Relativistic Point Dynamics and Einstein Formula as a Property of Localized Solutions of a Nonlinear Klein-Gordon Equation

Abstract

Einstein's relation E = M c 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.

Explicit mass renormalization and consistent derivation of radiative response of classical electron

“…Note that we derive this formula in a regime with acceleration where Newtonian definition of mass is applicable. A similar derivation of Einstein's formula in the regime without radiation is given in [2], [3]. Examples of accelerating localized solutions of KG equations are also given there.…”

“…where the star denotes the complex conjugation, m is a mass parameter, it is related to the mass of the charge, see [2], [3] and Section 3.1. It involves covariant derivatives…”

“…Now we formulate our assumptions (complete mathematical details in a similar situation can be found in [3], [7]). We introduce a small ball Ω = Ω R = Ω (r, R) = {x : |x −r (t)| ≤ R} centered at a point of the trajectory, below we take R → 0.…”

“…Note that this kind of assumptions is natural for a point-like behavior of a charge distribution, they are obviously fulfilled for delta-functions as in (55). Detailed mathematical formulations made in similar situations and the verification of fulfillment of this kind of conditions for non-trivial examples can be found in [3], [7]. The unlocalized quantities which involve only the EM fields A ν or A adν can make a non-vanishing contribution to the surface integrals, namely we assume that…”

“…This is in a contrast with the usual mass renormalization where a certain amount of energy is subtracted to obtain a finite energy; the subtracted energy is interpreted as the energy of cohesive forces and the finite energy is interpreted as the renormilized mass and its value is prescribed arbitrarily. Our method of determination of the limit dynamics can be considered as a refinement in the spirit of Ehrenfest theorem of Dirac's approach [10]; this method was applied in [2], [3] to derive Einstein's formula in regimes with acceleration in the case where there is no radiative response. Our derivation is explicit in the sense that not only the rest mass is calculated, but also it describes all effects of Poincaré cohesive forces relevant to the point charge dynamics, and in particular demonstrates that their dynamical effect cannot be reduced to one constant (the renormalized mass), but involves in addition another parameter which controls the relative magnitude of the radiative response.…”