# Photonbundle in two different inertial frames

1. Dec 5, 2014

### VVS

1. The problem statement, all variables and given/known data
A point P is emitting a bundle of photons. In inertial system I the opening angle of the bundle is $d\Omega$ which hits a plane at a distance R. The same process is being observed by an inertial frame I' moving wrt. I.
Show that: $R^2d\Omega=R'^2d\Omega'$

2. Relevant equations / The attempt at a solution
I know the general Lorentz boost is given by the boost matrix:
$L_v=\begin{bmatrix} \gamma & -\gamma v^T \\ -\gamma v & \mathbb{1}+\frac{\gamma^2}{1+\gamma}vv^T \end{bmatrix}$
In two dimensions I can choose $v=v\begin{bmatrix}cos(\theta) \\ sin(\theta)\end{bmatrix}$
Then I can apply the Lorentz boost onto the origin $\vec{0}$, at the center of the plane $\vec{R}$ and the edge of the light bundle at the plane $\vec{P}$.
To find the new angle $d\Omega'$ I can use the following equation $d\Omega'=\frac{\left|\vec{P'R'}\right|}{\left|\vec{R'0'}\right|}$
However there is a problem because of the relativity of simultaneity. I don't know how to go about it from here.

2. Dec 10, 2014