SEPARATION OF VARIABLES FOR THE DIRAC SQUARE EQUATION

1994 ◽  
Vol 03 (04) ◽  
pp. 739-746 ◽  
Author(s):  
V.G. BAGROV ◽  
V.V. OBUKHOV
Keyword(s):  
Jacobi Equation ◽  
Separation Of Variables ◽  
Complete Separation ◽  
Space Time ◽  
Riemann Space ◽  
Necessary Condition ◽  
Separation Procedure ◽  
Electromagnetic Potential ◽  
Curved Space ◽  
Hamilton Jacobi Equation

The problem of separation of variables for the Dirac square equation on a curved space-time in the presence of electromagnetic potential is considered. It is shown that the necessary condition for the separation of variables in the Dirac square equation is the complete separation of variables in the related Hamilton-Jacobi equation, i.e. the Riemann space should be Stäckel. The constructive scheme for separation procedure is presented.

scholarly journals Shapovalov Wave-Like Spacetimes

Symmetry ◽  
2020 ◽  
Vol 12 (8) ◽  
pp. 1372 ◽  
Author(s):  
Konstantin Osetrin ◽  
Evgeny Osetrin
Keyword(s):  
Jacobi Equation ◽  
Eikonal Equation ◽  
Separation Of Variables ◽  
Complete Classification ◽  
Complete Separation ◽  
Einstein Equations ◽  
Coordinate Systems ◽  
Test Particles ◽  
Hamilton Jacobi Equation

A complete classification of space-time models is presented, which admit the privileged coordinate systems, where the Hamilton–Jacobi equation for a test particle is integrated by the method of complete separation of variables with separation of the isotropic (wave) variable, on which the metric depends (wave-like Shapovalov spaces). For all types of Shapovalov spaces, exact solutions of the Einstein equations with a cosmological constant in vacuum are found. Complete integrals are presented for the eikonal equation and the Hamilton–Jacobi equation of motion of test particles.

scholarly journals The spacetime models with dust matter that admit separation of variables in Hamilton–Jacobi equations of a test particle

2016 ◽  
Vol 31 (06) ◽  
pp. 1650027 ◽  
Author(s):  
Konstantin Osetrin ◽  
Altair Filippov ◽  
Evgeny Osetrin
Keyword(s):  
Velocity Field ◽  
Energy Density ◽  
Test Particle ◽  
Jacobi Equation ◽  
Functional Form ◽  
Separation Of Variables ◽  
Complete Separation ◽  
Coordinate Systems ◽  
Jacobi Equations ◽  
Hamilton Jacobi Equation

The characteristics of dust matter in spacetime models, admitting the existence of privilege coordinate systems are given, where the single-particle Hamilton–Jacobi equation can be integrated by the method of complete separation of variables. The resulting functional form of the 4-velocity field and energy density of matter for all types of spaces under consideration is presented.

Hawking radiation of Reissner-Nordström de Sitter Black Hole with a Global Monopole derived from modified Hamilton-Jacobi equation

2020 ◽  
Author(s):  
Zhi-E Liu ◽  
Xia Tan ◽  
Yu-Zhen Liu ◽  
Bei Sha ◽  
Jie Zhang ◽  
...  
Keyword(s):  
Black Hole ◽  
Hawking Radiation ◽  
Jacobi Equation ◽  
Lorentz Violation ◽  
Physical Nature ◽  
Space Time ◽  
Global Monopole ◽  
De Sitter ◽  
Curved Space ◽  
Hamilton Jacobi Equation

The tunneling characteristics at cosmological horizon and event horizon of Reissner-Nordström de Sitter black hole with a global monopole are studied by using the modified Lorentz violation scalar field equation in curved space-time. Firstly, we get the modified Hamilton-Jacobi equation by semi-classical approximation, then the Hawking radiation and thermodynamical properties of Reissner-Nordström de Sitter black hole with a global monopole are computed based on the modified Hamilton-Jacobi equation. Our results show that Lorentz violation can lead to lower Hawking temperature and higher entropy of the black hole at the same time. This work can improve the understanding on the physical nature of Lorentz violation in curved space-time.

scholarly journals Separation of variables in Hamilton–Jacobi equation for a charged test particle in the Stackel spaces of type (2.1)

2020 ◽  
Vol 17 (14) ◽  
pp. 2050186
Author(s):  
Valeriy Obukhov
Keyword(s):  
Jacobi Equation ◽  
Separation Of Variables ◽  
Equations Of Motion ◽  
Type Equation ◽  
Parabolic Type ◽  
Complete Separation ◽  
Test Particles ◽  
Charged Test Particle ◽  
Charged Test ◽  
Hamilton Jacobi Equation

We can find all equivalence classes for electromagnetic potentials and space-time metrics of Stackel spaces, provided that the equations of motion of the classical charged test particles are integrated by the method of complete separation of variables in the Hamilton–Jacobi equation. Separation is carried out using the complete sets of mutually-commuting integrals of motion of type (2.1), whereby in a privileged coordinate system the Hamilton–Jacobi equation turns into a parabolic type equation.

Center of Mass Theorem and the Separation of Variables for Two-Body System on a Three-Dimensional Sphere

2020 ◽  
Vol 23 (3) ◽  
pp. 306-311
Author(s):  
Yu. Kurochkin ◽  
Dz. Shoukavy ◽  
I. Boyarina
Keyword(s):  
Constant Curvature ◽  
Jacobi Equation ◽  
Separation Of Variables ◽  
Center Of Mass ◽  
Three Dimensional ◽  
Relative Distance ◽  
Dimensional Sphere ◽  
Body System ◽  
Spaces Of Constant Curvature ◽  
Hamilton Jacobi Equation

The immobility of the center of mass in spaces of constant curvature is postulated based on its definition obtained in [1]. The system of two particles which interact through a potential depending only on the distance between particles on a three-dimensional sphere is considered. The Hamilton-Jacobi equation is formulated and its solutions and trajectory equations are found. It was established that the reduced mass of the system depends on the relative distance.

scholarly journals Hamilton–Jacobi Equation for a Charged Test Particle in the Stäckel Space of Type (2.0)

Symmetry ◽  
2020 ◽  
Vol 12 (8) ◽  
pp. 1289 ◽  
Author(s):  
Valeriy Obukhov
Keyword(s):  
Test Particle ◽  
Jacobi Equation ◽  
Separation Of Variables ◽  
Killing Vector ◽  
Field Equations ◽  
Tensor Fields ◽  
Charged Test Particle ◽  
Complete Sets ◽  
Charged Test ◽  
Hamilton Jacobi Equation

All electromagnetic potentials and space–time metrics of Stäckel spaces of type (2.0) in which the Hamilton–Jacobi equation for a charged test particle can be integrated by the method of complete separation of variables are found. Complete sets of motion integrals, as well as complete sets of killing vector and tensor fields, are constructed. The results can be used when studying solutions of field equations in the theory of gravity.

scholarly journals Symmetry operators and separation of variables in the (2 + 1)-dimensional Dirac equation with external electromagnetic field

2018 ◽  
Vol 15 (05) ◽  
pp. 1850085 ◽  
Author(s):  
A. V. Shapovalov ◽  
A. I. Breev
Keyword(s):  
Dirac Equation ◽  
Separation Of Variables ◽  
Flat Space ◽  
External Electromagnetic Field ◽  
Complete Separation ◽  
Electromagnetic Potential ◽  
First Order ◽  
Matrix Coefficients ◽  
Symmetry Operators ◽  
Differential Symmetry

We obtain and analyze equations determining first-order differential symmetry operators with matrix coefficients for the Dirac equation with an external electromagnetic potential in a [Formula: see text]-dimensional Riemann (curved) spacetime. Nonequivalent complete sets of mutually commuting symmetry operators are classified in a [Formula: see text]-dimensional Minkowski (flat) space. For each of the sets, we carry out a complete separation of variables in the Dirac equation and find a corresponding electromagnetic potential permitting separation of variables.

Symmetry operators for Maxwell’s equations on curved space-time

1992 ◽  
Vol 439 (1905) ◽  
pp. 103-113 ◽  
Keyword(s):  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Separation Of Variables ◽  
Sufficient Conditions ◽  
Symmetry Operator ◽  
Space Time ◽  
Conformal Killing ◽  
Curved Space ◽  
Background Space ◽  
Necessary And Sufficient

We derive necessary and sufficient conditions that a second-order co-variant differential operator be a symmetry operator of Maxwell’s equations in a general curved space-time background. It is found that such operators are naturally formulated in terms of conformal Killing vectors, tensors and spinors. Operators of this type play a role in the solution of Maxwell’s equations via separation of variables in the Kerr background space-time.

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