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# Polar decomposition of Lorentz transforms

Theorem : For any restricted Lorentz transformation $\Lambda \in \mathrm{SO}^+(n-1,1)$, and a given timelike vector $u_0$, there exists a unique boost $B \in \mathbb{R}^{n-1}$ and rotation $R \in \mathrm{SO}(n-1)$ such that

$$\Lambda = B R$$

such that $R$ is in the plane of $u_0$ and $R$'s plane is orthogonal to $u_0$.