Weyl-tensor for classical gravity

Could you construct an analogy to the Weyl-tensor (for quantifying the non-local field components) for classical gravity and for electrodynamics?

The Weyl-tensor is the trace-free part of the Riemann-curvature, and complements the Ricci-curvature as the trace of the Riemann-curvature, \(R_{\beta\nu} = g^{\alpha\mu}R_{\alpha\beta\mu\nu}\). The analogous quantity for the Riemann-curvature in classical gravity would be the tidal shear \(\partial_i\partial_j\Phi\) as the tensor of second derivatives of the potential \(\Phi\) (which is analogous to the metric \(g_{\mu\nu}\)). The trace of the tidal shear is \(\delta^{ij}\partial_i\partial_j\Phi = \Delta\Phi\) as an analogon to the Ricci-curvature (and is coupled in the Poisson-equation to the matter density, \(\Delta\Phi = 4\pi G\rho\)), such that the trace-free tidal shear \(\partial_i\partial_j\Phi - \delta_{ij}/3\:\Delta\Phi\) would correspond to the Weyl-tensor.