Generation of static solutions of a self-consistent system of Einstein-Klein-Gordon equations

A theorem is proved according to which a class of static solutions of a self-consistent system of Einstein-Klein-Gordon equations dependent on one arbitrary function is set in correspondence with a static solution of the Einstein equations with any given energy-momentum tensor T_ij. Two particular cases are examined as an illustration of this theorem. Methods of constructing the static solutions of a system of Einstein-Klein-Gordon equations with an ideal fluid energy-momentum tensor and a massive scalar field are indicated therein.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 11–14, June, 1989.     

A few properties of a certain class of degenerate space-times

The properties are studied of a class of space-times determined by assuming the shape of the metric formds ² including disposable coordinate functions. It has been found that this class includes degenerate space-times with geodetic, null, shear-free congruences with nonvanishing expansion. The theorem has been proved that this class of solutions of the Einstein equations can easily be expanded to solutions of Einstein-Maxwell equations with a fairly general electromagnetic field. For a selected subclass relations are given between the functions determining the metric form, and two new explicit solutions with arbitrary functions of the Einstein-Maxwell equations with a cosmological constant are found.On leave from the Institute of Theoretical Physics, Warsaw University, Warsaw, Poland.     

An exact solution of the Einstein-Maxwell equations referring to a magnetic dipole

There is a formal similarity between stationary exterior solutions of the Einstein equations and static magnetic solutions of the Einstein-Maxwell theory. This is particularly evident for axially symmetric fields, and one finds that the sets of equations governing the two cases can be transformed one into the other by simple transformations of the dependent variables.     

Time-dependent solutions with localized energy of the Einstein system of equations and a nonlinear scalar field

A soliton-like time-dependent solution in the form of a running wave is derived of a self-consistent system of the gravitational field equations of Einstein and Born-Infeld type of equations of a nonlinear scalar field in a conformally flat metric. This solution is localized in space and possesses a localized energy. It is shown that both the gravitational field and the nonlinearity of the scalar field are essential to the presence of such a localized solution. In recent years various classical particle models have been widely discussed which are static or time-independent solutions of nonlinear equations with localization in space and which possess a finite field energy. In particular, soliton solutions [1], solutions in the form of eddies [2], and so on have been derived and investigated. All these solutions were treated in a flat space-time. It is of interest to derive the analogous particle-like solutions with the gravitational field taken into account; in particular it is of interest to investigate the roles of the gravitational field in connection with the formation of localized objects. These problems have been discussed in [3] in the static case. We will present below a soliton-like time-dependent solution in the form of a solitary running wave as an example of the inter-action of a Born-Infeld type of nonlinear scalar field and an Einstein gravitational field in a conformally flat metric.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 12–17, May, 1979.     

Infinite hierarchies of exact solutions of the einstein and einstein-maxwell equations for interacting waves and inhomogeneous cosmologies

For space-times with two spacelike isometries, we present infinite hierarchies of exact solutions of the Einstein and Einstein-Maxwell equations as represented by their Ernst potentials. This hierarchy contains three arbitrary rational functions of an auxiliary complex parameter. They are constructed using the so-called "monodromy transform" approach and our new method for the solution of the linear singular integral equation form of the reduced Einstein equations. The solutions presented, which describe inhomogeneous cosmological models or gravitational and electromagnetic waves and their interactions, include a number of important known solutions as particular cases.     

A new construction technique for exact axially symmetric static electrovac solutions of the Einstein-Maxwell equations

Coordinates adapted to the full electrostatic symmetries of a charged axially symmetric static system are constructed and applied to the coupled Einstein-Maxwell equations. A new solution is presented and the framework for general solutions is developed. The Weyl solutions g₀₀(Φ) are readily extracted from the formalism.     

Axially symmetric electrovac solution

Using oblate and prolate spheroidal co-ordinates static axisymmetric solutions of the coupled Einstein-Maxwell equations are obtained in vacuo. The solutions are asymptotically flat and reduce to the electrically neutral solution of Zipoy when one of the constants vanishes. As a special case the solutions reduce to the ones generated from Bonnor's theorem. Our solutions are also shown to give arbitrarily large gravitational redshifts under certain conditions.The author wishes to thank Prof. S. Banerjee, The University of Burdwan, for helpful discussions and also the University Grants Commission for financial assistance. The useful suggestions of the referee are also acknowledged.     

Exact Interior Solutions for Charged Fluid Spheres

A new method is discussed to obtain the interior solution of Einstein-Maxwell equations for a charged static sphere from a known particular solutions of a similar kind. Beginning with a charged fluid interior solution reported by Patel and Pandya [11], a new interior Reissner-Nördstrom metric is obtained using this method and physical aspects of it are extensively discussed.     

Radiating black hole solutions in arbitrary dimensions

We prove a theorem that characterizes a large family of non-static solutions to Einstein equations in N-dimensional space-time, representing, in general, spherically symmetric Type II fluid. It is shown that the best known Vaidya-based (radiating) black hole solutions to Einstein equations, in both four dimensions (4D) and higher dimensions (HD), are particular cases from this family. The spherically symmetric static black hole solutions for Type I fluid can also be retrieved. A brief discussion on the energy conditions, singularities and horizons is provided.     

Static Spherical Charged Dust Electromagnetic Mass Models in Einstein-Cartan Theory

A set of solutions for the Einstein-Maxwell field equations with torsion and spin corresponding to a static spherical charged dust distribution has been obtained. This solution represents a charged body whose mass is purely of electromagnetic origin.     

On the algebraic type of space-times possessing a nonsingular Killing horizon

The characteristic initial-value problem is discussed for the Einstein and Einstein-Maxwell equations when initial data are specified on two branches of a bifurcate Killing horizon. In the latter case it is assumed that the principal null directions of the Maxwell tensor coincide with those of a Killing bivector. A theorem specifying necessary and sufficient conditions for such space-times to be Petrov typeD is derived and consequences of this with respect to Israel's theorem and Robinson's theorem are discussed.Part of this work was initiated while the author attended the summer school on global analysis held at the International Center for Theoretical Physics, Trieste, Italy (1972).     

Solutions of the Einstein-Maxwell equations with many black holes

In Newtonian gravitational theory a system of point charged particles can be arranged in static equilibrium under their mutual gravitational and electrostatic forces provided that for each particle the charge,e, is related to the mass,m, bye=G ^1/2 m. Corresponding static solutions of the coupled source free Einstein-Maxwell equations have been given by Majumdar and Papapetrou. We show that these solutions can be analytically extended and interpreted as a system of charged black holes in equilibrium under their gravitational and electrical forces.We also analyse some of stationary solutions of the Einstein-Maxwell equations discovered by Israel and Wilson. If space is asymptotically Euclidean we find that all of these solutions have naked singularities.Alfred P. Sloan Research Fellow, supported in part by the National Science Foundation.     

Black holes in N=8 supergravity

Solutions of the bosonic field equations of the ungauged, N=8 supergravity which describe black holes with no scalar hairs are obtained. It is found that, in contrast to the Einstein-Maxwell theory where a uniqueness theorem exists, there are two distinct families of black holes in N=8 supergravity. There are also two distinct generalizations of Majumdar-Papapertrou solutions which describe the static equilibrium of many black holes.     

General magnetized Weyl solutions: disks and motion of charged particles

We construct three families of general magnetostatic axisymmetric exact solutions of Einstein-Maxwell equations in spherical coordinates, prolate, and oblates. The solutions obtained are then presented in the system of generalized spheroidal coordinates which is a generalization of the previous systems. The method used to build such solutions is the well-known complex potential formalism proposed by Ernst, using as seed solutions vacuum solutions of the Einstein field equations. We show explicitly some particular solutions among them a magnetized Erez-Rosen solution and a magnetized Morgan-Morgan solution, which we interpret as the exterior gravitational field of a finite dislike source immersed in a magnetic field. From them we also construct using the well known “displace, cut and reflect” method exact solutions representing relativistic thin disks of infinite extension. We then analyze the motion of electrically charged test particles around these fields for equatorial circular orbits and we discuss their stability against radial perturbations. For magnetized Morgan-Morgan fields we find that inside of disk the presence of magnetic field provides the possibility of to find relativist charged particles moving in both prograde and retrograde direction.     

A NEW PROCEDURE GENERATING THE SOLUTIONS OF STATIONARY AXIALLY SYMMETRIC GRAVITATIONAL FIELD EQUATIONS

A new procedure used to generate a set of solutions of stationary axially symmetric gravitational field equations is presented. It is shown that the Chandrasekhar metric is the most simple member of our solutions for n=1.Making use of this procedure we obtained a new exact solution of static axially sym-metric Einstein-Maxwell equations.It represents the exterior field of an isolated mass m with electrical dipolemoment.     

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.     

Electromagnetic solutions of the field equations in presence of zeromass mesons

The field equations of general relativity with electromagnetic stress tensor and zeromass scalar meson field are investigated. The metric coefficients are assumed to be functions of three variables only. It is then shown that, if one assumes a functional relation between some one of the metric coefficients and the electromagnetic potentials, that one can find a solution of the coupled Einstein-Maxwell equations in terms of a solution of the Einstein equations with zeromass scalar meson field as source.     

Innere Reissner-Weyl-Lösung

With the aid of an invariance transformation of the Lagrangian we obtain a class of exact static solutions of the Einstein-Maxwell equations including perfect fluid. Application of the method to the interior and exterior Schwarzschild solution yields a corresponding solution with electromagnetic field (Reissner-Weyl solution). The boundary conditions of the resulting metric are automatically fulfilled.     

Generalized plane gravitational waves

The definition of plane gravitational waves is generalized to include the case in which rays are not orthogonal to the two-dimensional wave surfaces. All Einstein spaces and some new solutions of the Einstein-Maxwell equations of this type are given.     

Solutions of five-dimensional general relativity without spatial symmetry

Conformastationary solutions of the five-dimensional vacuum Einstein equations, depending on one or two harmonic potentials, are constructed. The solutions depending on one potential fall in three distinct classes. Solutions of two of these classes may be combined to yield a class of solutions depending on two potentials, which correspond to the Israel-Wilson-Perjès solutions of the Einstein-Maxwell theory. The asymptotically flat solutions of this class describe systems of rotating electric or magnetic monopoles.