Total Energy-momentum Tensor Research Articles

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

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.

On the relativistic dynamics of polarized systems. II

The classical equations of motion for electric and magnetic dipole atoms (or molecules) in an external electromagnetic field of force, derived in a previous paper on the basis of Møller's theory of the relativistic dynamics of systems with an internal angular momentum, are simplified by showing that certain terms, which contain an unphysical trembling motion (“Zitterbewegung”), are completely negligible with respect to the other terms in these equations. The resulting equations are used in order to derive the relativistic atomic energy-momentum tensor for a system, consisting of these dipole atoms. The field part of this tensor has exactly the same form as obtained before, but the material part is slightly different as a consequence of the simplification in the equations of motion. The same symmetrization procedure, as used in the preceding paper, can be applied to the total energy-momentum tensor. Radiative effects are neglected throughout the theory.

The relativistic energy-momentum tensor in polarized media: III. Statistical theory of the energy-momentum laws

From the atomic conservation laws of energy-momentum the corresponding macroscopic laws are derived with the help of a covariant averaging procedure. The total energy-momentum tensor is found as a statistical expression in terms of atomic quantities. It may be split into a field part T αβ (⨍) (α, β = 0, 1, 2, 3) containing the macroscopic fields and polarizations, which in the rest frame reads: T αβ (⨍) = 1 2 E 2+ 1 2 B 2 (E ∧ H) i (E∧H) i −E iD j−H iB j+( 1 2 E 2+ 1 2 B 2−M⋅B)g ij (i,j =1,2,3) and a material part T αβ ( m) which forms the relativistic generalization of the usual energy and momentum expressions.

The relativistic energy-momentum tensor in polarized media: IV. The macroscopic material energy-momentum tensor

The relativistic material energy-momentum tensor in polarized media, which contains bulk terms, fluctuation terms and correlation terms, is studied especially in the nonrelativistic limit. On this basis explicit forms of the energy-momentum and angular momentum laws are given. Contributions quadratic in the polarizations are proved to be contained in the correlation terms. For this reason a redefinition of the field and material part of the total energy-momentum tensor may be introduced.

The relativistic energy-momentum tensor in polarized media: VI. The difference between the energy-momentum tensors in the presence and in the absence of external fields

The difference of the relativistic energy-momentum tensors of a substance with and without fields is derived for the case of thermodynamic equilibrium and linear constitutive relations. It may be looked upon as the field part of the total energy-momentum tensor. The remainder is then the corresponding material part. The negative divergence of the field tensor is the ponderomotive force corresponding to pressure and internal energy defined at zero fields. It constitutes a relativistic generalization of the Helmholtz force. A comparison with the tensors of Minkowski and Abraham shows that these cannot be justified from a microscopic point of view.

Flat theory of gravitation

We construct a Lorentz-covariant theory of gravitation with linear field equations which accounts for all the observational facts,i.e., the three tests of general relativity, with only one adjustable parameter apart from the gravitational coupling constant. The equations of motion of the theory arise both from a variational principle and from the conservation law of the total energy-momentum tensor of the system. The conservation law of mass is also satisfied.