Hi,(adsbygoogle = window.adsbygoogle || []).push({});

I'm trying to understand a derivation / question for my revision. It has to do with the four momentum of a photon and doppler/redshifts.

1. The problem statement, all variables and given/known data

Show that ratio between emitted and received frequencies (for two observers E and R moving with velocities u and v respectively) is given by

[itex] \frac{\nu_E}{\nu_R} =\frac{\textbf{u} \bullet\textbf{k}}{\textbf{v}\bullet\textbf{k}}[/itex]

2. Relevant equations

My first issue is deriving the four-wavevector for a photon. I know this is related to the four-momentum by p = hbar * k. Which one do I derive first? Assuming I'm given the four-wavevector

[itex] k^a = \left( \omega / c, \vec{k}\right) [/itex]

Then I also proceed with the knowledge that scalar products of four-vectors are invariant under frame transformations

3. The attempt at a solution

Transform to emitter's frame (E). In this frame

[itex][u^{a}]=\left(c,0,0,0\right)[/itex]

and

[itex][k^{a}_E]=\left(\omega_E , \vec{k_E}\right)[/itex]

Taking the scalar product gives [itex]\textbf{u} \bullet \textbf{k}_{E} = \omega_{E}[/itex]

Performing similar steps gives

[itex]\textbf{u} \bullet \textbf{k}_{R} = \omega_{R}[/itex]

This bit I'm unclear on. [itex]\textbf{k}_E = \textbf{k}_R[/itex]

With this, we divide both scalar products giving the answer.

Elsewhere in my notes I have a comment saying that "metric is independent of 't', therefore [itex]\textbf{p}_0[/itex] at time of emission is the same as at the time of reception"

This is related so could someone comment on how this fits into the overall picture.

Thanks

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Four-Momentum of Photon / Doppler shift in General Relativity

Can you offer guidance or do you also need help?

**Physics Forums | Science Articles, Homework Help, Discussion**