I initially posted this question in the Beyond the Standard Model forum since diffeomorphism invariance is a key ingredient of loop quantum gravity but it was suggested that I post the question here. Why is Einstein's theory diffeomorphism invariant? A diffeomorphism is basically a map of the points of the manifold into other points of the manifold so the points are actually moved and the manifold gets distorted (which is totally different than a coordinate transformation in which the points are not changed but simply relabelled). At first, it seems impossible for GR to be invariant under an arbitrary mapping of the points. Consider a region of space very far from any mass/energy distribution. GR would say that this region is near flat. But if we can perform an arbitrary mapping of the points, what prevents us from transforming this region into a region where there is a nobn-zero curvature? (which would clearly violate Einstein's equations). So does it mean that there is some well-defined restriction on the allowed mappings? But this never seems to be mentioned... Or am I completely misunderstanding what diffeomorphism invariance means?