Jonathan wrote:
> In spherical and hyperbolic geometry we see spaces that are Euclidean on
> infinitesimal scales but become very non-euclidean on larger scales.
> However, the only non-euclidean geometry I know of that, like Euclidean
> geometry, is consistent on all scales is Lorentzian geometry, which is
> derived by flipping signs around in the Euclidean metric tensor.
> (Actually, come to think of it, I can think of Taxicab and Chebyshev
> geometry too). But other than that, are there any other non-Euclidean
> geometries that are the same on all scales? For instance, where the
> angles of a triangle add up to a constant other than 180, or where the
> ratio of the circumference of a circle to its diameter is something
> other than pi?

I think you'd have to be more specific than this to get at exactly what
it is you're asking. Non-Euclidean spaces include all kinds of other
weird cases that you probably don't want, such as fractal or other
non-differential spaces that aren't of much use here.

So first I'm guessing you're talking about a Riemannian manifold, which
requires differentiability (and thus "smoothness"). It's hard to
imagine a curved space(time) where the shapes of circles or triangles
are not distorted over distances, since that's what curvature _is_.
However, since you invoked Lorentzian geometry, I'm guessing you're
including four-dimensional spacetimes and thus also Minkowski space.

Since what you're asking about doesn't sound possible, I'll come as
close as I can, which is to guess that you're asking about maximally
symmetric four-dimensional space(time)s, which are spaces with the
maximum number of Killing vectors. These correspond to spaces that are
homogeneous and isotropic -- note, though, that that means that they're
homogeneous and isotropic _in all four coordinates_, so they must be so
not just in space but also in time. For metrics with signature (-+++),
this includes Minkowski space.

There do exist curved spaces that are maximally symmetric. There are
two cases: positively and negatively curved. These actually have
names: the de Sitter and anti-de Sitter space, respectively. These
correspond to exponentially expanding (or contracting) spaces. In
modern cosmology, they even have an application: They would be the
spacetimes resulting from a universe with no matter or energy, but a
nonzero vacuum energy that corresponds to a nonzero cosmological
constant (positive for de Sitter, negative for anti-de Sitter). If dark
energy explanations of vacuum energy are correct and the vacuum energy
is roughly constant and/or still dominates very late in the Universe,
the final stage of the Universe's evolution would look like a de Sitter
space, with vacuum energy dominant and matter and other energy
contributions negligible.

--
Erik Max Francis && max@... && http://www.alcyone.com/max/
San Jose, CA, USA && 37 18 N 121 57 W && AIM/Y!M/Skype erikmaxfrancis
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