**Required math: calculus**

**Required physics: special relativity**

Reference: Moore, Thomas A., *A General Relativity Workbook*, University Science Books (2013) – Chapter 3; Problem 3.7.

The Doppler effect can be derived from the Lorentz transformation of momentum for a photon. The energy and wavelength of a photon are related by Planck’s formula

where is Planck’s constant and is the wavelength.

In the rest frame of the photon’s source, we have

If we now transform to a frame moving at speed away from the source, we get

Thus the transformed energy is and the new wavelength is

which is the usual Doppler red-shift formula.

Another way of getting this is to consider the scalar product where is the four-velocity of an observer and is the four-momentum of a passing object. Since the scalar product is invariant under a Lorentz transformation, we can work it out in the rest frame of the observer, where . In this case, we get

That is, the scalar product is the energy of the object as seen by the observer.

In the case of the photon above, if we work out this product in the rest frame of the source, then in that frame and so

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