# General relativity: is curvature of spacetime really required or just a convenient representation?

I'm not really far into the general theory of relativity but already have an important question: are there formulations that can do without spacetime curvature and describe the general theory of relativity/all associated gravitational effects in global cartesian coordinates? Idea: Einstein chose spacetime curvature so that one had not to build gravitational effects into the rest of physics equations like Maxwell's equations. I assume that spacetime curvature provides a convenient way to apply physics laws not related to gravity unmodified locally because at small distances we may use special relativity as an approximation.

Please correct me everywhere I am wrong.

## 1 Answer

The curvature of spacetime is a property of the spacetime manifold itself, it is not related to any particular choice of coordinates. The very essence of relativity lies in the Einstein field equations, which, colloquially, tell you that the energy-matter content of spacetime determines its curvature and metric. There is no formulation of general relativity known to me that avoids the idea of curved spacetime.

But this is not worrying - generally, such a geometric theory of physics is the kind we want. For classical mechanics, such a geometric picture is given by the Hamiltonian formalism and its symplectic manifolds. For gauge theories describing the other fundamental forces except for gravity, the geometric picture is given by the theory of principal bundles and the (e.g. electromagnetic) physical fields are amenable to be described as the curvature of such bundles (and wonderfully analogous to formulating GR with jet and frame bundles). For gravity, the geometric picture is that it is spacetime itself that has curvature. We don't even want to get rid of a description in terms of curvature, generally speaking.

Just because you can locally choose frames such that, at least at one point, the metric is flat, and hence you have SR, this is different from saying that the curvature is not required - a spacetime is flat only when there is a choice of coordinates such that the metric is flat everywhere.

• I am reading your answer to is curvature of spacetime really required ?. You say, "We don't even want to get rid of a description in terms of curvature". Why won't we? Why not "imbed" this curved space in a flat space of higher dimension, and instead of curvature (that we'll have further to explain why is the space curved), we can speak of masses and forces of attraction between them? It's more materialistic. – Sofia Jan 13 '15 at 21:55
• @Sofia: Embedding curved spaces in flat space doesn't get rid of the fact that they are curved (a sphere is still curved after embedding it in $\mathbb{R}^3$). And I say that "we" don't want to get rid of the curvature because geometric theories like gauge theories and GR perfectly show how "forces" arise from an action principle - which is aesthetically satisfying and amenable to quantization. I see no advantage in trying to get away from geometric theories, since they are a success story (e.g., I think of gauge theories here, and how they solved the confusion that was the particle zoo). – ACuriousMind Jan 14 '15 at 17:32