# Lorentz Transformations and Aberration

Why does the azimuthal angle, $\phi$, remain unchanged between reference frames in special relativity?

Edit: Here is a link from "Radiative Processes in Astrophysics": http://i937.photobucket.com/albums/ad216/Kyanise/Capture-5.jpg.

I think this comes from the aberration formula, showing dependence only on the polar angle, $\theta$. The aberration formula is:

$\tan \theta = \frac{u_\perp}{u_{||}} =\frac{ u' \sin \theta '}{\gamma (u' \cos \theta ' + v)}$

where ' indicates a property of the moving frame.

But I'm not even sure why this is true. I know it comes from the transforms of the velocities, but why should $\tan \theta = \frac{u_\perp}{u_{||}}$ anyways? I mean $v$ can be in any direction, right? How come there is no dependence on $\phi$?

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For more intuitive approach for that, is if you are familiar with the basics of Electrodynamics (the source of special relativity historically) you may know that $\vec{H},\vec{E}$ are always in the same plane, and if the observer will move in a perpendicular orientation to that plane, this will cause those vectors to rotate (the angel between them will change) but they will remain in the same plane.
Ok right this was just confusing me since if we know that there is always a coordinate system in which we can boost in the $x$ direction, then it was a bit strange that we would need to derive a general equation for when $v$ is not in the $x$ direction –  Atreyu Dec 2 '12 at 1:08