Physical transformation associated with a Pseudo-Orthogonal matrix [closed]

Question

An orthogonal matrix $O$, which belongs to an orthogonal group, is characterized as $O^TO=I$. Let's take an example of a $2 \times 2$ orthogonal matrix, $$O = \begin{bmatrix} \phantom{-} \cos{\theta} & \sin{\theta} \\ - \sin{\theta} & \cos{\theta} \end{bmatrix}, ........(1)$$ Physically, The matrix $O$ represents the rotational transformation, which describes the rotation in a 2D plane. This is well known among the researchers.

Now coming to less-popular Matrices/Group:

A pseudo-orthogonal matrix $\cal{D}$, which belongs to a pseudo-orthogonal group, is characterized as $ \cal{D}^T \eta \cal{D}=\eta ,$ where $\eta$ is some constant metric.

Can anyone please give me a example of matrix $ \cal{D}$ like in Eq.(1)??.

What physical transformation represents the matrix $\cal{D}$ (e.g. $O$ represents the rotation in 2D plane) ??.

Answer 1

A well-known physical use of an indefinite/pseudo orthogonal group $O(p,q)$ is the Lorentz group $O(n,1)$ in SR.

Answer 2

After other's suggestions,

What I found is:

Orthogonal matrix $O$ (orthogonal group) which corresponds to a circular rotation. Similarly there exist a pseudo-orthogonal matrix $\cal D$ in pseudo orthogonal group, which corresponds to a hyperbolic rotation. $\cal D$ is as-

$$\cal D = \begin{bmatrix} \phantom{-} \cosh{\theta} & \sinh{\theta} \\ \sinh{\theta} & \cosh{\theta} \end{bmatrix},$$

which is pseudo-orthogonal as $\cal D^t \eta \cal D=\eta$, where $\eta$ = diag(1,-1).