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Vladimir O. Soloviev

Publications and source records attributed to Vladimir O. Soloviev.

At least 19 recordsLinked to original sources
arXiv2022

How Friedmann Shod Einstein

Exactly one hundred years ago, in May 1922, 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 immediately after finishing his seminal paper Friedmann wrote a book on GR for a wide audience. In that book he presented an impressive picture of the expanding Universe created from nothing.

physics.hist-ph↗
arXiv2020

The Canonical Structure of Bigravity

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

gr-qc↗
arXiv2020

Constraint Algebra in Bigravity

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-Rosen transform (characterized as "a complicated redefinition of the shift variable" according to the authors) appears here not as an ansatz but as fixing of a Lagrange multiplier. A comparison of this approach with the other ones is provided.

gr-qc↗
arXiv2016

The Friedmann cosmology: mountains and wells, as real and potential

The Universe dynamical equations discovered by Alexander Friedmann in 1922 are simple enough to be clear for undergraduate students or even for smart senior schoolkids. The background cosmology driven by the dust-like matter and by positive cosmological constant in General Relativity can be understood as a radial motion of a test particle in the superposition of Newtonian and anti-Hook potentials according to Newtonian mechanics. This analogy was mentioned by George Gamov in his book (1952). The present paper was written as a pedagogical and methodological text.

physics.gen-ph↗
arXiv2015

Hamiltonian cosmology in bigravity and massive gravity

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 work is to give an introduction both to the Hamiltonian formalism and to the cosmology of bigravity. We sketch three roads to the Hamiltonian of bigravity with the dRGT potential: the metric, the tetrad and the minisuperspace approaches.

hep-th↗
arXiv2014

Bigravity in tetrad Hamiltonian formalism and matter couplings

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 tensors. The Boulware-Deser ghost is then excluded by two second class constraints. In other case when the matter couples to a linear combination of two tetrads this ghost re-appears.

hep-th↗
arXiv2013

Canonical formalism of bigravity

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.

hep-th↗
arXiv2013

Bigravity in Kuchar's Hamiltonian formalism. 2. The special case

It is proved, that, in order to avoid the ghost mode in bigravity theory, it is sufficient to impose four conditions on the potential of interaction of the two metrics. First, the potential should allow its expression as a function of components of the two metrics' 3+1 decomposition. Second, the potential must satisfy the first order linear differential equations which are necessary for the presence of four first class constraints in bigravity. Third, the potential should be a solution of the Monge-Ampere equation, where the lapse and shift are considered as variables. Fourth, the potential must have a nondegenerate Hessian in the shift variables. The proof is based on the explicit derivation of the Hamiltonian constraints, the construction of Dirac brackets on the base of a part of these constraints, and calculation of other constraints' algebra in these Dirac brackets. As a byproduct, we prove that these conditions are also sufficient in the massive gravity case.

hep-th↗
arXiv2013

Bigravity in Kuchar's Hamiltonian formalism. 1. The general case

The Hamiltonian formalism of bigravity and massive gravity is studied here for the general form of the interaction potential of two metrics. In the theories equipped with two spacetime metrics it is natural to use the Kuchar approach, because then the role played by the lapse and shift variables becomes more transparent. We find conditions on the potential which are necessary and sufficient for the existence of four first class constraints. The algebra of constraints is calculated in Dirac brackets formed on the base of all the second class constraints. It is the celebrated algebra of hypersurface deformations. By fixing one metric we obtain a massive gravity theory free of first class constraints. Then we can use symmetries of the background metric to derive conserved quantities. These are ultralocal, if expressed in terms of the metric interaction potential. The special case of potential providing less number of degrees of freedom is treated in the companion paper.

hep-th↗
arXiv1999

Black hole entropy from Poisson brackets (demystification of some calculations)

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 unsatisfactory. The functionals used are really ``non-differentiable'' under required variations and also the standard Poisson brackets for these functionals are exactly zero so being unable to get any Virasoro algebra with a central charge. Nevertheless Carlip's calculations will be correct if we admit another definition for the Poisson bracket. This new Poisson bracket differs from the standard one in surface terms only and allows to work with ``non-differentiable'' functionals.

hep-th↗
arXiv1999

Bering's proposal for boundary contribution to the Poisson bracket

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 (or, saying it in other words, to another definition of the pairing between 1-forms and 1-vectors in the formal variational calculus). We extend the formula (initially suggested by Bering only for the ultralocal case with constant coefficients) onto the general non-ultralocal brackets with coefficients depending on fields and their spatial derivatives. The lack of invariance under changes of dependent variables (field redefinitions) seems a drawback of this proposal.

hep-th↗
arXiv1998

Divergences as a grading of the formal variational calculus

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 operations, such as the action of vector fields on functionals, the commutator of vector fields, the interior product of forms and vectors and the Schouten-Nijenhuis bracket are compatible with the grading. A definition of the adjoint graded operator is proposed and antisymmetric operators are constructed with the help of boundary terms. The fulfilment of the Jacobi identity for the new Poisson brackets is shown to be equivalent to vanishing of the Schouten-Nijenhuis bracket of the Poisson bivector with itself. It is demonstrated, as an example, that the second structure of the Korteweg-de Vries equation is not Hamiltonian with respect to the new brackets until special boundary conditions are prescribed.

math.DG↗
arXiv1998

Free Boundary Poisson Bracket Algebra in Ashtekar's Formalism

We consider the algebra of spatial diffeomorphisms and gauge transformations in the canonical formalism of General Relativity in the Ashtekar and ADM variables. Modifying the Poisson bracket by including surface terms in accordance with our previous proposal allows us to consider all local functionals as differentiable. We show that closure of the algebra under consideration can be achieved by choosing surface terms in the expressions for the generators prior to imposing any boundary conditions. An essential point is that the Poisson structure in the Ashtekar formalism differs from the canonical one by boundary terms.

gr-qc↗
arXiv1996

On the Difference between Admissible and "Differentiable" Hamiltonians

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. In the same time, boundary contributions to the variation of Hamiltonian are allowed. The Ashtekar formalism for gravity and hydrodynamics of the ideal fluid with a free surface in the Clebsch variables are treated as examples.

hep-th↗
arXiv1996

Boundary terms and their Hamiltonian dynamics

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. Examples from Einstein gravity and Yang-Mills gauge field theory are given.

hep-th↗
arXiv1995

Total Divergences in Hamiltonian Formalism of Field Theory

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.

hep-th↗
arXiv1995

Boundary values as Hamiltonian variables. II. Graded Structures

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 with respect to divergences. Bilinear operations, such as action of vector fields onto functionals, commutator of vector fields, interior product of forms and vectors and the Schouten-Nijenhuis bracket are compatible with the grading. A definition of the adjoint graded operator is proposed and skew-adjoint operators are constructed with the help of boundary terms. Fulfilment of the Jacobi identity for the new Poisson brackets is shown to be equivalent to vanishing of the Schouten-Nijenhuis bracket for Poisson bivector with itself. The simple procedure for testing this condition proposed by P. Olver is applicable with a minimal modification. It is demonstrated that the second structure of the Korteweg--de Vries equation is not Hamiltonian with respect to the new brackets.

q-alg↗