Загальні системи відліку і визначення енергії в теорії гравітації
Journal Title: Математичне моделювання - Year 2016, Vol 1, Issue 2
Abstract
GENERAL REFERENCE FRAMES AND DEFINITION OF ENERGY IN THE GRAVITATION THEORY Samokhvalov S.E. Abstract In the general relativity (GR) as the reference frame we understand an arbitrary coordinate system that defines in space-time a holonomic coordinate repère field. The disadvantage of this limitation in the choice of reference frames is the need to use the components of a metric tensor as potentials of the gravitational field, from which it is impossible to construct a scalar Lagrangian of a gravitational field that would depend on metric coefficients and their first derivatives. Such a tensor can be constructed in the gravitation theory in an orthonormal repère, where, as the potentials of the gravitational field, one use the coefficients of transition between orthonormal and coordinate repère. However, in the general case, an arbitrary affine repère field can be used as a reference frame, with a special case being a coordinate and orthonormal repère. In this paper, the gravitation theory in the affine repère (GTAR) is presented, the main relations of which are given in the general non-holonomic reference frame, that is, a relatively arbitrary affine repère field. The consequences of the generalized gauge translation invariance of theory in the general reference frame are analyzed, as well as the consequences of the theory invariance with respect to linear transformations of the reference fields corresponding to the transition between the general reference frames and implement the general principle of relativity in the GTAR. The expression for the mixed coordinate-repère energy-momentum tensor and the angular momentum tensor of the gravitational field and their superpotentials is given. The results of the work may prove to be useful in finding new solutions in the theory of gravity, and in the methods of so-called holographic renormalization of energy. References [1] Einstein A. Osnovi obschii teorii otnositelnosti [Basis of general relativity] // Sbornik nauchnih trudov – М.: «Nauka». – 1965. – С.452–504. [2] Szabados Laslo B. Quasi-local energy-momentum and angular momentum in general relativity: A Review Article. – Living Rev. Relativity. – 2004. – 7. – P.1–140. [3] Mǿller C. Conservation lows and absolute parallelism in general relativity // Mat. – Fys. Skr. K. Danske Vid Selsk. – 1961. – 1(10). – P.1–50. [4] Rodichev V.I. Theory of gravity in the orthogonal repère. – М.: «Nauka». – 1974. – 184 p. (in Russian) [5] Aldrovandi R., Pereira J.G. Teleparallel Gravity: An Introduction. – Heidelberg: Springer, 2013. – 212 p. [6] Samokhvalov S.E., Krikent A.I. Theory of gravity in the affine repère // Math. mod. – 2016. – №1(34). – С.14–15. (in Ukrainian) [7] Samokhvalov S.E. Consequences of the symmetry of the gauge theory of gravity // Math. mod. – 2001. – №1(6). – С.23–27. (in Ukrainian) [8] Samokhvalov S.E. Group-theoretic description of gauge fields // Theor. math. phys. – 1988. – 76 (1). – P.66–77. (in Russian) [9] Samokhvalov S.E. Group-theoretic description of Riemannian spaces // Ukr. math. mag. – 2003. – 55, №9. – С. 1238 – 1248. (in Ukrainian) [10] Samokhvalov S.E. Method Palatini in the affine repère // Probl. math. mod. – 2017. – Kamjanske. – С.21- 23. (in Ukrainian) [11] Landau L.D., Lifshic E.M. Field theory. – М.: «Nauka». – 1973. – 504 p. (in Russian) Noether E. Invariant Variation Problems // Transport theory and statistical physics. – 1971. – 1(3). – P.183– 207.
Authors and Affiliations
С. Є. Самохвалов
Keywords
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