= Newtonian tidal eigenvalues outside a point mass
{c}
{title2=$\operatorname{eig}T=(2,-1,-1)GM/r^3$}
For $\Phi=-GM/r$, the acceleration <tidal tensor> is
$$
T_{ij}=-\partial_i\partial_j\Phi=
\frac{GM}{r^3}(3n_i n_j-\delta_{ij}).
$$
Its radial eigenvalue is $2GM/r^3$ and its two transverse eigenvalues are $-GM/r^3$. This stretches a small freely falling cloud radially and compresses it transversely. The zero trace expresses the vacuum <Poisson equation> outside the source; within matter the trace is $-4\pi G\rho$.
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