Construct a conformal compactification, $\overline G$ of $G:=\Bbb R^{1,1}_{\gt 0}$ and/or provide a diagram of the conformal compactification of $G?$
Let $G$ have the metric tensor (not necessarily positive definite): $ds^2=\frac{dxdt}{xt}.$
This link, Penrose diagram, (under the heading "basic properties") states the relation between Minkowski coordinates $(x,t)$ and Penrose coordinates $(u,v)$ via $\tan(u \pm v)=x\pm t.$
So I have been playing around with trying to relate all three coordinates. I should note that $G$ has "null coordinates:"
I think that the coordinates for $G$ should simply be $(e^x,e^t).$ And then I'd get $\tan(e^u\pm e^v)=e^x\pm e^t.$ But this doesn't seem quite right.