# Unable to Understand a Step in Wald's General Relativity book

Refer to the following picture: How can one jump from ($$3.2.30$$) to $$(3.2.31)?$$

Here the passage:

$$\nabla_a R_c^a + \nabla_b R_c^b - \nabla_c R = 0$$ (3.2.30)
The a and b are dummy indices, the first two terms are the same.
$$2 \nabla_a R_c^a - \nabla_c R = 0$$
The covariant derivative $$\nabla_c$$ can be written as $$g_{ac} \nabla^a$$
In the first term the index a can be raised in the covariant derivative and lowered in the Ricci tensor, via the metric tensor.
$$2 \nabla^a R_{ca} - g_{ac} \nabla^a R = 0$$
Dividing by 2, moving the metric tensor in the second term after the covariant derivative (metric tensor compatibility) and collecting after the covariant derivative.
$$\nabla^a (R_{ca} - \frac{1}{2} g_{ac} R) = 0$$
As per definition of Einstein tensor, exchanging indices in the Ricci tensor (index symmetry) and relabelling the free index.
$$\nabla^a G_{ab}$$ (3.2.31)

Done.

Just substitute when in doubt:

$$\nabla^a G_{ac} = \nabla^a\left(R_{ac} - \frac{1}{2}Rg_{ac}\right) =\nabla^a R_{ac} - \frac{1}{2}\nabla_c R$$

So $$(3.2.30)$$ is:

$$2\nabla^a G_{ac}=0$$

Metric compatibility has been assumed here, of course.