Sergey G. FEDOSIN

The Integral Theorem of the Field Energy

The integral theorem of the vector field energy is derived in a covariant way, according to whichunder certain conditions the potential energy of the system’s field turns out to be half as large inthe absolute value as the field’s kinetic energy associated with the four-potential of the field andthe four-current of the system’s particles. Thus, the integral theorem turns out to be the analogueof 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 twoseemingly unrelated ways, either through the scalar potential of the field or through the stresenergy-momentum tensor of the field. In closed systems, the theorem formulation is simplifiedfor the electromagnetic and gravitational fields, which can act at a distance up to infinity. At thesame time for the fields acting locally in the matter, such as the acceleration field and the pressurefield, in the theorem formulation it is necessary to take into account the additional term withintegral taken over the system’s surface. The proof of the theorem for an ideal relativistic uniformsystem containing non-rotating and randomly moving particles shows full coincidence in allsignificant terms, particularly for the electromagnetic and gravitational fields, the accelerationfield and the vector pressure field.

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