#### Hurkyl

Staff Emeritus

Science Advisor

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But people don't know the reasons!

People think that astronomers are making all these additional assumptions about how the universe must look... but what they don't realize is that the exact opposite is true; astronomers are

*removing*assumptions.

Something that people don't realize is that the "obviously correct" idea of a Euclidean space is chock full with a lot of assumptions; space looks the same on small and large scales (I mean the geometry, not necessarily the content)... squares exist... perfectly parallel lines exist... spheres have insides and outsides... lines intersect in at most one point...

All in all, when keeping with the spirit of Euclid, Hilbert axiomized euclidean geometry using

**fifteen postulates**... and that's just for the geometry of the plane! People may remember that Euclid only had 5 postulates, but what they don't realize he also included some obvious rules which amount to additional assumptions, and Euclid was even incomplete! For example from Euclid's postulates you cannot prove that a line segment with one endpoint inside of a circle and one endpoint outside of a circle actually intersects the circle!

If we take a few steps back, we realize we're making a lot of assumptions, but then again, space certainly

*looks*euclidean, doesn't it? But we realize that we're only looking at the space near us. It would make sense, then, presuming the universe is Euclidean on small scales is a much more reasonable law than presuming the entire universe is Euclidean!

And that's just what astronomers do; instead of presuming the entire universe is Euclidean, they presume it simply looks Euclidean on small scales (and may or may not look Euclidean globally). The mathematician Riemann studied geometries that look Euclidean on small scales, so today we call such things "Riemann manifolds". (they're a special type of "differentiable manifold")

However, as often happens in mathematics, the above reasoning is not

*why*Riemann was studying his manifolds! He was active in a subject called

*differential geometry*... a subject that is interseted in studying the "intrinsic geometry" of curves in euclidean space. Riemann discovered the intrinsic geometry of

*any*smooth surface in euclidean space could be described with a set of coordinates that can be defined entirely within the surface and a (possibly) noneuclidean metric on those coordinates.

So we have come to an interesting discovery; we have presumed that space looks euclidean on small scales, which means that space can be described by a riemann manifold... however, riemann manifolds arose from the study of surfaces embedded in higher dimensional space! Becuase the two concepts have identical mathematical forms, all of the mathematics involved in the study of curves can be applied to the study of the universe.

Thus, when studying space, we use terms like "geodesics" and "curvature". These terms originally appeared when studying honest to goodness surfaces in euclidean spaces, but since the study of the universe has the same mathematical form as the study of surfaces, we use the same terms in our study of the universe.

General relativity was created using a very close relative of a riemann manifold; the construction instead presumes that the geometry space looks like the geometry of special relativity on small scales. Like with special relativity, Einstein imposed a single condition on this manifold and explored where the equations led. (The condition is that inertia and gravity are the same... presumed because the exact same quantity "mass" is used both in the classical description of inertia and the classical description of gravity and thought experiments that demonstrated the two concepts were sometimes indistinguishable, at least sometimes)

Even if Einstein's additional condition was incorrect, differential geometry is still the better way to describe the universe; our observations are necessarily small in scale, so we can only prove what the universe looks like on small scales. Of course, we're still presuming the small scale geometry looks the same everywhere, so differential manifolds aren't the perfect choice of study, but they're

*far*more reasonable than the assumption that space looks euclidean as a whole. Even if space

*is*euclidean as a whole, euclidean space is merely a special kind of differentiable manifold, we we would still have the correct description of space.