The algebraic Rainich conditions

It is well known that the Einstein-Maxwell field eguations are the Euler-Lagrange equations associated with a particular Lagrange density. It is also well known that, in a four-dimensional space, the Einstein-Maxwell field equations give rise to the Rainich conditions (which can be divided into two types, the algebraic and the differential Rainich conditions). In this note it is shown that the algebraic Rainich conditions are inevitably the consequence of every Euler-Lagrange equation associated with each member of a special class of Lagrange densities. However, in general, these Euler-Lagrange equations are not the Einstein-Maxwell field equations, although the Lagrange density associated with the latter is a particular member of this class.     

Maxwell fields onH-space

It is shown that theH space associated with a solutionM of the Einstein-Maxwell equations can be endowed with a self-dual Maxwell field which arises from the radiation component ofM's Maxwell field.     

Five-dimensional relativity theory

The five-dimensional relativity theory proposed by Kaluza is formulated covariantly for a Riemannian space containing a Killing geodesic vector field. From this five-dimensional space a four-dimensional physical space is extracted. The field equations in empty 5-space are essentially uniquely determined and correspond to the Einstein-Maxwell equations in 4-space. In the presence of a field in 5-space the field equations involve a tensor which is associated with energy, momentum, charge and current densities in 4-space. For a 5-space containing dust the field equations lead to particle motion described by the geodesic equations. The latter correspond in 4-space to the Lorentz equations of motion for particles with arbitrary ratios of charge to mass and also for certain entities (tachyons and luminons) unobserved hitherto.     

Spherically symmetric solutions of Einstein-Maxwell theory with null fluid

Solutions of the Einstein-Maxwell equations with the addition of terms representing charged null fluid emitting from a spherically symmetric body are found. One type of solution is a simple extension of that found by Bonnor and Vaidya while the other represents a null electromagnetic field with null electric current.     

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.     

A connection approach to the Einstein-Maxwell field equations

Scalar densities which are concomitants of the metric tensor, a symmetric affine connection and its first derivative, and the derivative of a vector field are examined. Rather simple demands are imposed on the corresponding Euler-Lagrange expressions. It is found that the associated field equations reduce essentially to the Einstein-Maxwell field equations.     

An electromagnetic interpretation of the Kerr-Vaidya metric

It is shown that at large distances from a rotating mass, the radiation may be associated with an Einstein-Maxwell null field with a non-zero null current.     

Einstein-Maxwell null fields with a null current distribution

Null Einstein-Maxwell charge-free fields such that the propagation vector is a scalar multiple of a gradient are not determined uniquely by the geometry of the space-time. The metric for space-times admitting such exceptional fields can always be transformed to the Wyman-Trollope form. This same result follows if there is a non-vanishing null current density associated with the field.     

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.     

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.     

Classes of accurate solution of the Einstein-Maxwell equations

The problem of integrating the Einstein-Maxwell equations with the additional condition of variable separation in the Hamilton-Jacobi equations is considered. The case where the integrals of motion form isotropic sets is studied in detail. Classes of accurate solution corresponding to the Einstein-Maxwell equations are obtained.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 33–36, December, 1981.     

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.     

Axially symmetric stationary electrovac solutions

Perjes and Israel and Wilson have given independently a new class of solutions of the sourcefree Einstein-Maxwell equations, which can be interpreted as the external gravitational and electromagnetic fields of a spinning source with unit specific charge. Starting from Zipoy's solutions in oblate and prolate spheroidal coordinates for the source-free gravitational field we generate some axially symmetric stationary solutions of the source-free Einstein-Maxwell equations by using Perjes' method. All these solutions become Euclidean at infinity. The asymptotic behavior and the singularity of the solutions are studied in order to gain some insight into the nature of the source. The solution in prolate spheroidal coordinates is found to contain closed timelike lines.     

Complex transformations of the curvature tensor

It is shown that complex transformations can be applied on the parameters and coordinates entering a known curvature tensor in order to generate new curvature tensors which, just as the seed tensor, possess the same symmetry properties and satisfy certain algebraic relationships following from the Einstein-Maxwell equations with cosmological constant.     

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.     

Charged Tomimatsu-Sato metrics

We show that the solutions of Einstein-Maxwell field equations can be constructed from the Tomimatsu-Sato solutions of Einstein's field equations for empty space simply by inspection when δ has integral values.     

An extension of the plane-symmetric electrovac general solution to Einstein equations

We present the general solution to Einstein-Maxwell equations representing plane-symmetric metrics associated with electromagnetic fields that are not fully plane-symmetric. There are two classes in the general solution, the first approaches Taub's static metric or Kasner's spatially homogeneous one as the electromagnetic field goes to zero, while the second approaches the fiat metric.     

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.     

Invariance of the Massless Field Equations under Changes of the Metric

It is shown that the Maxwell equations with sources, expressed in terms of the covariant tensor field F_ijand the current density four-vector Jⁱ, are invariant under the change of the metric g_ijby g_ij = g_ij+ l_il_jif l_iis a principal null direction of F_ijand that an analogous result holds in the case of the massless Klein-Gordon equation if l_iis null and orthogonal to the gradient of the field and in the case of the null dust equations if l_iis parallel to the dust four-velocity. An elementary proof of the following generalization of the Xanthopoulos theorem is also given: Let (g_ij, F_ij) be an exact solution of the Einstein-Maxwell equations and let l_ibe a principal null direction of F_ij, then (g_ij+ l_il_j, F_ij) is also an exact solution of the Einstein-Maxwell equations if and only if (l_il_j, 0) satisfies the Einstein-Maxwell equations linearized about the background solution (g_ij, F_ij). Furthermore, analogous theorems, where the source of the gravitational field is a massless Klein-Gordon field or null dust, are presented.     

Comments on the unification of electromagnetism and gravitation through “Generalized Einstein manifolds”

Akbar-Zadeh [J. Geom. Phys. 17 (1995) 342] has recently proposed a new geometric formulation of Einstein-Maxwell system with source in terms of what are called “Generalized Einstein manifolds”. We show that, contrary to the claim, Maxwell equations have not been derived in this formulation, and that the assumed equations can be identified only as source-free Maxwell equations in the proposed geometric setup. A genuine derivation of source-free Maxwell equations is presented within the same framework. We draw a conclusion that the proposed unification scheme can pertain only to source-free situations.