= Newtonian tidal curvature
{title2=$R^i{}_{0j0}=\partial_i\partial_j\Phi,\quad R_{00}=\nabla^2\Phi$}
Using $[\nabla_c,\nabla_d]V^a=R^a{}_{bcd}V^b$, the <Newton–Cartan gravitational connection> has only the displayed Hessian components and their antisymmetric partners in the last two indices. The <geodesic deviation> equation gives relative acceleration $-\partial_i\partial_j\Phi\,\delta x^j$. The <Poisson equation for Newtonian gravity> is therefore $R_{ab}=4\pi G\rho\,(dt)_a(dt)_b$. A spatially uniform gravitational acceleration has zero tidal curvature even if the connection coefficients do not vanish in the original coordinates.
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