$SO(2,1)$ and Hyperboloids

The polynomial $J(x,y,z)=x^2+y^2-z^2$ is a quadratic form of signature (2,1), because in diagonal form (already achieved) it has 2 positive coefficients and 1 negative one. This can be written in matrix form as

$$J(x,y,z) =\begin{bmatrix}x&y&z\end{bmatrix}\begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&-1\end{bmatrix}\begin{bmatrix}x\\ y\\ z\end{bmatrix}$$

We also use $J$ to represent this $3\times3$ matrix. Thus, with $\mathbf{x}=\begin{bmatrix}x\\ y\\ z\end{bmatrix}$, the quadratic form is $\mathbf{x}^T J\mathbf{x}$, where $\mathbf{x}^T$ is the transpose of the column vector $\mathbf{x}$.

The standard right circular cone is the zero set of $J$, namely, all $\mathbf{x}$ such that $\mathbf{x}^T J \mathbf{x}=0$. The surface defined by the equation $\mathbf{x}^T J \mathbf{x}= x^2+y^2-z^2 =-1$ is a smooth surface with two connected components, one in $z>0$ and one in $z<0$. The surface can be obtained by the revolution of a hyperbola around an axis of symmetry, and so is called a two-sheeted hyperboloid The intersections of planes with this hyperboloid can also be classified as ellipses, hyperbolas and parabolas, with the exception of some degenerate cases. Here is a picture of the upper sheet of the hyperboloid inside the cone.

The set of coordinate transformations in $\mathbb{R}^3$ that preserve the standard cone form a group called the orthogonal group of signature (2,1), denoted $O(2,1)$. These transformations correspond to $3\times3$ matrices $A$ such that $A^T JA=J$. These also preserve the two-sheeted hyperboloid $\mathbf{x}^T J\mathbf{x}=1$, although they may interchange the two sheets. The subgroup of transformations that also preserve each sheet of $\mathbf{x}^T J\mathbf{x}=1$ is known as the special orthogonal group of signature (2,1), written $SO(2,1)$. We shall see how this group is a version of the group of Möbius transformations that preserve the unit disk.

Since $A^T JA=J$, by taking determinants we can conclude that $\det A=\pm1$. The special orthogonal group $SO(2,1)$ can be shown to be the subgroup of $O(2,1)$ consisting of matrices $A$ with determinant 1.

We shall denote the upper sheet of the hyperboloid $\{(x,y,z)\mid z>0,\quad x^2+y^2+1=z^2\}$ by $\mathcal{H}$. We won't delve further into these questions, but it can be determined that the geodesics on the surface $\mathcal{H}$ with respect to the signature (2,1) metric are the intersections of $\mathcal{H}$ with planes $ax+by+cz=0$. This means that, if we measure the distance between two points by using the quadratic form $x^2+y^2-z^2$, then the shortest curve (``geodesics'') between two points $P$ and $Q$ on the hyperboloid is the intersection of the hyperboloid and the plane determined by the origin and $P$ and $Q$. These curves are always hyperbolas.