Cosmology: Static Universes and Negative Curvature

[Helios]

Dear Cosmology People,
Einstein added just enough ad hoc dust to a universe with hyperspherical topology and positive spatial curvature to bring about stasis. Is a similar thing possible with a pseudohyperspherical universe with negative curvature, something opposite of dust, that would have some exact value to attain stasis?

 

[Chalnoth]

No. The curvature is uniquely-determined by the density of matter/energy and the expansion rate. A universe that is static (i.e., the Hubble expansion is zero) has a positive curvature that is equal to the matter density (once you throw in the proper conversion factors). As long as the matter is made up of stuff that has $w > -1/3$, which normal matter and radiation do, it is guaranteed to recollapse on itself.

 

[Chronos]

Einstein added the cosmological constant to achieve a static universe. I'm not familiar with it being referred to as 'dust'.

 

[Helios]

yes, uniform dust is the description from wiki page "Static universe". The page goes on to say this model is unstable. I'm guessing there's no stable static universe in GR. I was interested if there was, even if some imaginary non-matter, non-radiation agent was necessary.

 

[PeterDonis]

Einstein added the cosmological constant to achieve a static universe. I'm not familiar with it being referred to as 'dust'.

I think what the OP might have meant to say is that Einstein started with de Sitter spacetime and added just enough dust (i.e., ordinary matter with zero pressure) to keep it from expanding. I think that gets the actual thought process Einstein followed backwards, though; as I understand it, he started with a closed FRW model at the moment of maximum expansion and then added enough cosmological constant to keep it from collapsing, as you say.

I'm guessing there's no stable static universe in GR. I was interested if there was, even if some imaginary non-matter, non-radiation agent was necessary.

I'm not sure you can do Einstein's trick with anti-de Sitter spacetime (i.e., a negative cosmological constant instead of a positive one) except in a trivial sense, because it will be collapsing with nothing else present, and the only thing I can see that you could add to it to keep it from collapsing would be a positive cosmological constant--but that just amounts to canceling out the cosmological constant and ending up with flat, empty Minkowski spacetime. Technically, that qualifies as a "stable, static" spacetime, but only because it's entirely empty.