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I'm looking at solutions to Einsten's equation with the cosmological constant included ,##G_{uv}=- 8 \pi GT_{uv} - \lambda g_{uv} ##, with the FRW metric at hand - i.e, ## R_{uv} ## has been

I have a question from these notes...http://arxiv.org/abs/gr-qc/9712019

they give the solution for ##\lambda <0 ##(and the only case of ##k=1##) as ## a= \sqrt{\frac{-3}{\lambda}}sin(\sqrt\frac{-\lambda}{3}t) ##

##\lambda >0 ##, for ##k=-1,0,1## are respectively:

## a= \sqrt{\frac{3}{\lambda}}sinh(\sqrt/frac{\lambda}{3}t) ##

## a \propto exp (\pm \sqrt{/frac{\lambda}{3}}t) ##

## a= \sqrt{\frac{3}{\lambda}}cosh(\sqrt\frac{\lambda}{3}t) ##

It then says that the ##k=-1,0,1 ##and ##\lambda >0 ## are all the same solutions, just in different coordinates.

The FRW metric is given in spherical polar coordinates. So when we solve for the solutions, how is it that they are in different coordinates? Would it be the use of various integration substitutions to solve, and what's given here, is the most easiest substitution that solves for each k, and so this easiest substituion has differed for each case?

__from this metric.__**computed****
**I have a question from these notes...http://arxiv.org/abs/gr-qc/9712019

they give the solution for ##\lambda <0 ##(and the only case of ##k=1##) as ## a= \sqrt{\frac{-3}{\lambda}}sin(\sqrt\frac{-\lambda}{3}t) ##

##\lambda >0 ##, for ##k=-1,0,1## are respectively:

## a= \sqrt{\frac{3}{\lambda}}sinh(\sqrt/frac{\lambda}{3}t) ##

## a \propto exp (\pm \sqrt{/frac{\lambda}{3}}t) ##

## a= \sqrt{\frac{3}{\lambda}}cosh(\sqrt\frac{\lambda}{3}t) ##

It then says that the ##k=-1,0,1 ##and ##\lambda >0 ## are all the same solutions, just in different coordinates.

**Question:**The FRW metric is given in spherical polar coordinates. So when we solve for the solutions, how is it that they are in different coordinates? Would it be the use of various integration substitutions to solve, and what's given here, is the most easiest substitution that solves for each k, and so this easiest substituion has differed for each case?

**Thanks in advance.**
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