On Wednesday, January 22, 2020 at 7:12:35 AM UTC-6, John Clark wrote:
>
> On Wed, Jan 22, 2020 at 12:48 AM Alan Grayson <[email protected]
> <javascript:>> wrote:
>
> *> **I strongly disagree that finite rates of expansion will result in an
>> open universe. I believe it will be a closed hyper-sphere, but I am open to
>> being wrong. AG *
>
>
> If empty space has a residual intrinsic energy of any value greater than
> zero, which General Relativity allows for and Quantum Mechanics demands,
> then the expansion of the universe will accelerate. If the universe is
> accelerating then it is open regardless of what its spatial shape is,
> regardless of how many degrees the angles of a triangle add up to (please
> remember the term "spatial shape" is not equivalent with the term "spacetime
> shape").
>
> And when you ask "How big is the universe?" you need to know exactly what
> you're really asking. You have nothing outside of the universe to compare
> it to so one answer would be "The universe is as big as the universe",
> but you may find that unsatisfying. What you really want to know is if
> the universe is open or closed, you want to know "If you keep going in
> one direction will you head out for infinity and keep getting further and
> further from your starting point for eternity, OR will you eventually hit
> some sort of wall or eventually start getting closer to your starting point
> ?"
>
> Today we have very good evidence the universe is accelerating and thus
> open. Independently we have moderately good evidence that Inflation
> occured. And we have pretty good evidence the universe is spatially pretty
> flat. We'll never be able to observationally prove it has exactly zero
> spatial curvature but if all you want to know is how big the universe is,
> that is to say if all you want to know is if it's open or closed, then how
> many degrees the angles of a triangle add up to is not important.
>
> John K Clark
>
The accelerated expansion does not imply an open universe. The dS spacetime
permits closed S^3 spherical spatial worlds. As I said above this depends
on how the hyperboloid is spatially sliced. With the FLRW metric
ds^2 = dt^2 - [exp(r√{3/Λ}}/(1 - kr^2]dS^2
is accelerated and closed for k = 1, flat and open for k = 0 and
saddle-hyperbolic shaped for k = -1.
LC
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