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The unit 2-sphere is often used, pedagogically, to help provide some visual intuition about topology, differential geometry and geometric objects and properties like curvature, geodesic, smoothness, Lie algebra, tangent vector spaces, singularity, homotopy group, Killing form, embedding, etc.

In the context of general relativity and cosmology, what are the limitations of usefulness of the 2-sphere in helping visual intuition about higher dimensions such as three or four and more general curved spaces and complicated geometries.

In particular when imagining the 2-sphere as a simple model of spacetime (not space) in what ways it will be misguiding say in differential geometry?

For example,

Given the Lorentzian metric with signature (-1, 1) on the 2-sphere, geodesics will behave differently from what we typically associate with the Riemannian geometry of the 2-sphere and great circles.

The question is how they behave or (look like intuitively)?

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    $\begingroup$ This question seems opinion-based and thus off-topic. The 2-sphere is a simple Riemannian manifold, but whether it is useful for understanding cosmology doesn’t have an objective answer. $\endgroup$
    – Ghoster
    Aug 25, 2023 at 23:08
  • $\begingroup$ A little edited, is it clear to answer objectively? $\endgroup$
    – VVM
    Aug 26, 2023 at 0:08

2 Answers 2

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As any analogy it can be useful, but the maths are not the same (between 2-sphere and 1+1 space-time). So we can not rely too much on it.

For example, a fly route between San Francisco and Washington DC, is a curve in a plane map. If we relate longitude as time and latitude as height, it can describe qualitatively the graph of a stone thrown upward.

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the Lorentzian metric with signature (-1, 1) on the 2-sphere

The assumption is wrong!

We can't get a Lorentzian metric on the 2-sphere.

So, the 2-sphere is useful for the visual intuition about the curvature of space only.

But the 2-sphere is useless and misleading for the visual intuition about the curvature of spacetime.

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