Equation of motion in a scalar model of gravity

TL;DR

This paper introduces a scalar gravity model with a Lorentz invariant field equation aiming to approximate Newtonian gravity, deriving particle fields and suggesting a link to Newton's law through singularity analysis.

Contribution

It proposes a novel scalar field equation for gravity, derives its static and moving particle solutions, and connects the model to Newton's law via singularity considerations.

Findings

Unique spherical symmetric static solution derived

Superposition of particle fields approximates multi-particle systems

Singularity analysis suggests Newton's law of force emerges from the model

Abstract

A scalar model of gravity is considered. We propose Lorentz invariant field equation $\square f = k\eta_{ab}f_{,a}f_{,b}$. The aim of this model is to get, approximately, Newton's law of gravity. It is shown that $f=-\frac 1k\ln(1-k\frac mr)$ is the unique spherical symmetric static solution of the field equation. $f$ is taken to be the field of a particle at the origin, having the mass $m$. The field of a particle moving with a constant velocity is taken to be the appropriate Lorentz transformation of $f$. The field $F$ of $N$ particles moving on trajectories ${\psi_j(t)}$ is taken to be, to first order, the superposition of the fields of the particles, where the instantaneous Lorentz transformation of the fields pertaining to the $j$-th particle is ${\dot\psi_j(t)}$. When this field is inserted to the field equation the outcome is singular at $({\psi_j(t)},t)$. The singular terms of the l.h.s. and of the r.h.s. are both $O(R^{-2})$. The only way to reduce the singularity in the field equation is by postulating Newton's law of force. It is hoped that this model will be generalized to system of equations that are covariant under general diffeomorphism.