Nonlinear spinor equations in general relativity

General relativistic nonlinear spinor equations are proposed which reduce in the linear approximation to the Dirac equations, and in the slightly nonlinear approximation reduce to the Ivanenko - Heisenberg equations. When written in a vector form, the nonlinear spinor equations take the form of the Einstein equations, in which matter is produced by spinor fields.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 121–125, March, 1977.The author thanks professor D. D. Ivanenko for his support and a number of useful observations.     

Non-local equations for general relativity

The field equations for two non-local variables, equivalent to the Einstein vacuum equations, are presented. These variables are the holonomy operator associated with special paths and the light cone cut function.

Starting from these equations, one shows via a perturbation argument that a single, fourth-order equation for the cut function can be derived. This single equation encodes the entire conformal structure of a vacuum space—time. The same perturbation technique yields, via quadratures, solutions to the vacuum Einstein equations to any order.     

Heisenberg equations of motion for spin-1/2 wave equation in general relativity

The Heisenberg equations of motion for the spin-1/2 wave equation in general relativity are obtained by a covariant procedure. They are found to be similar to the equations of motion for a classical pole-dipole test-particle in general relativity. The identification is complete when the Heisenberg equations are taken to be satisfied by the respective expectation values.     

Heisenberg equations of motion for spin-1/2 wave equation in general relativity

The Heisenberg equations of motion for the spin-1/2 wave equation in general relativity are obtained by a covariant procedure. They are found to be similar to the equations of motion for a classical pole-dipole test-particle in general relativity. The identification is complete when the Heisenberg equations are taken to be satisfied by the respective expectation values.     

Nonlinear acoustic wave equations with fractional loss operators

Fractional derivatives are well suited to describe wave propagation in complex media. When introduced in classical wave equations, they allow a modeling of attenuation and dispersion that better describes sound propagation in biological tissues. Traditional constitutive equations from solid mechanics and heat conduction are modified using fractional derivatives. They are used to derive a nonlinear wave equation which describes attenuation and dispersion laws that match observations. This wave equation is a generalization of the Westervelt equation, and also leads to a fractional version of the Khokhlov-Zabolotskaya-Kuznetsov and Burgers' equations.     

Nonlinear equations of the gravitational field in the special theory of relativity

It is proposed that the nonlinearity of the field be taken into account with the help of a method which essentially consists of the fact that the structure of the Lagrangian, expressed in terms of the potential of the field and its derivatives, is not known a priori, but is obtained from a solution of the self-action equation in phase space in which the Lagrangian is the unknown. This equation has a solution and the Lagrangian turns out to be a nonpolynomial function with respect to the field potential. The gravitational field equations following from the variational principle have a similar structure to the equations of general relativity and coincide with them in the linear approximation. The equations of other fields taking into account gravitation, as well as the equation of motion of a test particle in a gravitational field, are constructed.     

引力波与广义相对论

介绍了引力波的广义相对论理论基础.介绍如何通过双星周期的时间改变间接检测引力辐射,如何利用引力波的偏振效应直接检测引力波.     

Quasi-local conservation equations in general relativity

A set of exact quasi-local conservation equations is derived from the Einstein's equations using the first-order Kaluza–Klein formalism of general relativity in the (2,2)-splitting of 4-dimensional spacetime. These equations are interpreted as quasi-local energy, momentum, and angular momentum conservation equations. In the asymptotic region of asymptotically flat spacetimes, it is shown that the quasi-local energy and energy-flux integral reduce to the Bondi energy and energy-flux, respectively. In spherically symmetric spacetimes, the quasi-local energy becomes the Misner–Sharp energy. Moreover, on the event horizon of a general dynamical black hole, the quasi-local energy conservation equation coincides with the conservation equation studied by Thorne et al. We discuss the remaining quasi-local conservation equations briefly.     

Nonlinear equations for a slow magnetogasdynamic quasi-monochromatic wave in a medium with dispersion and dissipation

Evolutionary equations are obtained in the vicinity of a singularity for a slow wave with allowance for the quadratic nonlinearity, dispersion, and dissipation. It is demonstrated that the propagation of quasi-monochromatic waves in a current-conducting gas—liquid mixture is described by ordinary differential equations for the amplitudes in the aforementioned region.     

Nonlinear wave equation for torsion

Specializing the geometry of a Riemann-Cartan space-time U₄ to the case of a completely antisymmetric torsion tensor (axial vector torsion, ¹K_μ) a set of nonlinear wave equations and constraints are established relating the torsion of the U₄ geometry to an axial vector source current. This current obeys an anomaly relation similar to the axial current in spinor electrodynamics.     

On wave equations for viscous fluids

This paper deals with wave equations describing small-amplitude disturbances in horizontally stratified, continuously varying, viscous fluids; gradients of the static pressure and of the coefficient of viscosity are neglected. A set of equations in first-order matrix form, which describes coupled longitudinal and transverse disturbances, is treated by the methods ofClemmow andHeading and ofHeading.The work of this paper could be extended in a number of ways; for example, the effect of a gravitational field could be included, and the coefficient of viscosity could be allowed to vary with position.     

Speed selection for coupled wave equations

We discuss models for coupled wave equations describing interacting fields, focusing on the speed of travelling wave solutions. In particular, we propose a general mechanism for selecting and tuning the speed of the corresponding (multi-component) travelling wave solutions under certain physical conditions. A number of physical models (molecular chains, coupled Josephson junctions, propagation of kinks in chains of adsorbed atoms and domain walls) are considered as examples.     

A wave automaton for Maxwell's equations

This paper presents an extension to electromagnetic fields of the wave automaton, which was introduced in recent years for describing wave propagation in inhomogeneous media. Using elementary processes obeying a discrete Huygens' principle and satisfying fundamental symmetries such as time reversal and reciprocity, this new wave automaton is capable of modeling Maxwell's equations in 3+1 dimensions. It supplements the methods that were developed early for scalar and spinor fields. Received 19 July 2001     

Nonlinear coupled-mode equations

Nonlinear coupled-mode equations governing the modal coupling of a two-mode coupled system (such as twin core couplers) are integrable; power swapping in such a system follows a periodical manner and can be expressed analytically. When three or more modes (for systems such as multiple-core couplers) are involved, the nonlinear coupled-mode equations are no longer integrable and chaotic power swapping is expected. A numerical approach is required, in general, to solve such nonlinear coupled systems involving the coupling of three or more modes. We find, however, that for certain structural configurations, such as triple-core couplers with the cores arranged in the shape of an isosceles triangle, the nonlinear coupled-mode equations for multiple-core couplers can be solved analytically under a resonant condition. The analytical solution indicates that power swapping among, for example, the three cores placed in the shape of an isosceles triangle can be aperiodic at high power, although power may flow from core to core periodically at low power.     

Nonlinear heat equations

The modified Smoluchowski equation, coupled to a temperature field, leads to a pair of nonlinear heat equations obeying the first and second laws of thermodynamics. We obtain a solution representing a particle under gravity, moving in a slab and maintained in stasis away from the Gibbs state by a temperature gradient. A two-state atom in a potential in isothermal conditions is described by coupled equations satisfying detailed balance. It is shown that the free energy is a monotonic decreasing function of time.     

Cauchy problems for the conformal vacuum field equations in general relativity

Cauchy problems for Einstein's conformal vacuum field equations are reduced to Cauchy problems for first order quasilinear symmetric hyperbolic systems. The “hyperboloidal initial value” problem, where Cauchy data are given on a spacelike hypersurface which intersects past null infinity at a spacelike two-surface, is discussed and translated into the conformally related picture. It is shown that for conformal hyperboloidal initial data of classH ^S,s≧4, there is a unique (up to questions of extensibility) development which is a solution of the conformal vacuum field equations of classH ^S. It provides a solution of Einstein's vacuum field equations which has a smooth structure at past null infinity.