The integral theorem of the field energy

Abstract

The integral theorem of the vector field energy is derived in a covariant way, according to which under certain conditions the potential energy of the system’s field turns out to be half as large in the absolute value as the field’s kinetic energy associated with the four-potential of the field and the four-current of the system’s particles. Thus, the integral theorem turns out to be the analogue of the virial theorem, but with respect to the field rather than to the particles. Using this theorem, it becomes possible to substantiate the fact that electrostatic energy can be calculated by two seemingly unrelated ways, either through the scalar potential of the field or through the stres energy-momentum tensor of the field. In closed systems, the theorem formulation is simplified for the electromagnetic and gravitational fields, which can act at a distance up to infinity. At the same time for the fields acting locally in the matter, such as the acceleration field and the pressure field, in the theorem formulation it is necessary to take into account the additional term with integral taken over the system’s surface. The proof of the theorem for an ideal relativistic uniform system containing non-rotating and randomly moving particles shows full coincidence in all significant terms, particularly for the electromagnetic and gravitational fields, the acceleration field and the vector pressure field .