Equation of motion in a scalar model of gravity

Abstract

A scalar model of gravity is considered. We propose Lorentz invariant field equation $\square f = kη_{ab}f_{,a}f_{,b}$. 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 ${ψ_j(t)}$ 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 ${\dotψ_j(t)}$. When this field is inserted to the field equation the outcome is singular at $({ψ_j(t)},t)$. 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.