Non-uniqueness of potential field inversion

Remarks

Since $\Delta_y (1/\norm{x - y}) = -4\pi\,\delta_x$, two integrations by parts give $V_{\Delta\varphi} = -4\pi G\varphi$ on all of $\R^3$, and $\varphi$ vanishes outside its support. If $\rho = \Delta\varphi$ were $0$, then $\varphi$ would be harmonic with compact support, hence $0$. So adding $\Delta\varphi$ to any density changes the body but not its exterior gravity field: gravity data alone cannot determine the density. Example: by the shell theorem, a homogeneous ball of mass $M$ has the same field outside it as a thin homogeneous spherical shell of mass $M$ with the same centre. Uniqueness returns under extra hypotheses. For instance, P. S. Novikov proved a uniqueness theorem for bodies of the same known constant density. If two such bodies are star-shaped with respect to a common point and have the same exterior potential, they coincide. The same non-uniqueness holds for magnetic fields, since they are also governed by the Laplace equation outside the sources.

Going deeper

For a student

This is the basic obstruction in the inverse problem of potential theory. The kernel of the forward map contains all Laplacians of compactly supported test functions, an infinite-dimensional space. For square-integrable densities on a regular domain, the exterior field determines only the orthogonal projection of $\rho$ onto the harmonic functions in $L^2(\Omega)$. Inversions usually choose one solution by regularisation, for example the one of minimal norm. Related results are the uniqueness theorems for the exterior Dirichlet problem and the geodetic boundary value problem. They recover the field from boundary data but say nothing about the sources inside.