 Tidal field

A tidal field describes the spatial variation of gravitational acceleration across an extended body. With $g_i=-{\partial }_i\Phi$, a small separation $\xi$ gives relative acceleration $\delta g_i=T_{ij}{\xi }_j$, where $T_{ij}=-{\partial }_i{\partial }_j\Phi$.

For a point mass, a radial separation is stretched at rate coefficient $2GM/r^3$, while transverse separations are compressed with coefficient $-GM/r^3$. The common acceleration can be removed by choosing a freely falling frame; these relative accelerations remain. This makes tides a probe of gravitational inhomogeneity rather than merely the strength of the local acceleration.