# Lorentz transformations of the polarization vector

Let $\bf{n}'$ be a unit vector in the direction of a wavevector in the plasma rest frame and $\bf{B'}$ be a unit vector along the magnetic field in the plasma rest frame. The electric field of a linearly polarized electromagnetic wave is directed along the unit vector $\hat{e}=\bf{n'} \times \hat{B'}$, and the magnetic field of the wave is along the unit vector $\hat{b'}=\bf{b}\times \hat{e'}$, such that the Poynting flux along is $\hat{e'}\times \hat{b'}$ directed along $\bf{n'}$. We give a Lorentz boost to the explosion frame to find the electric field $e$ there, normalize it to unity, and project e on some given direction (e.g. , along the projection of the flow axis on the plane of the sky).

Fields in the wave expressed in terms of the direction of a photon in the explosion frame $\bf{n}$ is

$$\hat{e'} ~=~\frac{\bf{b} \times \hat{B'}}{\Gamma (1-\bf{n}\cdot\bf{v})}+\frac{1+\Gamma (1-\bf{n}\cdot \bf{v})}{(1+\Gamma(1-\bf{n}\cdot \bf{v}))}$$

where $\Gamma=\frac{1}{\sqrt{1-v^2/c^2}}$ is the bulk Lorentz factor. Would some one help to deduce the above equation?

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Can you give some more context to the terms plasma, explosion, and the like? – Emilio Pisanty Jul 3 '12 at 14:45
I really don't know the answer to this, but I suspect it might help to start thinking about how a quantity like $\vec{A}\times \vec{B}$ where $\vec{A}$ and $\vec{B}$ are standard 3 -vectors, transforms under regular rotations, and thus build up the intuition to Lorentz transformations. – DJBunk Jul 3 '12 at 15:03
@ Emilio Pisanty, I was trying to derive this from the Ref: M. Lyutikov, V. I. Pariev, and R. D. BlandfordPOLARIZATION OF PROMPT GAMMA-RAY BURST EMISSION: EVIDENCE FOR ELECTROMAGNETICALLY DOMINATED OUTFLOW, , The Astrophysical Journal, 597:998–1009, 2003 November 10. – kallo Jul 3 '12 at 23:10