This section details the mathematical description of the gravitational field and its derivatives outside a spherically symmetric mass distribution of mass $M$. The gravitational potential $\phi$ at a distance $r$ from the center is given by the standard Newtonian formula, $\phi(x)=-\frac{G M}{r}$. The gravitational field (or acceleration) $g$ is then derived from the potential as its negative gradient, $g=-\nabla \phi$, which results in the inverse-square law expression, $g=-\frac{G M}{r^2} \hat{r}$. Finally, the tidal tensor $T_{i j}$, which describes the tidal forces (stretching and squeezing), is calculated as the negative second derivative of the potential, $T_{i j}=-\frac{\partial^2 \phi}{\partial x^* \partial x^j}$, yielding the expression $T_{i j}=G M\left(\frac{3 x^i x^j}{r^j}-\frac{\delta_{i j}}{r^3}\right)$, where $\delta_{i j}$ is the Kronecker delta.
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