Witten's gauge field equations and an infinite-dimensional Grassmann manifold

Witten's gauge fields are interpreted as motions on an infinite-dimensional Grassmann manifold. Unlike the case of self-dual Yang-Mills equations in Takasaki's work, the initial data must satisfy a system of differential equations since Witten's equations comprise a pair of spectral parameters. Solutions corresponding to (anti-) self-dual Yang-Mills fields are characterized in the space of initial data and in application, some Yang-Mills fields which are not self-dual, anti-self-dual nor abelian can be constructed.     

关于稳态轴对称真空引力场方程的两个扩展解

将Ehlers变换应用于Ernst方程的Schwarzschild解和Kerr解，通过引入Boyer-Lindquist坐 标变换以及相关的参数代换，得到了Ernst方程的两个扩展解. 当所含参数L=0时，其中一个扩展解退化为Schwarzschild解，另一个退化为Kerr解.当参数|L|M时，如果取近似1-LM2≈1，则这两个扩展解分别退化为已知的NUT-Taub解和Kerr-NUT解.这一结果表明 NUT-Taub解和Kerr-NUT解中所含的参数l并非能任意取值，它的取值要受到引力源质量M的限 制，即要求|l|M. 关键词： Ehlers变换群 Ernst方程 Boyer-Lindquist坐标变换     

A new class of asymptotically flat stationary axisymmetric vacuum gravitational fields

We present the full four-dimensional metric for a stationary axisymmetric solution of the vacuum Einstein equations related to the Ernst potential which we obtain by application of anarbitrary number of rank-zero HKX transformations to the general static Weyl solution. The metric function is determined in a completely algebraically way. We perform a suitable Ehlers transformation to ensure asymptotic flatness and give the expression for the total mass of this asymptotically flat solution by analyzing the behavior of the solution on the positive z axis.     

Soliton geometry and the vacuum gravitational field equations

A soliton geometry is introduced on manifolds with arbitrary dimensions. The usual soliton connection 1-form defined by Crampin et al. is recovered when the soldering form is a 0-form. It is shown that Einstein's vacuum field equations admit a soliton connection and a soldering 1-form. An associated linear equation with a spectral parameter of Einstein's vacuum field equations are found and some properties of this equation are explored. An example of a Bäcklund transformation is also given.     

Singular manifold analysis of the Einstein vacuum field equations

The Painlevé property of the vacuum Einstein field equations is investigated. It is observed that the field equations possess this property when spacetime admits commuting, nonnull two Killing vector fields.     

The general stationary gravitational vacuum field of cylindrical symmetry

The general stationary vacuum gravitational field of cylindrical symmetry as recently found by Davies and Caplan is even static. The possible Petrov types of the Riemann tensor areI,D orO. In spacelike infinity the spacetime becomes necessarily flat.     

Double-Alekseev inverse scattering method in the stationary axi-symmetric vacuum gravitation field equations

We present a new improvement to the Alekseev inverse scattering method. This improved inverse scattering method is extended to a double form, followed by the generation of some new solutions of the double-complex Kinnersley equations. As the double-complex function method contains the Kramer-Neugebauer substitution and analytic continuation, a pair of real gravitation soliton solutions of the Einstein’s field equations can be obtained from a double N-soliton solution. In the case of the flat Minkowski space background solution, the general formulas of the new solutions are presented.      

Functional method for solving gravitational field equations

A new method for solving the equations of a gravitational field (G field), which is based on functional transformations (FT), which replace holonomic transformations (HT), is discussed. The method is equivalent to nonholonomic transformations (NT) of the field functions and makes it possible to use a known solution to construct a new set of solutions in the same coordination, which contains a fixed number (up to 12) of arbitrary functions. The FT method can be used to solve the Cauchy problem for a non-steady-state G field, to eliminate singularities in the field, etc.This article is published as a matter for discussion.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 50–55, February, 1979.     

On the stationary axisymmetric Einstein-Maxwell equations

This paper is concerned with the class of exterior stationary axisymmetric solutions of the Einstein-Maxwell equations that arise from sources in which the mass is proportional to the charge and the angular momentum is proportional to the magnetic moment. With the use of Ernst's formulation the Einstein-Maxwell equations for this class are reduced to two coupled equations for two unknowns.     

Conformally symmetric vacuum solutions of the gravitational field equations in the brane-world models

A class of exact solutions of the gravitational field equations in the vacuum on the brane are obtained by assuming the existence of a conformal Killing vector field, with non-static and non-central symmetry. In this case, the general solution of the field equations can be obtained in a parametric form in terms of the Bessel functions. The behavior of the basic physical parameters describing the non-local effects generated by the gravitational field of the bulk (dark radiation and dark pressure) is also considered in detail, and the equation of state satisfied at infinity by these quantities is derived. As a physical application of the obtained solutions we consider the behavior of the angular velocity of a test particle moving in a stable circular orbit. The tangential velocity of the particle is a monotonically increasing function of the radial distance and, in the limit of large values of the radial coordinate, tends to a constant value, which is independent on the parameters describing the model. Therefore, a brane geometry admitting a one-parameter group of conformal motions may provide an explanation for the dynamics of the neutral hydrogen clouds at large distances from the galactic center, which is usually explained by postulating the existence of the dark matter.     

Orbital effect in the stationary axisymmetric field

This paper uses an elegant mathematical method to calculate the orbital effects in the axisymmetric field created by the spinning mass with electric charge and a large number of magnetic monopoles. In comparison with that in the Reissner-Nordstroem （R-N） field, the correction terms caused by the spinning mass decrease the advanced effect as the revolution direction of the test particle coincides with that of the Kerr field, however, the correction terms caused by the spinning charged mass increase the advance effect as the revolution direction of the test particle coincides with that of the Kerr-Newman-Kasnya （KNK） field. Generalizing the effect in the axisymmetric field, it obtains interesting results by discussing the parameters of the celestial body, these parameters provide a feasible experimental verification of the general relativity.     

Exact solutions of noncommutative vacuum Einstein field equations and plane-fronted gravitational waves

We construct a class of exact solutions of the noncommutative Einstein field equations in the vacuum, which are noncommutative analogues of the plane-fronted gravitational waves in classical gravity.     

Extended double soliton solution families for the statically axisymmetric self-dualSU(2) gauge field equations

By using the extended double complex function method, the statically axisymmetric self-dualSU(2) gauge field equations and, in turn, the Belinsky-Zakharov solution-generating technique are generalized to extended double forms. The restriction on “soliton index” in the original solution-generating technique is eliminated so that for each positive integer, we can obtainphysical soliton solutions of the statically axisymmetric self-dualSU(2) gauge field equations in pairs. Some sufficient conditions are given for seed solutions with which the corresponding scattering wave function can be written out directly. As examples, some soliton solution families are given, most solutions of which are new.     

Space-time algebra for the generalization of gravitational field equations

The Maxwell–Proca-like field equations of gravitolectromagnetism are formulated using space-time algebra (STA). The gravitational wave equation with massive gravitons and gravitomagnetic monopoles has been derived in terms of this algebra. Using space-time algebra, the most generalized form of gravitoelectromagnetic Klein–Gordon equation has been obtained. Finally, the analogy in formulation between massive gravitational theory and electromagnetism has been discussed.     

On the chiral connection between the ferromagnet,the axisymmetric gravitational problem and the SU(2) vacuum gauge field

A single chiral structure is shown to underly several three-dimensional field equations of physics, e.g., the ferromagnet, the stationary axi-symmetric gravitational problem, and the vacuum SU(2) gauge field in the Coulomb gauge. This connection invites cross-fertilization of exact solutions. As illustrations, exact finite energy solutions are obtained for the ferromagnet via the corresponding Weyl, Kerr, and Tominatsu-Sato solutions of Einstein's gravitational equations. Comments are made on the “local gauge invariance” of the ferromagnet and the gravitational field.     

A solution of the Maxwell equations for any static,axisymmetric vacuum space-time

We study Maxwell's equations in a static, axisymmetric vacuum space-time. For any such background metric a solution is found, which far from all masses goes over into a homogeneous magnetic field parallel to the axis of symmetry.     

On a general static axisymmetric solution of the Einstein vacuum equations

It is shown that a generalization of the procedure given in [1] for construction of the gravitational multipoles leads to the same Newtonian limit as the generalized Erez-Rosen solution [2].     

Calculation of the Shirokov effect for circular orbits in an axisymmetric static gravitational field

In the post-Newtonian approximation of the metric theory of gravitation we calculate the frequencies of Shirokov oscillations for equatorial circular geodesics in an asymptotically flat static space-time which satisfies the conditions of axial symmetry and symmetry with respect to the plane of the equator.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 78–81, April, 1990.We thank Yu. S. Vladimirova and the participants of his seminar for discussions of the results of our work.     

Exact one-body solution of the gauge-covariant gravitational field equations

Dirac has recently introduced a new theory of gravity which includes his large numbers hypothesis. The exact one-body vacuum solution consistent with multiplicative matter creation is obtained and discussed.Supported by a grant from the National Research Council of Canada.