How do I apply a Lorentz boost to the wave four-vector for relativistic Doppler?

[ChloeQG]

I was working through a derivation of the relativistic Doppler effect for light and got stuck on the transformation of the wave four-vector. My textbook just states the final frequency shift, but when I try to get there myself by applying a Lorentz boost to the four-vector components, the signs in my time-like component keep getting messy.

[PatrickA]

I remember chasing a sign issue when I tried to treat the wave four vector as having a time part omega over c The trick for me was to lock the metric in my head and write the boost as t prime is gamma times t minus v x over c squared Then k zero prime is gamma times k zero minus beta times k one When I did that the Doppler piece lined up with the textbook result

[Matthew.R]

One concrete thing I did was plug in a photon heading along the boost direction and watch the numbers I saw the frequency transform to gamma times the original frequency times one minus beta It matched once I used k one equal to omega over c times the direction along the boost

[Dennis.J]

Do you think the confusion comes from the sign convention for the time component or from how the wave four vector is defined

[Richard92]

Sometimes I wonder if using angular frequency instead of ordinary frequency changes the sign bookkeeping I tried to stay with angular frequency because the planck relation feels cleaner but the four vector still wants the omega component Not sure if that helps here