# Hyperbolic Geometry - Parabolic Matrix?

In lecture we defined for hyperbolic geometry using the Lorentz model on the upper sheet of the two sheeted hyperboloid: $$Para_x=\begin{bmatrix} 1 + \frac{x^2}{2} & -\frac{x^2}{2} & x\\ \frac{x^2}{2} & 1 - \frac{x^2}{2} & x \\ x & -x & 1 \end{bmatrix}$$ where $Para$ stands for Parabolic.

For the sake of context, as an exercise (non assessed) we were asked to show that $Para_x \in O^{+}(1, 2)$, where $O^{+}(1, 2)=\left\{A|a_{11}>0, A^{T}JA=J\right\}$, $a_{11}$ is the first entry of the matrix $A$ and $$J=\begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$

My question is, what exactly is this $Para_x$? I've tried searching online for more information but I can't find this matrix defined anywhere. In case it helps, I've inputted this matrix into WolframAlpha, so some properties of the matrix can be found here.

• Better tags for this would also be appreciated if any such tags exist. – Irregular User Nov 25 '15 at 23:28
• This is the general form for a parabolic element of the Lorentz group. – Archaick Nov 25 '15 at 23:34
• Thank you very much for your answer. It turns out that my searches should have contained "Lorentz" instead of "hyperbolic". – Irregular User Nov 25 '15 at 23:39