The tidal tensor is a rank-two tensor that quantifies the differential acceleration ($da$) between two nearby points in a gravitational field, describing the tendency to stretch or compress objects; this effect is known as the tidal force. Mathematically, the differential acceleration is related to the tensor and the small separation vector ( $d x$ ) by the equation $d a^i=T_j^i d x^j$. The tensor itself is derived by performing a firstorder Taylor expansion of the gravitational field ($g$) around a point, which reveals that $T_j^i$ is simply the gradient of the gravitational field, expressed as $T_j^i=\frac{\partial g^i}{\partial x^j}$. The physical tidal force is thus understood not as a new fundamental force, but as a result of the non-uniformity of the gravitational field.
<aside> ❓