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?
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