= 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 $Gm(3\mathbf R\mathbf R^{\mathsf T}/R^5-\mathbf1/R^3)$.
The <tidal tensor> is the <Jacobian matrix> of the <Newtonian gravitational field>, rather than the <gravitational acceleration> itself. Its entries have units $\mathrm{s^{-2}}$: multiplying by a <displacement> produces an <acceleration>. For a smooth <Newtonian gravitational potential>, the <tidal tensor> is a <symmetric matrix>, since mixed <partial derivatives> commute. A spatially uniform <gravitational acceleration> has zero <tidal tensor>, however large that <acceleration> is.
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