Deriving the line element in homogenous isotropic space

TheMan112

If the Ricci-scalar [tex]R[/tex] is constant for a given spatial hypersurface, then the curvature of that region should be homogenous and isotropic, right?

A homogenous and isotropic hypersurface (disregarding time) has by definition the following line element (due to spherical symmetry):

[tex]d\sigma^2 = a^2 \left(\frac{1}{1-kr^2} dr^2 + r^2(d \theta^2 + sin^2(\theta) d \Phi^2) \right)[/tex]

Where k = -1, 0 or +1 and a is constant.

Why [tex]\frac{1}{1-kr^2} dr^2[/tex] ?

This is apparently very important as the value of k determines the evolution of the universe, but I don't know how to come to this line element.

Mentz114

The three cases come from three possible solutions of the Robertson-Walker metric. A good description is given here

http://www.jb.man.ac.uk/~jpl/cosmo/RW.html

and even Wiki on FLRW is not bad

http://en.wikipedia.org/wiki/Friedma...-Walker_metric

I hope this helps, I don't know if you've seen this material before.