Zafar Turakulov | None - Academia.edu (original) (raw)
Videos by Zafar Turakulov
We demonstrate transforamtions of interior of a circle, which conserve the closure phenomenon men... more We demonstrate transforamtions of interior of a circle, which conserve the closure phenomenon mentioned in the Poncelet theorem. Conditions of the theorem are simplified without loss of generality. First, the outer ellipse is reduced to a circle and second, center of the inner one lies on a certain ray. Under these restrictions the transformations under consideration form a three-parameter group. Note that the complete family of ellipses whose center lie on a given ray possesses only four arbitrary parameters, consequently, the closure conditions specifies only one constraint.
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Papers by Zafar Turakulov
Туракулов З. Я. 202PREPRINTS.RU. https://doi.org/10.24108/preprints-3113171
Приведено полное решение задачи о движении заряженной частицы в магнитном поле тока, текущего в б... more Приведено полное решение задачи о движении заряженной частицы в магнитном поле тока, текущего в бесконечном прямом проводе. Приведена также прямая проверка полученного результата.
Новое семейство стационарных осесимметричных вакуумов типа семейства Керра. PREPRINTS.RU. https://doi.org/10.24108/preprints-3113077, 2024
Второе стационарное осесимметричное решение уравнения Эйнштейна, допускающее полное разделение ос... more Второе стационарное осесимметричное решение уравнения Эйнштейна, допускающее полное разделение основных ковариантных уравнений Зафар Туракулов 23 ноября 2023 г. Аннотация Получено второе стационарное осесимметричное решение уравнения Эйнштейна, допускающее полное разделение основных ковариантных уравнений. До сих пор считалось, что трехпараметрическое решение, известное как решение "Керра-НУТ является единственным решением такого рода. Такое утверждение сделано, в частности, в наших работах [1, 2]. В отличие от решения Керра-НУТ, описывающего кроме прочего, двухпараметрическое семейство контурных кротовых нор, представленное ниже описывает только кротовые норы "обычного" типа.
We have obtained the most general solution of the Einstein vacuum equation for the axially symmet... more We have obtained the most general solution of the Einstein vacuum equation for the axially symmetric stationary metric in which the Klein - Gordon equation is separable. It has four parameters of which the two are familiar mass and rotation (or NUT-like parameter) while the other two are new and dimensionless, and can survive only when the rotation parameter is non-zero. The general solution includes the Kerr black hole which would now have two new hairs arising at the cost of asymptotic flatness
We have obtained the most general solution of the Einstein vacuum equation for the axially symmet... more We have obtained the most general solution of the Einstein vacuum equation for the axially symmetric stationary metric in which both the Hamilton-Jacobi equation for particle motion and the Klein-Gordon equation are separable. It has four parameters of which the two are familiar mass and rotation while the other two are new and dimensionless. In the absence of the rotation parameter, the solution reduces to the Schwarzschild solution with a deficit angle. This and the other considerations strongly suggest that it may describe a rotating black hole with a cosmic string. This is a new prediction of Einstein's theory which is at the same footing as the prediction of a rotating black hole. It is well-known that the Kerr solution is the unique two parameter solution of the Einstein vacuum equation for asymptotically flat axially symmetric stationary spacetime admitting regular smooth horizon. The two parameters represent mass and angular momentum of a rotating black hole.
Geomagnetism and Aeronomy, 2020
The paper presents an analytical model of a magnetic tube, which is described as a cylindrically ... more The paper presents an analytical model of a magnetic tube, which is described as a cylindrically symmetric equilibrium state of two-component plasma. The plasma flow is represented as a superposition of two charged, compressible plasma components that only interact via their common electromagnetic field. The total electric charge density is assumed to be equal to zero. The system is in equilibrium under the influence of three forces: magnetic, centrifugal, and pressure gradient. The presented model is idealized and limited by the condition that the surrounding plasma is homogeneous and motionless. Despite this, the presented model, with the given parameters, makes it possible to calculate the main parameters of the magnetic flux tube.
NATO ASI Series, 1997
Foundations of the standard theory of radiation by moving charge are analysed. The well-known dif... more Foundations of the standard theory of radiation by moving charge are analysed. The well-known difficulty concerned with energy conservation law is considered. An exact solution of Maxwell equations obtained by the method of variables separation, which expressses the field of a charge describing hyperbolic motion, is presented. Unlike the Lienard-Wiechert potentials for this case of motion, it displays abcence of radiation and, hence, does not lead to any difficulties mentioned above. It is concluded that exact solutions of Maxwell equations constitute the only correct approach to electromagnetic field of a moving charge. The new classical theory of radiation that may be composed on the basis of exact solutions seems to be purely geometric.
Academia Letters
The phenomenon of magnetic burst is analysed from the point of view of classical electrodynamics.... more The phenomenon of magnetic burst is analysed from the point of view of classical electrodynamics. For this end, we introduce a simplified model of coil assuming that its winding disappears at some moment of time and releases the field which thus is free to run away to the surrounding space. A model of coil which is infinite straight solenoid in form is considered for which mathematical description of the the phenomenon is built. Electromagnetic spectrum of the burst is obtained.
Possible scenario of formation and analytic solution of Vlasov equation for the final state of a ... more Possible scenario of formation and analytic solution of Vlasov equation for the final state of a self-gravitating collisionless system are proposed. The system is assumed to possess axial symmetry and zero total kinetic energy in the initial state. It is shown that under these initial conditions evolution of the velocity distribution goes a special way and after all finalizes with formation of phase density of special form. It turns out that for this form of the phase density the Vlasov equation is solvable. A particular solution is obtained which describes a spindle-shaped cluster of gravitating particles.
Plateau problem and its generalizations appear in various fields of physics. Hamilton-Jacobi like... more Plateau problem and its generalizations appear in various fields of physics. Hamilton-Jacobi like equation for the Plateau problem is proposed whose solutions, undersome conditions, provide foliations of the space with minimal surfaces. Solutionsof this equation obtained in cylinder coordinates gave some trivial foliations withplanes as minimal surfaces and non-trivial ones as families of helicoids, catenoids andhelicatenoids which contains them as limiting cases. Solution obtained in sphericalcoordinates gave a foliation unknown before with leafs resembling seashells in form.
Main properties and explicit form of Green's functions for covariant scalar equations are... more Main properties and explicit form of Green's functions for covariant scalar equations are discussed. It is shown that equations for Green's functions has or has no meaning depending on signature of the space. In spaces with Euclidean signature the equation is meaningful and in spaces with pseudo-Euclidean signature the same equation contains contradiction hence, is meaningless. Correspondingly, Green's functions exist for Poisson's and simplified Yukawa equations and do not for D'Alembert's and Klein-Gordon equations. Impact of nonexistence of Green's functions for these equations on CFT and QFT is discussed.
Pramana, 2009
In mechanics of the mass point passage from one frame of reference to another moving with velocit... more In mechanics of the mass point passage from one frame of reference to another moving with velocity u consists in subtracting this vector from the velocity of the particle. In general case the vector u is not constant, as, for example, when passing through a rotating frame, this operation creates inertial forces. Analysis of this phenomenon from the point of view of Lagrangian and Hamiltonian mechanics is interesting from the general relativistic point of view due to Einstein's principle of equivalence. We show that the vector u plays the role of vector potential which, however, essentially differs from vector potential known in classical electrodynamics. Comparative analysis of the two kinds of vector potentials is completed.
In particular cases of stationary and stationary axially symmetric space-time passage to non-rela... more In particular cases of stationary and stationary axially symmetric space-time passage to non-relativistic limit of Einstein equation is completed. For this end the notions of absolute space and absolute time are introduced due to stationarity of the space-time under consideration. In this construction absolute time is defined as a function ttt on the space-time such that prtt\prt_tprtt is exactly the Killing vector and the space at different moments is presented by the surfaces t=cont=\con t=con. The space-time metric is expressed in terms of metric of the 3-space and two potentials one of which is exactly Newtonian gravitational potential Phi\PhiPhi, another is vector potential vecA\vec AvecA which, however, differs from vector potential known in classical electrodynamics. In the first-order approximation on Phi/c2\Phi/c^2Phi/c2, ∣vecA∣/c|\vec A|/c∣vecA∣/c Einstein equation is reduced to a system for these functions in which left-hand sides contain Laplacian of the Newtonian potential, derivatives of the vector potential and cur...
Propagation of electromagnetic wave in media with electric permittivity � and magnetic permeabili... more Propagation of electromagnetic wave in media with electric permittivity � and magnetic permeability μ depending on one of Cartesian coordinates is studied. It is shown that under these conditions Maxwell equations reduce to one-dimensional Schr¨odinger equation. Reduction is demonstrated in one particular case of �(z) and μ(z) specified as same function which grows monotonically as z 1/3 Since both functions have . opposite sign in half-spaces z > 0 and z < 0, passage of the wave would expose negative refraction. Unlike interface, on which � and μ behave as step functions, a layer −a ≤ z ≤ −a between half-spaces in which these characteristics are constant, provides conditions under which theoretically, negative refraction can be described in terms of smooth functions. Qualitative solution is obtained by the WKB method, which yields the depth of maximal penetration of the wave inside the layer.
It is shown that the notion of "element of current" which was introduced as a vector an... more It is shown that the notion of "element of current" which was introduced as a vector analogue of the notion of point-like electric charge, contradicts the charge conservation law and hence, does the field equation for the vector potential. Till now, this notion played the role of sufficient condition of applicability of the method of Green's functions to the equation for the vector potential, therefore the contradiction found shows that application of this method to magnetostatics has no foundation. An example of current is found, for which the integral of Green's function can be taken analytically so that the result can be verified straightforwardly. The straightforward verification shows that in this particular case the method yields an apparently erroneous result that allows us to conclude that Green's function for the field equation for the vector potential does not exist.
An example of mechanical system whose configuration space is direct product of a curved space and... more An example of mechanical system whose configuration space is direct product of a curved space and the local group of rotations, is presented. The system is considered as a model of spinning particle moving in the space. The Hamiltonian formalism for this system and possible method for its quantization are discussed. It is shown that the Hamilton equations coincide with the Papapetrou equations for spinning test-particle in general relativity.
Modern Physics Letters A, 2001
We present a stationary axially symmetric two-parameter vacuum solution which could be considered... more We present a stationary axially symmetric two-parameter vacuum solution which could be considered as "dual" to the Kerr solution. It is obtained by removing the mass parameter from the function of the radial coordinate and introducing a dimensionless parameter in the function of the angle coordinate in the metric functions. It turns out that it is in fact the massless limit of the Kerr–NUT solution.
We demonstrate transforamtions of interior of a circle, which conserve the closure phenomenon men... more We demonstrate transforamtions of interior of a circle, which conserve the closure phenomenon mentioned in the Poncelet theorem. Conditions of the theorem are simplified without loss of generality. First, the outer ellipse is reduced to a circle and second, center of the inner one lies on a certain ray. Under these restrictions the transformations under consideration form a three-parameter group. Note that the complete family of ellipses whose center lie on a given ray possesses only four arbitrary parameters, consequently, the closure conditions specifies only one constraint.
4 views
Туракулов З. Я. 202PREPRINTS.RU. https://doi.org/10.24108/preprints-3113171
Приведено полное решение задачи о движении заряженной частицы в магнитном поле тока, текущего в б... more Приведено полное решение задачи о движении заряженной частицы в магнитном поле тока, текущего в бесконечном прямом проводе. Приведена также прямая проверка полученного результата.
Новое семейство стационарных осесимметричных вакуумов типа семейства Керра. PREPRINTS.RU. https://doi.org/10.24108/preprints-3113077, 2024
Второе стационарное осесимметричное решение уравнения Эйнштейна, допускающее полное разделение ос... more Второе стационарное осесимметричное решение уравнения Эйнштейна, допускающее полное разделение основных ковариантных уравнений Зафар Туракулов 23 ноября 2023 г. Аннотация Получено второе стационарное осесимметричное решение уравнения Эйнштейна, допускающее полное разделение основных ковариантных уравнений. До сих пор считалось, что трехпараметрическое решение, известное как решение "Керра-НУТ является единственным решением такого рода. Такое утверждение сделано, в частности, в наших работах [1, 2]. В отличие от решения Керра-НУТ, описывающего кроме прочего, двухпараметрическое семейство контурных кротовых нор, представленное ниже описывает только кротовые норы "обычного" типа.
We have obtained the most general solution of the Einstein vacuum equation for the axially symmet... more We have obtained the most general solution of the Einstein vacuum equation for the axially symmetric stationary metric in which the Klein - Gordon equation is separable. It has four parameters of which the two are familiar mass and rotation (or NUT-like parameter) while the other two are new and dimensionless, and can survive only when the rotation parameter is non-zero. The general solution includes the Kerr black hole which would now have two new hairs arising at the cost of asymptotic flatness
We have obtained the most general solution of the Einstein vacuum equation for the axially symmet... more We have obtained the most general solution of the Einstein vacuum equation for the axially symmetric stationary metric in which both the Hamilton-Jacobi equation for particle motion and the Klein-Gordon equation are separable. It has four parameters of which the two are familiar mass and rotation while the other two are new and dimensionless. In the absence of the rotation parameter, the solution reduces to the Schwarzschild solution with a deficit angle. This and the other considerations strongly suggest that it may describe a rotating black hole with a cosmic string. This is a new prediction of Einstein's theory which is at the same footing as the prediction of a rotating black hole. It is well-known that the Kerr solution is the unique two parameter solution of the Einstein vacuum equation for asymptotically flat axially symmetric stationary spacetime admitting regular smooth horizon. The two parameters represent mass and angular momentum of a rotating black hole.
Geomagnetism and Aeronomy, 2020
The paper presents an analytical model of a magnetic tube, which is described as a cylindrically ... more The paper presents an analytical model of a magnetic tube, which is described as a cylindrically symmetric equilibrium state of two-component plasma. The plasma flow is represented as a superposition of two charged, compressible plasma components that only interact via their common electromagnetic field. The total electric charge density is assumed to be equal to zero. The system is in equilibrium under the influence of three forces: magnetic, centrifugal, and pressure gradient. The presented model is idealized and limited by the condition that the surrounding plasma is homogeneous and motionless. Despite this, the presented model, with the given parameters, makes it possible to calculate the main parameters of the magnetic flux tube.
NATO ASI Series, 1997
Foundations of the standard theory of radiation by moving charge are analysed. The well-known dif... more Foundations of the standard theory of radiation by moving charge are analysed. The well-known difficulty concerned with energy conservation law is considered. An exact solution of Maxwell equations obtained by the method of variables separation, which expressses the field of a charge describing hyperbolic motion, is presented. Unlike the Lienard-Wiechert potentials for this case of motion, it displays abcence of radiation and, hence, does not lead to any difficulties mentioned above. It is concluded that exact solutions of Maxwell equations constitute the only correct approach to electromagnetic field of a moving charge. The new classical theory of radiation that may be composed on the basis of exact solutions seems to be purely geometric.
Academia Letters
The phenomenon of magnetic burst is analysed from the point of view of classical electrodynamics.... more The phenomenon of magnetic burst is analysed from the point of view of classical electrodynamics. For this end, we introduce a simplified model of coil assuming that its winding disappears at some moment of time and releases the field which thus is free to run away to the surrounding space. A model of coil which is infinite straight solenoid in form is considered for which mathematical description of the the phenomenon is built. Electromagnetic spectrum of the burst is obtained.
Possible scenario of formation and analytic solution of Vlasov equation for the final state of a ... more Possible scenario of formation and analytic solution of Vlasov equation for the final state of a self-gravitating collisionless system are proposed. The system is assumed to possess axial symmetry and zero total kinetic energy in the initial state. It is shown that under these initial conditions evolution of the velocity distribution goes a special way and after all finalizes with formation of phase density of special form. It turns out that for this form of the phase density the Vlasov equation is solvable. A particular solution is obtained which describes a spindle-shaped cluster of gravitating particles.
Plateau problem and its generalizations appear in various fields of physics. Hamilton-Jacobi like... more Plateau problem and its generalizations appear in various fields of physics. Hamilton-Jacobi like equation for the Plateau problem is proposed whose solutions, undersome conditions, provide foliations of the space with minimal surfaces. Solutionsof this equation obtained in cylinder coordinates gave some trivial foliations withplanes as minimal surfaces and non-trivial ones as families of helicoids, catenoids andhelicatenoids which contains them as limiting cases. Solution obtained in sphericalcoordinates gave a foliation unknown before with leafs resembling seashells in form.
Main properties and explicit form of Green's functions for covariant scalar equations are... more Main properties and explicit form of Green's functions for covariant scalar equations are discussed. It is shown that equations for Green's functions has or has no meaning depending on signature of the space. In spaces with Euclidean signature the equation is meaningful and in spaces with pseudo-Euclidean signature the same equation contains contradiction hence, is meaningless. Correspondingly, Green's functions exist for Poisson's and simplified Yukawa equations and do not for D'Alembert's and Klein-Gordon equations. Impact of nonexistence of Green's functions for these equations on CFT and QFT is discussed.
Pramana, 2009
In mechanics of the mass point passage from one frame of reference to another moving with velocit... more In mechanics of the mass point passage from one frame of reference to another moving with velocity u consists in subtracting this vector from the velocity of the particle. In general case the vector u is not constant, as, for example, when passing through a rotating frame, this operation creates inertial forces. Analysis of this phenomenon from the point of view of Lagrangian and Hamiltonian mechanics is interesting from the general relativistic point of view due to Einstein's principle of equivalence. We show that the vector u plays the role of vector potential which, however, essentially differs from vector potential known in classical electrodynamics. Comparative analysis of the two kinds of vector potentials is completed.
In particular cases of stationary and stationary axially symmetric space-time passage to non-rela... more In particular cases of stationary and stationary axially symmetric space-time passage to non-relativistic limit of Einstein equation is completed. For this end the notions of absolute space and absolute time are introduced due to stationarity of the space-time under consideration. In this construction absolute time is defined as a function ttt on the space-time such that prtt\prt_tprtt is exactly the Killing vector and the space at different moments is presented by the surfaces t=cont=\con t=con. The space-time metric is expressed in terms of metric of the 3-space and two potentials one of which is exactly Newtonian gravitational potential Phi\PhiPhi, another is vector potential vecA\vec AvecA which, however, differs from vector potential known in classical electrodynamics. In the first-order approximation on Phi/c2\Phi/c^2Phi/c2, ∣vecA∣/c|\vec A|/c∣vecA∣/c Einstein equation is reduced to a system for these functions in which left-hand sides contain Laplacian of the Newtonian potential, derivatives of the vector potential and cur...
Propagation of electromagnetic wave in media with electric permittivity � and magnetic permeabili... more Propagation of electromagnetic wave in media with electric permittivity � and magnetic permeability μ depending on one of Cartesian coordinates is studied. It is shown that under these conditions Maxwell equations reduce to one-dimensional Schr¨odinger equation. Reduction is demonstrated in one particular case of �(z) and μ(z) specified as same function which grows monotonically as z 1/3 Since both functions have . opposite sign in half-spaces z > 0 and z < 0, passage of the wave would expose negative refraction. Unlike interface, on which � and μ behave as step functions, a layer −a ≤ z ≤ −a between half-spaces in which these characteristics are constant, provides conditions under which theoretically, negative refraction can be described in terms of smooth functions. Qualitative solution is obtained by the WKB method, which yields the depth of maximal penetration of the wave inside the layer.
It is shown that the notion of "element of current" which was introduced as a vector an... more It is shown that the notion of "element of current" which was introduced as a vector analogue of the notion of point-like electric charge, contradicts the charge conservation law and hence, does the field equation for the vector potential. Till now, this notion played the role of sufficient condition of applicability of the method of Green's functions to the equation for the vector potential, therefore the contradiction found shows that application of this method to magnetostatics has no foundation. An example of current is found, for which the integral of Green's function can be taken analytically so that the result can be verified straightforwardly. The straightforward verification shows that in this particular case the method yields an apparently erroneous result that allows us to conclude that Green's function for the field equation for the vector potential does not exist.
An example of mechanical system whose configuration space is direct product of a curved space and... more An example of mechanical system whose configuration space is direct product of a curved space and the local group of rotations, is presented. The system is considered as a model of spinning particle moving in the space. The Hamiltonian formalism for this system and possible method for its quantization are discussed. It is shown that the Hamilton equations coincide with the Papapetrou equations for spinning test-particle in general relativity.
Modern Physics Letters A, 2001
We present a stationary axially symmetric two-parameter vacuum solution which could be considered... more We present a stationary axially symmetric two-parameter vacuum solution which could be considered as "dual" to the Kerr solution. It is obtained by removing the mass parameter from the function of the radial coordinate and introducing a dimensionless parameter in the function of the angle coordinate in the metric functions. It turns out that it is in fact the massless limit of the Kerr–NUT solution.
Modern Physics Letters A, 2002
We have obtained the general solution of the Einstein vacuum equation for the axially symmetric s... more We have obtained the general solution of the Einstein vacuum equation for the axially symmetric stationary metric in which both the Hamilton-Jacobi equation for particle motion and the Klein - Gordon equation are separable. It can be interpreted to describe the gravitational field of a rotating dyon, a particle endowed with both gravoelectric (mass) and gravomagnetic (NUT parameter) charges. Further, there also exists a duality relation between the two charges and the radial and the polar angle coordinates which keeps the solution invariant. The solution can however be transformed into the known Kerr - NUT solution indicating its uniqueness under the separability of equations of motion.
Масштабная инвариантность и магнитное поле постоянного тока в бесконечном-образном проводе Зафар ... more Масштабная инвариантность и магнитное поле постоянного тока в бесконечном-образном проводе Зафар Туракулов 28 марта 2023 г. Аннотация Получено выражение для магнитного поля постоянного тока в бесконечном V-образном проводе как пример масштабно-инвариантной системы тока и его магнитного поля.
It was traditionally believed that whenever there is any oscillation of charge and current densit... more It was traditionally believed that whenever there is any oscillation of charge and current densities, there will be electromagnetic waves they emit. In this work examples of oscillations of this sort are presented when oscillations of this sort do not emit any electromagnetic waves.
A simple mathematical representation of the phenomenon of magnetic explosion is proposed. Accordi... more A simple mathematical representation of the phenomenon of magnetic explosion is proposed. According to it, a current density which produces magnetic field, disappears at a certain moment of time, so that the field is not longer confined by it. As the field is released, it passes from stationary state to the state of free propagation in the space. This representation underlies a certain mathematical theory of magnetic explosion. In this work, some details of the theory are outlined.
In particular cases of stationary and stationary axially symmetric space-time passage to non-rela... more In particular cases of stationary and stationary axially symmetric space-time passage to non-relativistic limit of Einstein equation is completed. For this end the notions of absolute space and absolute time are introduced due to stationarity of the space-time under consideration. In this construction absolute time is defined as a function t on the space-time such that ∂ t is exactly the Killing vector and the space at different moments is presented by the surfaces t = const. The space-time metric is expressed in terms of metric of the 3-space and two potentials one of which is exactly Newtonian gravitational potential Φ, another is vector potential A which, however, differs from vector potential known in classical electrodynamics. In the first-order approximation on Φ/c 2 , | A|/c Einstein equation is reduced to a system for these functions in which left-hand sides contain Laplacian of the Newtonian potential, derivatives of the vector potential and curvature of the space and the right-hand sides do 3-dimensional stress tensor and densities of mass and energy.