Physlib.Relativity.LorentzGroup.Restricted.FromBoostRotation

For an element $\Lambda$ of the restricted Lorentz group $\text{SO}^+(1, d)$ in $d$ spatial dimensions, this function returns the associated Lorentz velocity $\mathbf{v}$. This velocity is derived from the vector components of the Lorentz transformation $\Lambda$, specifically corresponding to the boost component of the transformation.

For any natural number $d$, the map $\text{toVelocity}: \text{SO}^+(1, d) \to \text{Velocity}_d$, which assigns each element of the restricted Lorentz group its associated Lorentz velocity $\mathbf{v}$, is continuous with respect to the standard topologies.

For an element $\Lambda$ of the restricted Lorentz group $\text{SO}^+(1, d)$, this function returns its associated spatial rotation $R$. The rotation is calculated by multiplying $\Lambda$ on the left by the inverse of the generalized boost $B(\mathbf{v})$ corresponding to the velocity $\mathbf{v}$ of the transformation, such that $R = B(\mathbf{v})^{-1} \Lambda$. This map effectively extracts the rotational component from the decomposition of a Lorentz transformation into a boost and a rotation.

For any natural number $d$, the mapping $\text{toRotation}: \text{SO}^+(1, d) \to \text{SO}(d)$, which assigns to each element $\Lambda$ of the restricted Lorentz group its associated spatial rotation $R$ (defined such that $\Lambda = B(\mathbf{v})R$ for a boost $B(\mathbf{v})$), is continuous with respect to the subspace topologies inherited from the respective spaces of real matrices.

For a given spatial dimension $d$, there is a homeomorphism between the restricted Lorentz group $SO^+(1, d)$ and the Cartesian product of the space of Lorentz velocities $\text{Velocity}_d$ and the special orthogonal group $SO(d)$. The map assigns to each restricted Lorentz transformation $\Lambda \in SO^+(1, d)$ a pair $(\mathbf{v}, R)$, where $\mathbf{v}$ is the velocity vector extracted from the transformation's boost component and $R$ is the corresponding spatial rotation. The inverse of this map decomposes any restricted Lorentz transformation into the product $\Lambda = B(\mathbf{v})R$, where $B(\mathbf{v})$ is the generalized boost corresponding to velocity $\mathbf{v}$ and $R$ is a rotation in $SO(d)$.