On the naturalness of Einstein’s equation

We compute all 2-covariant tensors naturally constructed from a semi-Riemannian metric which are divergence-free and have weight greater than −2.As a consequence, it follows a characterization of the Einstein tensor as the only, up to a constant factor, 2-covariant tensor naturally constructed from a semi-Riemannian metric which is divergence-free and has weight 0 (i.e., is independent of the unit of scale). Since these two conditions are also satisfied by the energy–momentum tensor of a relativistic space–time, we discuss in detail how these theorems lead to the field equation of General Relativity.