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So for ## \lambda > 0 ##, these 3 solutions for ## k=-1,0,1 ## differ only by a coordinate transformation, and describe the same space-time.

## \lambda <0 ##: ## k=-1 ## :

## a=\sqrt{\frac{-3}{\Lambda}}sin(\sqrt {\frac{-\Lambda}{3}}t) ##

## \lambda>0 ## :

## k=-1 ##: ## a=\sqrt{\frac{3}{\Lambda}}sinh(\sqrt{\frac{\Lambda}{3}}t) ## \\

## k=0 ##: ## a\propto e^{(\pm\sqrt{\frac{\Lambda}{3}}t)} ## \\

## k=1 ##: ##a=\sqrt{\frac{3}{\Lambda}}cosh(\sqrt{\frac{\Lambda}{3}}t) ##

I've read on a few sources that the scale factor grows exponentially, but surely this is just in a particular coordinate chart, as it is only apparent , above, in the ##k=0## solution.

Probably a stupid question, but different coordinate charts suggest different dynamics of ##a(t)##, so which is the 'true' relation?

I.e- Is the de-sitter space geometrically open, closed or flat?

Anti de-sitter space geometrically? I'm guessing it's called anti for a reason

The exponential solution for ##k=0## has a ## \pm ##, how is this possible , don't they describe completely different things - contraction and expansion?

## \lambda <0 ##: ## k=-1 ## :

## a=\sqrt{\frac{-3}{\Lambda}}sin(\sqrt {\frac{-\Lambda}{3}}t) ##

## \lambda>0 ## :

## k=-1 ##: ## a=\sqrt{\frac{3}{\Lambda}}sinh(\sqrt{\frac{\Lambda}{3}}t) ## \\

## k=0 ##: ## a\propto e^{(\pm\sqrt{\frac{\Lambda}{3}}t)} ## \\

## k=1 ##: ##a=\sqrt{\frac{3}{\Lambda}}cosh(\sqrt{\frac{\Lambda}{3}}t) ##

**Questions**I've read on a few sources that the scale factor grows exponentially, but surely this is just in a particular coordinate chart, as it is only apparent , above, in the ##k=0## solution.

Probably a stupid question, but different coordinate charts suggest different dynamics of ##a(t)##, so which is the 'true' relation?

I.e- Is the de-sitter space geometrically open, closed or flat?

Anti de-sitter space geometrically? I'm guessing it's called anti for a reason

The exponential solution for ##k=0## has a ## \pm ##, how is this possible , don't they describe completely different things - contraction and expansion?

**Thanks .**
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