🧄Derivation and Calculation of the Gravitational Tidal Tensor

The Gravitational Tidal Tensor ( T), derived from the negative second spatial derivatives of the gravitational potential ( $\phi$ ), describes the differential acceleration experienced by two adjacent particles in a gravitational field. This tensor is fundamental to understanding tidal effects, as it relates the change in acceleration (da) linearly to the particle separation vector ( $d x$ ) via $d a^i=T_j^i d x^j$. Notably, the tensor is symmetric and, for a spherical mass distribution, its components $T_j^i= G M\left[\frac{3 x^i x^j}{r^j}-\frac{\delta_{j i}}{r^3}\right]$ reveal the dual nature of tidal forces: the off-diagonal terms are responsible for the shearing and stretching effects lateral to the mass center, while the diagonal terms govern the radial compression and stretching along the line of centers.

Derivation and Calculation of the Gravitational Tidal Tensor | Proof and Derivationviadean on Notion