Solving Scalar Curvature for Homogenous & Isotropic FLRV Metric

[Elliptic]

Homework Statement

Find the equation of scalar curvature for homogenous and isotropic space with FLRV metric.

Homework Equations

$R=6(\frac{\ddot{a}}{a}+\left( \frac{\dot{a}}{a}\right )^2+\frac{k}{a^2})$

The Attempt at a Solution

$G_{AB}=R_{AB}-\frac{1}{2}Rg_{AB}$

 

[WannabeNewton]

That's not really much of an attempt to be honest :p

What did you get when you calculated the Ricci curvature for the FLRW metric? Just plug the metric into the formulas.

 

[Elliptic]

If I strart from this point:
$B_{\mu\nu}+\lambda g_{\mu\nu}B=0 / \cdot g^{\mu\nu} \\ R(1+4\lambda)=0$
what next?

 

[Elliptic]

That's not really much of an attempt to be honest :p

What did you get when you calculated the Ricci curvature for the FLRW metric? Just plug the metric into the formulas.

Any help?

 

[WannabeNewton]

I can't really understand your notation. Why not just calculate it directly? $R = g^{\mu\nu}R_{\mu\nu} = g^{\mu\nu}R^{\alpha}{}{}_{\mu\alpha\nu}$. The FLRW metric is diagonal and extremely simply in the usual form so the computation shouldn't be so bad.

 

[tia89]

I can't really understand your notation. Why not just calculate it directly? $R = g^{\mu\nu}R_{\mu\nu} = g^{\mu\nu}R^{\alpha}{}{}_{\mu\alpha\nu}$. The FLRW metric is diagonal and extremely simply in the usual form so the computation shouldn't be so bad.

With the FLRW metric actually you should be able to use directly the definition of $R_{\mu\nu}$ and then take out the scalar as here above.
Anyway try and look in any GR book (e.g. Carroll or others). It is done quite everywhere.