Wednesday, July 12, 2017

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?