S. Ye. Samokhvalov


In this paper, we study the change in the energy-momentum of a gravitational field when the general reference frame is changed. From a mathematical point of view, the general reference frame is described by the affine reference field on the space-time manifold, and the transitions between them form the gauge general linear group . Under the transformations of the group  the Lagrangian of the theory, obtains a divergent term: its canonical transformation takes place. Consequently, the gravitational equations do not change, however, the total energy of the system and its superpotential are changes. This phenomenon is called the kinematic renormalization of energy and can be used to deprive the theory of singularities.

The paper also examines individual cases of the selection of reference frames, in particular the so-called holonomic reference frames used in the general theory of relativity, as well as non-holonomic pseudoorthonormal reference frames.

In the case of holonomic reference frames, it is possible to associate with them coordinate systems  whose coordinate rays will set the reference frames. In contrast to the Euler coordinates , which number the points of space-time and whose change does not lead to a change in any physical quantities, the Lagrangian coordinates  carry the physical content, since their change leads to a change in the physical reference frame, as a result of which certain physical quantities change, such as the energy-momentum of the system, and so on. In the case of  coordinate transformations there are gauge translations in space-time in a fixed general reference frame belonging to the group . In the case of Lagrange coordinates  transformations the transformations of holonomic reference frames occur, and such transformations belong to another group, namely the group , forming its subgroup, which is isomorphic to the group of diffeomorphisms due to restriction by holonomic reference frames. In the case of the identification of the Lagrangian coordinates with the Eulerian , the energy-momentum tensor becomes Einstein energy-momentum complex of gravitational field, and its superpotential becomes Freud superpotential.

Obtained results open up wide opportunities for depriving the theory of gravity from singularities by choosing the appropriate general reference frames.


general reference frame; general theory of relativity; energy-momentum complex of gravitational field; superpotential; gauge transformations


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Самохвалов С.Є. Загальні системи відліку і визначення енергії в теорії гравітації / Самохвалов С.Є. // Математичне моделювання. – 2016. – №2(35). – С.19-23.



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Copyright (c) 2019 S. Ye. Samokhvalov

ISSN (print) 2519-2884

ISSN (online) 2617-8389