Relativistic point dynamics and Einstein formula as a property of localized solutions of a nonlinear Klein-Gordon equation

TL;DR

This paper demonstrates that localized solutions of a nonlinear Klein-Gordon equation exhibit relativistic dynamics consistent with Einstein's energy-mass relation, bridging classical and relativistic particle models.

Contribution

It proves that solutions concentrating along a trajectory obey relativistic Newton's law with mass satisfying Einstein's relation, linking field theory to particle dynamics.

Findings

Localized solutions follow relativistic Newton's law.

Energy concentration implies Einstein's mass-energy relation.

The model applies to a wide class of accelerating motions.

Abstract

Einstein's relation E=Mc^2 between the energy E and the mass M is the cornerstone of the relativity theory. This relation is often derived in a context of the relativistic theory for closed systems which do not accelerate. By contrast, Newtonian approach to the mass is based on an accelerated motion. We study here a particular neoclassical field model of a particle governed by a nonlinear Klein-Gordon (KG) field equation. We prove that if a solution to the nonlinear KG equation and its energy density concentrate at a trajectory, then this trajectory and the energy must satisfy the relativistic version of Newton's law with the mass satisfying Einstein's relation. Therefore the internal energy of a localized wave affects its acceleration in an external field as the inertial mass does in Newtonian mechanics. We demonstrate that the "concentration" assumptions hold for a wide class of rectilinear accelerating motions.