Electromagnetic Duality in General Relativity

By resolving the Riemann curvature relative to a unit timelike vector into electric and magnetic parts, we consider duality relations analogous to those in electromagnetic theory. It turns out that the duality transformation implies the Einstein vacuum equation without the cosmological term. The vacuum equation is invariant under interchange of active and passive electric parts, giving rise to the same vacuum solutions but with the opposite sign for the gravitational constant. Further, by modifying the equation it is possible to construct interesting dual solutions to vacuum as well as to flat spacetimes.     

Solutions of the einstein equations with zero coupling. II

The possibility of obtaining new exact solutions of the Einstein equations in vacuum and of the Einstein-Maxwell equations from known solutions through a zero-coupling transformation is analyzed. The changes in the algebraic properties resulting from this transformation are analyzed for spaces which allow a congruence of isotropic geodesies with no distortion (shear) or rotation (curl). Examples of the derivation of new exact solutions are considered.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii Fizika, No. 11, pp. 68–76, November, 1971.     

On torsion-free vacuum solutions of the model of de Sitter gauge theory of gravity (II)

It is shown that all torsion-free vacuum solutions of the model of de Sitter (dS) gauge theory of gravity are the vacuum solutions of Einstein field equations with the same positive cosmological constant. Furthermore, for the gravitational theories with more general quadratic gravitational Lagrangian (F ² + T ²), the torsion-free vacuum solutions are also the vacuum solutions of Einstein field equations.     

Conformal Continuations in Gravitation Theory with Lagrangian <Emphasis Type="Italic">F</Emphasis>(<Emphasis Type="Italic">R</Emphasis>)

Static spherically-symmetric vacuum solutions of gravitation theory equations with Lagrangian f(R) are examined, where R is a scalar curvature and f is an arbitrary function. Equations of f(R)-theories are reduced to the Einstein scenario — general relativity theory (GRT) equations with a source in the form of a scalar field with potential — with the use of the well-known conformal transformation. The necessary and sufficient conditions of existence of solutions admitting conformal continuations are formulated. This means that the central singularity of the Einstein scenario is mapped into a regular sphere S_trans of the Jordan scenario (that is, into the manifold corresponding to the initial formulation of the theory), and a solution of the field equations can be smoothly continued through it. The value of curvature R on the sphere S_trans corresponds to an extremum of the function f(R). Concrete examples are considered. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 46–51, September, 2005.     

A family of Jordan-Brans-Dicke Kerr solutions

A family of solutions of the vacuum Jordan-Brans-Dicke or scalar-tensor gravitational field equations is given. This family reduces to the Kerr rotating solution of the vacuum Einstein equations when the scalar field is constant. The family does not have spherical symmetry when the rotation is zero and the scalar field is not constant. The method used to generate the new solutions can also be used to obtain vacuum Jordan-Brans-Dicke solutions from any given vacuum stationary, axisymmetric solution.     

A class of Einstein–Weyl spaces associated to an integrable system of hydrodynamic type

HyperCR Einstein–Weyl equations in 2+1 dimensions reduce to a pair of quasi-linear PDEs of hydrodynamic type. All solutions to this hydrodynamic system can in principle be constructed from a twistor correspondence, thus establishing the integrability. Simple examples of solutions including the hydrodynamic reductions yield new Einstein–Weyl structures.     

Double-dual solutions of generalized theories of gravitation

Solutions to the Stephenson-Yang theory of gravity and its generalizations are discussed. By considering the inclusion of a cosmological term in the action spherically symmetric static solutions are presented that do not fall into the vacuum Einstein class. A simple double-duality ansatz is responsible for all the solutions that are discussed.On leave from the Physics Department, Middle East Technical University, Ankara, Turkey.     

Generation of static solutions of the self-consistent system of Einstein-Maxwell equations

A theorem is proved, according to which to each solution of the Einstein equations with an arbitrary momentum-energy tensor in the right hand side there corresponds a static solution of the self-consistent system of Einstein-Maxwell equations. As a consequence of this theorem, a method is established of generating static solutions of the self-consistent system of Einstein-Maxwell equations with a charged grain as a source of vacuum solutions of the Einstein equations.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 98–102, February, 1988.     

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.     

Comment on conformally Ricci-flat perfect fluids of Petrov-typeN

It is shown that vacuum solutions of the Einstein field equations, which are of Petrov-typeN, cannot be conformally transformed into nonvacuum perfect fluid space-times.     

Regularity of harmonic maps with prescribed singularities

In this paper, we studied the regularity problem for harmonic maps into hyperbolic spaces with prescribed singularities along codimension two submanifolds. This is motivated from one of Hawking's conjectures on the uniqueness of Kerr solutions among all axially symmetric asymptotically flat stationary solutions to the vacuum Einstein equation in general relativity.Research partially supported by a NSF grant DMS-8907849.Research partially supported by a NSF grant     

Common solutions of the einstein and Brans-Dicke theories

It is shown that non-trivial solutions common to the vacuum field equations of the Einstein and of the Brans-Dicke theories necessarily representpp-waves and the set of all common solutions is precisely the set of allpp-wave solutions of the Einstein equations. The form of the associated scalar field is found and is shown to be singular when ω相似文献   

Axisymmetric regular multiwormhole solutions in five-dimensional general relativity

A class of regular, asymptotically flat solutions to the five-dimensional vacuum Einstein equations with a two-parameter Abelian isometry group is constructed, under the additional assumption of axial symmetry in three-dimensional space. The possibility of interpreting these multiwormhole solutions as multiparticle systems is discussed.     

Linearization procedure for cylindrically and axially symmetric space-times

An investigation of the vacuum Einstein gravitational field equations for cylindrically and axially symmetric space-times is presented which leads to an equivalent differential system involving a simple nonlinearity only. The case when this equivalent system is linear is analyzed in detail and two methods for generating solutions of the Einstein vacuum equations are set up. As a result, in the axially symmetric case the linearity of the equivalent system characterizes completely the Kramer-Neugebauer transforms of Papapetrou line elements. Accordingly, Weyl solutions are shown to generate exhaustively both Lewis and van Stockum solutions. Analogous results are obtained also in the cylindrically symmetric case.     

Higher-dimensional vacuum solutions of Einstein's field equations

Some general solutions of the (general)D-dimensional vacuum Einstein field equations are obtained. The four-dimensional properties of matter are studied by investigating whether the higher-dimensional vacuum field equations reduce (formally) to Einstein's four-dimensional theory with matter. It is found that the solutions obtained give rise to an induced four-dimensional cosmological perfect fluid with a (physically reasonable) linear equation of state.     

The York map is a canonical transformation

The York mapping from the space of freely chosen conformal data to the space of constraint-satisfying physical data is shown to be a canonical transformation for both the vacuum Einstein theory and the Einstein-Maxwell theory.     

On torsion-free vacuum solutions of the model of de Sitter gauge theory of gravity

It is shown that all vacuum solutions of Einstein field equation with a positive cosmological constant are the solutions of a model of dS gauge theory of gravity. Therefore, the model is expected to pass the observational tests on the scale of solar systems and explain the indirect evidence of gravitational wave from the binary pulsars PSR1913+16.      

Necessary and sufficient conditions for trivial solutions in supergravity

We determine the conditions necessary for a solution of the supergravity field equations with infinitesimal spin-3/2 field to be a pure gauge transformation of an Einstein vacuum field. The analysis depends on the Petrov classification of the curvature tensor and uses two-component spinor calculus. For general type I, the type II, and typeD, the necessary conditions found are also shown to be sufficient, and the explicit form of the gauge transformation can be given.Work supported in part by the Einstein Memorial Foundation.