Gen. Rel. - Simple tensor stuff

[quasar987]

Homework Statement

This is a problem that is solved in my course notes, but I don't follow how it's done.

The problem is to find the general solution to Einstein's equations for a Robertson-Walker cosmology with a stress-energy tensor corresponding to the fluctuations of the void in QFT. This only means that $$T_{\mu\nu}=\Lambda g_{\mu\nu}$$ where lambda is a scalar.

So it start with

$$G_{\mu\nu}=8\pi GT_{\mu\nu}$$

$$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=8\pi G \Lambda g_{\mu\nu}$$

And then the next line is

$$R_{\mu\nu}=8\pi G(\Lambda - \frac{1}{2}g_{\mu\nu}T^{\lambda}_{\lambda})$$

Since when is

$$R=-8\pi GT^{\lambda}_{\lambda}$$

?

 

[Dick]

Since:

$$R^\lambda_\lambda-\frac{1}{2}g^\lambda_\lambda R=8\pi G T^\lambda_\lambda$$

and $$g^\lambda_\lambda=\delta^\lambda_\lambda=4$$ and $$R^\lambda_\lambda=R$$.

That's since when.