= Weak-field tidal curvature in a stationary metric
{title2=$R^i{}_{0j0}=-\tfrac12\partial_i\partial_jg_{00}$}
Use signature $(-,+,+,+)$, $x^0=ct$ and $R^\rho{}_{\sigma\mu\nu}=\partial_\mu\Gamma^\rho{}_{\nu\sigma}-\partial_\nu\Gamma^\rho{}_{\mu\sigma}+\cdots$. For a weak stationary metric, the <Christoffel symbol> satisfies $\Gamma^i{}_{00}=-\tfrac12\partial_ig_{00}$ to first order. Differentiation gives the displayed <Riemann curvature tensor> component; products of connections are second order. For $g_{00}=-1+2GM/(c^2r)$ it is $GM(\delta_{ij}-3x_ix_j/r^2)/(c^2r^3)$. Its radial eigenvalue is negative and its transverse eigenvalues positive, giving radial stretching and transverse compression in the <geodesic deviation> equation with a positive curvature term on the left.
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