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V. V. Obukhov

Publications and source records attributed to V. V. Obukhov.

At least 19 recordsLinked to original sources
arXiv2025

Exact solutions of Maxwell vacuum equations in Petrov homogeneous non-null spaces

The classification of exact solutions of Maxwell vacuum equations for pseudo-Riemannian spaces with spatial symmetry (homogeneous non-null spaces of Petrov) in the presence of electromagnetic fields invariant with respect to the action of the group of space motions is summarized. A new classification method is used, common to all homogeneous zero spaces of Petrov. The method is based on the use of canonical reper vectors and a on the use a new approach to the systematization of solutions. The classification results are presented in a form more convenient for further use. Using the previously made refinement of the classification of Petrov spaces, the classification of exact solutions of Maxwell vacuum equations for spaces with the group of motions $G_3(VIII )$ is completed.

gr-qc↗
arXiv2025

Classification of Einstein spaces with Stackel metric of type (3.0)

The classification of the Einstein spaces with the Stackel metric of the (3.0) has been done. These spaces are invariant under the action of the three-parameter abelian group of motions and belong to the first type Bianchi spaces. Thus the classification of vacuum and electrovacuum Stackel spaces of all types is completed and the complete list of metrics of such spaces in privileged coordinate systems is given.

math.DG↗
arXiv2024

Classification of Petrov Homogeneous Spaces

In this paper the final stage of the Petrov classification is carried out. As it is known, the Killing vector fields specify infinitesimal transformations of the group of motions of space $V_4$. In the case when in the homogeneous space $V_4$ the group of motions $G_3$ acts simply transitive, the geometry of the non-isotropic hypersurface is determined by the geometry of the transitivity space $V_3$ of the group $G_3$. In this case, the metric tensor of the space $V_3$ can be given by a nonholonomic reper consisting of three independent vectors $\ell_{(a)}^α$, which define the generators of the group $G_3$ of finite transformations in the space $V_3$. The representation of the metric tensor of $V_4$ spaces by means of vector fields $\ell_{(a)}^α$ has a great physical meaning and allows to simplify substantially the equations of mathematical physics in such spaces. Therefore, the Petrov classification should be complemented by the classification of vector fields $\ell_{(a)}^α$ connected to Killing vector fields. For homogeneous spaces this problem has been largely solved. A complete solution of this problem is presented in the present paper, where the Petrov classification for homogeneous spaces in which the group $G_3$, which belongs to type $VIII$ according to the Petrov classification, acts simply transitively, is refined. In addition, the complete classification of vector fields $\ell_{(a)}^α$ for spaces $V_4$ in which the group $G_3$ acts simply transitivity on isotropic hypersurfaces.}

gr-qc↗
arXiv2024

Classification of the non-null electrovacuum solution of Einstein-Maxwell equations with three-parameter abelian group of motions

The classification of the Stackel spaces of the electrovacuum of the type (3.0) has been done. These spaces are invariant under the action of the three-parameter abelian group of motions and belong to the first type Bianchi spaces. In the case of a non-zero cosmological term, the metrics and potentials contain solutions of a nonlinear ordinary differential equation of the second order. When the cosmological term equals zero, the metrics and the components of the electromagnetic field tensor are expressed through elementary functions. Thus the classification of the electrovacuum Stackel spaces of all types is completed and complete list of these spaces is constructed.

gr-qc↗
arXiv2023

Solutions of Maxwell equations for admissible electromagnetic fields, in spaces with simply transitive four-parameter groups of motions

All non-equivalent solutions of vacuum Maxwell equations are found for the case when space-time manifolds admit simply transitive four-parameter groups of motions $G_4(N)$. The potentials of the admissible electromagnetic fields admit the existence of the algebra of motion integrals of the Hamilton-Jacobi and Klein-Gordon-Fock equations which is isomorphic to the algebra of the group operators for the same group $G_4(N)$

math-ph↗
arXiv2023

Comments on the article "M.O. Katanaev, Complete separation of variables in the geodesic Hamilton-Jacobi equation in four dimensions, Physica Scripta (2023), 98, 104001"

The present note is a feedback on the article by M.O. Katanaev in "Physica Scripta" (2023, 98, 104001), where, in our opinion, a distorted view of the classical theory of separation of variables in the Hamilton-Jacobi equation is given. We show that the metrics given in this paper, addmiting separation of variables, are special cases of V.N. Shapovalov metrics ("Siberian Mathematical Journal", 1979, 20, 790), finally obtained in the 70s of the last century. The results of the article in question, unlike the original ones, do not have scientific novelty and contain omissions and incorrect statements.

gr-qc↗
arXiv2023

Hamilton-Jacobi and Klein-Gordon-Fock equations for a charged test particle in space-time with simply transitive four-parameter groups of motions

Metric components of potentials of admissible electromagnetic fields in spaces with simply transitive four-parameter motion group are found. The components of frame vectors corresponding to the components of the metric tensors found by Petrov are given. The results obtained complement the coordinate-free classification given by Magazev et al. Previously, admissible electromagnetic fields were found for the case when three- and four-parameter groups of motions act on hypersurfaces of spacetime. Thus, non-equivalent sets of potentials for all electromagnetic fields that admit three- and four-parameter groups of motions are known now.

math-ph↗
arXiv2023

Exact solutions of Maxwell equations in homogeneous spaces with the group of motions $G_3(VIII)$

The problem of classification of exact solutions of Maxwell's vacuum equations for admissible electromagnetic fields and homogeneous space-time with the group of motions $G_3(VIII)$ according to the Bianchi classification is considered. All non-equivalent solutions are found. The classification problem for remaining groups of motions $G_3(N)$ has already been solved in the other papers. That is why all non-equivalent solutions of empty Maxwell equations for all homogeneous spaces with admissible electromagnetic fields are known now.

gr-qc↗
arXiv2022

Maxwell equations in homogeneous spaces with solvable groups of motions

The classification of exact solutions of Maxwell vacuum equations for the case when the electromagnetic fields and metrics of homogeneous spaces are invariant with respect to the motion group G(VII) is completed. All non-equivalent exact solutions of Maxwell vacuum equations for electromagnetic fields and spaces with such symmetry have been obtained. The vectors of the canonical frame of a homogeneous space of type VII according to the Bianchi classification, and the electromagnetic field potentials have been found.

math-ph↗
arXiv2022

Maxwell's equations in homogeneous spaces for admissible electromagnetic fields

Maxwell's vacuum equations are integrated for admissible electromagnetic fields in homogeneous spaces. Admissible electromagnetic fields are those for which the space group generates an algebra of symmetry operators ( integrals of motion ) that is isomorphic to the algebra of group operators. Two frames associated with the group of motions are used to obtain systems of ordinary differential equations to which Maxwell's equations reduce. The solutions are obtained in quadratures. The potentials of the admissible electromagnetic fields and the metrics of the spaces contained in the obtained solutions depend on six arbitrary time functions, so it is possible to use them to integrate field equations in the theory of gravity.

gr-qc↗
arXiv2022

Algebras of integrals of motion for the Hamilton-Jacobi and Klein-Gordon-Fock equations in spacetime with a four-parameter groups of motions in the presence of an external electromagnetic field

The algebras of the integrals of motion of the Hamilton-Jacobi and Klein-Gordon-Fock equations for a charged test particle moving in an external electromagnetic field in a spacetime manifold are found. The manifold admits a four-parameter groups of motions that act nontransitively on the spacetime. All admissible electromagnetic fields for which such algebras exist are found. In the case of an arbitrary n-dimensional Riemannian space on which the group of motions acts, it is proved that the admissible field does not deform the algebra of symmetry operators of the free Hamilton-Jacobi and Klein-Gordon-Fock equations. In addition, the system of differential equations, which must be satisfied by the potentials of the admissible electromagnetic field, have been investigated for compatibility.

math-ph↗
arXiv2022

Algebra of the symmetry operators of the Klein-Gordon-Fock equation for the case when groups of motions $G_3$ act transitively on null subsurfaces of spacetime

The algebras of the symmetry operators for the Hamilton-Jacobi and Klein-Gordon-Fock equations are found for a charged test particle moving in an external electromagnetic field in a spacetime manifold, on the isotropic (null) hypersurface of which a three-parameter groups of motions act transitively.on the isotropic (null) hypersurface of which a three-parameter groups of motions act transitively. We have found all admissible electromagnetic fields for which such algebras exist. We have proved that an admissible field does not deform the algebra of symmetry operators for the free Hamilton-Jacobi and Klein-Gordon-Fock equations. The results complete the classification of admissible electromagnetic fields in which the Hamilton-Jacobi and Klein-Gordon-Fock equations admit algebras of motion integrals that are isomorphic to the algebras of operators of $r$-parametric groups of motions of spacetime manifolds if $(r \leq 4)$.

gr-qc↗
arXiv2021

Algebra of symmetry operators for Klein-Gordon-Fock equation

All external electromagnetic fields in which the Klein-Gordon-Fock equation admits the first-order symmetry operators are found, provided that in the space-time $V_4$ a group of motion $G_3$ acts simply transitively on a non-null subspace of transitivity $V_3$. It is shown that in the case of a Riemannian space $V_n$, in which the group $G_r$ acts simply transitively, the algebra of symmetry operators of the $n$-dimensional Klein-Gordon-Fock equation in an external admissible electromagnetic field coincides with the algebra of operators of the group $G_r$.

math-ph↗
arXiv2020

Separation of variables in Hamilton-Jacobi and Klein-Gordon-Fock equations for a charged test particle in the Stackel spaces of type (1.1)

All equivalence classes for electromagnetic potentials and space-time metrics of Stackel spaces, provided that Hamilton-Jacobi equation and Klein-Gordon-Fock equation for a charged test particle can be integrated by the method of complete separation of variables are found. The separation is carried out using the complete sets of mutually-commuting integrals of motion of type (1.1). Whereby in a privileged coordinate system the given equations turn into parabolic type equations. Hence, these metrics can be used as models for describing plane gravitational waves.

gr-qc↗
arXiv2018

Inflation in Terms of a Viscous van der Waals Coupled Fluid

We propose to describe the acceleration of the universe by introducing a model of two coupled fluids. We focus on the accelerated expansion at the early stages. The inflationary expansion is described in terms of a van der Waals equation of state for the cosmic fluid, when account is taken of bulk viscosity. We assume that there is a weak interaction between the van der Waals fluid and the second component (matter). The gravitational equations for the energy densities of the two components are solved for a homogeneous and isotropic Friedmann-Robertson-Walker universe, and analytic expressions for the Hubble parameter are obtained. The slow-roll parameters, the spectral index, and the tensor-to-scalar ratio are calculated and compared with the most recent astronomical data from the Planck satellite. Given reasonable restriction on the parameters, the agreement with observations is favorable.

gr-qc↗
arXiv2016

Inflationary Universe with a Viscous Fluid Avoiding Self-Reproduction

We consider a universe with a bulk viscous cosmic fluid, in a flat Friedmann-Lemaitre-Robertson-Walker geometry. We derive the conditions for the existence of inflation, and those which at the same time prevent the occurrence of self-reproduction. Our theoretical model gives results which are in perfect agreement with the most recent data from the PLANCK surveyor.

gr-qc↗
arXiv2016

Inflationary Cosmology Leading to a Soft Type Singularity

A remarkable property of modern cosmology is that it allows for a special case of symmetry, consisting in the possibility of describing the early-time acceleration (inflation) and the late-time acceleration using the same theoretical framework. In this paper we consider various cosmological models corresponding to a generalized form for the equation of state for the fluid in a flat Friedmann -Robertson-Walker universe, emphasizing cases where the so-called type IV singular inflation is encountered in the future. This is a soft (non-crushing) kind of singularity. Parameter values for an inhomogeneous equation of state leading to singular inflation are obtained. We present models for which there are two type IV singularities, the first corresponding to the end of the inflationary era and the second to a late time event. We also study the correspondence between the theoretical slow-roll parameters leading to type IV singular inflation and the recent results observed by the Planck satellite.

gr-qc↗