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.

• 1. Fedosin S.G. About the cosmological constant, acceleration field, pressure field and energy. Jordan Journal of Physics. Vol. 9, No. 1, pp. 1-30 (2016). doi:10.5281/zenodo.889304.
• 2. Fedosin S.G. The Principle of Least Action in Covariant Theory of Gravitation. Hadronic Journal, Vol. 35, No. 1, pp. 35-70 (2012). doi:10.5281/zenodo.889804.
• 3. Fedosin S.G. The procedure of finding the stress-energy tensor and vector field equations of any form. Advanced Studies in Theoretical Physics. Vol. 8, pp. 771-779 (2014). doi: 10.12988/astp.2014.47101.
• 4. Fedosin S.G. The virial theorem and the kinetic energy of particles of a macroscopic system in the general field concept. Continuum Mechanics and Thermodynamics, Vol. 29, Issue 2, pp. 361-371 (2016). doi:10.1007/s00161-016-0536-8.
• 5. Fedosin S.G. Estimation of the physical parameters of planets and stars in the gravitational equilibrium model. Canadian Journal of Physics, Vol. 94, No. 4, pp. 370-379 (2016). doi:10.1139/cjp-2015-0593.
• 6. Fedosin S.G. Relativistic Energy and Mass in the Weak Field Limit. Jordan Journal of Physics. Vol. 8, No. 1, pp. 1-16 (2015). doi:10.5281/zenodo.889210.
• 7. Fedosin S.G. The Integral Energy-Momentum 4-Vector and Analysis of 4/3 Problem Based on the Pressure Field and Acceleration Field. American Journal of Modern Physics, Vol. 3, No. 4, pp. 152-167 (2014). doi:10.11648/j.ajmp.20140304.12.
• 8. Sergey Fedosin. The physical theories and infinite hierarchical nesting of matter, Volume 1, LAP LAMBERT Academic Publishing, pages: 580, ISBN-13: 978-3-659-57301-9. (2014).
• 9. Fedosin S.G. The Hamiltonian in Covariant Theory of Gravitation. Advances in Natural Science, Vol. 5, No. 4, pp. 55-75 (2012). doi:10.3968%2Fj.ans.1715787020120504.2023.
• 10. Fedosin S.G. The Gravitational Field in the Relativistic Uniform Model within the Framework of the Covariant Theory of Gravitation. International Letters of Chemistry, Physics and Astronomy, Vol. 78, pp. 39-50 (2018). doi:10.18052/www.scipress.com/ILCPA.78.39.