# Non-linear Perturbations of Minkowski Spacetime

I am reading some of the following paper on the bounded $$L^2$$ conjecture in general relativity where it mentions non-linear perturbations of the Minkowski metric in the context of quasilinear wave equations.

Is it possible for any spacetime manifold (at least in principle) to be represented by a non-linear perturbation of flat Minkowski space?

The answer is negative (if I correctly understand the question) for topological reasons: you cannot change the topology of a spacetime simply changing the metric. There is no chance, for instace, to pass this way from Minkowski spacetime (diffeomorphic to $$\mathbb{R}^4$$) to Kruskal (diffeomorphic to $$\mathbb{R}^2\times \mathbb{S}^2$$).