Equations of motion for a (non-linear) scalar field model as derived from the field equations

TL;DR

This paper introduces a new algorithm for deriving equations of motion from field equations, demonstrated on a nonlinear scalar field model, revealing Newtonian attraction between singularities.

Contribution

A novel algorithm leveraging the specific form of Einstein's equations to derive equations of motion from field equations, applicable to N-body problems.

Findings

Derivation of equations of motion from field equations using the new algorithm.

Application to a nonlinear Lorentz invariant scalar field model.

Resulting equations reproduce Newton's law of attraction.

Abstract

The problem of derivation of the equations of motion from the field equations is considered. Einstein's field equations have a specific analytical form: They are linear in the second order derivatives and quadratic in the first order derivatives of the field variables. We utilize this particular form and propose a novel algorithm for thederivation of the equations of motion from the field equations. It is based on the condition of the balance between the singular terms of the field equation. We apply the algorithm to a nonlinear Lorentz invariant scalar field model. We show that it results in the Newton law of attraction between the singularities of the field moved on approximately geodesic curves. The algorithm is applicable to the $N$-body problem of the Lorentz invariant field equations.