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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?

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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 $\mathbb{R}^2\times \mathbb{S}^2$).

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