# Lorentz Transformations and Aberration

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

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 $\phi$?

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