Tidal acceleration ($a_{\mathrm{rel}}^a=-R^a{}_{bcd}{U}^b{\xi }^c{U}^d$)

The relative acceleration of neighboring freely falling bodies is measured by geodesic deviation, comparing their infinitesimal separation $\xi$ in a parallel transport frame along a reference timelike geodesic with tangent $U$. With $R^a{}_{bcd}$ defined by the covariant derivative commutator, it has the displayed form. In the Newtonian limit the corresponding tidal tensor is the spatial gradient of the gravitational acceleration. A single freely falling body can remove local gravitational acceleration by its frame choice, whereas relative tidal acceleration measures curvature.

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 ${\Gamma }^i{}_{00}=-\frac{1}{2}{\partial }_i{g}_{00}$ to first order. Differentiation gives the displayed Riemann curvature tensor component; products of connections are second order. For $g_{00}=-1+2GM/(c^{2}r)$ it is $GM({\delta }_{ij}-3x_i{x}_j/r^2)/(c^2{r}^3)$. 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.