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M. O. Katanaev

Publications and source records attributed to M. O. Katanaev.

At least 19 recordsLinked to original sources
arXiv2026

Black hole formation by a scalar field

The Liouville solution in General Relativity with a scalar field is discussed. This solution is invariant with respect to global Lorentz transformations, and dependence on time cannot be removed. If the scalar field potential is exponential and unbounded from below, the Liouville solution describes the formation of spherically symmetric black hole. The event horizon is a sphere, which appears with infinitesimal radius at a finite moment of time and afterwards expands with the velocity of light to infinity. A distant observer can measure the geometric defect at the point where the horizon appears. It is similar to the defect produced by the monopole or spherical dislocation of space-time. Comparison with the Schwarzschild solution yields the mass function which is proportional to the time squared.

gr-qc↗
arXiv2024

Geodesics and Global Properties of the Liouville Solution in General Relativity with a Scalar Field

One parameter family of exact solutions in General Relativity with a scalar field has been found using the Liouville metric. The scalar field potential has exponential form. This model is interesting, because, in particular, the solution corresponding to the naked singularity provides smooth extension of the Friedmann universe with accelerated expansion through the zero of the scale factor back in time. All geodesics are found explicitly. Their analysis shows that the Liouville solutions are global ones: every geodesic is either continued to infinite value of the canonical parameter in both directions or ends up at the singularity at its finite value.

gr-qc↗
arXiv2024

Was there a Big Bang?

New one parameter family of exact solutions in General Relativity with a scalar field is found. The metric is of Liouville type which admits complete separation of variables in the geodesic Hamilton-Jacobi equation. This solution exists for the exponential potential for a scalar field and is invariant with respect to global Lorentz transformations. It describes, in particular, evolution of the space-time with the naked singularity. Solutions corresponding to the naked singularity provide accelerating expansion of the homogeneous and isotropic Universe, and can be smoothly continued along geodesics to infinite past without Big Bang.

gr-qc↗
arXiv2024

Geometrical methods in mathematical physics

We give detailed exposition of modern differential geometry from global coordinate independent point of view as well as local coordinate description suited for actual computations. In introduction, we consider Euclidean spaces and different structures on it; rotational, Lorentz, and Poincare groups; special relativity. The main body of the manuscript includes manifolds, tensor fields, differential forms, integration, Riemannian and Lorentzian metrics, connection on vector and frame fiber bundles, affine geometry, Lie groups, transformation groups, homotopy and fundamental group, coverings, principal and associated fiber bundles, connections on fiber bundles, Killing vector fields, geodesics and extremals, symplectic and Poisson manifolds, Clifford algebras, principle of least action, canonical formalism for constrained systems. Applications of differential geometry in quantum theory (adiabatic theorem, Berry phase, Aharonov-Bohm effect), general relativity and geometric theory of defects are described. We give introduction to general relativity and its Hamiltonian formulation; describe scalar, spinor, electromagnetic and Yang-Mills fields. Riemannian and Lorentzian surfaces with one Killing vector field are discussed in detail, and their global structure is described using conformal block method. We also classified all global vacuum solutions of the Einstein equations, which have the form of warped product metrics of two surfaces. The manuscript is not a textbook, and intended for efficient reader.

math-ph↗
arXiv2023

Complete separation of variables in the geodesic Hamilton-Jacobi equation

We consider a (pseudo)Riemannian manifold of arbitrary dimension. The Hamilton-Jacobi equation for geodesic Hamiltonian admits complete separation of variables for some (separable) metrics in some (separable) coordinate systems. Separable metrics are very important in mathematics and physics. The Stäckel problem is: ``Which metrics admit complete separation of variables in the geodesic Hamilton-Jacobi equation?'' This problem was solved for inverse metrics with nonzero diagonal elements, in particular, for positive definite Riemannian metrics long ago. However the question is open for indefinite metrics having zeroes on diagonals. We propose the solution. Separable metrics are divided into equivalence classes characterised by the number of commuting Killing vector fields, quadratic indecomposable conservation laws for geodesics, and the number of coisotropic coordinates. The paper contains detailed proofs, sometimes new, of previous results as well as new cases. As an example, we list all canonical separable metrics in each equivalence class in two, three, and four dimensions. Thus the Stäckel problem is completely solved for metrics of any signature in any number of dimensions.

gr-qc↗
arXiv2023

Complete separation of variables in the geodesic Hamilton--Jacobi equation in four dimensions

We list all metrics of arbitrary signature in four dimensions which admit complete separation of variables in the Hamilton--Jacobi equation for geodesic Hamiltonians. There are only ten classes of separable metrics admitting commuting Killing vector fields, indecomposable quadratic conservation laws, and coisotropic coordinates. Canonical separable metrics parameterized by several (up to twelve) arbitrary functions of single coordinates are written explicitly. The full set of independent conservation laws in involution for each canonical metrics is also found.

gr-qc↗
arXiv2021

Nonrelativistic limit of the bosonic string

We propose the action for the nonrelativistic string invariant under general coordinate transformations on the string worldsheet. The Hamiltonian formulation for the nonrelativistic string is given. Particular solutions of the Euler-Lagrange equations are found in the time gauge.

hep-th↗
arXiv2021

Gravity with dynamical torsion

We propose four simple Lagrangians for gravity models with dynamical torsion which are free from ghosts and tachyons. The torsion propagates two massive or massless particles of spin 1^\pm and 0^\pm besides the massless graviton 2^+ propagated by metric.

gr-qc↗
arXiv2021

Spherically symmetric 't Hooft-Polyakov monopoles

A general analytic spherically symmetric solution of the Bogomol'nyi equations is found. It depends on two constants and one arbitrary function on radius and contains the Bogomol'nyi-Prasad-Sommerfield and Singleton solutions as particular cases. Thus all spherically symmetric 't Hooft-Polyakov monopoles with massless scalar field and minimal energy are derived.

physics.gen-ph↗
arXiv2021

Disclinations in the geometric theory of defects

In the geometric theory of defects, media with a spin structure, for example, ferromagnet, is considered as a manifold with given Riemann--Cartan geometry. We consider the case with the Euclidean metric corresponding to the absence of elastic deformations but with nontrivial ${\mathbb S}{\mathbb O}(3)$-connection which produces nontrivial curvature and torsion tensors. We show that the 't Hooft--Polyakov monopole has physical interpretation in solid state physics describing media with continuous distribution of dislocations and disclinations. The Chern--Simons action is used for the description of single disclinations. Two examples of point disclinations are considered: spherically symmetric point "hedgehog" disclination and the point disclination for which the $n$-field has a fixed value at infinity and essential singularity at the origin. The example of linear disclinations with the Franc vector divisible by $2π$ is considered.

cond-mat.mtrl-sci↗
arXiv2021

On the existence of the global conformal gauge in string theory

The global conformal gauge is playing the crucial role in string theory providing the basis for quantization. Its existence for two-dimensional Lorentzian metric is known locally for a long time. We prove that if a Lorentzian metric is given on a plain then the conformal gauge exists globally on the whole ${\mathbb R}^2$. Moreover, we prove the existence of the conformal gauge globally on the whole worldsheets represented by infinite strips with straight boundaries for open and closed bosonic strings. The global existence of the conformal gauge on the whole plane is also proved for the positive definite Riemannian metric.

physics.gen-ph↗
arXiv2020

Global properties of warped solutions in General Relativity with an electromagnetic field and a cosmological constant. II

We consider general relativity with cosmological constant minimally coupled to the electromagnetic field and assume that the four-dimensional space-time manifold is a warped product of two surfaces with Lorentzian and Euclidean signature metrics. Field equations imply that at least one of the surfaces must be of constant curvature leading to the symmetry of the metric (``spontaneous symmetry emergence''). We classify all global solutions in the case when the Lorentzian surface is of constant curvature. These solutions are invariant with respect to the Lorentz SO(1,2) or Poincare IO(1,1) groups acting on the Lorentzian surface.

gr-qc↗
arXiv2020

The 't Hooft-Polyakov monopole in the geometric theory of defects

The 't Hooft-Polyakov monopole solution in Yang-Mills theory is given new physical interpretation in the geometric theory of defects. It describes solids with continuous distribution of dislocations and disclinations. The corresponding densities of Burgers and Frank vectors are computed. It means that the 't Hooft-Polyakov monopole can be seen, probably, in solids.

physics.gen-ph↗
arXiv2020

Gauge parameterization of the $n$-field

We propose gauge parameterization of the three-dimensional $n$-field using orthogonal SO(3)-matrix which, in turn, is defined by a field taking values in the Lie algebra so(3) (rotational-angle field). The rotational-angle field has an additional degree of freedom, which corresponds to the gauge degree of freedom of rotations around the $n$-field. As a result, we obtain a gauge model with local SO(2)=U(1) symmetry that does not contain a U(1) gauge field.

physics.gen-ph↗
arXiv2019

Global properties of warped solutions in General Relativity with electromagnetic field and cosmological constant

We consider general relativity with cosmological constant minimally coupled to electromagnetic field and assume that four-dimensional space-time manifold is the warped product of two surfaces with Lorentzian and Euclidean signature metrics. Einstein's equations imply that at least one of the surfaces must be of constant curvature. It means that the symmetry of the metric arises as the consequence of equations of motion (`spontaneous symmetry emergence'). We give classification of global solutions in two cases: (i) both surfaces are of constant curvature and (ii) the Riemannian surface is of constant curvature. The latter case includes spherically symmetric solutions (sphere S^2 with SO(3)-symmetry group), planar solutions (two-dimensional Euclidean space R^2 with IO(2)-symmetry group), and hyperbolic solutions (two-sheeted hyperboloid H^2 with SO(1,2)-symmetry). Totally, we get 37 topologically different solutions. There is a new one among them, which describes changing topology of space in time already at the classical level.

physics.gen-ph↗
arXiv2017

Chern--Simons term in the geometric theory of defects

The Chern--Simons term is used in the geometric theory of defects. The equilibrium equations with $δ$-function source are explicitly solved with respect to the $SO(3)$ connection. This solution describes one straight linear disclination and corresponds to the new kind of geometrical defect: it is the defect in the connection but not the metric which is the flat Euclidean metric. This is the first example of a disclination described within the geometric theory of defects. The corresponding angular rotation field is computed.

math-ph↗
arXiv2016

Killing vector fields and a homogeneous isotropic universe

Some basic theorems on Killing vector fields are reviewed. In particular, the topic of a constant-curvature space is examined. A detailed proof is given for a theorem describing the most general form of the metric of a homogeneous isotropic space-time. Although this theorem can be considered to be commonly known, its complete proof is difficult to find in the literature. An example metric is presented such that all its spatial cross sections correspond to constant-curvature spaces, but it is not homogeneous and isotropic as a whole. An equivalent definition of a homogeneous and isotropic space-time in terms of embedded manifolds is also given.

gr-qc↗