Stress Energy Tensor - Confusion about signs and metrics

[Allday]

Hi everyone. I'm working on deriving Friedmanns Equations from the Einstein Field Equations. I've got the '00' components worked out but I'm having some trouble with the spatial indices 'ii' of the stress energy tensor $T_{ii}$. I'm the FLRW metric with c=1 and signature (-,+,+,+) so that $g_{\mu \nu} = (-1,a^2,a^2,a^2)$. My question is what is the stress energy tensor for a perfect fluid and how does it change with raised and lowered indices. I know how the metric operates to raise and lower indices but I don't know what to start with for $T_{ii}$. For example, I've seen

$$ T^{\alpha \beta} = diag(\rho,P,P,P) $$

but also the same thing for $T^{\alpha}_{\beta}$. The trouble comes in with factors of the scale factor $a$ when I try to calculate the diagonal spatial components for Einsteins Field Equations. For example if $T^{\alpha \beta} = diag(\rho,P,P,P)$ then $T_{ii}$ gets two factors from the metric meaning $T_{ii} = P a^4$. Is that correct? Any help would be appreciated.

thanks,
Allday

 

[yenchin]

This could be helpful: http://equatorfreq.wordpress.com/2010/08/13/signs-in-einsteins-equation/

 

[Allday]

Thanks yenchin, those notes do help. It seems I want to use the form
$$ T_{\alpha \beta} = (\rho + P) u_{\alpha} u_{\beta} + P g_{\alpha \beta} $$
along with the fact that the spatial components of $u_{\alpha}$ are zero so that with my choice of metric sign convention
$$ T_{i i} = P a^2 $$