Tensor calculus for general relativity question.

[iampeterr]

Use the metic that Einstein proposed in the first cosmological model based on general relativity.

ds² = -dt² + (dr²) / (1 - Kr²) + r²(dθ² + sin²θd$\phi$²) where K > 0

Show that the stress energy tensor is that of a static, spatially uniform perfect fluid and determine ρ and p in terms of G and K. If the universe contains only cold matter (denoted by subscript m, with p_m << ρ_m) and vacuum energy (denoted by subscript v, with p_v = -ρ_v), what is the ratio of ρ_v / ρ_m?

could someone help me out? its something I've been working on for a while now and end up in some weird mess.

 

[stevendaryl]

Use the metic that Einstein proposed in the first cosmological model based on general relativity.

ds² = -dt² + (dr²) / (1 - Kr²) + r²(dθ² + sin²θd$\phi$²) where K > 0

Show that the stress energy tensor is that of a static, spatially uniform perfect fluid and determine ρ and p in terms of G and K. If the universe contains only cold matter (denoted by subscript m, with p_m << ρ_m) and vacuum energy (denoted by subscript v, with p_v = -ρ_v), what is the ratio of ρ_v / ρ_m?

could someone help me out? its something I've been working on for a while now and end up in some weird mess.

Every time I try to compute the curvature tensor, I end up with pages of mathematics that always has an error.

 

[iampeterr]

which part do you get up to ?