Bootstrapping Newtonian gravity

Abstract

A nonlinear equation obtained by adding gravitational self-interaction terms to the Poisson equation for Newtonian gravity is here employed in order to analyze a static spherically symmetric homogeneous compact source of given proper mass and radius and the outer vacuum. The main feature of this picture is that, although the freedom of shifting the potential by an arbitrary constant is of course lost, the solutions remain qualitatively very close to the Newtonian behavior. We also notice that the negative gravitational potential energy is smaller than the proper mass for sources with small compactness, but for sources that should form black holes according to general relativity, the gravitational potential energy becomes of the same order of magnitude of the proper mass, or even larger. Moreover, the pressure overcomes the energy density for large values of the compactness, but it remains finite for finite compactness; hence there exists no Buchdahl limit. This classical description is meant to serve as the starting point for investigating quantum features of (near) black hole configurations within the corpuscular picture of gravity in future developments.