Doppler shift derivation from Lorentz Transformations

[RyanP]

Homework Statement

Show how one can obtain the Doppler transformation for the frequency of a receding
source just using the Lorentz transformations for the energy (where E=h).

Homework Equations

Relativistic transformations for momentum and energy:
E = γ(E' + vp'_x)
pc/E = v/c = β

The Attempt at a Solution

E = γ(E' + vp'_x)
since p = Eβ/c, E = γ(E' + Eβ²) = γE' (1+β²).

Which then would imply that = γ'(1+β²). But I have in my notes that the observed frequency = γ' (1-β). Where did I go wrong?

 

[vela]

The speed $\beta = \frac{pc}{E} = 1$ for electromagnetic radiation. It's not the same as $v$, the speed between the two reference frames.

 

[RyanP]

The speed $\beta = \frac{pc}{E} = 1$ for electromagnetic radiation. It's not the same as $v$, the speed between the two reference frames.

That makes sense, thanks.