon is moving at angle relative to the axis, then assuming it is moving in the plane, its momentum is

Since , we use the Lorentz transformation for an observer moving at speed along the axis to get for (be careful not to confuse the symbol (Greek lowercase nu) with ):

In the special case where the photon is moving along the axis, and the formula becomes

If , this formula gives a red-shift, since . If , the direction of motion of the observer relative to the light is reversed and we get a blue-shift. Note that it is impossible for the Doppler shift to reduce a photon’s frequency to zero, since this would require , and relativity forbids anything except massless particles (photons, mainly) from moving at the speed of light.

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[...] direction to the photon, the photon will appear blue-shifted towards a higher frequency, by the Doppler factor (assuming ) [...]

[...] Note that we could have just applied the Lorentz transformation to the final form of ; we didn’t need to work out the transformations on each photon separately. However it’s interesting to see how the two photons transform. The energy of the photon travelling to the right is reduced by a factor of , while that of the photon moving to the left is increased by a factor of . Since the speeds of the photons remain unchanged (the speed of light), this change in energy is reflected in a change of their wavelengths. This is of course the Doppler effect. [...]

[...] Doppler effect can be derived from the Lorentz transformation of momentum for a photon. The energy and wavelength [...]

[...] on the ground is effectively moving towards the light source with speed so the light will appear blue-shifted, with the relation between the emitted wavelength and observed wavelength [...]