# Local curvature in spacetime and spacetime symmetries?

Correct me if I'm wrong, but as far as I understand fundamental symmetries like the Lorentz invariance and the general covariance principle do not hold globally in a curved spacetime. But because it can be approximated into a flat Minkowskian space, the invariances hold just fine locally.

However, what would happen if that curvature was also present at local scales? Would the general covariance as well as the local Lorentz invariance still hold? Would they hold in highly curved spacetimes, such as the Gödel universe metric?

• In think the definition of "local" is the region where Minkowski space works.
– JEB
Mar 18 at 22:15
• @JEB could you elaborate that a bit more? Mar 19 at 3:23
• Schutz proves the "local-flatness theorem" in the curved manifolds chapter of his GR book. Carroll gives a sketch of the proof here around Eq. 2.34. In a sense, at any point on a Riemannian manifold there are coordinates where the metric looks like Minkowski space, and the first partial derivative of the metric components vanish. This is how "local" is typically defined in GR. Mar 19 at 4:14
• @Aiden but can there be spacetimes where there is no way to have a flat "minkowskian" region? Mar 19 at 9:09
• @vengaq The theorem states you can always find coordinates at any point where the spacetime is locally Minkowskian. Its proof isn’t from physics, but the coordinate degrees of freedom inherent to your smooth manifold. In other words, if you want to have local regions that don’t look like Minkowski space, you cannot work on a (pseudo-) Riemannian manifold which is where GR is typically defined. You would need an alternative theory on a different kind of space. Mar 19 at 14:55