= Tidal tensor
{title2=$T_{jk}$}
The acceleration tidal tensor is $T_{jk}=\partial_k g_j=-\partial_j\partial_k\Phi$ for <gravitational acceleration> $\mathbf g=-\nabla\Phi$. It describes differential acceleration $\Delta\mathbf g\simeq T\Delta\mathbf r$. Some authors instead call the positive <Hessian matrix> $\partial_j\partial_k\Phi$ the tidal tensor, reversing this sign convention. For a point mass and the positive potential convention $\mathbf g=\nabla\psi$, the tensor is $Gm(3\mathbf R\mathbf R^{\mathsf T}/R^5-\mathbf1/R^3)$.
Back to article page
