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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 $\zeta^{1,1}.$

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    $\begingroup$ You should edit the question so that it's self-contained: people should be able to understand it just from this post. $\endgroup$
    – Javier
    Apr 16 '20 at 19:35
  • $\begingroup$ 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. $\endgroup$ Apr 16 '20 at 20:21
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    $\begingroup$ Actually, what is your question? $\endgroup$ Apr 16 '20 at 20:26
  • $\begingroup$ @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. $\endgroup$ Apr 16 '20 at 20:51