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.     

Electrodynamics in an expanding universe

An algebraic form of the energy momentum tensor of the electromagnetic field is derived in terms of two scalars and two mutually orthogonal vector fields. Upon inserting this tensor into the field equations, solutions of the co-determined Einstein-Maxwell equations are obtained. The line element used is that corresponding to a conformal flat universe, whose form is then uniquely determined by the field equations. The case of a charged fluid is also considered and it is found that the particular form of the velocity field chosen limits the choice of the possible equation of state connecting the pressure and density distributions.     

The uniqueness of the Einstein-Maxwell field equations

The most general Lagrange density (which is a concomitant of the metric tensor together with a vector field and its first derivatives) for which the associated Euler-Lagrange equations are precisely Maxwell's equations is obtained. Although it is more general than the Lagrangian which is commonly used, it still has essentially the same energy momentum tensor.     

Biframe bundle geometry and an extension of RMW theory: Application to a charged perfect fluid

An extension of the original Rainich-Misner-Wheeler (RMW) theorem to include Einstein-Maxwell spacetimes with geometrical sources has recently been accomplished by generalizing the geometrical arena from the linear frame bundleLM to the bundle of biframesL ² M. The assumptions of a Riemannian connection one-form onLM and a general connection one-form onL ² M necessarily implies the existence of a difference formK. We provide new algebraic and differential conditions on an arbitrary triple (M, g, K), in addition to those already imposed by the generalization of the RMW theorem, which guarantee the form of the coupled Einstein-Maxwell field equations associated with a charged perfect fluid spacetime. All physical quantities associated with these field equations, namely the Maxwell field strength, the mass-energy density, the pressure, the electric and magnetic charge to mass ratios, and the unit four velocity of the fluid, can be recovered from the geometry as they are constructible entirely from the metricg, the difference formK, and their derivatives.     

Biframe bundle geometry: An extension of Rainich-Misner-Wheeler theory to include sources

Recently the original theory of Rainich, Misner, and Wheeler (RMW) has been shown to have a natural reformulation in terms of a new principal fiber bundle, namely the bundle of biframesL ² M over spacetime. We extend this new formalism further and show that the original RMW program can be generalized to include Einstein-Maxwell spacetimes with geometrical sources. The assumptions of a Riemannian connection one-form on the linear frame bundleLM and a general connection one-form onL ² M necessarily imply the existence of a difference formK. A generalization of the standard RMW theorem is developed which provides the necessary and sufficient conditions on an arbitrary triple (M, g, K) in order for this triple to be an Einstein-Maxwell spacetime with geometrical sources. All sources for the field equations associated with such spacetimes are geometrical, as they are constructible from the metricg, the difference formK, and their derivatives. The extension of the RMW program presented here introduces a second complexion vector, in addition to the standard RMW complexion vector, and the formalism reduces, in the special case of no sources, to the standard RMW program.     

The uniqueness of the Einstein-Yang-Mills equations

The most general gauge-invariant Lagrange density (concomitant of the metric tensor together with the gauge potentials of a gauge and its first derivatives) for which the associated Euler-Lagrange equations are precisely Yang-Mills equations is obtained. It is more general than the Lagrangian which is commonly used, but it still has essentially the same energy momentum tensor.     

New geometrical approach to Rainich-Misner-Wheeler theory

We introduce a new principal fiber bundle, the bundle of biframes, associated with the geometry of bivectors on spacetime. It is shown that the biframe bundle is a natural geometric arena for modeling the already unified theory of Rainich, Misner, and Wheeler (RMW). The structure equations for the bitorsion inherent in the biframe bundle lead to a generalization of Rainich's algebraic conditions for electromagnetic-type stress tensors which includes sources in a natural way. Besides the usual complexion vector of the RMW theory, an additional new complexion-type vector is found. The generalized algebraic conditions reduce to the usual RMW conditions in the special case of no sources.     

Helmholtz conditions,covariance, and invariance identities

This paper is concerned with the problem of finding a multiplier matrixg which converts a prescribed system of second-order ordinary differential equations to the Euler-Lagrange form. Sufficient conditions for the existence of a multiplier matrix are given in the form of an infinite system of linear algebraic equations, provided the entries ofg may be regarded as components of a (0, 2) symmetric tensor field. As an application, conditions for the local existence of a metric tensor compatible with a given torsion-free connection are deduced.     

Can electro-magnetic field,anisotropic source and varying Λ be sufficient to produce wormhole spacetime?

It is well known that solutions of general relativity which allow for traversable wormholes require the existence of exotic matter (matter that violates weak or null energy conditions (WEC or NEC)). In this article, we provide a class of exact solution for Einstein-Maxwell field equations describing wormholes assuming the erstwhile cosmological term Λ to be space variable, viz., Λ=Λ(r). The source considered here not only a matter entirely but a sum of matters i.e. anisotropic matter distribution, electromagnetic field and cosmological constant whose effective parts obey all energy conditions out side the wormhole throat. Here violation of energy conditions can be compensated by varying cosmological constant. The important feature of this article is that one can get wormhole structure, at least theoretically, comprising with physically acceptable matters.     

The neutrino energy-momentum tensor and the Weyl equations in curved space-time

In general, a first order Lagrangian gives rise to second order Euler-Lagrange equations. However, there are important examples where the associated Euler-Lagrange equations are of first order only, the Weyl neutrino equations being of this type. In this paper we therefore consider first order spinor Lagrangians which give rise to firstorder Euler-Lagrange equations. Specifically, the most general first order spinor field equations of rank one in curved space-time which are derivable from a first order Lagrangian of the same type are explicitly constructed. Subject to a certain restriction, the Weyl neutrino equation is the only possibility. Furthermore, if the spinor field satisfies the Weyl neutrino equation, then the associated energy momentum tensor is the conventional neutrino energymomentum tensor.     

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.     

Rainich theory for type D aligned Einstein–Maxwell solutions

The original Rainich theory for the non-null Einstein–Maxwell solutions consists of a set of algebraic conditions and the Rainich (differential) equation. We show here that the subclass of type D aligned solutions can be characterized just by algebraic restrictions.     

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.     

A class of algebraically general solutions of the Einstein-Maxwell equations for non-null electromagnetic fields

The Einstein-Maxwell field equations for non-null electromagnetic fields are studied under the conditions that the null tetrad is parallelly propagated along both principal null congruences. It is shown that the resulting spacetime solutions are necessarily algebraically general. The twist-free solution found in a previous article is shown to be the most general twist-free solution. An expansionfree solution with twist and shear is also found.     

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.     

On Relativistic Generalization of Gravitational Force

In relativistic theories, the assumption of proper mass constancy generally holds. We study gravitational relativistic mechanics of point particle in the novel approach of proper mass varying under Minkowski force action. The motivation and objective of this work are twofold: first, to show how the gravitational force can be included in the Special Relativity Mechanics framework, and, second, to investigate possible consequences of the revision of conventional proper mass concept (in particular, to clarify a proper mass role in the divergence problem). It is shown that photon motion in the gravitational field can be treated in terms of massless refracting medium, what makes the gravity phenomenon compatible with SR Mechanics framework in the variable proper mass approach. Specifically, the problem of point particle in the spherical symmetric stationary gravitational field is studied in SR-based Mechanics, and equations of motion in the Lorentz covariant form are obtained in the relativistic Lagrangean problem formulation. The dependence of proper mass on potential field strength is derived from the Euler-Lagrange equations as well. One of new results is the elimination of conventional 1/r divergence, which is known to be not removable in Schwarzschild gravitomechanics. Predictions of particle and photon gravitational properties are in agreement with GR classical tests under weak-field conditions; however, deviations rise with potential field strength. The conclusion is made that the approach of field-dependent proper mass is perspective for development of SR gravitational mechanics and further studies of gravitational problems.     

Geometrodynamics of electromagnetic fields in the Newman-Penrose formalism

The already unified field theory of Rainich, Misner, and Wheeler is rederived in the spin-coefficient formalism of Newman and Penrose. Conditions equivalent to the Rainich algebraic conditions are obtained by classifying the tracefree Ricci tensor according to its principal null directions. The case of a null electromagnetic field is also treated fully. Necessary and sufficient conditions are given for a Riemannian geometry to have an electromagnetic field, null or non-null, as its source.Supported in part by the National Research Council of Canada.     

Some new radiating Kerr-Newman solutions

Three exact non-static solutions of Einstein-Maxwell equations corresponding to a field of flowing null radiation plus an electromagnetic field are presented. These solutions are non-static generalizations of the well known Kerr-Newman solution. The current vector is null in all the three solutions. These solutions are the electromagnetic generalizations of the three generalized radiating Kerr solutions discussed by Vaidya and Patel. The solutions discussed by us describe the exterior gravitational fields of rotating radiating charged bodies. Many known solutions are derived as particular cases.     

4-Dimensional black holes from Kaluza-Klein theories

In this paper we consider generalizations in 4 dimensions of the Einstein-Maxwell equations which typically arise from Kaluza-Klein theories. We specify conditions such that stationary solutions lead to non-linear-models for symmetric spaces. Using both this group theoretic structure and some properties of harmonic maps we are able to generalize many of the known existence and uniqueness theorems for black holes in Einstein-Maxwell theory to this more general setting.     

The Hamiltonian formulation of regular rth-order Lagrangian field theories

A Hamiltonian formulation of regular rth-order Lagrangian field theories over an m-dimensional manifold is presented in terms of the Hamilton-Cartan formalism. It is demonstrated that a uniquely determined Cartan m-form may be associated to an rth-order Lagrangian by imposing conditions of congruence modulo a suitably defined system of contact m-forms. A geometric regularity condition is given and it is shown that, for a regular Lagrangian, the momenta defined by the Hamilton-Cartan formalism, together with the coordinates on the (r−1)st-order jet bundle, are a minimal set of local coordinates needed to express the Euler-Lagrange equations. When r is greater than one, the number of variables required is strictly less than the dimension of the (2r−1)st order jet bundle. It is shown that, in these coordinates, the Euler-Lagrange equations take the first-order Hamiltonian form given by de Donder. It is also shown that the geometrically natural generalization of the Hamilton-Jacobi procedure for finding extremals is equivalent to de Donder's Hamilton-Jacobi equation. Research supported by the Natural Sciences and Engineering Research Council.