#017: Is space flat?

Category: General relativity - Tags: Differential geometry, Riemannian geometry

In general relativity, space-time is curved, but it is locally flat. The question is whether, at a fixed time, space is globally flat.

Mathematical problem. For a pseudo-Riemannian manifold with signature (-,+,+,+), the metric tensor can always be diagonalized at each point. This is what is meant by “space is locally flat”. In general, the metric tensor cannot be diagonalized on the whole space. We can do that only if the curvature vanishes.

The question is whether the spatial part of the metric tensor can be diagonalized. That is, \(g_{\alpha\beta}\) would not be diagonalized, but the spatial part would be such that \(g_{ij} = \delta_{ij}\).

If it can’t be done globally, it is still interesting if it can be done on an open neighborhood of each point.

Physical significance. A way to picture space-time curvature is to imagine a rectangular grid all around yourself, which gets distorted as time flows. More concretely, imagine a rectangular grid formation of spaceships. In the absence of gravity, the formation would stay rectangular, while in the presence of gravity the spaceships would drift apart.

The question is whether this picture works in all cases. That is, can we always put our formations in a rectangular grid at a particular time?