In the de Sitter cosmology the curvature of the spatial surface is
determined by how it is embedded in the hyperboloid. Below is a diagram of
such a hyperboloid of 2 dimensions where the spatial surface is the green
curve. In this situation the spatial surface is infinite. If the spatial
surface is embedded as a circle, such as the black circle around the
coordinates q_1 and q_4, the spatial surface is a sphere S^3. Finally the
hyperbolic spatial surface occurs if it is embedded along a more vertical
direction.
This is based on the idea the de Sitter manifold is embedded in 5
dimensions with metric s = t^2 - u^2 - x^2 - y^2 - z^2, and the constraint
s = 1 results in the dS metric. The metric reduced to 4 dimensions is
ds^2 = dt^2 - cosh(t√{3/Λ})dS^2
for dS^2 the spatial metric. The FLRW metric approximates this for large
time with exp(t√{3/Λ}) ≈ cosh(t√{3/Λ}).
LC
[image: de Sitter space hyperboloid.png]
On Monday, January 20, 2020 at 8:55:06 AM UTC-6, John Clark wrote:
>
> I counted, the post I am now responding to had 10 iterated quotes, that's
> quotes of quotes of quotes of quotes of quotes of quotes of quotes of
> quotes of quotes. People are too lazy to trim anything and that's why in long
> threads the signal to noise ratio declines exponentially and soon becomes
> almost
> unreadable. And that's why I'm starting a new one.
>
> On Mon, Jan 20, 2020 at 7:49 AM Alan Grayson <[email protected]
> <javascript:>> wrote:
>
> *> What I am established is that flatness is incompatible with a universe
>> which had a beginning. So if it's flat, it never had a beginning; or else
>> it did, and is closed, hyper-spherical in shape. AG*
>
>
> Are you talking about spatial curvature or spacetime curvature? By
> "curvature" do you mean the angles of a triangle add up to something other
> than 180 degrees, or do you mean if you keep going in one direction you
> will eventually end up where you started? They are not necessarily the same
> thing.
>
> If the universe once expanded faster than the speed of light (as inflation
> hypothesizes) then it's conceivable the angles of a triangle could add up
> to be more than 180 degrees, so the universe would have a positive spatial
> curvature like a sphere does, and yet you'd be going further and further
> from your starting point into infinity and never return. And as long as
> there had been a faster than light expansion at some point in the
> universe's history if the angles of a triangle added up to less than 180
> degrees then the universe would have negative spatial curvature, like the
> shape of a saddle does, and you'd still never return to your starting place.
>
> John K Clark
>
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