Doppler shift derivation from Lorentz Transformations

AI Thread Summary
The discussion focuses on deriving the Doppler transformation for the frequency of a receding source using Lorentz transformations for energy. The key equation presented is E = γ(E' + vp'x), leading to the relationship E = γE' (1 + β²). A participant notes a discrepancy between their derived frequency expression and the expected observed frequency, γ' (1 - β). Clarification is provided that the speed β, defined as β = pc/E, equals 1 for electromagnetic radiation, distinguishing it from the relative speed v between reference frames. The conversation emphasizes the importance of correctly applying Lorentz transformations in the context of relativistic physics.
RyanP
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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?
 
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The speed ##\beta = \frac{pc}{E} = 1## for electromagnetic radiation. It's not the same as ##v##, the speed between the two reference frames.
 
vela said:
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.
 
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