(not sure if this belongs here, in StackOverflow or in the physics StackExchange, but I believe my issue is with the math so I'm posting here.)
I've recently been reading the book Dichronauts by Greg Egan and I decided to make a little simulation to understand the world more. The book takes place in a universe where the distance metric is $d^2=\Delta x^2+\Delta y^2 - \Delta u^2$ where $u$ is basically just the Z coordinate (named as such due to the fact that it's a second timelike dimension, and $u$ is the letter after $t$). This has the effect of stretching out space as one rotates. I'll be simplifying to $d^2=\Delta x^2-\Delta u^2$ since my simulation only uses 2 dimensions. Doing this in both directions creates a hyperbolic rotation, which is the following matrix:
$\begin{bmatrix}\cosh\theta & -\sinh\theta \\ \sinh\theta & \cosh\theta\end{bmatrix}$
which looks like this:
Hyperbolic rotation in both directions
From what I understand, this is rotating in two timelike directions at once. In the story, however, they're rotating in 2 spacelike and 1 timelike direction (or, since we're in 2D, I've reduced it to 1 spacelike and 1 timelike.) This is what I'm unsure about; I believe the matrix looks like this:
$\begin{bmatrix}\sin\theta & -\sinh\theta \\ \cos\theta & \cosh\theta\end{bmatrix}$
thereby rotating $\hat{\imath}$ by $\theta$ and hyperbolically rotating $\hat{\jmath}$ by $\theta$. This looks like this:
This seems consistent with the transformations described in the book (and on Greg Egan's website) but I'm still confused about a few things. I thought that a hyperbola (which I've shown in purple) should be rotationally invariant, however when rotating it, it appears to stretch into a set of straight lines.
Hyperbola stretching into straight lines
Did I do my math wrong or is this just what rotational invariance looks like when we try to project this space into a euclidean one?
Also, in the book rotation by more than 45 degrees is impossible. I expected that as I increase the angle, things would just get more stretched out endlessly, similar to how doing purely hyperbolic rotation actually never rotates more than 45 degrees, just stretching things forever. However, when I try that here, space just sort of flips endlessly.
Flipping behavior at greater than 45 degrees
Is this just an artifact of how I'm using hyperbolic rotations to represent Lorentz transformations here or is this something wrong with the matrix I'm using?
I found this answer a while ago but forgot to post it until now. The correct matrix is:
$\begin{bmatrix}\cosh\theta & \sinh\theta \\ \sinh\theta & \cosh\theta\end{bmatrix}$
which exhibits the right properties.