Required math: calculus
Required physics: special relativity
Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 3; Problem 3.7.
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