Ricci-tensor for classical gravity

Following up on last week’s question: Can you construct an analogue of the Ricci-tensor for classical gravity? Would it correspond to the actual Ricci-tensor in the weak field limit for a stationary source? Is there an analogous object for electrodynamics as well?

On can in fact set up many analogies between quantities in general relativity, electrodynamics and classical gravity/electrostatics (I don’t treat them differently because they share the same field equation up to a sign). The metric \(g_{\mu\nu}\) is analogous to the potentials \(A^\mu\) or \(\Phi\), the derivatives \(\partial_\alpha g_{\mu\nu}\) of the metric are the accelerations \(F^{\mu\nu}\) or \(g^i = -\partial^i\Phi\) and the curvature corresponds to differential accelerations \(\partial^\alpha F^{\mu\nu}\) or \(\partial^j g^i = -\partial^j\partial^i\Phi\). The Ricci-tensor is the local part of curvature, which is equated to the source of the field, \(g^{\alpha\mu}R_{\alpha\beta\mu\nu}\), for which you have counterparts as \(\eta_{\alpha}F^{\mu\nu}\) and in particular \(\delta_{ij}\partial^i\partial^j\Phi = \Delta\Phi\).

Of course the classical Poisson-equation only holds for static gravitational fields and sources, and only comprises spatial derivatives, so the analogy is a bit stretched. But extending it to a relativistic field equation, \(\Delta = \delta_{ij}\partial^i\partial^j\rightarrow \eta_{\mu\nu}\partial^\mu\partial^\nu = \Box\) would in fact be the wave equation in the slow-motion, weak field limit in traceless, transverse gauge.