general relativity - What is the geometric interpretation of the Einstein tensor $R_{mu nu} - frac{1}{2} g_{mu nu} R$

The Riemann curvature tensor $R_{\mu \nu \rho \sigma}$ has the geometric interpretation of giving how much parallel transport fails to close around tiny loops. The Ricci tensor $R_{\mu \nu}$ the Riemann curvature averaged over all directions, as in, if there is negative curvature in some direction there must be positive curvature in another if $R_{\mu \nu} = 0$.

What is the geometric interpretation of the Einstein tensor $$G_{\mu \nu} = R_{\mu \nu} - \frac{1}{2} g_{\mu \nu} R?$$ Is there a way to understand $$\nabla^\mu G_{\mu \nu} = 0$$ Intuitively?