On the relativistic dynamics of polarized systems. IV: The classical equations of motion with radiation reaction and the energy-momentum tensor for electric and magnetic dipole atoms and molecules

Abstract

The classical equations of motion for electric and magnetic dipole atoms (or molecules) in an external electromagnetic field of force, treated in papers I and II on the basis of Møller's theory of the relativistic dynamics of systems with an internal angular momentum, are extended to the case that the reaction of radiation on the atoms is taken into account. To this end Møller's theory, which is valid only for finite systems (total energy-momentum tensor zero outside a finite region in space for arbitrary fixed time), is modified in order to be applicable to the case of radiating atoms (or molecules). Dirac's method of splitting retarded fields covariantly into self-parts (half sum of retarded and advanced fields) and “radiative” parts (half difference of retarded and advanced fields) is applied to the sub-atomic fields. It is proved that the sub-atomic self-force density can be written as minus the divergence of a symmetric four-tensor, which is zero outside the atomic system and which, added to the sub-atomic material energy-momentum tensor, may be interpreted as the total energy-momentum tensor of the finite atomic system. With the help of the latter tensor the atomic mass, intrinsic angular momentum and centre of gravity are defined, using Møller's theory. The influence of the “radiative” part of the field on the centre of gravity motion of the atoms and the change of their intrinsic angular momentum is then analysed. The equation of motion and the intrinsic angular momentum balance equation, obtained for radiating charged dipole atoms are used in order to derive the relativistic atomic energy-momentum tensor for a system consisting of a large number of these atoms. In contrast with the tensors in the previous papers, this tensor is no longer symmetrical. The treatment of the present paper could be extended to include the case in which the atoms also possess electric quadrupole moments.