When calculations of spacetime curvature are rendered for a given situation, the resultant curvatures are obtained with respect to Minkowski spacetime.... Flat. What evidence exists that the underlying spacetime is, in fact, flat, and what would be the ramifications if the underlying spacetime is curved?
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$\begingroup$ It isn't clear what you are asking. When we calculate curvature we are calculating the Riemann curvature tensor and this is not relative to anything, except possibly in the sense that the Riemann tensor is zero for a flat spacetime. If you're asking what would happen if our universe was not (spatially) flat the answer is that there would be a cosmological constant. $\endgroup$– John RennieCommented Dec 24, 2020 at 7:20
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2$\begingroup$ Your premise is entirely incorrect. $\endgroup$– m4r35n357Commented Dec 24, 2020 at 9:28
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1$\begingroup$ This may just be a misunderstanding of the concept of curvature applied to general manifolds (and the difference between intrinsic and extrinsic curvature). In general relativity we're usually talking about Riemann curvature en.wikipedia.org/wiki/Riemann_curvature_tensor - we don't need to make reference to an embedded space to talk about that (i.e. it's intrinsic). We're not saying anything about 'underlying' spacetime, but we understand the geometry of flat space, and we can calculate how much a manifold fails to be isometric to a hypothetical flat space. $\endgroup$– EletieCommented Dec 24, 2020 at 9:31
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2 Answers
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Spacetime curvature is not calculated with respect to a reference spacetime (Minkowski or other). Curvature only depends on the properties of your given spacetime (metric and connection).
I don't know what you mean by "underlying spacetime". Do you mean that we calculate the extrinsic curvature of a spacetime embedded in some higher-dimensional Minkowski spacetime? This is not what we usually do in GR.
What might be the cause of your confusion is that we assume that the signature of spacetime is that of Minkowski. This is an assumption of GR, and reflects the fact that freely falling observers experience locally flat Minkowski spacetime.
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You are mistaken. There is no fixed background geometry in GR. Calculations of curvature are made using only internal measurements (which is obvious, since we cannot go out of the universe and measure things from there). Gauss and Riemann showed that it is possible to measure curvature and other geometric features of a manifold only by doing internal measurements in the manifold. For example on Earth, you can prove its non-flatness by making careful geodesy measurements (eg. the sum of the angles of a large triangular region is more than 180 degrees). You do not need to go out in space to do that. What is used in GR is thus the intrinsic curvature calculated with the Rieman tensor. There is also the concept of extrinsic curvature (for example a piece of paper slightly bent), but in GR we are not interested in that. It would not make sense physically.
