# Manipulation of Einstein's equation

In my lecture notes, there is the following consequence of Einstein's equations, (it follows on to have multiple consequences, so I don't think there's a mistake in the notes):

$$R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} = 8\pi GT_{\mu\nu},$$

$$\implies -R = 8\pi G g^{\mu\nu}T_{\mu\nu} = 8\pi GT.$$

However I can't see why this is the case, my working is as follows.

Take the trace of Einstein's equation through multiplication by the inverse metric:

$$R_{\mu\nu}g^{\mu\nu} - \frac{1}{2}Rg_{\mu\nu}g^{\mu\nu} = 8\pi GT_{\mu\nu}g^{\mu\nu},$$

$$\implies R - \frac{1}{2}R = 8\pi GT,$$

$$\implies \frac{1}{2}R = 8\pi GT.$$

Help would be much appreciated in deciphering where I've gone wrong.

• $g^{\mu \nu} g_{\mu\nu}\neq 1$ – nwolijin May 19 at 11:29

$${g_{\mu \nu }}{g^{\mu \nu }} = 4, \text{not} ~ 1.$$
This follows from the definition of the inverse metric, i.e. $${g_{\mu \lambda }}{g^{\lambda \nu }} = \delta _\mu ^\nu$$, and $$\delta _\mu ^\mu =1+1+1+1=4$$. For higher $$D$$-dimensions, you can also find:
$${g_{\mu \nu }}{g^{\mu \nu }} = D.$$