Using the relation $g^i=-\frac{\partial \phi}{\partial x}$ and the derivation $T_j^i=\frac{\partial g^i}{\partial x^i}$, the expression for the tidal tensor in terms of the potential $\phi$ is:

$$ T_j^i=\frac{\partial}{\partial x^j}\left(-\frac{\partial \phi}{\partial x^i}\right)=-\frac{\partial^2 \phi}{\partial x^i \partial x^j} $$

The relation used in this derivation is:

• The gravitational field (acceleration) is the negative gradient of the potential: $g^i=-\frac{\partial \phi}{\partial x}$.
• The tidal tensor is the gradient of the gravitational field: $T_j^i=\frac{\partial g^i}{\partial x^j}$.

Note: The resulting tensor $T_{i j}=-\frac{\partial^2 \phi}{\partial x^i \partial x^j}$ is minus the Hessian matrix of the gravitational potential. The order of the partial derivatives can be swapped because $\phi$ is a smooth function, confirming that the tidal tensor is symmetric ( $T_{i j}=T_{j i}$ ).

What is the expression for the tidal tensor in terms of the gravitational potential-L.mp4