Complete Separation Of Variables Research Articles

Conformally-flat Stäckel spaces of type (2.0)

Conformally-flat models of space-time that admit privileged coordinate systems allowing complete separation of variables in the Hamilton–Jacobi equation of type (2.0) are considered. An explicit form of metrics for the given spaces is derived in the privileged coordinate systems. Models with dust matter and cosmological constant are constructed for the metrics obtained.

Dynamics of toroidal spiral strings around five-dimensional black holes

We examine the separability of the Nambu-Goto equation for test strings in a shape of toroidal spiral in a five-dimensional Kerr-AdS black hole. In particular, for a `{\it Hopf loop}\rq string which is a special class of the toroidal spiral strings, we show the complete separation of variables occurs in two cases, Kerr background and Kerr-AdS background with equal angular momenta. We also obtain the dynamical solution for the Hopf loop around a black hole and for the general toroidal spiral in Minkowski background.

Hidden symmetries and black holes

The paper contains a brief review of recent results on hidden symmetries in higher dimensional black hole spacetimes. We show how the existence of a principal CKY tensor (that is a closed conformal Killing-Yano 2-form) allows one to generate a `tower' of Killing-Yano and Killing tensors responsible for hidden symmetries. These symmetries imply complete integrability of geodesic equations and the complete separation of variables in the Hamilton-Jacobi, Klein-Gordon, Dirac and gravitational perturbation equations in the general Kerr-NUT-(A)dS metrics. Equations of the parallel transport of frames along geodesics in these spacetimes are also integrable.

Conformally flat Stäckel spaces in the Brans–Dicke theory

A problem of classification of the conformally flat Stäckel spaces admitting complete separation of variables in the Hamilton–Jacobi equation is studied within the framework of the Brans–Dicke scalar-tensor gravitational theory. Solutions to the field equations of the theory are found for the conformally flat Stäckel spaces of type (1.1), and explicit forms of the metric and scalar field are presented.

Stationary strings in the spacetime of rotating black holes in five-dimensional minimal gauged supergravity

We examine the separability properties of the equation of motion for a stationary string near a rotating charged black hole with two independent angular momenta in five-dimensional minimal gauged supergravity. It is known that the separability problem for the stationary string in a general stationary spacetime is reduced to that for the usual Hamilton-Jacobi equation for geodesics of its quotient space with one dimension fewer. Using this fact, we show that the ``effective metric'' of the quotient space does not allow the complete separability for the Hamilton-Jacobi equation, albeit such a separability occurs in the original spacetime of the black hole. We also show that only for two special cases of interest the Hamilton-Jacobi equation admits the complete separation of variables and therefore the integrability for the stationary string motion in the original background, namely, when the black hole has zero electric charge or it has an arbitrary electric charge but two equal angular momenta. We give the explicit expressions for the Killing tensors corresponding to these cases. However, for the general black hole spacetime the effective metric of the quotient space admits a conformal Killing tensor. We construct the explicit expression for this tensor.

Higher-dimensional black holes: hidden symmetries and separation of variables

In this paper, we discuss hidden symmetries in rotating black hole spacetimes. We start with an extended introduction which mainly summarizes results on hidden symmetries in four dimensions and introduces Killing and Killing–Yano tensors, objects responsible for hidden symmetries. We also demonstrate how starting with a principal CKY tensor (that is a closed non-degenerate conformal Killing–Yano 2-form) in 4D flat spacetime one can ‘generate’ the 4D Kerr–NUT–(A)dS solution and its hidden symmetries. After this we consider higher-dimensional Kerr–NUT–(A)dS metrics and demonstrate that they possess a principal CKY tensor which allows one to generate the whole tower of Killing–Yano and Killing tensors. These symmetries imply complete integrability of geodesic equations and complete separation of variables for the Hamilton–Jacobi, Klein–Gordon and Dirac equations in the general Kerr–NUT–(A)dS metrics.

Infection fronts in contact disease spread

We analyze the epidemic spread via a contact infection process in an immobile population within the Susceptible-Infected-Removed (SIR) model. We present both the results of stochastic simulations assuming different numbers of individuals (degrees of freedom) per cell as well as the solution of the corresponding deterministic equations. For the last ones we show that the appropriate system of nonlinear partial differential equations (PDE) allows for a complete separation of variables and present the approximate analytical expressions for the infection wave in different ranges of parameters. Comparing these results with the direct Monte-Carlo simulations we discuss the domain of applicability of the PDE models and their restrictions.

Hidden Symmetries of Higher-Dimensional Black Hole Spacetimes

The paper contains a brief review of recent results on hidden symmetries in higher dimensional black hole spacetimes. We show how the existence of a principal CKY tensor (that is a closed non-degenerate conformal Killing-Yano 2-form) allows one to generate a `tower' of Killing-Yano and Killing tensors responcible for hidden symmetries. These symmetries imply complete integrability of geodesic equations and the complete separation of variables in the Hamilton-Jacobi, Klein-Gordon and Dirac equations in the general Kerr-NUT-(A)dS metrics.

Conformally flat spaces admitting complete separation of variables in the eikonal equation

A classification of conformally flat space-times, admitting complete separation of variables in the eikonal equation, is carried out. The metric tensors, corresponding to different complete sets of integrals of motion, are found. With these metrics, as an example, some solutions to the Einstein equations with a cosmological constant are obtained. For the space-times obtained, examples of integration of the eikonal and scalar field equations are presented.

Kinetics of fluctuational deliquescence

Deliquescence is the very first stage of heterogeneous unary condensation on solid soluble nuclei; at this stage, a solid nucleus dissolves in a liquid film formed by the molecules of the vapor condensing on the nucleus. As recently shown [Djikaev et al., J. Phys. Chem. B 105, 7708 (2001)], deliquescence in the atmosphere may occur in a fluctuational mode at relative humidities lower than the “deliquescence point” (at this point deliquescence becomes barrierless). Developing the kinetics of fluctuational deliquescence, we derive a two-dimensional kinetic equation, governing the time evolution of the droplet distribution. When solid soluble nuclei consist of molecules of a single species, in the vicinity of the saddle point of the free energy surface of deliquescence this equation has the form of the kinetic equation of binary nucleation. A quasisteady-state solution for this equation is obtained by using the method of complete separation of variables. An expression for the average deviation of the stable variable of state of a droplet from its equilibrium value is also derived. It shows that the film of liquid solution, forming around the nucleus during its deliquescence, is not in equilibrium neither with the vapor nor with the solid core. A recipe is proposed to construct the time evolution of the quasisteady state caused by the depletion of undeliquesced nuclei in the system. The theoretical results are illustrated by numerical calculation for the deliquescence of model particles in a water vapor.

The quantum mechanical two-Coulomb-centre problem inthe Dirac equation framework

The asymptotic expansions (at small and large internuclear distancesR) of the eigenvalues (potential curves) E(R) of thetwo-Coulomb-centre problem are obtained. The Dirac equation with anaxially symmetrical potential, not allowing complete separation ofvariables, is solved analytically by the Wentzel-Kramers-Brillouinapproach and boundary-layer method. In the framework of thisscheme, the relativistic two-Coulomb-centre wavefunction isconstructed. The first two terms of the asymptotic (at largeinternuclear distance) behaviour of the exchange interactionpotential for an ion with an atom are calculated.

Conformally Steckel metrics in Einstein spaces

Anisotropic conformally Steckel metrics that satisfy Einstein vacuum equations containing a Λ term are examined in the paper. The metrics allow the integration of the isotropic geodetic Hamilton-Jacobi equations by complete variable separation.

Kinetic theory of nonisothermal binary nucleation: the stage following thermal relaxation

A kinetic theory is constructed for a nonisothermal binary nucleation at the stage following the thermal relaxation of nuclei. The three-dimensional kinetic equation to be solved reaches beyond the framework of the Fokker–Planck approximation even if one of two components has a large value of condensation heat. It is shown that, by successively applying the method of Enskog–Chapman and the method of complete separation of variables to that kinetic equation, one can reduce the problem of constructing the three-dimensional kinetic theory to the well-investigated problem of solving an one-dimensional kinetic equation of first-order phase transition, in the nonstationary case as well as in the stationary one. For the steady state, the main characteristics of nucleation, including the nucleation rate, are found. Theoretical results are numerically evaluated for the nucleation in ethanol–hexanol system and compared with predictions of classical (isothermal) theory and experimental data.

Gravitational instantons from minimal surfaces

Physical properties of gravitational instantons which are derivable from minimal surfaces in three-dimensional Euclidean space are examined using the Newman-Penrose formalism for Euclidean signature. The gravitational instanton that corresponds to the helicoid minimal surface is investigated in detail. This is a metric of Bianchi type , or E(2), which admits a hidden symmetry due to the existence of a quadratic Killing tensor. It leads to a complete separation of variables in the Hamilton-Jacobi equation for geodesics, as well as in Laplace's equation for a massless scalar field. The scalar Green function can be obtained in closed form, which enables us to calculate the vacuum fluctuations of a massless scalar field in the background of this instanton.

Nontrivial conformally steckel metrics in einstein spaces

All isotropic conformally Steckel metrics are derived that satisfy the Einstein vacuum equations containing a Λ term and do not reduce to the purely Steckel type. The metrics allow the integration of the Hamilton-Jacobi equation for a massless particle by complete variable separation.

Integration of Dirac equation in Riemannian spaces with five-dimensional group of motions

In the present article a classification of Riemannian spaces with five-dimensional group of motion is described from the point of view of a solution of the Dirac equation. A class of spaces is identified in which the Dirac equation does not admit a complete separation of variables, and exact solutions of the Dirac equation are obtained in these spaces by means of the method of noncommutative integration.

Integration of the Einstein–Dirac equations

The problem of exact integration of the Einstein–Dirac equation is studied. To solve this set of equations metrics admitting complete separation of variables in the Dirac equation are considered. At the first stage Stäckel spaces of types (3.1) and (3.0) are studied. For the first case obtained solutions contain arbitrary functions depending on one variable only (five real functions for the Einstein–Weyl equations and four real functions for the Einstein–Dirac ones). For the second one all solutions belonging to the diagonal metrics of Bianchi type 1 are found.

Dynamics of scalar fields in the background of rotating black holes.

A numerical study of the evolution of a massless scalar field in the background of rotating black holes is presented. First, solutions to the wave equation are obtained for slowly rotating black holes. In this approximation, the background geometry is treated as a perturbed Schwarzschild spacetime with the angular momentum per unit mass playing the role of a perturbative parameter. To first order in the angular momentum of the black hole, the scalar wave equation yields two coupled one-dimensional evolution equations. In this approximation, the late time dynamics of a massless scalar field exhibit the same power-law behavior as in the case of a Schwarzschild background. Solutions to the wave equation are also obtained for rapidly rotating black holes. In this case, owing to higher order terms in the angular momentum, the wave equation does not admit complete separation of variables and yields a two-dimensional evolution equation. The study shows that, depending on initial conditions, the late-time dynamics of a massless scalar field is dominated by the lowest-allowed mode with respect to $l$ for a fixed value of $m$.

Isomonodromic quantization of dimensionally reduced gravity

We present a detailed account of the isomonodromic quantization of dimensionally reduced Einstein gravity with two commuting Killing vectors. This theory constitutes an integrable “midisuperspace” version of quantum gravity with infinitely many interacting physical degrees of freedom. The canonical treatment is based on the complete separation of variables in the isomonodromic sectors of the model. The Wheeler-DeWitt and diffeomorphism constraints are thereby reduced to the Knizhnik-Zamolodchikov equations for SL (2, R) . The physical states are manifestly invariant under the full diffeomorphism group. An infinite set of independent observables a la Dirac exists both at the classical and the quantum level. Using the discrete unitary representations of SL(2, R) , we construct explicit quantum states. However, the problem of satisfying the additional constraints associated with the coset space SL(2, R)/SO(2) remains open. We briefly discuss the possible implications of our results for string theory.

On 2 D quantum gravity coupled to a σ-model

This contribution is a review of the method of isomonodromic quantization of dimensionally reduced gravity. Our approach is based on the complete separation of variables in the isomonodromic sector of the model and the related ``two-time" Hamiltonian structure. This allows an exact quantization in the spirit of the scheme developed in the framework of integrable systems. Possible ways to identify a quantum state corresponding to the Kerr black hole are discussed. In addition, we briefly describe the relation of this model with Chern Simons theory.