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In Riemannian geometry (i.e. GR) each point has some local invariants associated to it, a famous one being curvature. "Local" here means that it only depends on
by joppy 5y ago
In Riemannian geometry (i.e. GR) each point has some local invariants associated to it, a famous one being curvature. "Local" here means that it only depends on the metric in an arbitrarily small disc/ball around the point. So the answer is no, in GR you can't always change coordinates so that space is locally flat. (This is in contrast to symplectic geometry for example, where each point does look locally like the model space).
- ithinkso 5y agoOk, yeah. We obviously can't get rid of a curvature. I think SE answer [1] explains what I assumed incorrectly to mean 'locally flat'. I meant it so that you can make a metric look like Minkowski metric at a point p, basically existence of normal coordinates. Of course second derivatives (and thus curvature) won't vanish in the neighborhood. Case closed, thank you. [1] https://physics.stackexchange.com/a/157019 https://physics.stackexchange.com/a/157019