= Time-orientation-preserving Lorentz group in one spatial dimension
{title2=$O^{\uparrow}(1,1)$}
For $J=\operatorname{diag}(1,-1)$, the <matrices> satisfying $A^TJA=J$ and $A_{11}>0$ form the time-orientation-preserving <Lorentz group> in one spatial dimension. They are exactly $A=H(t)D_\varepsilon$, where $H(t)=\begin{pmatrix}\cosh t&\sinh t\\\sinh t&\cosh t\end{pmatrix}$, $D_\varepsilon=\operatorname{diag}(1,\varepsilon)$ and $\varepsilon=\pm1$. Preservation of the <Minkowski metric> makes the first column a future-pointing unit timelike <vector>; <Minkowski inner product> orthogonality determines the second up to sign. The identity $D_\varepsilon H(s)=H(\varepsilon s)D_\varepsilon$ gives the multiplication law $(t,\varepsilon)(s,\eta)=(t+\varepsilon s,\varepsilon\eta)$ and inverse $(-\varepsilon t,\varepsilon)$.
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