Physlib.Relativity.LorentzGroup.Restricted.Basic

For a given natural number $d$ representing the number of spatial dimensions, the restricted Lorentz group is defined as the subgroup of the Lorentz group $\mathcal{L}$ consisting of transformations $\Lambda$ that are both proper and orthochronous. Specifically, an element $\Lambda \in \mathcal{L}$ belongs to this subgroup if it satisfies the conditions $\det(\Lambda) = 1$ and $\Lambda_{00} \ge 0$.

For any natural number $d$ representing the number of spatial dimensions, if $\Lambda$ is an element of the restricted Lorentz group $\mathcal{L}_+^\uparrow$, then $\Lambda$ is orthochronous, which means its $(0,0)$-component satisfies $\Lambda_{00} \ge 0$.

For any natural number $d$ representing the number of spatial dimensions, the restricted Lorentz group $\mathcal{L}_+^\uparrow$ (consisting of Lorentz transformations $\Lambda$ that are proper, $\det \Lambda = 1$, and orthochronous, $\Lambda_{00} \ge 0$) is a normal subgroup of the Lorentz group $\mathcal{L}$.

Let $\mathcal{L}$ be the Lorentz group for $d$ spatial dimensions. Let $\mathcal{L}_+^\uparrow$ denote the restricted Lorentz group, which is the subgroup of $\mathcal{L}$ consisting of transformations $\Lambda$ that are both proper ($\det \Lambda = 1$) and orthochronous ($\Lambda_{00} \ge 0$). If $\mathcal{L}_+^\uparrow$ is a connected space under the subspace topology, then $\mathcal{L}_+^\uparrow$ is equal to the connected component of the identity in $\mathcal{L}$.