How do you calculate the "boosts" for $\zeta^{3,1}?$ [closed]

I have the metric for $$\zeta^{3,1}$$ as $$g=\frac{dudv}{uv}-\frac{dr^2}{r^2}-\frac{dw^2}{w^2}.$$

A substitution yields: $$g=dudv-dr^2-dw^2,$$ which is $$\Bbb R^{3,1}$$ in null/light-cone coordinates.

So this tells me that the isometry group, $$\text{Iso}(\zeta^{3,1})$$ is isomorphic to the Poincaré group.

How do you calculate the "boosts" for $$\zeta^{3,1}?$$

I've been able to do this for $$\zeta^{1,1}.$$

• You should edit the question so that it's self-contained: people should be able to understand it just from this post. Apr 16 '20 at 19:35
• Your objectives do not appear consistent. Lorentz transforms apply on a flat space time. Einstein's equation determines curvature of a 4-dimensional manifold. Its solutions are not trivial. Apr 16 '20 at 20:21
• Actually, what is your question? Apr 16 '20 at 20:26
• @geocalc33 - It seems to me that the conceptual problem here is that you want to have some variant of the GR equations that make your space a vacuum solution, yet this is a totally under-determined problem and not really physics. Or, you are interested in getting a formula for the metric of the Minkowski space in your coordinates, in which case the GR equations just describe flat space. Apr 16 '20 at 20:51