Consistently Implementing the Fields Self-Energy in Newtonian Gravity

TL;DR

This paper explores how to consistently incorporate the gravitational field's self-energy into Newtonian gravity, clarifying misconceptions and connecting to general relativity.

Contribution

It presents a consistent method for including self-energy in Newtonian gravity, addressing previous inconsistencies and linking to relativistic theory.

Findings

Simple addition of energy term is inconsistent

A consistent prescription for self-energy is proposed

Connection to general relativity is discussed

Abstract

We consider in a pedagogical fashion alterations to Newtonian gravity due to the postulate that all energy corresponds to active gravitational mass when applied to the self-energy of the gravitational field. We show why a simple addition of ${1\over c^2}$ times the gravitational field energy to the matter density in Newton's field equation is inconsistent. A consistent prescription is shown and discussed. The connection to general relativity is pointed out.