Approximating compact objects in bootstrapped Newtonian gravity: use of the canonical potential

TL;DR

This paper develops a perturbative approach using a canonical potential to approximate compact objects in a non-relativistic, higher-order gravity theory, revealing differences between ADM and proper mass due to non-linearities.

Contribution

It introduces a canonical formulation of bootstrapped Newtonian gravity enabling perturbative solutions for compact objects, highlighting mass differences from non-linear couplings.

Findings

Canonical potential simplifies analysis of compact objects.

Mass differences between ADM and proper mass are demonstrated.

Perturbative solutions effectively approximate non-relativistic compact objects.

Abstract

We consider compact objects in a classical and non-relativistic generalisation of Newtonian gravity, dubbed bootstrapped Newtonian theory, which includes higher-order derivative interaction terms of the kind generically present in the strong-field regime of gravity. By means of a field redefinition, the original bootstrapped Newtonian action is written in a canonical Newtonian form with non-linear source terms. Exact analytic solutions remain unattainable, but we show that perturbative solutions of the canonical theory can be efficiently used to derive approximate descriptions of compact objects. In particular, using the canonical potential, we can more directly and generally show that the Arnowitt-Deser-Misner mass differs from the (Newtonian) proper mass due to the non-linear couplings in the theory. A few examples of sources with different density profiles are explicitly reanalysed in this framework.