Consider a Lorentzian manifold $(\zeta,g)$ with metric $g=\frac{dudv}{uv}-\frac{dr^2}{r^2}-\frac{dw^2}{w^2}.$
Clearly $\zeta$ is diffeomorphic to Minkowski space, $\Bbb R^{3,1}.$ $\zeta$ also has zero curvature like Minkowski space.
Is $\zeta$ a solution to the Einstein field equations?
I know that Lorentzian manifolds are solutions to the Einstein field equations. But I'm wondering if $\zeta$ would still be a solution to the equations or rather some variant of the equations.