Physlib.Relativity.LorentzGroup.Rotations

For a given natural number $d$ representing the number of spatial dimensions, the subgroup of rotations in the Lorentz group $\mathcal{L}$ consists of those Lorentz transformations $\Lambda$ such that the time-time component of the matrix is $\Lambda_{0,0} = 1$ and the transformation is proper, i.e., $\det \Lambda = 1$.

For any Lorentz transformation $\Lambda$ in the Lorentz group with $d$ spatial dimensions, $\Lambda$ belongs to the subgroup of rotations if and only if its time-time component is $\Lambda_{0,0} = 1$ and it is a proper transformation (i.e., $\det \Lambda = 1$).

For any Lorentz transformation $\Lambda$ in the Lorentz group with $d$ spatial dimensions, the matrix transpose $\Lambda^T$ is an element of the rotation subgroup if and only if $\Lambda$ itself is an element of the rotation subgroup.

For a given number of spatial dimensions $d$, this definition establishes a group isomorphism between the special orthogonal group $SO(d)$ and the subgroup of rotations in the Lorentz group $\mathcal{L}$. Under this isomorphism, a matrix $A \in SO(d)$ is mapped to the $(1+d) \times (1+d)$ Lorentz transformation matrix $\Lambda$ defined by the block form: \[ \Lambda = \begin{pmatrix} 1 & \mathbf{0} \\ \mathbf{0} & A \end{pmatrix} \] where $1$ is the time-time component and $A$ corresponds to the spatial transformation. The inverse map extracts the $d \times d$ spatial block from a Lorentz rotation.

For a given number of spatial dimensions $d$, the map $\text{ofSpecialOrthogonal} : SO(d) \to \text{Rotations}_d$, which embeds the special orthogonal group $SO(d)$ into the rotation subgroup of the Lorentz group $\mathcal{L}$ via the block form \[ A \mapsto \begin{pmatrix} 1 & \mathbf{0} \\ \mathbf{0} & A \end{pmatrix}, \] is continuous with respect to the subspace topologies inherited from the spaces of real matrices.

For any natural number $d$, let $\text{Rotations}(d)$ be the subgroup of rotations within the Lorentz group and $SO(d)$ be the special orthogonal group of $d \times d$ real matrices. The inverse of the group isomorphism $\text{ofSpecialOrthogonal}$, which maps a Lorentz rotation $\Lambda \in \text{Rotations}(d)$ to its corresponding $d \times d$ spatial block $A \in SO(d)$, is continuous with respect to the subspace topologies inherited from the respective matrix spaces.

For any natural number $d$ representing the number of spatial dimensions, the subgroup of rotations in the Lorentz group is a subgroup of the restricted Lorentz group $\mathcal{L}_+^\uparrow$.

For any rotation $\Lambda$ in the subgroup of rotations of the Lorentz group with $d$ spatial dimensions, the image of the unit temporal basis vector $e_0$ (the basis vector corresponding to the time dimension) under the transformation $\Lambda$ is equal to $e_0$. That is, $\Lambda e_0 = e_0$.