\subsection{velocity addition}
\label{ss:velocityaddition}
\begin{eqnarray*}
1/(\gamma\gamma^\prime)^2 &=& (1-{\beta^\prime}^2)(1-\beta_z^2)\\
&=&  1-\beta_z^2-{\beta^\prime}^2 +{\beta^\prime}^2\beta_z^2\\
\beta_0 &=& \frac{\gamma^\prime}{\gamma_0}({\beta^\prime}^2 + \gamma^2\beta_z^2)^{1/2}\\
1/\gamma_0^2 &=& 1-\beta_0^2 =  1-\frac{{\gamma^\prime}^2}{\gamma_0^2}({\beta^\prime}^2 + \gamma^2\beta_z^2)\\
\rightarrow 1 &=& \gamma_0^2 - {\gamma^\prime}^2({\beta^\prime}^2 + \gamma^2\beta_z^2)\\
 \gamma_0^2 &=& 1+ {\gamma^\prime}^2({\beta^\prime}^2 + \gamma^2\beta_z^2)\\
 \gamma_0^2 &=& 1+ {\gamma^\prime}^2(\frac{{\gamma^\prime}^2-1}{{\gamma^\prime}^2} + \gamma^2\frac{\gamma^2-1}{\gamma^2})\\
 \gamma_0^2 &=& 1+ ({\gamma^\prime}^2-1) + {\gamma^\prime}^2(\gamma^2-1)\\
 \gamma_0^2 &=& {\gamma^\prime}^2\gamma^2\\
\end{eqnarray*}
