tidal shear invariants

the gravitational shear field could be quantified by the trace \(\sum_{i}\partial^2_{ii}\Phi\), which is proportional to the density \(\rho\) due to Poisson’s equation \(\Delta\Phi=4\pi G\rho\) or by the quantity \(\sum_{ij}\partial^2_{ij}\Phi\partial^2_{ji}\Phi\). in what way do the two quantifiers differ? can you construct more of them? what would be their relation to the Ricci-scalar \(R\) or the invariant curvature \(R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\) in general relativity?