Decomposition of restricted Lorentz transformation matrix

Question

Please can someone give the proof of all restricted Lorentz transformation matrix can uniquely decomposed into product of pure rotation and pure boost? Or at least give some hint how to proceed to prove it?

Answer 1

If I understood your question, what you should do is write the Lorentz transformation $\Lambda$ as the exponential of a matrix $e^{\mathbb{L}}$, as is the form of the transformation of a Lie group and from the generic condition of all Lorentz transformation $\Lambda^T\mathbb{G}\Lambda=\mathbb{G}$ you should arrive to prove that $(\mathbb{G}\mathbb{L})^T=-\mathbb{G}\mathbb{L}$, where $\mathbb{G}\doteq(G_{\alpha\beta})$ is the metric tensor (while $\mathbb{L}$ matrix has components of the form ${{\mathbb{L}}^\alpha}_\beta$). You should conclude that the matrix has this form $$ \mathbb{L} = \begin{pmatrix} 0&{\mathbb{L}^0}_1&{\mathbb{L}^0}_2&{\mathbb{L}^0}_3 \\ {\mathbb{L}^0}_1&0&{\mathbb{L}^1}_2&{\mathbb{L}^1}_3 \\ {\mathbb{L}^0}_2&-{\mathbb{L}^1}_2&0&{\mathbb{L}^2}_3 \\ {\mathbb{L}^0}_3&-{\mathbb{L}^1}_3&-{\mathbb{L}^2}_3&0 \end{pmatrix} $$ and that's it. The matrices associated to ${\mathbb{L}^0}_1,{\mathbb{L}^0}_2,{\mathbb{L}^0}_3$ parameters are the generators of the boosts, the others represent the generators of rotations.

Edit: \begin{gather*} \boldsymbol{\beta} \doteq \left( {\mathbb{L}^0}_1, {\mathbb{L}^0}_2, {\mathbb{L}^0}_3 \right), \, \boldsymbol{\alpha} \doteq \left( -{\mathbb{L}^1}_2, {\mathbb{L}^1}_3, -{\mathbb{L}^2}_3 \right) \\ \mathbb{K}_1 \doteq \begin{pmatrix} 0&1&0&0 \\ 1&0&0&0 \\ 0&0&0&0 \\ 0&0&0&0 \end{pmatrix}, \, \mathbb{K}_2 \doteq \begin{pmatrix} 0&0&1&0 \\ 0&0&0&0 \\ 1&0&0&0 \\ 0&0&0&0 \end{pmatrix}, \, \mathbb{K}_3 \doteq \begin{pmatrix} 0&0&0&1 \\ 0&0&0&0 \\ 0&0&0&0 \\ 1&0&0&0 \end{pmatrix} \\ \mathbb{J}_1 \doteq \begin{pmatrix} 0&0&0&0 \\ 0&0&0&0 \\ 0&0&0&-1 \\ 0&0&1&0 \end{pmatrix}, \, \mathbb{J}_2 \doteq \begin{pmatrix} 0&0&0&0 \\ 0&0&0&1 \\ 0&0&0&0 \\ 0&-1&0&0 \end{pmatrix}, \, \mathbb{J}_3 \doteq \begin{pmatrix} 0&0&0&0 \\ 0&0&-1&0 \\ 0&1&0&0 \\ 0&0&0&0 \end{pmatrix} %\end{drcases} \\ \mathbb{L} \equiv \beta^i\mathbb{K}_i +\alpha^i\mathbb{J}_i \end{gather*} As you can see $\boldsymbol{\alpha}$ are the parameters of rotation, $\boldsymbol{\beta}$ of boost. The following commutation relations hold \begin{gather*} \begin{cases} \left[ \mathbb{K}_i, \mathbb{K}_j \right] = -{\epsilon_{ij}}^k \mathbb{J}_k \\ \left[ \mathbb{J}_i, \mathbb{J}_j \right] = {\epsilon_{ij}}^k \mathbb{J}_k \\ \left[ \mathbb{J}_i, \mathbb{K}_j \right] = {\epsilon_{ij}}^k \mathbb{K}_k \end{cases} \end{gather*}