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\pagestyle{myheadings}
\markright{{\sf Page 11.17.6 of EM2 notes, Nov 95}}
\begin{document}

\noindent{\Large\textbf{Boost and rotation}}
\smallskip

We want to write a generic Lorentz matrix $\Lambda $ as the product $RB,$
where $R$ is a rotation and $B$ a boost. The generic boost matrix is
(Jackson, eq. (11.98)): 
\[
B_{\;\beta }^\alpha =\left( 
\begin{array}{cc}
\gamma  & -\gamma v^j \\ 
-\gamma v^i & \delta _{\;j}^i+\gamma ^2v^iv^j/(\gamma +1)
\end{array}
\right) 
\]
where $v^i$ are the Cartesian components of the boost velocity $\mathbf{v}$.
On the other hand, $R$ is of the form 
\[
R_{\;\beta }^\alpha =\left( 
\begin{array}{cc}
1 & 0 \\ 
0 & R_{\;j}^i
\end{array}
\right) 
\]
where $R_{\;j}^i$ is a three-dimensional rotation matrix. The problem is to
determine $B$ and $R,$ given $\Lambda .$

We can keep the indices consistently placed (up or down) by rewriting $B$ in
the form 
\[
B_{\;\gamma }^\beta =\left( 
\begin{array}{cc}
\gamma & \gamma v_k \\ 
-\gamma v^j & \delta _{\;k}^j-\gamma ^2v^jv_k/(\gamma +1)
\end{array}
\right) 
\]
where $v_i=-v^i$ are covariant components. We find 
\[
\left( 
\begin{array}{cc}
\Lambda _{\;0}^0 & \Lambda _{\;k}^0 \\ 
\Lambda _{\;0}^i & \Lambda _{\;k}^i
\end{array}
\right) =R_{\;\beta }^\alpha B_{\;\gamma }^\beta =\left( 
\begin{array}{cc}
\gamma & \gamma v_k \\ 
-\gamma R_{\;j}^iv^j & R_{\;k}^i-\gamma ^2R_{\;j}^iv^jv_k/(\gamma +1)
\end{array}
\right) 
\]
hence $\gamma =\Lambda _{\;0}^0,\;v_k=\Lambda _{\;k}^0/\gamma
,\;R_{\;j}^iv^j=-\Lambda _{\;0}^i/\gamma ,$ and 
\[
R_{\;k}^i=\Lambda _{\;k}^i+\gamma ^2R_{\;j}^iv^jv_k/(\gamma +1)=\Lambda
_{\;k}^i-\gamma \,\Lambda _{\;0}^iv_k/(\gamma +1) 
\]
or explicitly 
\[
R_{\;k}^i=\Lambda _{\;k}^i-\Lambda _{\;0}^i\Lambda _{\;k}^0/(\gamma +1) 
\]

\end{document}
