A Physical Deduction of an Equivalent Landau–Lifshitz Equation of Motion in Classical Electrodynamics. A New Expression for the Large Distance Radiation Rate of Energy

A new scheme is proposed in order to deduce an equation of motion for a spinless charged point particle leading to an equivalent Landau–Lifshitz equation of motion. Consequently Larmor’s formula must be substituted by a new expression for the large distance radiation rate of energy. A constraint appears on the applicability of the Maxwell electromagnetic tensor. The particular case of a sudden force is analyzed in order to show the physical results predicted by the new model. A geometrical rearrangement of the energy explains the balance.     

A Mechanical Model for Analyzing the Runaway Solutions in the Radiation Reaction Problem

In order to understand the rise of runaway solutions in the radiation reaction problem a mechanical model is used. An alternative demonstration of Daboul’s theorem, through Hurwitz’s criterion, is given. The origin of runaway solutions in electrodynamics is discussed. They arise when the particle has a negative mechanical mass or when approximations are used in the equation of motion. In the 1-dimensional mechanical model an exact and linear equation of motion for the particle is obtained, the corresponding exact solution is again runaway when the mechanical mass is negative. The exact solution is not runaway when the mechanical mass is positive. However, the use of approximations leads to an equation of motion which has runaway solutions. It is exhibited that the use of approximations in the 3-dimensional mechanical model is completely necessary because the general equation of motion for the particle is non-linear. The analysis of this case proceeds in a very similar way to the one carried out in electrodynamics. This means that the number of dimensions also plays an important role in the analysis.     

The Universe Accelerated Expansion using Extra-dimensions with Metric Components Found by a New Equivalence Principle

Curved multi-dimensional space-times (5D and higher) are constructed by embedding them in one higher-dimensional flat space. The condition that the embedding coordinates have a separable form, plus the demand of an orthogonal resulting space-time, implies that the curved multi-dimensional space-time has 4D de-Sitter subspaces (for constant extra-dimensions) in which the 3D subspace has an accelerated expansion. A complete determination of the curved multi-dimensional spacetime geometry is obtained provided we impose a new type of “equivalence principle”, meaning that there is a geodesic which from the embedding space has a rectliniar motion. According to this new equivalence principle, we can find the extra-dimensions metric components, each curved multi-dimensional spacetime surface’s equation, the energy-momentum tensors and the extra-dimensions as functions of a scalar field. The generic geodesic in each 5D spacetime are studied: they include solutions where particle’s motion along the extra-dimension is periodic and the 3D expansion factor is inflationary (accelerated expansion). Thus, the 3D subspace has an accelerated expansion.     

Asymmetric Bethe-Salpeter Equation for Pairing and Condensation

The Martin-Schwinger hierarchy of correlations are reexamined and the three-particle correlations are investigated under various partial summations. Besides the known approximations of screened, ladder and maximally crossed diagrams the pair-pair correlations are considered. It is shown that the recently proposed asymmetric Bethe-Salpeter equation to avoid unphysical repeated collisions is derived as a result of the hierarchical dependencies of correlations. Exceeding the parquet approximation we show that an asymmetry appears in the selfconsistent propagators. This form is superior over the symmetric selfconsistent one since it provides the Nambu-Gorkov equations and gap equation for fermions and the Beliaev equations for bosons while from the symmetric form no gap equation results. The selfenergy diagrams which account for the subtraction of unphysical repeated collisions are derived from the pair-pair correlation in the three-particle Green’s function. It is suggested to distinguish between two types of selfconsistency, the channel-dressed propagators and the completely dressed propagators, with the help of which the asymmetric expansion completes the Ward identity and is Φ-derivable.     

Relativistic compact objects in isotropic coordinates

We present a matrix method for obtaining new classes of exact solutions for Einstein’s equations representing static perfect fluid spheres. By means of a matrix transformation, we reduce Einstein’s equations to two independent Riccati-type differential equations for which three classes of solutions are obtained. One class of the solutions corresponding to the linear barotropic-type fluid with an equation of statep =γρ is discussed in detail.     

The Hamilton-Jacobi equation revisited

A new analysis of the nature of the solutions of the Hamilton-Jacobi equation of classical dynamics is presented based on Caratheodory’s theorem concerning canonical transformations. The special role of a principal set of solutions is stressed, and the existence of analogous results in quantum mechanics is outlined.     

Explicit solutions for relativistic acceleration and rotation

The Lorentz transformations are represented by Einstein velocity addition on the ball of relativistically admissible velocities. This representation is by projective maps. The Lie algebra of this representation defines the relativistic dynamic equation. If we introduce a new dynamic variable, called symmetric velocity, the above representation becomes a representation by conformal, instead of projective maps. In this variable the relativistic dynamic equation for systems with an invariant plane becomes a non-linear analytic equation in one complex variable. We obtained explicit solutions for the motion of a charge in uniform, mutually perpendicular electric and magnetic fields. By assuming the Clock hypothesis and using these solutions, we were able to describe the space-time transformations between two uniformly accelerated and rotating systems. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005.     

带强迫项变系数组合KdV方程的无穷序列复合型类孤子新解

为了获得变系数非线性发展方程的无穷序列复合型新解,研究了G′(ξ)G(ξ)展开法.通过引入一种函数变换,把常系数二阶齐次线性常微分方程的求解问题转化为一元二次方程和Riccati方程的求解问题.在此基础上,利用Riccati方程解的非线性叠加公式,获得了常系数二阶齐次线性常微分方程的无穷序列复合型新解.借助这些复合型新解与符号计算系统Mathematica,构造了带强迫项变系数组合KdV方程的无穷序列复合型类孤子新精确解.     

Accuracy of long-term forecasting geosynchronous satellite motion

The accuracy of forecasting geosynchronous satellite motion for 242-year term of forecast using a numerical model of artificial satellite motion taking into account the main perturbing factors including nonsphericity of the Earth’s gravitational field, attraction by the Moon and the Sun, tides inside the Earth, direct light pressure with allowance for the Earth’s shadow effect, and the Pointing–Robertson effect is considered. It is demonstrated that in this case, perturbations of the Earth’s gravitational field harmonics up to the 27th order must be considered. For regular motions, the maximum error in forecasting the geosynchronous satellite position ranges from 0.14 to 2400 km, the error in forecasting the long semiaxis ranges from 0.013 to 1100 m, and the error in forecasting the subsatellite point longitude ranges from 0.069″ to 3.4° depending on the libration amplitude. The accuracy of forecasting depends on the libration amplitude: the less the libration amplitude, the higher the accuracy of forecasting. For quasi-random trajectories, the integration period for which the errors in forecasting do not exceed values obtained for libration motion is determined by the frequency and proximity of the trajectory to unstable stationary points. For the examples considered, this period is about 200 years. The estimated MEGNO factor confirms the efficiency of the numerical model of artificial satellite motion used to investigate the stochastic properties of geosynchronous satellite motion.     

Proof of absence of spooky action at a distance in quantum correlations

I prove that there is no spooky action at a distance and nonlocal state-reduction during measurements on quantum entangled systems. The prediction of quantum theory as well as experimental results are in conflict with the concept of nonlocal state-reduction, as conclusively shown here under very general assumptions. This has far-reaching implications in the interpretation of quantum mechanics in general, and demands a radical change in its present interpretation of measurements on entangled multiparticle systems. Motivated by these results we re-examine Bell’s theorem for correlations of entangled systems and find that the correlation function used by Bell fails to incorporate phase correlations at source. It is the use of such an unphysical correlation function, and not failure of locality, that leads to the Bell’s inequalities.     

Stability and collapse of localized solutions of the controlled three-dimensional Gross-Pitaevskii equation

On the basis of recent investigations, a newly developed analytical procedure is used for constructing a wide class of localized solutions of the controlled three-dimensional (3D) Gross-Pitaevskii equation (GPE) that governs the dynamics of Bose-Einstein condensates (BECs) in the presence of a spatio-temporally varying external potential. The controlled 3D GPE is decomposed into a two-dimensional (2D) linear Schr?dinger equation (called the `transverse equation’) and a one-dimensional (1D) nonlinear Schr?dinger equation (called the `longitudinal equation’), constrained by a variational condition for the controlling potential. The latter corresponds to the requirement for the minimization of the control operation in the transverse plane. Then, the above class of localized solutions are constructed as the product of the solutions of the transverse and longitudinal equations. A consistency condition between the transverse and longitudinal solutions yields a relationship between the transverse and longitudinal restoring forces produced by the external trapping potential well through a `controlling parameter’ (i.e. the average, with respect to the transverse profile, of the nonlinear inter-atomic interaction term of the GPE). It is found that the longitudinal profile supports localized solutions in the form of bright, dark or grey solitons with time-dependent amplitudes, widths and centroids. The related longitudinal phase is varying in space and time with time-dependent curvature radius and wavenumber. In turn, all the above parameters (i.e. amplitudes, widths, centroids, curvature radius and wavenumbers) can be easily expressed in terms of the controlling parameter. It is also found that the transverse profile has the form of Hermite-Gauss functions (depending on the transverse coordinates), and the explicit spatio-temporal dependence of the controlling potential is self-consistently determined. On the basis of these exact 3D analytical solutions, a stability analysis is carried out, focusing our attention on the physical conditions for having collapsing or non-collapsing solutions.     

Two-electron bound states in a continuum in quantum dots

A bound state in a continuum (BIC) might appear in open quantum dots for the variation in the dot’s shape. By means of the equations of motion of the Green’s functions, we investigate the effect of strong intradot Coulomb interactions on that phenomenon within the framework of the impurity Anderson model. The equation that the imaginary part of the poles of the Green’s function equals zero yields the condition for BICs. As a result, we show that the Coulomb interactions replicate the single-electron BICs into two-electron ones. The text was submitted by the authors in English.     

Unphysical and physical(?) solutions of the Lorentz-Dirac equation

A simple proof of a weak version of Eliezer's theorem on unphysical solutions of the Lorentz-Dirac equation is given. This version concerns a free particle scattered by a spatially localized electric field in one space dimension. (The solutions are also solutions in three space dimensions.) It establishes that for certain physically reasonable localized fields, all solutions which are free (i.e., unaccelerated) before they enter the field have unbounded proper acceleration and velocity asymptotic to that of light in the future. For any given initial velocity, the fields yielding this unphysical behavior can be arbitrarily weak. The result is then extended to a class of static fields which need not be spatially localized, including Coulomb fields. For this case the same conclusion is obtained omitting the assumption that the particle be free in the past.The rest of the paper discusses solutions to the localized field problem which are assumed free in the future rather than the past. Some strange features of these solutions are noted. The possibility of experimentally detecting deviations from the Coulomb law for a particle obeying the Lorentz-Dirac equation is discussed.     

Extended bodies with quadrupole moment interacting with gravitational monopoles: reciprocity relations

An exact solution of Einstein’s equations representing the static gravitational field of a quasi-spherical source endowed with both mass and mass quadrupole moment is considered. It belongs to the Weyl class of solutions and reduces to the Schwarzschild solution when the quadrupole moment vanishes. The geometric properties of timelike circular orbits (including geodesics) in this spacetime are investigated. Moreover, a comparison between geodesic motion in the spacetime of a quasi-spherical source and non-geodesic motion of an extended body also endowed with both mass and mass quadrupole moment as described by Dixon’s model in the gravitational field of a Schwarzschild black hole is discussed. Certain “reciprocity relations” between the source and the particle parameters are obtained, providing a further argument in favor of the acceptability of Dixon’s model for extended bodies in general relativity.     

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. Omsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 24–28, August, 1997.     

Green’s functions for spin half field theory in Rindler space

The solutions of Dirac equation in different regions of the complete extension of Rindler space are obtained near the event horizons and in the asymptotic limits. Continuity of these solutions across the event horizons is established. The Green’s functions are written down in the two causally disconnected regions, continued in the future (F) and past (P) regions using the techniques a la Boulware and a consistent scheme of Green’s functions in all regions is exhibited.     

On a class of exactly solvable models of autowaves in active media with diffusion

A class of autowave models in the form of nonlinear diffusion equations, closely related with the Liouville equation and two-dimensionalized Toda chains, is investigated. Exact solutions of these equations are constructed and analyzed. Pis’ma Zh. éksp. Teor. Fiz. 65, No. 3, 285–290 (10 February 1997)     

Relativistic breakdown of a non-linear classical field theory

It is shown that the relativistic generalization of a certain classical, non-linear, scalar, field theory displays unphysical behavior not found for the non-relativistic case. An explicit, unphysical solution of the relativistic equation is given. The different properties of relativistic and non-relativistic particle densities affect the behavior of the solutions in the respective cases.     

Anisotropic Bulk Viscous Cosmological Models with?Variable G and Λ

This paper deals with Bianchi-I, Kantowski Sachs and Bianchi-III anisotropic cosmological models of the universe, filled with a bulk viscous cosmic fluid, in the presence of variable gravitational and cosmological constants. A new set of exact solutions of Einstein’s field equation have been obtained in both truncated and full causal theories. Physical behaviour of the models has also been discussed.