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Publications (49)
In Friedmann–Lobachevsky space-time with a radius of curvature slowly varying over time, we study numerically the problem of motion of a particle moving in the Cornell potential. The mass of the particle is taken to be a reduced mass of the charmonium system. In contrast to the similar problem in flat space, in Lobachevsky space the Cornell potenti...
В 2022 году исполнилось 100 лет работе Александра Александровича Фридмана, в которой он получил главный вывод из общей теории относительности: Вселенная подчиняется уравнению движения, она была рождена и развивается. Фридман изучил все основные решения уравнения и сценарии эволюции мира. Эйнштейн считал главной ошибкой своей жизни, то что он упусти...
Foreward to the Bolyai-Gauss-Lobachevsky 2022 special issue, published in ${\it Symmetry}$, in lieu of the 12${}^{\rm th}$ International Conference on Non-Euclidean Geometry, ``BGL-2022''.
Nearly 2300 years ago, the Greek mathematician Euclid of Alexandria laid down the basis of the geometry now known from the textbooks and used in everyday life [...]
Exactly one hundred years ago, Alexander Friedmann discovered that General Relativity (GR) predicts non-stationary Universe. His equation describing the evolution of the Universe is now named after him. In this paper we briefly recall the human and scientific aspects of this revolutionary change in our picture of the world. We also recall that imme...
Bigravity is one of the most natural modifications of General Relativity (GR), as it is based on the equivalence principle. However, its canonical structure appears rather complicated because of the unusual form of the interaction between two metrics. As a consequence, there are different approaches that are difficult to compare in detail. This wor...
The constraint algebra is derived in the second order tetrad Hamiltonian formalism of the bigravity. This is done by a straightforward calculation without involving any insights, implicit functions, and Dirac brackets. The tetrad approach is the only way to present the bigravity action as a linear functional of lapses and shifts and the Hassan–Rose...
This work is motivated by an intention to make the theory of bigravity more comprehensible. Bigravity is a modification of the General Relativity (GR), maybe even the most natural one because it is based on the equivalence principle. The Hamiltonian formalism in tetrad variables transparently demonstrates the structure of bigravity
The constraint algebra is derived in the 2nd order tetrad Hamiltonian formalism of the bigravity. This is done by a straightforward calculation without involving any insights, implicit functions, and Dirac brackets. The tetrad approach is the only way to present the bigravity action as a linear functional of lapses and shifts, and the Hassan-Rosen...
This article discusses cosmology, bimetric gravity and their possible interplay.
The non-Euclidean geometry created by Bolyai, Lobachevsky and Gauss has led to a new physical theory—general relativity. In due turn, a correct mathematical treatment of the cosmological problem in general relativity has led Friedmann to a discovery of dynamical equations for the universe. And now, after almost a century of theoretical and experime...
Celebrating the centenary of general relativity theory, we must recall that Friedmann’s discovery of the equations of evolution of the Universe became the strongest prediction of this theory. These equations currently remain the foundation of modern cosmology. Nevertheless, data from new observations stimulate a search for modified theories of grav...
This article is written as a review of the Hamiltonian formalism for the bigravity with de Rham–Gabadadze–Tolley (dRGT) potential, and also of applications of this formalism to the derivation of the background cosmological equations. It is demonstrated that the cosmological scenarios are close to the standard ΛCDM model, but they also uncover the d...
The number of degrees of freedom in bigravity theory is found for a potential of general form and also for the potential proposed by de Rham, Gabadadze, and Tolley (dRGT). This aim is pursued via constructing a Hamiltonian formalismand studying the Poisson algebra of constraints. A general potential leads to a theory featuring four first-class cons...
In the Hamiltonian language we provide a study of flat-space cosmology in
bigravity and massive gravity constructed mostly with de Rham, Gabadadze,
Tolley (dRGT) potential. It is demonstrated that the Hamiltonian methods are
powerful not only in proving the absence of the Boulware-Deser ghost, but also
in solving other problems. The purpose of this...
Theory of bigravity is one of approaches proposed to solve the dark energy problem of the Universe. It deals with two metric tensors, each one is minimally coupled to the corresponding set of matter fields. The bigravity Lagrangian equals to a sum of two General Relativity Lagrangians with the different gravitational coupling constants and differen...
Тетрадный формализм используется для вычисления квадратного корня из матрицы, возникающего в потенциале де Рам-Габададзе-Толи. В минимальном случае получены уравнения связи и их алгебра. Показано, что число гравитационных степеней свободы соответствует безмассовому и массивному полям гравитации. Дух Бульвара-Дезера исключается благодаря двум связям...
We use the tetrad formalism to calculate a matrix square root occurring in the de Rham-Gabadadze-Tolley potential. In the minimal case, we obtain the constraints and their algebra. We show that the number of gravitational degrees of freedom corresponds to the massless and massive gravity fields. The Boulware-Deser ghost is eliminated because of two...
The tetrad approach is used to resolve the matrix square root appearing in
the dRGT potential. Constraints and their algebra are derived for the minimal
case. It is shown that the number of gravitational degrees of freedom
corresponds to one massless and one massive gravitational fields when two sorts
of matter separately interact with two metric t...
We construct the Hamiltonian formalism of bigravity for the potential of a
general form. We find conditions on this potential and prove that under these
conditions the formalism is equivalent to the one constructed with the
celebrated dRGT-potential.
We develop the Hamiltonian formalism of bigravity and bimetric theories
for the general form of the interaction potential of two metrics. When
studying the role of lapse and shift functions in theories with two
metrics, we naturally use the Kuchař formalism in which these
functions are independent of the choice of the space-time coordinate
system....
In the first part of this work (arxiv:1211.6530) it was shown that the
Hamiltonian formalism of bigravity with the ultralocal potential of a general
form, under some conditions imposed on the potential, had 4 first class
constraints and their algebra was the algebra of hypersurface deformations. In
this article we suppose that the potential also sa...
We provide a space-time covariant Hamiltonian treatment for a finite-range
gravitational theory. The Kuchar approach is used to demonstrate the bimetric
picture of space-time in its most transparent form. This Hamiltonian formalism
is applied for the straightforward realization of the Poincar\'e algebra in
Dirac brackets. It uncovers the simplest f...
We obtain the Poincare group generators by proper choice of arbitrary functions present in the Relativistic Theory of Gravitation (RTG) Hamiltonian. Their Dirac brackets give the Poincare algebra in accordance with the fact that RTG has 10 integrals of motion. Comment: 8 pages, Russian
The Hamiltonian of the Relativistic Theory of Gravitation (RTG) with nonzero graviton mass is derived. Scalar field is taken as a matter source. The second class constraints are excluded and Dirac brackets are obtained. There are no first class constraints in the theory. The Poincare group generators are found by specifying the family of hypersurfa...
We discuss the most interesting approaches to derivation of the Bekenstein-Hawking entropy formula from a statistical theory.
An application of the approach to Hamiltonian treatment of boundary terms proposed in previous articles of this series is considered. Here the Hamiltonian formalism is constructed and the role of standard boundary conditions is revealed for a inviscid compressible fluid with surface tension which moves in a field of the Newtonian gravitational pote...
It is shown that the Poisson bracket with boundary terms recently proposed by
Bering (hep-th/9806249) can be deduced from the Poisson bracket proposed by the
present author (hep-th/9305133) if one omits terms free of Euler-Lagrange
derivatives ("annihilation principle"). This corresponds to another definition
of the formal product of distributions...
Recently it has been suggested by S. Carlip that black hole entropy can be derived from a central charge of the Virasoro algebra arising as a subalgebra in the surface deformations of General Relativity in any dimension. Here it is shown that the argumentation given in Section 2 of hep-th/9812013 and based on the Regge-Teitelboim approach is unsati...
It is shown that the new formula for the field theory Poisson brackets arise naturally in the extension of the formal variational calculus incorporating divergences. The linear spaces of local functionals, evolutionary vector fields, functional forms, multi-vectors and differential operators become graded with respect to divergences. The bilinear o...
It is shown that the Regge-Teitelboim criterion for fixing the unique boundary contribution to the Hamiltonian compatible with free boundary conditions should be modified if the Poisson structure is noncanonical. The new criterion requires cancellation of boundary contributions to the Hamiltonian equations of motion. At the same time, boundary cont...
We consider the algebra of spatial diffeomorphisms and gauge transformations in the framework of the canonical formalism for
general relativity in the Ashtekar and ADM variables. Modifying the Poisson bracket of fields by including surface terms in
accordance with our previous proposal allows us to consider all local functionals as admissible. We s...
It is described how the standard Poisson bracket formulas should be modified in order to incorporate integrals of divergences into the Hamiltonian formalism and why this is necessary.
It is shown how to extend the formal variational calculus in order to incorporate integrals of divergences into it. Such a generalization permits to study nontrivial boundary problems in field theory on the base of canonical formalism.
In this report it is proposed to generalize the definition of Poisson brackets in order to treat spatial integrals of divergences as Hamiltonians which generate a kind of Hamiltonian equations on the boundary. Nonlinear Schrodinger equation is used as an illustrative example.
It is shown that the new Poisson brackets proposed in Part I of this work (J. Math. Phys. 34, 5747(hep-th/9305133)) arise naturally in an extension of the formal variational calculus incorporating divergences. The linear spaces of local functionals, evolutionary vector fields, functional forms, multi-vectors and differential operators become graded...
Attention is paid to the fact that in the field theory a commutator of functional derivatives may differ from zero by surface integrals. Ashtekar's formalism is a primer which demonstrates that transformations looking locally as canonical can lead to the appearance of surface terms in symplectic form of the field theory. The prescription for delta-...
When an asymptotically flat spacetime is considered the new canonical formulation of General Relativity invented by Ashtekar requires a due account of surface integrals which are necessary to realize the Poincare algebra. In particular, the Poisson brackets should be redefined to get reasonable results.
The ordinary Poisson brackets in field theory do not fulfil the Jacobi identity if boundary values are not reasonably fixed by special boundary conditions. We show that these brackets can be modified by adding some surface terms to lift this restriction. The new brackets generalize a canonical bracket considered by Lewis, Marsden, Montgomery and Ra...
Attention is paid to the fact that in field theory a commutator of functional derivatives may differ from zero by surface integrals. Ashtekar's formalism is a primer which demonstrates that transformations looking locally canonical can lead to the appearance of surface terms in a symplectic form of field theory. The prescription for the delta-funct...
