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The interpretation usually given for Gauss' Theorema Egregium is that a being living on a surface with intrinsic curvature can detect that curvature just by walking around the surface and measuring angles and distances. With the advent of general relativity many years later, it became evident that the our 3+1 dimensional universe need not be flat, and the Theorema Egregium is the basis for several experimental efforts to measure the curvature of the universe.

Question: Did it ever occur to Gauss or other geometers of that era to measure the curvature of the universe?

[Bonus: If not, who was the first to propose that the universe might be curved? Einstein?]

I imagine there may have been a conceptual difficulty due to the fact that most differential geometry done in the early days was restricted to curves and surfaces, rather than 3-manifolds. But even in lieu of the proper mathematical machinery, I imagine someone may have realized that an analogous sort of curvature to the Gauss curvature of surfaces should be available for 3D spaces. Did anyone propose to measure such properties of space?

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Supposedly yes, but... Two quotes from the links:

One of the most famous stories about Gauss depicts him measuring the angles of the great triangle formed by the mountain peaks of Hohenhagen, Inselberg, and Brocken for evidence that the geometry of space is non-Euclidean. It's certainly true that Gauss acquired geodetic survey data during his ten-year involvement in mapping the Kingdom of Hanover during the years from 1818 to 1832, and this data included some large "test triangles", notably the one connecting the those three mountain peaks, which could be used to check for accumulated errors in the smaller triangles. It's also true that Gauss understood how the intrinsic curvature of the Earth's surface would theoretically result in slight discrepancies when fitting the smaller triangles inside the larger triangles, although in practice this effect is negligible, because the Earth's curvature is so slight relative to even the largest triangles that can be visually measured on the surface. Still, Gauss computed the magnitude of this effect for the large test triangles because, as he wrote to Olbers, "the honor of science demands that one understand the nature of this inequality clearly". (The government officials who commissioned Gauss to perform the survey might have recalled Napoleon's remark that Laplace as head of the Department of the Interior had "brought the theory of the infinitely small to administration".) It is sometimes said that the "inequality" which Gauss had in mind was the possible curvature of space itself, but taken in context it seems he was referring to the curvature of the Earth's surface.
There is no clear documentary evidence that Gauss was actually seeking evidence of non-Euclidean geometry of physical space. Indeed, doubt has been cast by some experts on this idea: mathematician John Conway [MathForum, 1998] pointed out that a departure from Euclidean geometry large enough to be measurable on the scale of the Earth would result in massive distortions on an astronomical scale, and would have been evident long before Gauss made his measurements . Moreover, Bühler, in his biography of Gauss [Bühler, 1981], dismisses as a myth the idea that Gauss was measuring the curvature of space. He considered that the purpose of the great triangle was to act as a control to check the consistency of the measurements of the smaller triangles within it.

Another quote from the second link:

The first person to publicly propose an actual test of the geometry of space was apparently Lobachevski, who suggested that one might "investigate a stellar triangle for an experimental resolution of the question." The "stellar triangle" he proposed was the star Sirius and two different positions of the Earth at 6-month intervals. This was used by Lobachevski as an example to show how we could place limits on the deviation from flatness of actual space, based on the fact that, in a hyperbolic space of constant curvature, there is a limit to how small a star's parallax can be, even the most distant star.
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