There is an innumerable quantity of questions and answers on this site about spatial, spacetime, and temporal curvature. None of these answer my question.
Unfortunately, many use the terms in different, sometimes confusing ways, that make it unclear what we exactly mean by our universe's spatial curvature, spacetime curvature, and temporal curvature.
Our universe has spacetime curvature, so the spacetime version of the Pythagorean theorem doesn’t hold. Our universe does not have a Minkowski metric. But our universe does not appear to have any measurable spatial curvature, so in only the three spatial dimensions the Pythagorean theorem does hold.
Very nice explanation.
You need to be cautious about treating a time curvature and spatial curvature separately because this split is not observer-independent. and the answer is that at least two principal curvatures must be non-zero. So you cannot find a geometry/coordinate system where the curvature is only in the time coordinate.
Now, this is where it gets a little confusing. Our universe has no spatial curvature. But our spacetime does. So the curvature must be in the temporal dimension? But this says we cannot find a coordinate system where the curvature is only in the temporal dimension. So the connection between the statement that our universe and our spacetime is curved, and that there is no spatial curvature, is not trivial.
Our spacetime has intrinsic curvature. But it is not in the spatial dimensions (there is no spatial curvature), so it has to manifest in the temporal dimension?
Just to clarify, one of the answers specifically says our universe does not have spatial curvature (talks about spatial curvature separately), and the other answer talks about possible existing temporal curvature (mentioning that you have to be cautious to treat spatial and temporal curvature separately), but then they both talk about spacetime curvature.
I am looking for a connection between the spatial, temporal, and spacetime curvature.
- What is the connection between spatial, temporal, and spacetime curvature?