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 Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2014, Issue 2(35), Pages 180–198 (Mi vsgtu1291)

This article is cited in 2 scientific papers (total in 2 papers)

Theoretical Physics

Gauge-invariant Tensors of 4-Manifold with Conformal Torsion-free Connection and their Applications for Modeling of Space-time

L. N. Krivonosova, V. A. Luk'yanovb

a Nizhny Novgorod State Technical University, Nizhnii Novgorod, 603600, Russian Federation
b Zavolzhsk Branch of Nizhny Novgorod State Technical University, Zavolzhsk, Nizhegorodskaya obl., 606520, Russian Federation

Abstract: We calculated basic gauge-invariant tensors algebraically expressed through the matrix of conformal curvature. In particular, decomposition of the main tensor into gauge-invariant irreducible summands consists of 4 terms, one of which is determined by only one scalar. First, this scalar enters the Einstein's equations with cosmological term as a cosmological scalar. Second, metric being multiplied by this scalar becomes gauge invariant. Third, the geometric point, which is not gauge-invariant, after multiplying by the square root of this scalar becomes gauge-invariant object — a material point. Fourth, the equations of motion of the material point are exactly the same as in the general relativity, which allows us to identify the square root of this scalar with mass. Thus, we obtained an unexpected result: the cosmological scalar coincides with the square of the mass. Fifth, the cosmological scalar allows us to introduce the gauge-invariant 4-measure on the manifold. Using this measure, we introduce a new variational principle for the Einstein equations with cosmological term. The matrix of conformal curvature except the components of the main tensor contains other components. We found all basic gauge-invariant tensors, expressed in terms of these components. They are 1- or 3-valent. Einstein's equations are equivalent to the gauge invariance of one of these covectors. Therefore the conformal connection manifold, where Einstein's equations are satisfied, can be divided into 4 types according to the type of this covector: timelike, spacelike, light-like or zero.

Keywords: conformal connection, gauge group, Einstein's equations, cosmological term
Author to whom correspondence should be addressed

DOI: https://doi.org/10.14498/vsgtu1291

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Bibliographic databases:

UDC: 514.82:514.774.2
MSC: Primary 53C07, 81T13; Secondary 51B20, 83C05, 53Z05
Original article submitted 10/XII/2013
revision submitted – 21/II/2014

Citation: L. N. Krivonosov, V. A. Luk'yanov, “Gauge-invariant Tensors of 4-Manifold with Conformal Torsion-free Connection and their Applications for Modeling of Space-time”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2(35) (2014), 180–198

Citation in format AMSBIB
\Bibitem{KriLuk14} \by L.~N.~Krivonosov, V.~A.~Luk'yanov \paper Gauge-invariant Tensors of 4-Manifold with Conformal Torsion-free Connection and their Applications for Modeling of Space-time \jour Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.] \yr 2014 \vol 2(35) \pages 180--198 \mathnet{http://mi.mathnet.ru/vsgtu1291} \crossref{https://doi.org/10.14498/vsgtu1291} \zmath{https://zbmath.org/?q=an:06968887} \elib{https://elibrary.ru/item.asp?id=22813989} 

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This publication is cited in the following articles:
1. L. N. Krivonosov, V. A. Luk'yanov, “The structure of the main tensor of conformally connected torsion-free space. Conformal connections on hypersurfaces of projective space”, J. Math. Sci., 231:2 (2018), 189–205
2. L. N. Krivonosov, V. A. Lukyanov, “Konformnaya svyaznost so skalyarnoi kriviznoi”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 28:1 (2018), 22–35
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