![]() ![]() In other words, we use the mathematics of curved spacetime, but we don't actually describe anything directly in terms of Riemannian curvature. Time is a single dimension, and a one-dimensional subspace does not have any Riemannian curvature. As says in comments, "temporal curvature" does not make any sense. This is far too cumbersome for even mathematicians to think about, but what is certainly true is that you cannot separate it nicely into space curvature and time curvature. I think the essential problem lies in the difference between the mathematical meaning of curvature, and the way in which we actually describe a manifold, or a curved space (or spacetime).Īlthough we describe the universe as having spacetime curvature (which is mathematically true), curvature refers to the Riemann curvature tensor, which is a rank-4 tensor, meaning that it has $4^4 =256$ components, of which (due to various symmetries) $20$ are independent. What is the connection between spatial, temporal, and spacetime curvature?.I am looking for a connection between the spatial, temporal, and spacetime curvature. 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. But it is not in the spatial dimensions (there is no spatial curvature), so it has to manifest 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. 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. Now, this is where it gets a little confusing. How do spatial curvature and temporal curvature differ? So you cannot find a geometry/coordinate system where the curvature is only in the time coordinate. You need to be cautious about treating a time curvature and spatial curvature separately because this split is not observer-independent.Īnd the answer is that at least two principal curvatures must be non-zero. Our universe does not have a Minkowski metric.īut our universe does not appear to have any measurable spatial curvature, so in only the three spatial dimensions the Pythagorean theorem does hold. Our universe has spacetime curvature, so the spacetime version of the Pythagorean theorem doesn’t hold. 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. There is an innumerable quantity of questions and answers on this site about spatial, spacetime, and temporal curvature. ![]()
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