Page 11.17.6 of EM2 notes, Nov 95 1

Boost and rotation

We want to write a generic Lorentz matrix Λ as the product RB, where R is a rotation and B a boost. The generic boost matrix is (Jackson, eq. (11.98)):

B_{β}^{α} = ( \begin{matrix}γ & −γv^{j} \\ −γv^{i} & δ_{j}^{i} + γ^{2}v^{i}v^{j}/(γ + 1) \end{matrix} )

where v^{i} are the Cartesian components of the boost velocity v. On the other hand, R is of the form

R_{β}^{α} = ( \begin{matrix}1 & 0 \\ 0 & R_{j}^{i} \end{matrix} )

where R_{j}^{i} is a three-dimensional rotation matrix. The problem is to determine B and R, given Λ.

We can keep the indices consistently placed (up or down) by rewriting B in the form

B_{γ}^{β} = ( \begin{matrix}γ & γv_{k} \\ −γv^{j} & δ_{k}^{j} − γ^{2}v^{j}v_{k}/(γ + 1) \end{matrix} )

where v_{i} = −v^{i} are covariant components. We find

( \begin{matrix}Λ_{0}^{0} & Λ_{k}^{0} \\ Λ_{0}^{i} & Λ_{k}^{i} \end{matrix} ) = R_{β}^{α}B_{γ}^{β} = ( \begin{matrix}γ & γv_{k} \\ −γR_{j}^{i}v^{j} & R_{k}^{i} − γ^{2}R_{j}^{i}v^{j}v_{k}/(γ + 1) \end{matrix} )

hence γ = Λ_{0}^{0}, v_{k} = Λ_{k}^{0}/γ, R_{j}^{i}v^{j} = −Λ_{0}^{i}/γ, and

R_{k}^{i} = Λ_{k}^{i} + γ^{2}R_{j}^{i}v^{j}v_{k}/(γ + 1) = Λ_{k}^{i} − γ Λ_{0}^{i}v_{k}/(γ + 1)

or explicitly

R_{k}^{i} = Λ_{k}^{i} − Λ_{0}^{i}Λ_{k}^{0}/(γ + 1)