The gravitational tidal tensor, denoted as $T_j^i$, is a rank-two tensor that describes the differential acceleration between two nearby particles in a gravitational field. It quantifies the tidal effect-the relative force that stretches or compresses objects. The relationship is expressed as:
$$ d a^i=T_j^i d x^j $$
where $d a$ is the difference in acceleration (tidal acceleration) and $d x$ is the small displacement vector separating the two particles.