Photons,   Relativity,   Doppler shift

The "usual" doppler shift is given by v = cΔλ/λ = cz    or β = v/c = Δλ/λ =z

where Δλ = λº-λ is "observed-laboratory" wavelength,
z = Δλ/λ is the "redshift parameter" often seen in cosmology and
β = v/c is often seen in relativity notes.
rearranging, we get another less usual form   λº = λ(1+β)   or, since ν=c/λ,   ν=νº(1+β)

How do we derive the Doppler formula?

First, recall a few photon facts:

(Relativistic) energy, E=hν,
3momentum, p=E/c=hν/c=h/λ
Next, recall the 4momentum Pμ = (pº, pi) = (E/c, pi)
so the photon 4momentum in simplest form is just
Pμ=(E/c,E/c,0,0)
and we note in passing that PμPμ = m²c² = 0 -- the photon is massless.

If we boost along the x-axis we get

L(β)Pºμ = Pμ
where L(β) is the inverse Lorentz transform
| γ  +γβ|
|+γβ   γ|
(γ[Eº/c+βEº/c], γ[Eº/c+βEº/c], 0, 0) = (E/c,E/c, 0, 0)
so
E/c = γEº/c(1+β)    (Hogg's 6.39 with Q=E/c)
or
ν = νºγ(1+β)
in wavelength terms
λº = λγ(1+β)

Which apart from the γ is the same as the "usual" formula. The dilation factor γ comes from the fact that the moving atomic clock (spectral line source) "runs slow".

If the source is moving at an angle θ to the line of sight we see only the component βcosθ and the formulae are so modified:

λº = λγ(1+βcosθ)  for the relativistic formula.
---The 4momentum is eg, (E/c, E/c cos θ, E/c sin θ, 0)

But it is interesting to note that if θ=90° (moving at right angles or transverse to the line of sight) the usual formula predicts no spectral shift while the relativistic formula gives a redshift of γ. This can be observed in the laboratory and is known as the transverse Doppler shift.

More interesting is the fact that if the source is moving towards the observer then there is some speed at which the dilation (γλ so always a redshift) just cancels the expected blueshift and at higher speed the approaching source will be REDshifted!

eg, for 45° approach, θ=135°,  λº/λ=γ(1-√2/2β)=1 when β=2√2/3