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I have some questions about the curvature of space (NB not of spacetime) near a planet like Earth. Unambiguously defining space curvature requires choice of a coordinate system, so I choose the Swarzschild system. Here are my questions:
Thank you very much for any answers.
Andrew
- Would constant-time hypersurfaces under the Swarzschild metric exhibit spatial curvature near Earth?
- Given current equipment accuracy, how large a triangle would be needed to reliably indicate the existence of such curvature by observing the angle excess of the triangle? If the triangle needs to be so large as to not fit within the Earth's atmosphere, I imagine using three laser beams making the sides of the triangle, fired from space stations that are at the triangle's vertices.
- Has such an experiment been conducted, and did it verify the spatial curvature of the hypersurfaces?
- If the answer to 1 is Yes, is there any coordinate system S that produces a foliation under which there is a nonzero open interval ##I = (t1,t2)## of time and a nonzero open volume ##V## of space such that, for all ##t\in(t1,t2)## the region ##\{t\}\times V## is spatially flat? In short, is there a coordinate system that can flatten out the spatial curvature within a nonzero volume and a finite time interval?
- Is there a more appropriate coordinate system than the Swarzschild for this question? If so, what is it and what would the answers to 1-3 be for that?
Thank you very much for any answers.
Andrew