Lorentz transformations that preserve the direction of time are called "orthochronous." [Lorentz group]
I want to reformulate the orthochronous transformations in $\Bbb R^{1,1}:=$ (2-dim. Minkowski space), to a different space I call $\zeta_\Bbb {R^2}.$ $\zeta-$ space, is simply Minkowski space, $M-$space, but in a different coordinate system $-$ a log-log coordinate system.
For more context see: Geometry of transformed spacetimes? and Find sequential orthographic projections, linking three different manifolds of dimension $n=1,2,3$.
The lines of constant time in $\zeta$ are $\ln(x)\ln(\phi)=St$ for $t\in\Bbb R.$
This follows from: Transforming algebraic equations. Specifically, setting $u=\ln(x)$ and $v=\ln(\phi),$ let's us recover rectangular hyperbolas. This is what I mean when I say "$\zeta-$space is simply $M-$space in a log-log coordinate system."
$S$ is a generator, and generates the lines of constant time in $\zeta-$space. See this post for more on $S$ as a generator: Example of an equation that generates solutions?.
Manipulating this equation we arrive at $\phi_{St}(x)=e^{\frac{St}{\ln(x)}}$ where $\phi_{St}$ just specifies the parameter $t$ and the generator $S.$
$so(1,1)=\{2\times2~ \mathscr{matrices }~X|e^{tX}\in SO(1,1)~ \forall t\}$ is the Lie algebra of the Lorentz group.
$X$ consists of all matrices $\begin{pmatrix} 0 & a \\ a & 0 \end{pmatrix},$ where $a$ is an arbitrary real number.
So then what is this lie algebra in $\zeta-$space? I'm not exactly sure how the Lie algebra reacts to this change in coordinates.
Is it just, $so(1,1)=\{2\times2~ \mathscr{matrices }~Y|e^{tY}\in SO(1,1)~ \forall t\}$
for $Y=\exp(X)?$
