The source provides a comprehensive derivation and calculation of the Gravitational Tidal Tensor ($T$), which is a rank two tensor defining the differential acceleration ($d\vec{a}$) experienced by two closely separated particles in a gravitational field, known as the tidal effect. This relationship is formally expressed as $d a^i=T_j^i d x^j$. By using a first-order Taylor series approximation for the gravitational field ($\vec{g}$), the tensor components are initially found to be $T_j^i=\frac{\partial g^i}{\partial x^j}$. Since the gravitational field is the negative gradient of the potential ($\vec{g}=-\nabla \phi$), $T$ is ultimately defined as the negative of the Hessian matrix of the gravitational potential: $T_j^i=-\frac{\partial^2 \phi}{\partial x^j \partial x^i}$, confirming its symmetry. The final section computes this tensor for the specific case of movement outside a spherical mass distribution where $\phi(\vec{x})=-\frac{G M}{r}$, resulting in the compact final expression for its components: $T_j^i=G M\left[\frac{3 x^i x^j}{r^5}-\frac{\delta_{i j}}{r^3}\right]$.

Black Holes to Ocean Tides Decoding the Gravitational Tidal Tensor-L.mp4