Determining Conformal Map from Conformal Factor

Question

I am working on a project and need to apply a certain (Lorentzian) conformal transformation, $\psi:\mathbb{R}^2\mapsto \mathbb{R}^2$, to a figure which I have generated numerically in Mathematica. I know that the new metric will be conformally flat. Hence, I know that the pullback metric will be $g_\text{ab}=\psi^*\eta_\text{ab}=\phi(t,x)^2 \eta_\text{ab}$ for some conformal factor $\phi(t,x)$. Fortunately, I know this conformal factor everywhere at least numerically. Is there an easy way to determine either the conformal map $\psi:\mathbb{R}^2\mapsto \mathbb{R}^2$ or its inverse given that I know the conformal factor, $\phi(t,x)$, at every point?

Answer 1

I figured it out. The Lorentz conformal transformations in the plane are actually very restrictive. See this paper. The trick is to work in null coordinates.